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How to do multitask learning in PyTorch. In PyTorch, multitask learning can be achieved by using a multitask loss function to simultaneously optimize multiple tasks. One common approach is to use multiple loss functions, with each one corresponding to a task, and then combine these loss functions by weighting them to create the final loss function. Below is a simple example code: import torch import torch.nn as nn import torch.optim as optim # 定义多任务损失函数 class MultiTaskLoss(nn.Module): def __init__(self, task_weights): super(MultiTaskLoss, self).__init__() self.task_weights = task_weights def forward(self, outputs, targets): loss = 0 for i in range(len(outputs)): loss += self.task_weights[i] * nn.CrossEntropyLoss()(outputs[i], targets[i]) return loss # 定义模型 class MultiTaskModel(nn.Module): def __init__(self): super(MultiTaskModel, self).__init__() self.fc1 = nn.Linear(10, 5) self.fc2 = nn.Linear(5, 2) def forward(self, x): x = self.fc1(x) output1 = self.fc2(x) output2 = self.fc2(x) return [output1, output2] # 定义数据和标签 data = torch.randn(1, 10) target1 = torch.LongTensor([0]) target2 = torch.LongTensor([1]) # 创建模型和优化器 model = MultiTaskModel() criterion = MultiTaskLoss([0.5, 0.5]) # 两个任务的损失函数权重均为0.5 optimizer = optim.SGD(model.parameters(), lr=0.01) # 训练模型 outputs = model(data) loss = criterion(outputs, [target1, target2]) In the example above, we defined a multi-task model with two tasks and a corresponding multi-task loss function, where the weights for the loss functions of the two tasks are both 0.5. During training, we calculate the loss between the model’s output and the target values, and update the model parameters based on the total loss. This allows us to achieve multi-task learning. Leave a Reply 0	{"url":"https://www.silicloud.com/blog/how-to-do-multitask-learning-in-pytorch/","timestamp":"2024-11-12T16:31:37Z","content_type":"text/html","content_length":"128408","record_id":"<urn:uuid:e3e4982e-6df8-4aca-a5f1-9fd1c19dc7ce>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00441.warc.gz"}
Math formatting "Pro tips" edition: Boxing answers! - "?" \| Socratic Math formatting "Pro tips" edition: Boxing answers! - "?" 1 Answer I want to share a cool technique I plan on using in my answers from now on. I've been addicted to boxing formulas and solutions ever since I was in secondary school, so I figure it was about time to start doing it on Socratic as well. So here's what I came up with for doing that. Let's say I want to box something classic like $a {x}^{2} + b x + c = 0$ The two side lines are covered by the vertical line symbol, $\|$, the one I assume many people are using for absolute value. \| ax^2 + bx + c = 0 \| $\| a {x}^{2} + b x + c = 0 \|$ The bottom line is covered by the underline function, ul(), which looks like this ul(ax^2 + bx + c = 0) $\underline{a {x}^{2} + b x + c = 0}$ The top line is covered by the bar() function, which looks like this bar(ax^2 + bx + c = 0) $\overline{a {x}^{2} + b x + c = 0}$ So basically all you have to do is put all these three things together to get bar( ul( \| ax^2 + bx + c = 0 \| )) $\overline{\underline{\| a {x}^{2} + b x + c = 0 \|}}$ And there you have it, a boxed answer! Now, if you bring color into the mix, things can get really interesting. To write all that in color simply write it in a color()() function color(green)(bar( ul( \| ax^2 + bx + c = 0 \| ))) $\textcolor{g r e e n}{\overline{\underline{\| a {x}^{2} + b x + c = 0 \|}}}$ You can get creative with it, too. Let's say that you want to have a black frame and a blue text. You can write bar( ul( \| color(blue)( (2pi)/7 ) \| )) $\overline{\underline{\| \textcolor{b l u e}{\frac{2 \pi}{7}} \|}}$ Here's a tricky one. Get the text black and the frame red, for example. To do that, you will have to use color(black)() to break the red text. color(red)( bar( ul( \| color(black)((2pi)/7) \| )) #color(red)( bar( ul( \| color(black)((2pi)/7) \| ))# Pretty cool, right? :D ADDENDUM - CREATING A BIT OF SPACE Notice that the outline of the box is really close to the text. If you want to enlarge the box, you can use this cool trick. color(red)( bar( ul( \| color(white)(a/a) color(black)( ax^2 + bx + c = 0 ) color(white)(a/a) \| ))) $\textcolor{red}{\overline{\underline{\| \textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{a {x}^{2} + b x + c = 0} \textcolor{w h i t e}{\frac{a}{a}} \|}}}$ Here I used color(white)(a/a) to create some space between the edges of the box and the text. Here's how that would look with color(green)(a/a) color(red)( bar( ul( \| color(green)(a/a) color(black)( ax^2 + bx + c = 0 ) color(green)(a/a) \| ))) $\textcolor{red}{\overline{\underline{\| \textcolor{g r e e n}{\frac{a}{a}} \textcolor{b l a c k}{a {x}^{2} + b x + c = 0} \textcolor{g r e e n}{\frac{a}{a}} \|}}}$ Impact of this question 2656 views around the world	{"url":"https://socratic.org/questions/math-formatting-pro-tips-edition-boxing-answers-color-white#234919","timestamp":"2024-11-09T10:02:43Z","content_type":"text/html","content_length":"42226","record_id":"<urn:uuid:46460290-fac5-45eb-b4b4-bd833d5b5aff>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00798.warc.gz"}
AccuBook: How to calculate capacity \| Accurx Help Centre For the lead organisation setting up clinics, we've provided an easy way to calculate how many patients in total you will be able to vaccinate in each clinic you have set up. Below, you will see some examples of some clinics: We calculate capacity in the following way: Length of session x Patients per hour* = Total capacity. e.g. For the clinic on the 20th December in the picture above, this clinic is 5 hours long. The practice has said they can see 9 patients per hour, therefore the total amount of patients they will be able to vaccinate is 45 patients. We've also added an extra 500% allowance for bookings as we don't expect all patients invited to want the vaccine or contact the practice to be vaccinated - therefore, if you have the capacity for 50 patients for one clinic, you can actually invite 250 patients for booking. However, please note the upload limit per CSV file is 500 patients. We don't count 'to manually book' patients in this calculation, to free up more capacity for self-service patients. This needs to be the total amount of patients that ALL the vaccinators will be able to vaccinate in one hour. For example, if you have two nurses vaccination - one that can vaccine 6 patients an hour, and another that can vaccinate 8 an hour - the total will be 14 patients per hour. Please note that clinics set up in this way do not include breaks - you will need to calculate capacity accordingly. You will also need to make sure your session length matches up with the length you're setting for each individual appointment. For example, if each of your appointments will last 15 minutes, it's best not to set a session to 70 minutes, for example, as you will have 10 minutes at the end that won't be one full appointment. ๐ จThe most important thing to draw your attention to is to remind practices that theyโ ll want to start slow. Start by uploading small groups of patients first. You can always upload more later. If more patients are invited than there are available slots, patients still receive an SMS invite but will be unable to book if others book it out before them. To avoid this, we recommend dividing slots up fairly between the participating practices, and inform them how many patients can be invited. ๐ จ If you still have any questions or concerns, feel free to chat with us using the green message bubble in the bottom right-hand corner of this page. ๐	{"url":"https://support.accurx.com/en/articles/4733072-accubook-how-to-calculate-capacity","timestamp":"2024-11-06T01:51:18Z","content_type":"text/html","content_length":"61932","record_id":"<urn:uuid:ec593207-62b6-4b98-924b-6c2d184c7aa4>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00334.warc.gz"}
Primitive Roots of Unity \| Brilliant Math & Science Wiki Let $n$ be a positive integer. A primitive $\ n^\text{th}$ root of unity is an $n^\text{th}$ root of unity that is not a $k^\text{th}$ root of unity for any positive $ k<n.$ That is, $ \zeta $ is a primitive $n^\text{th}$ root of unity if and only if \[ \zeta^n=1, \text{and } \zeta^k \ne 1 \text{ for any positive integer } k < n. \] There are four $ 4^\text{th}$ roots of unity given by $ \pm 1, \pm i.$ Two of these, namely $ \pm i,$ are primitive. The other two are not: $ 1^1 = 1$ and $ (-1)^2=1.$ The primitive $n^\text{th}$ roots of unity are the complex numbers \[ e^{2\pi i k/n} : 1\le k \le n, \text{gcd}(k,n)=1. \] There are $ \phi(n)$ primitive $n^\text{th}$ roots of unity, where $ \phi(n)$ is Euler's totient function. Let $ \zeta_n = e^{2\pi i/n}.$ Recall that the $n^\text{th}$ roots of unity are the $n$ distinct powers $ \zeta_n^k = e^{2\pi i k/n} : 1 \le k \le n.$ So it remains to show that $ \ zeta_n^k$ is primitive if and only if $k$ and $n$ are coprime. The key fact is that $ \zeta_n$ is a primitive $n^\text{th}$ root of unity, since its first $n$ powers are distinct. A standard order argument shows that $ \zeta_n^a = 1$ if and only if \ ( a\|n,\) since writing $ n = aq+r,$ $ 0 \le r < n,$ gives $\zeta_n^r = \zeta_n^{n-aq} = 1,$ but this is impossible unless $ r=0.$ If $ \text{gcd}(k,n) =d,$ then it is easy to check that $ (\zeta_n^k)^{n/d} = 1,$ so if $ d>1 $ then $ \zeta_n^k$ is not primitive. $\big($In fact, it is not hard to show that $ \ zeta_n^k$ is a primitive $\big(\frac nd\big)^\text{th}$ root of unity.$\big)$ If $ \text{gcd}(k,n)=1,$ and $ (\zeta_n^k)^r=1,$ then $n\|kr,$ but $n$ and $k$ are coprime so $n\|r.$ This shows that $ \zeta_n^k$ is a primitive $n^\text{th}$ root of unity, because the first positive power of $\zeta_n^k$ that equals $ 1$ is $ (\zeta_n^k)^n.$ Euler's totient function counts the number of positive integers $k \le n$ that are coprime to $n,$ which are precisely the exponents that produce primitive $n^\text{th}$ roots of unity, so this is the number of primitive $n$th roots of unity. $_\square$ Classify the $12^\text{th}$ roots of unity by their multiplicative order. Let $ \zeta_{12} = e^{2\pi i/12}.$ Then the powers of $ \zeta_{12}$ are classified according to the GCD of the exponent and $ 12:$ as in the above proof, $ \zeta_{12}^k$ is a $ \left(\ frac{12}{\gcd (12,k)}\right)^\text{th}$ root of unity. $ \zeta_{12}, \zeta_{12}^5, \zeta_{12}^7, \zeta_{12}^{11} $ are primitive $ 12^\text{th}$ roots of unity. $($Note $ \phi(12)=4.)$ $ \zeta_{12}^2, \zeta_{12}^{10} $ are primitive $6^\text{th}$ roots of unity. $ \zeta_{12}^3, \zeta_{12}^9$ are primitive $4^\text{th}$ roots of unity. $ \zeta_{12}^4, \zeta_{12}^8$ are primitive $3^\text{rd}$ roots of unity. $ \zeta_{12}^6 = -1$ is a primitive $2^\text{nd}$ root of unity. $\zeta_{12}^{12} = 1 $ is a primitive $1^\text{st}$ root of unity. Note that the orders are the divisors of $ 12.$ There are $ \phi(d)$ primitive roots of order $ d,$ for each $ d\|12.$ Since $ \sum\limits_{d\|12} \phi(d) = 12,$ this accounts for all of the $12^\text{th}$ roots of unity. $_\square$ The above arguments show that the powers of a primitive $n^\text{th}$ root of unity enumerate all the primitive $d^\text{th}$ roots of unity, for all the divisors $d$ of $n.$ \[ \ell = mn\] \[ \ell = \gcd(m,n)\] \[ \ell = \text{lcm}(m,n)\] None of these choices Let $ \zeta_m$ be a primitive $m^\text{th}$ root of unity, and let $ \zeta_n$ be a primitive $ n^\text{th}$ root of unity. Then $ \zeta_m\zeta_n$ is a primitive $\ell^\text{th}$ root of unity for some positive integer $ \ell.$ What can we say about $ \ell$ in general? Clarification: In the answer choices, $\gcd(\cdot) $ and $\text{lcm}(\cdot) $ denote the greatest common divisor function and the lowest common multiple function, respectively. The product of the primitive $n^\text{th}$ roots of unity is $ 1$ unless $ n=2.$ This is because the set of primitive $n^\text{th}$ roots of unity, $n\ge 3,$ can be partitioned into pairs \ ( \big\{ \zeta^k, \zeta^{n-k} \big\}, \) which multiply to give $ 1.$ $($For $ n=2$ this fails because $ \zeta^1 $ and $ \zeta^{2-1} $ coincide.$)$ The sum of the primitive $n^\text{th}$ roots of unity is $ \mu(n),$ where $ \mu$ is the Möbius function; see that wiki for a proof. What is the sum of all primitive $2015^\text{th}$ roots of unity, $w$, meaning that 2015 is the smallest positive integer $n$ such that $w^n=1$? Extra Credit Question: What is the sum of the primitive $2009^\text{th}$ roots of unity? In fact, there is an elegant formula for the sum of the $k^\text{th}$ powers of the primitive $n^\text{th}$ roots of unity: The sum of the $k^\text{th}$ powers of the primitive $n^\text{th}$ roots of unity is \[ \mu(r) \frac{\phi(n)}{\phi(r)}, \] where $ r = \frac{n}{\gcd (n,k)}.$ The extremal examples of the theorem are when $ \text{gcd}(n,k) =1$ and when $ \text{gcd}(n,k)=n.$ When $ \text{gcd}(n,k)=1,$ taking $k^\text{th}$ powers permutes the primitive $n^\text{th}$ roots of unity, so the sum should still be $ \mu(n).$ Indeed, $ r=n,$ so $ \mu(r)\frac{\phi (n)}{\phi(r)} = \mu(n).$ When $ \text{gcd}(n,k)=n,$ the powers are all $ 1,$ so the sum is $ \phi(n).$ In this case $ r=1,$ so $ \mu(r)\frac{\phi(n)}{\phi(r)} = \phi(n)$ as desired. Here is an outline: the sum $ \sum \zeta^k $ is a sum of primitive $r^\text{th}$ roots of unity, and it runs over all of them. But there are repetitions: the sum has $\phi(n)$ terms and there are $\phi(r)$ primitive $r^\text{th}$ roots of unity, so they are each counted $ \frac{\phi(n)}{\phi(r)}$ times. The sum of the primitive $r^\text{th}$ roots of unity is $\mu(r),$ so the result follows. $_\square$ where the sum is taken over all primitive $2015^\text{th}$ roots of unity $\omega$. The $n^\text{th}$ roots of unity form a cyclic group under multiplication, generated by $ \zeta_n = e^{2\pi i/n}.$ This group is isomorphic to the additive group ${\mathbb Z}/n$ of the integers mod $n,$ under the isomorphism $ \zeta_n^k \mapsto k \pmod n.$ In this context, the primitive $n^\text{th}$ roots of unity correspond via this isomorphism to the other generators of the group. This is the set of multiplicative units $ ({\mathbb Z}/n)^*$ $($which has $\phi(n)$ elements$).$ The $12^\text{th}$ roots of unity are isomorphic to $ {\mathbb Z}/12.$ The primitive $12^\text{th}$ roots of unity, $ \zeta_{12}, \zeta_{12}^5, \zeta_{12}^7,$ and $\zeta_{12}^{11},$ correspond to the elements $1,5,7,11$ in ${\mathbb Z}/12,$ respectively. These are the elements which generate $ {\mathbb Z}/12 $ additively. E.g. the multiples of $5$ are \[ 5,10,3,8,1,6,11,4,9,2,7,0, \, 5,10,\ldots. \] Every element in ${\mathbb Z}/12$ is on that list. The polynomial \[ \prod_{\zeta \text{ a primitive } n\text{th root of unity}} (x-\zeta) \] is a polynomial in $x$ known as the $n$th cyclotomic polynomial. It is of great interest in algebraic number theory. For more details and properties, see the wiki on cyclotomic polynomials.	{"url":"https://brilliant.org/wiki/primitive-roots-of-unity/","timestamp":"2024-11-13T12:48:42Z","content_type":"text/html","content_length":"57993","record_id":"<urn:uuid:279219aa-856d-41ac-a8fe-4f6ca4aacac9>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00293.warc.gz"}
Calorie per Kilogram Units of measurement use the International System of Units, better known as SI units, which provide a standard for measuring the physical properties of matter. Measurement like latent heat finds its use in a number of places right from education to industrial usage. Be it buying grocery or cooking, units play a vital role in our daily life; and hence their conversions. unitsconverters.com helps in the conversion of different units of measurement like cal/kg to mJ/mg through multiplicative conversion factors. When you are converting latent heat, you need a Calorie per Kilogram to Millijoule per Milligram converter that is elaborate and still easy to use. Converting cal/kg to Millijoule per Milligram is easy, for you only have to select the units first and the value you want to convert. If you encounter any issues to convert Calorie per Kilogram to mJ/mg, this tool is the answer that gives you the exact conversion of units. You can also get the formula used in cal/kg to mJ/mg conversion along with a table representing the entire conversion.	{"url":"https://www.unitsconverters.com/en/Cal/Kg-To-Mj/Mg/Utu-8805-8818","timestamp":"2024-11-03T15:39:57Z","content_type":"application/xhtml+xml","content_length":"111318","record_id":"<urn:uuid:72baf95c-d6ef-4ace-a969-91b511171598>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00074.warc.gz"}
Prepare Figures and Axes for Graphs Behavior of MATLAB Plotting Functions MATLAB^® plotting functions either create a new figure and axes if none exist, or reuse an existing figure and axes. When reusing existing axes, MATLAB • Clears the graphics objects from the axes. • Resets most axes properties to their default values. • Calculates new axes limits based on the new data. When a plotting function creates a graph, the function can: • Create a figure and an axes for the graph and set necessary properties for the particular graph (default behavior if no current figure exists) • Reuse an existing figure and axes, clearing and resetting axes properties as required (default behavior if a graph exists) • Add new data objects to an existing graph without resetting properties (if hold is on) The NextPlot figure and axes properties control the way that MATLAB plotting functions behave. How the NextPlot Properties Control Behavior MATLAB plotting functions rely on the values of the figure and axes NextPlot properties to determine whether to add, clear, or clear and reset the figure and axes before drawing the new graph. Low-level object-creation functions do not check the NextPlot properties. They simply add the new graphics objects to the current figure and axes. This table summarizes the possible values for the NextPlot properties. NextPlot Figure Axes new Creates a new figure and uses it as the current figure. Not an option for axes. add Adds new graphics objects without clearing or resetting the current figure. (Default) Adds new graphics objects without clearing or resetting the current axes. replacechildren Removes all axes objects whose handles are not hidden before adding new objects. Does Removes all axes child objects whose handles are not hidden before adding new graphics objects. not reset figure properties. Equivalent to clf. Does not reset axes properties. Equivalent to cla. replace Removes all axes objects and resets figure properties to their defaults before adding Removes all child objects and resets axes properties to their defaults before adding new new objects. Equivalent to clf reset. objects. Equivalent to cla reset. (Default) Plotting functions call the newplot function to obtain the handle to the appropriate axes. The Default Scenario Consider the default situation where the figure NextPlot property is add and the axes NextPlot property is replace. When you call newplot, it: 1. Checks the value of the current figure's NextPlot property (which is, add). 2. Determines that MATLAB can draw into the current figure without modifying the figure. If there is no current figure, newplot creates one, but does not recheck its NextPlot property. 3. Checks the value of the current axes' NextPlot property (which is, replace), deletes all graphics objects from the axes, resets all axes properties (except Position and Units) to their defaults, and returns the handle of the current axes. If there is no current axes, newplot creates one, but does not recheck its NextPlot property. 4. Deletes all graphics objects from the axes, resets all axes properties (except Position and Units) to their defaults, and returns the handle of the current axes. If there is no current axes, newplot creates one, but does not recheck its NextPlot property. hold Function and NextPlot Properties The hold function provides convenient access to the NextPlot properties. When you want add objects to a graph without removing other objects or resetting properties use hold on: • hold on — Sets the figure and axes NextPlot properties to add. Line graphs continue to cycle through the ColorOrder and LineStyleOrder property values. • hold off — Sets the axes NextPlot property to replace Use the ishold to determine if hold is on or off. Control Behavior of User-Written Plotting Functions MATLAB provides the newplot function to simplify writing plotting functions that conform to the settings of the NextPlot properties. newplot checks the values of the NextPlot properties and takes the appropriate action based on these values. Place newplot at the beginning of any function that calls object creation functions. When your function calls newplot, newplot first queries the figure NextPlot property. Based on the property values newplot then takes the action described in the following table based on the property Figure NextPlot Property Value newplot Function No figures exist Creates a figure and makes this figure the current figure. add Makes the figure the current figure. new Creates a new figure and makes it the current figure. replacechildren Deletes the figure's children (axes objects and their descendants) and makes this figure the current figure. replace Deletes the figure's children, resets the figure's properties to their defaults, and makes this figure the current figure. Then newplot checks the current axes' NextPlot property. Based on the property value newplot takes the action described in the following table. Axes NextPlot Property Value newplot Function No axes in current figure Creates an axes and makes it the current axes add Makes the axes the current axes and returns its handle. replacechildren Deletes the axes' children and makes this axes the current axes. replace Deletes the axes' children, reset the axes' properties to their defaults, and makes this axes the current axes.	{"url":"https://au.mathworks.com/help/matlab/creating_plots/preparing-figures-and-axes-for-graphics.html","timestamp":"2024-11-13T13:17:44Z","content_type":"text/html","content_length":"75657","record_id":"<urn:uuid:55ede004-b676-412b-8940-84976413d125>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00254.warc.gz"}
Minimum Time to Make Rope Colorful \| Technical Interview Study Guide There is are approaches that uses or greedy instead but I will not cover those. Consecutive balloons do not count the popped balloons, so popping the middle R in RRR will still violate the criteria of having no consecutive colors. Let's start with an arbitrary rope with balloons: RRRGGBB and an arbitrary cost of [2, 1, 5, 3, 4, 1, 2]. From hand-simulation, we know that the ultimate "optimal" answer is XXRXGBX where X are the popped balloons. For every segment of like colors, we never pop the highest cost balloon to achieve the lowest possible cost within the segment (the mathematical proof is relatively easy to come up with) When encountering a same color, a decision between popping the current balloon or the previously seen, unpopped, same balloon should be made, with the decision made to be the smallest of the two (see (1)) For every segment of same color, once a balloon encountered is of a different color, there's no need to pop anymore of the previous color so the color tracked can be changed Recurrence relation $dp(i) = \begin{cases} dp(i-1), r[i] eq r[prev]\\ dp(i-1)+\min(time[i], time[prev]) \end{cases}$ prev is needed alongside the recurrence. Processing from left to right, $dp(i)$ can be interpreted as the minimum time to make the rope $r[:i+1]$ colorful. This means we are solving for the prefix of the array. def min_time(r, time): n = len(time) t = 0 prev = 0 for i in range(1, n): if r[i] == r[prev]: t += min(time[i], time[prev]) if time[prev] < time[i]: prev = i # since we pop the prev balloon, we have a new prev prev = i # we don't need to track the same color anymore return t	{"url":"https://interviews.woojiahao.com/problems-guide/dynamic-programming-roadmap/linear-sequence/minimum-time-to-make-rope-colorful","timestamp":"2024-11-02T15:19:21Z","content_type":"text/html","content_length":"261155","record_id":"<urn:uuid:11cc4d06-db38-434b-8075-84f9506a25ca>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00681.warc.gz"}
Intersection Between Elements One question that regularly comes up is how to determine whether certain building elements intersect. There are basically two approaches to this, either querying the elements for their pure geometry and making use of pure geometrical intersection methods, or using the higher-level FindReferencesByDirection method, which works on the building elements themselves. Here is a typical version of this Question: I need to find out if any two elements in a project are interfering with each other. Searching the Revit API help, I found that the Intersect methods on the Autodesk.Revit.Geometry.GeometryObject can be used to calculate intersections. So it seems that I need to find out the type of geometry for the given building element and then call the relevant intersection method. Is this correct? Is there any sample code available demonstrating this? Is there any API method which accepts two elements and finds out whether they are interfering? Answer: There is no direct API method that calculates the intersection between two Revit building elements. One possibility is to query them for their geometry elements, and then use the geometry Intersect methods to determine intersection points. Here is some sample code which shows how to find the intersection points of two geometry lines, excerpted from the Revit SDK sample CreateTruss: private XYZ GetIntersection( Line line1, Line line2 ) IntersectionResultArray results; SetComparisonResult result = line1.Intersect( line2, out results ); if( result != SetComparisonResult.Overlap ) throw new InvalidOperationException( "Input lines did not intersect." ); if( results == null \|\| results.Size != 1 ) throw new InvalidOperationException( "Could not extract line intersection point." ); IntersectionResult iResult = results.get_Item( 0 ); return iResult.XYZPoint; A more advanced and complex use of the pure geometrical intersection methods is provided by the AreSolidsCut method defined by the Revit SDK RoomsRoofs sample. While there is no direct method to determine whether two building elements intersect, one could implement such a method based on the FindReferencesByDirection method, as we suggested in the discussion on the analytical support tolerance. Several SDK samples demonstrate its use.	{"url":"http://jeremytammik.github.io/tbc/a/0399_find_element_intersection.htm","timestamp":"2024-11-12T14:01:51Z","content_type":"text/html","content_length":"4542","record_id":"<urn:uuid:3fc7ff77-1c90-4747-a4ea-1700479392dd>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00559.warc.gz"}
What is three-tenths of three dollars fifty cents? - Answers A half-dollar is $0.50 ( fifty cents); three half-dollars is 3(.50) = $1.50 ( one dollar and fifty cents) three hundred fifty dollars and seventy cents or three hundred fifty point seven or three hundred fifty and seven tenths Three hundred fifty-nine and thirty-three hundredths orfor US dollars, three hundred fifty-nine dollars and thirty-three cents. The number is spelled "three hundred fifty" (and no hundredths) The US currency value would be "three hundred and fifty dollars" (and no cents). Seventy-three and five tenths or seventy-three dollars and fifty cents. The numeral 351.00 is "three hundred fifty-one" (and no hundredths). The currency value $351.00 is three hundred fifty-one dollars (and no cents). Nine hundred eighty-three and fifty-three hundredths. For US currency: Nine hundred eighty-three dollars and fifty-three cents	{"url":"https://math.answers.com/math-and-arithmetic/What_is_three-tenths_of_three_dollars_fifty_cents","timestamp":"2024-11-06T18:36:09Z","content_type":"text/html","content_length":"155672","record_id":"<urn:uuid:9d8fcc16-1ed1-4c45-9ff5-faa0f9fa1221>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00488.warc.gz"}
The subsidy amount is the difference between the total cost of all utility services, considering the social standard, and the required payment amount. To determine the required payment, the average monthly total income of the household is multiplied by the mandatory payment percentage. The mandatory payment percentage is also calculated using a formula. For this, the average monthly total income is divided by the number of family members to determine the income per person. Then, this amount is divided by the subsistence minimum. The result is divided by 2 (the basic income coefficient for subsidy allocation), and then multiplied by 15% (the basic rate for service	{"url":"https://rating.net.ua/en/interesno-1/kak-rasschityivaetsya-razmer-zhilischnoj-1","timestamp":"2024-11-08T14:03:45Z","content_type":"text/html","content_length":"31477","record_id":"<urn:uuid:d7772bfe-b3cf-433f-971d-6d51b1f1f3c6>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00271.warc.gz"}
Distance Relays Types and their Applications Distance Relays Types and their Applications: In applying relays to a transmission system it is necessary to state the relay characteristic in the same terms that the system conditions are stated. This is especially true of Distance Relays Types. If the relay characteristics are thought of in terms of volts and amperes, then the system conditions should be stated in the same terms. With the distance relays, however, it is difficult to think in terms of volts and amperes, because these values vary widely for the same relay response. This relay response is some function of the ratio between volts and amperes and for any given value of the ratio there may exist an infinite number of values of the volts and amperes. It, therefore, simplifies matters greatly to think of the distance relay response in terms of the ratio of volts to amperes or in other words to think of impedance, reactance, resistance or combination thereof to which the relay responds. However in designing Distance Relays Types it is necessary to think in terms of the volts, amperes and phase angle to which it must respond because these are the quantities which actually operate the contact actuating parts. Relay Types and Their Application: Principal Distance Relays Types are: 1. impedance; 2. reactance; 3. admittance (mho); 4. ohm; and 5. offset mho. The common types compare two input quantities either in magnitude or in phase. Any of the relay characteristics can be obtained either by an amplitude comparator or by a phase comparator. Impedance Replay: This is a device which measures distance by comparing the fault current I with the voltage V across the fault loop. It is simple in this case to have an amplitude comparator and the balanced beam type structure is most common. The equation for the amplitude comparator at threshold as derived already in Eq. (4.4) is If the constants are so adjusted that the input signals are Substituting these conditions in the above equation, we get The characteristics when plotted on the R-X plane is shown in Fig. (4.16) which is a circle with origin as its centre; signifying that a simple impedance relay would operate for any value of impedance lying within the circle. The characteristic also depicts that the relay is not directional and it is essential to provide a directional relay along with an impedance relay. The combined characteristics of an impedance and directional relay are shown in Fig. (4.17) where DD represents the directional relay characteristic and the operating region is the shaded portion. Reactance Replay: All other relays except the impedance relay are conveniently obtained by A phase comparator. The basic equation for a phase comparator already derived in Eq. (4.6) at threshold is Using the input signals in the following manner Substituting these conditions in the above equation, we have Now if Î¸ is Ï€/2 the above equation reduces to the reactance form, i.e. When plotted on the R-X diagram the characteristic is represented by a straight line parallel to the horizontal axis it as in Fig. (4.18). With some predetermined setting of value of X, the relay will measure any value of reactance below the setting. A reactance relay responds only to the reactive component of system impedance; consequently it is unaffected by fault arc resistance. However, when fault resistance is such a high value that load and fault current magnitudes are comparable the reach of the relay is modified by the value of the load and its power factor and may either overreach or underreach. Voltage-restrained starting relays are used in a reactance measuring scheme to give directional response and to prevent operation on load. The reactance relay as seen by Eq. (4.15) is a particular case of an ohm relay, in which the angle of compensation Î¸ is 90Â°. Admittance (MHO) Relay: If the signals S[1]Â and S[2]Â given to the phase comparator are Substituting these conditions in Eq. (4.6) we have This represents the admittance or the mho characteristic and when plotted on the R-X diagram is a circle passing through the origin and when plotted on the G-B diagram is a straight line as illustrated in Fig. (4.19). The circle passing through the origin makes it inherently directional. With such a characteristic the relay measures distances in one direction only. From the expression (4.16) it is evident that the relay is inoperative if the voltage falls to zero, because both terms contain V. A memory circuit may be used to prevent the immediate decay of voltage applied to the relay terminals when a closeup three-phase short circuit occurs. This enables high speed mho protection to operate correctly on closeup faults, provided that the protected circuit is energised before the short circuit is applied. OHM Relay: As explained earlier a reactance relay is a particular case of an ohm relay. Its characteristic is represented by Eq. (4.14) when plotted on the R-X plane is a straight line (Fig. (4.20)). The ohm relay is used as a supplementary element to modify the operating region of the other types of measuring elements. OFFSET MHO Relay: Let the signals S[1]Â and S[2]Â given to the phase comparator be Substituting these conditions we have This equation represents a circle, with centre at (K[2]â€”K[4])/2Kâˆ Î¸ on the R-X plane, the radius being of magnitude (K[2]Â + K[4])/2K.Â The offset threshold characteristic is shown in Fig.	{"url":"https://www.eeeguide.com/distance-relays-types/","timestamp":"2024-11-03T16:48:46Z","content_type":"text/html","content_length":"227607","record_id":"<urn:uuid:88e6a07f-87f7-4b53-b997-6b2b2d9efbd4>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00555.warc.gz"}
Amortization of premium on bonds payable To record bond interest payment.This entry records $1,000 interest expense on the $100,000 of bonds that were outstanding for one month. Valley collected $5,000 from the bondholders on May 31 as accrued interest and is now returning it to them. For the years in which you own the bond for all 12 months, you simply take amortization of 12 times the monthly amount. Which statements are true about amortization of bond premiums? Which statements are TRUE about amortization of bond premiums? Amortization of bond premiums reduces reported interest income each year, but it does not represent a cash loss, since the issuer pays the stated interest amount. The primary advantage of premium bond amortization is that it is a tax deduction in the current tax year. In that case, the premium paid on the bond can be amortized, or in other words, a part of the premium can be utilized towards reducing the amount of taxable income. Also, it leads to reducing the cost basis of the taxable bond for premium amortized in each period. Assume instead that Lighting Process, Inc. issued bonds with a coupon rate of 9% when the market rate was 10%. The bond purchaser would be willing to pay only $9,377 because Lighting Process, Inc. will pay $450 in interest every six months ($10,000 × 9% × 6/ 12), which is lower than the market rate of interest of $500 every six months. The total cash paid to investors over the life of the bonds is $19,000, $10,000 of principal at maturity and $9,000 ($450 × 20 periods) in interest throughout the life of the bonds. If you choose to amortize the bond premium, then you can offset your interest income from the bond by the annual amount of amortized bond premium. I write a separate line with negative entries forBond Amortization and typically also include the name of the 1099 that I am matching up with,. Whether you are starting your first company or you are a dedicated entrepreneur diving into a new venture, Bizfluent is here to equip you with the tactics, tools and information to establish and run your ventures. By striking “, or by a foreign personal holding company, as defined in section 552” after “section 584” and by striking “, or foreign personal holding company” after “common trust fund”. The difference of $7,580 between the face value of bond of $100,000 and the proceeds of $92,420 represents the discount on bond. Founded in 1993 by brothers Tom and David Gardner, The Motley Fool helps millions of people attain financial freedom through our website, podcasts, books, newspaper column, radio show, and premium investing services. Bond Amortization Methods Hereby elects, pursuant to IRC Sec. 171, to amortize bond premium during the taxable year and for all subsequent taxable years. The first taxable year to which this election applies is the taxpayer’s calendar year ending . As the discount is amortized, the discount on bonds payable account’s balance decreases and the carrying value of the bond increases. The amount of discount amortized for the last payment is equal to the balance in the discount on bonds payable account. Our systems have detected unusual traffic activity from your network. Please complete this reCAPTCHA to demonstrate that it’s you making the requests and not a robot. If you are having trouble seeing or completing this challenge, this page may help. If you continue to experience issues, you can contact JSTOR support. • First, calculate the bond premium by subtracting the face value of the bond from what you paid for it. • In most cases, it is the investor’s decision to convert the bonds to stock, although certain types of convertible bonds allow the issuing company to determine if and when bonds are converted. • Intrinsically, a bond purchased at a premium has a negative accrual; in other words, the basis amortizes. • This procedure ensures that after the discount or premium is fully amortized, the investment account will reflect the bond’s maturity value. FASB made targeted changes Thursday to the rules governing accounting for amortization of premiums for purchased callable debt securities. Bonds are secured when specific company assets are pledged to serve as collateral for the bondholders. Since bond buyers will receive more at maturity than they paid at purchase, they treat bond discounts as gains. Individuals amortize discounts using either the straight-line method or constant yield method. They must amortize original issue discounts but can choose not to amortize market discounts and instead recognize these as an ordinary gain at bond maturity or sale. Corporations amortize bond discounts using the straight-line method or the effective yield method. Step 4 For example, assume you amortize a bond’s discount by $100 annually and pay $500 in annual interest. These unsecured bonds require the bondholders to rely on the good name and financial stability of the issuing company for repayment of principal and interest amounts. A subordinated debenture bond means the bond is repaid after other unsecured debt, as noted in the bond agreement. The amortized bond premium is offset against the interest income of the bond. • Regardless of when the bonds are physically issued, interest starts to accrue from the most recent interest date. • For all other entities, the amendments are effective for fiscal years beginning after Dec. 15, 2019, and interim periods within fiscal years beginning after Dec. 15, 2020. • This is not the case; however, you must follow certain guidelines when it comes to reporting negative amounts on your balance sheet if you choose to take them into account in determining net • Based on the remaining payment schedule of the bond and A’s basis in the bond, A’s yield is 8.07 percent, compounded annually. • Full BioCierra Murry is an expert in banking, credit cards, investing, loans, mortgages, and real estate. Subtract the annual amortization of the premium from the amount of unamortized premium on your balance sheet to calculate your unamortized premium remaining. Continuing with the example, assume you have yet to amortize $2,000 of the bond’s premium. Subtract $200 from $2,000 to get $1,800 in unamortized premium remaining. Subtract the annual amortization of a bond’s premium to the annual interest you paid to bondholders to calculate total annual interest expense. How to Amortize a Bond Premium All such information is provided solely for convenience purposes only and all users thereof should be guided accordingly. In this entry, Cash is debited for $600, which is the full 6 months’ interest payment ($12,000 x 0.05). “ The amendments made by this paragraph shall apply to obligations issued after September 27, 1985. “ with reference to the amount payable on maturity , in the case amortizing bond premium of any bond described in subsection which is acquired after December 31, 1957, and”. For purposes of the preceding sentence, the term “taxable bond” means any bond the interest of which is not excludable from gross income. The brokerage house you used to purchase the bond should be able to provide you with all the information you need about how often, and when, interest payments occur. • The coupon rate of interest is 10% and has a market rate of interest at 8%. • These tools are designed to help you understand the official document better and aid in comparing the online edition to the print edition. • On May 1, 1999, C purchases for $130,000 a taxable bond maturing on May 1, 2006, with a stated principal amount of $100,000, payable at maturity. • Any amount you cannot deduct because of this limit can be carried forward to the next accrual period. • The journal entries made by Lighting Process, Inc. to record its issuance at par of $10,000 ten‐year bonds with a coupon rate of 10% and the semiannual interest payments made on June 30 and December 31 are as shown. Although the accrual period ends on August 1, 1999, the qualified stated interest of $5,000 is not taken into income until February 1, 2000, the date it is received. Likewise, the bond premium of $645.29 is not taken into account until February 1, 2000. The adjusted acquisition price of the bond on August 1, 1999, is $109,354.71 (the adjusted acquisition price at the beginning of the period ($110,000) less the bond premium allocable to the period ($645.29)). These final regulations adopt the rule in the temporary and proposed regulations. You choose to amortize the premium on taxable bonds by reporting the offsetting amortized bond premium on your income tax return for the first tax year for which you want the choice to apply. You must attach a statement to your return showing how you figured your offset amount. How to Calculate an Amortized Bond Premium Helstrom attended Southern Illinois University at Carbondale and has her Bachelor of Science in accounting. The articles and research support materials available on this site are educational and are not intended to be investment or tax advice. How do you amortize a bond premium straight line? In the straight-line method of amortization of bond discount or premium, bond discount or premium is charged equally in each period of the bond's life. When the coupon rate on a bond is lower than the market interest rate, the bond is issued at a discount to par value. Interest is typically paid twice per year, at the end of June and at the end of December. However, check with the specifics about your bond.If there’s https://www.bookstime.com/ five years left until the bond matures, and you bought the bond at the beginning of the year, then there are most likely 10 interest payments left . Problems with New IRS Bond Premium Amortization Rules This takes into account the basis of the bond’s yield to maturity, determined by using the bond’s basis and compounding at the close of each accrual period. Note that your broker’s computer system just might do this for you automatically. In order to calculate the premium amortization, you must determine the yield to maturity of a bond. The yield to maturity is the discount rate that equates the present value of all coupons and principal payments to be made on the bond to its initial purchase price. Those who invest in taxable premium bonds typically benefit from amortizing the premium, because the amount amortized can be used to offset the interest income from the bond. Create a journal entry that decreases the account “premium on bonds payable” with a debit when interest is paid semi-annually. Decrease cash for the interest paid to the lender with a credit, and debit the account “interest expense” for the difference. Alternatively, if the coupon rate is higher than the market interest rate, the bond is issued at a premium to its par value. The issue price and face value are equal only when market interest rate and the coupon rate are equal. Amortizable Bond Premium refers to the cost of premium paid above the face value of a bond. The face value of a bond is also called “par value”, it is the original cost of a stock or the amount paid to the holder of a bond. The bond premium is a part of a bond’s cost basis and is amortized over the remaining life of the bond. The premium is a gain for the bond issuer and loss to the buyer. To get the current interest expense, you’ll use the yield at the time you purchased the bond and the book value. For example, if you purchased a bond for $104,100 at an 8% yield, then the interest expense is $8,328 ($104,100 x 8%). Remember, though, that interest is paid twice per year so you need to divide that number by two, giving you $4,164. Stakeholders said this accounting results in the recognition of too much interest income before a borrower calls the debt security, followed by the recognition of a loss on the call date. See Table 3 for interest expense and carrying value calculations over the life of the bond using the straight‐line method of amortization . The deductible portion of investment interest expense is calculated on Form 4952. This is because we paid an amount lower than the face value of the bond at issue date but will get the full face value at maturity. According to the effective interest rate method, the adjustment reflects the reality better. In other words, it reflects what the change in the bond price would be if we assumed that the market discount rate doesn’t change. The Level 1 CFA Exam is approaching, so we have to keep up the pace. Today, let’s discuss the methods of amortizing bond discount or premium. When you first purchase the bond, the book value is the same as the amount you paid for it. For example, if you purchased a bond for $104,100, then the book value is $104,100.The book value will decrease every time you receive an interest payment. The acquisition of a zero coupon debt instrument at a premium and with a negative yield was not contemplated when the prior regulations were revised in 1997 . Using the straight-line method of amortization, the seller evenly spreads the “premium” on its books over the life of the bond. This amortization reduces or offsets the interest expense incurred from the bond on the seller’s books. When bonds are sold at a discount or a premium, the interest rate is adjusted from the face rate to an effective rate that is close to the market rate when the bonds were issued. Therefore, bond discounts or premiums have the effect of increasing or decreasing the interest expense on the bonds over their life. Under these conditions,it is necessary to amortize the discount or premium over the life of the bonds by using either the straight-line method or the effective interest method. CPA Exam Cost & Fees Amortization of the discount may be done using the straight‐line or the effective interest method. Currently, generally accepted accounting principles require use of the effective interest method of amortization unless the results under the two methods are not significantly different. If the amounts of interest expense are similar under the two methods, the straight‐line method may be used. Sellers can either accumulate the interest income in a suspense account and then close it at maturity, or they can use the proportionate method, which is to debit cash for the full interest expense on each coupon date. Paying straight-line amortization of bond discount or premium over the life of the bond is very complicated and not recommended. Calculating bond premium amortization using the straight-line method couldn’t be simpler. First, calculate the bond premium by subtracting the face value of the bond from what you paid for it. This spreads out the gain or loss over the remaining life of the bond instead of recognizing the gain or loss in the year of the bond’s redemption. The constant-yield method will give you a smaller amortization amount than the straight-line method in early years, with the constant-yield amortization figure growing in later years. That puts it at a overall disadvantage to the straight-line method from the taxpayer’s standpoint, which might be one reason why tax laws were changed to have newer bonds use the less favorable method. When understanding the tax effect of purchasing a bond at a premium, remember that the premium becomes a part of the investor’s cost basis for the bond. Calculate the total amount of interest you’ll receive if you hold the bond until maturity. You can do that by multiplying the interest payments times the number of payments left. In the case of a bond , the amount of the amortizable bond premium for the taxable year shall be allowed as a deduction. The interest income on a debt-investment purchased at a discount must also be similarly higher than the interest income received. Your Account An individual bond buyer amortizes bond premium by applying the constant yield method. You subtract the annual amortized amount from interest income and deduct any excess amortized premium as an itemized expense. Corporations normally use straight-line amortization or the effective interest method to amortize bond premium. Bond issuers debit the amortized amount to the premium on bonds payable account and credit the interest income account monthly.	{"url":"https://techplusjm.com/amortization-of-premium-on-bonds-payable/","timestamp":"2024-11-02T20:40:16Z","content_type":"text/html","content_length":"445523","record_id":"<urn:uuid:b1bc51f3-ab78-4628-b956-2c0047cbd61c>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00609.warc.gz"}
PhysicsScotland.co.uk - Interference ( Amplitude ) In order to cause light to show Interference, a coherent (same Frequency, Wavelength, Speed and constant Phase difference) light source must be used. Division of Amplitude A light wave that has been "split" in terms of amplitude is simply the separating the wave into two (or more) waves whose energy, the sum of which is the Energy of the original beam. This "splitting" is done by light being partially transmitted and partially reflected when incident on a surface between two media of different refractive index. Thin Film Interference In the above diagram, a single beam of monochromatic light is incident on the glass at an angle to the normal. Note - As a single light beam is used, all the reflected and transmitted rays will be coherent. Also, this diagram is not representative of size, the actual thickness of material is equivalent to oil on water. At the air-glass boundary :- Some light is reflected from the surface and follows a path towards A and some light is refracted into the glass and follows a path D1-D2 At the second glass-air boundary ( D2 ) :- Some light is reflected back into the glass and some of the light is refracted and follows a path towards B At the original air-glass boundary :- The reflected ray is partially refracted and follows a path towards A and the rest of the ray is reflected back into the glass. At the second glass-air boundary :- The doubly reflected ray is refracted and follows a path towards B This means that an observer at both A and B has two rays approaching it, with a path difference between them, and may observe an interference pattern. Maxima and Minima for Thin Film Interference - Reflected Rays The above diagram shows only light that would arrive at A the original full diagram. Each ray (1 and 2) undergo different paths and different types of reflection such that :- Ray 1 - This ray is reflected at the transition to a higher refractive index, and therefore undergoes a phase change of π Ray 2 - This ray is reflected at the transition to a lower refractive index, and therefore undergoes no phase change. This ray does travel a longer optical path length of 2nt, where n is the refractive index of the material. In order for constructive Interference to be seen, the two separate rays must arrive in phase. In order for this to happen, the increased Optical Path Length must cancel out the π phase change caused by reflection :- 2nt = (m + 1/2) λ Where m is an integer In order for destructive Interference to be seen, the two separate rays must arrive out of phase. In order for this to happen, the increased Optical Path Length must not cancel out the π phase change caused by reflection :- 2nt = m λ Where m is an integer Note - This is the OPPOSITE of what we would normally expect for constructive and destructive bands, make sure you understand why!! Maxima and Minima for Thin Film Interference - Transmitted Rays The above diagram shows only light that would arrive at B the original full diagram. Each ray ( 3 and 4 ) undergo different paths and different types of reflection / refraction such that :- Ray 3 - This ray passes through the block, refracting twice, with a phase change of 0 Ray 4 - This ray reflects twice within the block, but as the transitions are to a lower refractive index, and therefore undergoes no phase change. The ray does travel an additional Optical Path length of 2nt, however. In order for constructive Interference to be seen, the two separate rays must arrive in phase. In order for this to happen, the increased Optical Path Length must give zero phase change :- 2nt = m λ Where m is an integer In order for destructive Interference to be seen, the two separate rays must arrive out of phase. In order for this to happen, the increased Optical Path Length must cause a phase change of π :- 2nt = (m + 1/2) λ Where m is an integer Note - This is the EXACTLY what we would normally expect for constructive and destructive bands, make sure you understand why!! By comparing the above conditions for maxima and minima, it can be shown that for a set thickness (t), the transmitted ray shows always the opposite of the reflected ray. This is because energy must be conserved, and as the Amplitude of the wave is being "split", when on ray is at maximum, the other must be at minimum. Example 1 - A sheet of Mica is 4.8x10-6 m thick. Light of Wavelength 512 nm is shone onto the sheet of Mica. When viewed from above, which of the three choices below will be seen :- 1.Total constructive interference 2. Total destructive interference 3. Partial constructive interference The refractive index for Mica is 1.58 for Red light 1.60 for Green light 1.62 for Violet light In order to complete this question we must first decide what type of rays are being observed (reflected or transmitted) then apply the correct formula :- In this question as the light source and observer are both above the Mica, it is the reflected rays that are being observed. For reflected rays destructive interference is given by - 2nt = m λ 2 x 1.6 x 4.8x10-6 = m x 512x10-9 m = 30 as m is an integer destructive interference is observed. The embedded website below allows a simulation of thin film interfence on trasmission through a soap film :- Non-Reflective Coatings When a high quality camera lens is viewed under bright lights, it often has a slightly purple hue to it. The lens has been treated with a non-reflective coating in order to reduce the visible reflected light by utilising thin film reflection. The process of coating a lens in this way is called Blooming, giving a "Bloomed Lens". A very thin layer of (generally) magnesium fluoride (n ~ 1.38) has been added onto the lens (n ~ 1.50). Examinable Derivation - Non- Reflective Coating In the above diagram the ray of light is incident first upon the magnesium fluoride, then the transmitted ray is incident upon the glass lens itself. As both the surfaces have a higher refractive index, both reflected rays have a phase change of π. Following the above formula for destructive interference in reflected rays, it can be shown that :- path difference = λ/2 Optical Path in Fluoride = 2nd Where n = refractive index of coating therefore : 2nd = λ/2 If this is rearranged to solve for the thickness of the coating (d), then the following formula can be found :- Note - as can be seen from the formula, for a given thickness, only one specific Wavelength is completely removed through destructive interference. Partial cancellation occurs for the other visible Wavelengths. As green is the colour chosen to be completely removed (centre of the visible spectrum), the remaining red and blue light being reflected combine to give a slightly purple hue to the Example 1 - What thickness of magnesium fluoride (n = 1.38) must be applied to a glass lens (n = 1.50) in order to make the lens non-reflective at λ = 520 nm ? d = (520x10-9) / (4 x 1.38) d = 9.42x10-8 m d = 94.2 nm The diagram below shows the set up for an experiment to view Wedge Fringes, a series of linear maxima and minima generated by interference of light passing through two flat pieces of glass at an angle to one another :- Note - The angle between the two plates has been greatly increased in the above diagrams for clarity, in most experiment work, approximately a sheet of paper thickness is the difference between the plates. In order to derive the formulae below, it is assumed that the angle between the plates is approximately zero. Non Examinable Derivation - Wedge Fringes In the above diagrams light is incident from above onto two thin glass plates, separated by a very small thickness (t). When viewed from above, this gives the Optical path difference between the rays as equal to 2t. There is a Phase change only at the reflection at P (a boundary of higher refractive index), therefore this ray has a Phase change of π. As there is a phase change, the formula to express Destructive interference is given by :- 2t = m λ where it is assumed that the material between the plates is air (n = ) By looking at the right hand diagram above, it can be shown that the separation between two sets of dark fringes can be expressed by λ/2. This gives the formula for fringe separation as :- Where :- D - the separation of the the plates L - the length of the plates The fringe spacing can then be shown as :- Example 2 - Two glass slides of length 60mm are placed on top of each other and separated by a sheet of paper at one end. Monochromatic light of 512nm is shone onto the plates and an interference pattern is observed. A Travelling microscope is used to measure the separation of 10 fringes, giving a value of 2.56x10-3 m. What is the thickness of the sheet of paper? Δx = (2.56x10-3) / 10 = 2.56x10-4 m λ = 512x10-9 m L = 60mm = 60x10-3 m D = ? Δx = λL / 2D (2.56x10-4) = ((512x10-9 )x(60x10-3)) / (2xD) D = 0.06x10-3 m Application of Wedge Fringes By the use of the above formula, it is possible to use Wedge Fringes to measure the size of very small objects accurately :- 1. By using Microscope slides and a human hair to separate them, it is possible to find the diameter of a human hair. 2. If a crystal is placed between the ends of the plates and heated, it is possible to measure the thermal expansion by observing the increase in fringe separation.	{"url":"https://www.physicsscotland.co.uk/classes/advanced-higher-physics/interference-amplitude","timestamp":"2024-11-12T08:59:14Z","content_type":"text/html","content_length":"510077","record_id":"<urn:uuid:ebf03f54-0730-4469-aa41-9e2e46026115>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00181.warc.gz"}
What Is Flight of Stairs \| How Many Flight of Stairs Per FloorWhat Is Flight Of Stairs \| How Many Flight Of Stairs Per Floor What Is Flight of Stairs \| How Many Flight of Stairs Per Floor What Is a Flight of Stairs? The flight of stairs is a set of steps between the two floors or the two landings. A flight of stairs is also known as a stairwell, stairway, just stair, or staircase. Flight of Stair Flight is a continuous series of steps between the floor and landing, or between landing. A flight should possess no fewer than 3 risers and no more than 15 risers in it otherwise it might be risky, especially for the young or the elderly. Stairs can be defined as series of steps suitably arranged to connect various floors of a building. It may similarly be defined as an arrangement of treads, risers, stringers, newel posts, handrails, and balusters, so designed and constructed to give easy and quick access to the various floors. It must be designed to transmit certain loads, which are comparable to those utilized for the design of the floors. Stairs may be created of Timber, Bricks, Stone, Steel, or Reinforced Cement How to Calculate flight of stairs To calculate the number of steps in a flight of stairs, you need to know the height of the landing, the height of each step, and the length of the tread. The formula for calculating the number of steps is: Steps = (Height of landing) / (Height of step) The height of each step is determined by the rise, which is the vertical distance between each step, and the tread, which is the horizontal surface area on which you step. The standard rise is typically around 7 inches and the standard tread is around 11 inches. Once you have calculated the number of steps, you can determine the total length of the flight of stairs by multiplying the number of steps by the length of the tread. This will give you the total distance that the staircase will run, which is important for determining the amount of space required for the staircase and for ensuring that it meets local building codes and safety regulations. Read More: Components of Staircase \| Parts of Staircase Stairs Require In A Stairs Flight The flight is a continuous series of stairs between the landings. If there are too many stairs (or steps) in a single flight (without existing broken up by landings) it can be boring to walk up, disorientating to walk down, and if there is a fall, increases the risk that there will be a considerable injury. Most of the flights of stairs average out at 12 steps or 13 steps but it depends on the height of the staircase, the location of the stairs (as stair height regulations vary between public and private buildings and between countries), and the intention of a staircase (as fire escapes have more specific rules than other types of stairs). Some Facts About Stairwell The excellent height of rising 17 cm but the height of rising of steps in ancillary rooms may be up to 22 cm. To design the stair, you have to understand the tread-to-rise ratio. The ratio of rise and tread is indicated as, m = a/s m = ratio of rising and tread a = width of tread s = height of rising. We construct any stair after specifying the ratio of rising and tread in advance. The staircases have traditionally been constructed of wood, stone, or marble, and iron or steel. The usage of steel and reinforced concrete has created it possible to design better-looking staircases and add aesthetics to them by creating good curves and additional features. Escalators and ladders are considered unique types of stairs. There are some options for the stair too. These are elevators (also known lifts), stair lifts, inclined moving walkways, and ramps. Total Steps In A Flight There are various ways to compute the number of steps you will require, but the simplest is to take the whole ceiling height, including the width of the joists helping the floor above and the thickness of the subfloor. • A flight of stairs in a house with 8 feet ceilings will have 14 steps if the rise of each step is 7 ¾ inches. • If ceilings are 9 feet, then a flight of stairs will have 16 steps if the rise of each step is 7 ¾ inches. • If the ceiling is 10 feet, then a flight of stairs will have 17 steps if the rise of each step is 7 ¾ inches. Total Steps For 8 Feet Ceilings For example, your house has 8 feet ceilings, with 10 inches-wide joists supporting the floor above. 10 inches wide joists are 9¼ inches wide. You similarly have a ¾ inches subfloor on top of those Therefore, the total rise from the ground to the top floor of your flight of stairs is 8’ 10”. Transform that total into inches to get 106 inches. We will utilize the maximum permitted stair rise per the IRC which is 7.75 inches. Divide 106 inches by 7.75 inches and you get 13.7 inches. Therefore, you will need 14 steps. Divide 106 inches by 14 to discover the rise per step and you get 7.6 rises per step. How Long Is A Stairs Flight? The length of stairs can differ widely since tread depths frequently observe a wider variance than riser height. Since vertical drop is more of a safety concern per step than tread depth, builders frequently manipulate tread depth to fit horizontal space. The lowest tread depth of a residential step is 10 inches. Each tread must possess a nosing that is at least ¾ inches beyond the front perimeter of the step and no more than 1 ½ inches. There is no extreme tread length. Computing the length of a flight of stairs involves recognizing the depth of your tread and the length of the stair tread nosing. The lowest tread nosing is ¾inches. Therefore, if you have 10 inches treads, subtract ¾inches from true horizontal span so, 9¼ inches. Read More: Dog-Legged Staircase \| Dog-Legged Staircase Design \| Dog Leg Stairs How Tall Is A Flight Of Stairs? A flight of stairs is generally the height of the ceiling plus the framing and subfloor of the floor above where the stairs end. In a space with 8 feet ceilings, a flight of stairs is anywhere from 8 feet 8 inches to just over 9 inches high. Houses with ceilings that are 9 feet or 10 feet will have stairs that are either 10 feet or 11 feet, respectively. Similarly, the real height depends on the floor framing and, to a lesser extent, the subfloor on the top floor. To compute the stair height, take the ceiling height and add the width of the joists that frame the ground where the height of the stairs terminates. Generally, this is either 8 inches, 10 inches, or 12 inches dimensional. Then you have to add the thickness of the subfloor generally between ½inches and ¾inches. Most of the stairs are framed during construction and are computed without the finish floor in place, which is why we just compute the subfloor in the computations. The landing at the top of a flight of stairs does not compute as a stair “tread”. There is often one additional riser than stair tread because a flight of stairs starts up with a riser and ends with a riser. You May Also Like	{"url":"https://civiconcepts.com/blog/flight-of-stairs","timestamp":"2024-11-04T18:16:41Z","content_type":"text/html","content_length":"496741","record_id":"<urn:uuid:0cdd4cf6-1eb7-44fc-8e4c-a0c18256406e>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00216.warc.gz"}
Strategic Human Resource Management and Corporate Performance 1. Introduction In 2005 the European Commission (2005) explicitly stated that ‘‘well developed human resources in R & D are the cornerstone of advancement in scientific knowledge, technological progress, enhancing the quality of life, ensuring the welfare of European citizens and contributing to Europe’s competitiveness’’. This acknowledgement derives from already existing EU research policies. With the coming of information and knowledge period, human capital is dominating the direction and process of the company. Therefore, there is a problem to be solved. How can we ensure the steady accumulation of human capital in the continuous development of the company? The accumulation of human capital can be done in two ways. On the one hand, each person’s intelligence and skills will improve in economic activities without the pressure of production. On the other hand, we can accumulate the human capital through learning-by-doing in work, attending training courses, having foreign exchange learning. In contemporary society, when employees take part in the company, plenty of companies will use the second way to improve their skills. So how do we manage human capital in organizations? By acknowledging this situation，more and more scholars study human capital through the following four aspects. With regular and random networks being considered, an agent model has been introduced to investigate the influence of the agents’ social network [1] . By overcoming the traditional fragmentation of specific research field, the case of the Theme Teams has been analyzed to develop the researchers’ human capital [2] . A brief overview of human capital research that summarizes its evolution and current areas of emphasis has been first demonstrated [3] . Consequently, it is necessary to generalize “traditional” network theory by developing a frame work and associated tools to study multilayer systems in a comprehensive fashion [4] . We find no empirical analysis of the relationship for human capital and multilayer networks. However, we can study the relationship between them from the following four aspects. 1) We need to calculate the human capital of each department in ICM. And through the objective link of human capital, we can establish the network model based on the human capital. 2) Based on Question one, we need to establish a mathematical model to describe the dynamic process of human capital network. The dynamic process consists of organizational churn and direct and indirect effects on the organization’s productivity. Next, we analyze our organization’s budget requirements for talent management in terms of σ for both recruiting and training over the next 2 years through the dynamic model. 3) By using the dynamic model in Question two, we can study whether ICM can sustain position’s integrity rate of 80% or not when the annual churn rate for all positions reaches 25%? How about churn rate of 35%? And we analyze the costs of these higher turnover rates and the indirect effects caused by these high churn rates. 4) We should simulate the impact on the junior managers and experienced supervisors when churn rate arrives at 30%. And then we have to explain the impact of the HR health condition of the organization in that situation to the ICM HR supervisor. 2. The Overall Analysis of the Problem Building an organization filled with good, talented, well-trained people is one of the keys to success. In order to achieve this, an organization not only needs to recruit and hire the best candidates, but also they need to retain good people, keep them properly trained and place them in proper positions, and eventually target new hires to replace those leaving the organization. All members in the team play a specific role. Therefore, the departure of individuals from an organization leaves important informational and functional components missing that need to be replaced. Through analysis, we can know that there is an objective link among employees. So we can construct a complex network model to study the impact on the entire complex relationship network system of ICM when staff turnover occurs. What’s more, we can take advantage of large complex network analysis tool to dynamically analyze and deal with the whole complex network model. And we can eventually give advice to the human resource management of ICM. At last, we show our research ideas through this flow chart (Figure 1). 3. Symbol Descriptions (Table 1) Table 1. Variables and definitions. Chart source: organize by the author. 4. Human Capital Network Model of ICM Organization 4.1. The VP Branch to Demonstrate the Complex Network For question one, to make it easier to demonstrate the method of building a Figure 1. The flow chart of the analysis of problems. Chart source: organize by the author. complex network, we select a branch of ICM including each level department as an example, so that we can analyze the general situation of human capital network in ICM. We define the total cost of company’s paying for employees, average annual salary rate of employees and the value of the employees’ returning to the company, and then we build their interrelationships to calculate the human capital value. 4.2. The Calculation of Each Department’s Human Capital Value 4.2.1. Definition of the Human Capital Value We believe that companies will pay a certain costs when hiring employees. Besides, we also need to spend some expenditure on training, education, health care and other related investment items in order to make employees give better service to our company. This part of expense together with recruitment cost forms the total cost of company’s spending on employees. Employees in different positions will produce different values. A part of the values will feed back to employees themselves, while the rest of the values will return to our company. It is worth mention that the part of values returning to company should be greater than the total cost of company’s paying for employees to ensure the input-output regulations, which is shown in Figure 2. According to the mentioned above, we can obtain the relational expression below. The return of employees + Average annual salary = Human capital For simplicity, we set $b+As=x$ that $b>a$ . We define two rates to represent the employees’ satisfaction rate and company’s satisfaction rate. ${\alpha }_{ij}=\frac{As}{x}$$\left(i=1,2,3,4,5;j=1,2,3,4,5,6,7\right)$ , Figure 2. The allocation of human capital. Chart source: organize by the author. Among the equation, i represents the order of hierarchy, j represents the order of department in the floor of i. $\beta =\frac{b-a}{a}$ , when employees are not satisfied with their present condition in our company, they will possibly leave the company. So, employees’ satisfaction can be expressed as follows, ${\alpha }_{ij}=1-\delta$ . According to the relationships above, the human capital value can be expressed as follows, $x=\frac{{\alpha }_{ij}\beta +a}{1-{\alpha }_{ij}}$ . 4.2.2. The Assumption of the Position Status of Each Department We assume that we have already known the position status of each department in the hierarchical structure example (rigid structure mentioned in the title) we have selected just now. In modern society, we pursue for maximizing the work efficiency and minimize the cost of human capital investment in the enterprise, so the setting of position in each department is relatively stable. Combining with the actual business administration science, we make reasonable assumptions on the quantity of position of departments in each hierarchy of the selected branch, which is showed as Table 2 and Table 3. We will figure out the human capital value of each department in the selected branch based on the calculation formula of the human capital value. As for the calculation of a, we should combine with Table 1 in the title, and convert the time cost on the recruitment into opportunity cost measured by money, which can be shown as follows, $a={n}_{ij}×\left(m×\frac{As}{12}+Mc+At\right)$ . 4.3. Fuzzy Synthetic Evaluation Model of Euclidean Distance’s Calculation Fuzzy mathematics is the mathematics researching and treating with fuzziness Table 2. The personnel assignment of the VP branch (a). Chart source: organize by the author. Table 3. The personnel assignment of the VP branch (b). Chart source: organize by the author. phenomena. With the development of the society, the problems that people research and solve by math are increasingly complex, which is hard to become precise [5] . So it’s hard to research and describe the internal link between different matters by accurate math. The human capital in the enterprise management is an evaluation problem involved in the multiple factors. In addition, there will be inevitable fuzziness problem in the process of evaluation. Therefore, we should make fuzzy comprehensive evaluation of human capital value and then figure out the close degree between different departments, so that we can draw the complicated relation among various departments. 4.3.1. Membership We can eliminate the subjectivity occurred in the process of evaluation and the fuzziness phenomena encountered objectively by fuzzy comprehensive evaluation. The fuzzy comprehensive evaluation is usually carried out as follows. Establish judgment sets namely $U=\left\{{u}_{1},{u}_{2},{u}_{3},\cdots ,{u}_{n}\right\}$ . For example, if we want to ascertain the evaluation result of human capital value, we can describe the judgment sets as $U=\left\{\text{proximate},\text{approximate},\text{dis-approximate}\right\}.$ We respectively describe the membership degree of each element relative to judgment sets U with membership degree, and consequently reach the evaluation matrix of single factor. ${D}_{i}=\left\{\begin{array}{cc}\begin{array}{l}{v}_{i1}\\ {v}_{i2}\\ \text{\hspace{0.17em}}⋮\\ {v}_{ik}\end{array}& \left[\begin{array}{cccc}{S}_{11}^{i}& {S}_{12}^{i}& \cdots & {S}_{1n}^{i}\\ {S}_ {21}^{i}& {S}_{22}^{i}& \cdots & {S}_{2n}^{i}\\ ⋮& ⋮& \ddots & ⋮\\ {S}_{k1}^{i}& {S}_{k2}^{i}& \cdots & {S}_{kn}^{i}\end{array}\right]\end{array}\right\}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\ text{\hspace{0.17em}}\left(i=1,2,\cdots ,m\right)$ Among the evaluation matrix, ${S}_{ij}^{i}$ represents the first grade indicator of i’s second grade of l’s membership degree of opinion rating, i represents the amount of the first grade indicator, k represents the second grade’s amount of the first grade of i and n represents the amount of judging concentration comments. The meaning and method for evaluating of ${S}_{lj}^{i}$ are as follows, ${S}_{lj}^{i}=\frac{{v}_{ijk}}{\underset{k=1}{\overset{n}{\sum }}{v}_{ijk}}=\frac{{v}_{ijk}}{{v}_{ij1}+{v}_{ij2}+\cdots +{v}_{ijn}}$ . Above all, we should judge every factor and then aggregate several remarks of every sub factor ${v}_{ij}$ by statistics. The remarks consist of ${u}_{1}$ grade remark with the amount of ${v}_{ij1}$ , ${u}_{2}$ grade remark with the amount of ${v}_{ij2}$ and ${u}_{n}$ grade remark with the amount of ${v}_{ijn}$ . So the degree of sub factor floor’s indicator attaching to the ${u}_{k}$ grade remark is membership degree. Thus the membership degree’s vector quantity of the sub factor floor’s indicator ${v}_{ij}$ is as follows, ${S}_{l}^{i}=\left({s}_{l1}^{i},{s}_{l2}^{i},\cdots ,{s}_{ln}^{i}\right)$ . 4.3.2. First Class Fuzzy Comprehensive Evaluation We use fuzzy operator to ascertain fuzzy relation matrix $R={\left({R}_{1},{R}_{2},\cdots ,{R}_{n}\right)}^{\text{T}}$ , among the matrix ${R}_{i}=\left({w}_{1}^{i},{w}_{2}^{i},\cdots ,{w}_{k}^{i}\right)\left[\begin{array}{cccc}{S}_{11}^{i}& {S}_{12}^{i}& \cdots & {S}_{1n}^{i}\\ {S}_{21}^{i}& {S}_{22}^{i}& \cdots & {S}_{2n}^{i}\\ ⋮& ⋮& \ddots & ⋮\\ {S}_{k1}^{i}& {S}_{k2}^{i}& \cdots & {S}_{kn}^{i}\end{array}\right]=\left({r}_{i1},{r}_{i2},\cdots ,{r}_{in}\right)\left({w}_{1}^{i},{w}_{2}^{i},\cdots ,{w}_{k}^{i}\right)$ is the ranking weight vector of the first grade indicator’s subordination to the second grade indicator. 4.3.3. The Opinion Rating of the Evaluation Object According to the maximum membership principle, we ascertain the opinion rating of the evaluation object. If ${e}_{k}=\mathrm{max}\left({e}_{1},{e}_{2},\cdots ,{e}_{k},\cdots ,{e}_{n}\right)$ , then ${e}_{k}$ is the vector quantity’s K’th component. According to the maximum membership principle of fuzzy mathematics, the evaluation result of the evaluation object belongs to grade of K. At last, we figure out the Euclidean distance between different grades. ${d}_{ij}=1-\sqrt{\frac{{\sum }_{k}^{K}{\left({a}_{ik}-{v}_{ki}\right)}^{2}}{K}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(i=1,2,3,4,5;j=1,2,3,4,5,6,7\right)$ From the calculation of steps mentioned above, we obtain the close degree basing on the human capital value of selected various department, the tidied results is showed as Table 4. 4.4. The VP’s Branch Network of Human Capital We establish the fuzzy synthetic evaluation model to calculate Euclidean distance. Besides, by the programming of Pajek, we use the powerful visualization tools to build three-dimensioned reference frame. We can see it clearly as follows (Figure 3). 5. Dynamic Complex Network Model of Human Capital In the first task, we have established the human capital network model. For enterprises, human capital flows into the (I), through the (II) and outflows (III) of three kinds of main forms. We can see this process in the model below (Figure 4). In this model, the ICM and its individuals have experienced three stages all the process. ICM has the right to fire employees. Individuals also have rights to be a member of ICM. But for ICM staff, when their satisfaction is lower than a certain value, they have the right to choose to leave. At this time, the churn rate for all positions will rise. As is known to us, it is not easy to estimate the interpersonal relationship. When people leave for other jobs, other employees will be influenced. It will cause higher staff turnover rate and reduce the efficiency of organization. Therefore, human capital network can be described as a dynamic process, Table 4. The Euclidean distance of staff offices. Chart source: organize by the author. Figure 3. The VP’s Branch network of human capital. Chart source: organize by the author. Figure 4. The human capital flow process. we can build dynamic complex networks to study the flow of the organization, and the effects of the flow on the efficiency of organization. 5.1. Time-Varying Complex Dynamical Network Model and Preliminaries In this section, we consider a complex dynamical network consisting of N nonlinearly coupled different nodes, with each node being an ${n}_{i}$ -dimensional $\left(i=1,2,\cdots ,N\right)$ dynamical system [6] . The proposed time-varying dynamical network is described as, ${\stackrel{˙}{x}}_{i}={f}_{i}\left({x}_{i}\right)+\underset{\begin{array}{l}j=1\\ je i\end{array}}{\overset{N}{\sum }}{c}_{ij}\left(t\right){H}_{i}\left({x}_{i}\right)\left({\phi }_{j}\left({x}_{j}\ right)-{\phi }_{i}\left({x}_{j}\right)\right)+{G}_{i}\left({x}_{i}\right){u}_{i}$$\left(i=1,2,\cdots ,N\right)$(1) where ${x}_{i}={\left({x}_{i1},{x}_{i2},\cdots ,{x}_{i{n}_{i}}\right)}^{\text{T}}\in {\Re }^{{n}_{i}},{u}_{i}\in {\Re }^{m}$ are the state vector and the control input of node i, respectively. For $i =1,2,\cdots ,N$ , ${f}_{i}\left({x}_{i}\right)\in {\Re }^{{n}_{i}}$ are sufficiently smooth nonlinear vector fields, ${\phi }_{i}\left({x}_{i}\right):{\Re }^{{n}_{i}}\to {\Re }^{{n}_{1}}$ are sufficiently smooth nonlinear vector mappings, ${H}_{i}\left({x}_{i}\right)\in {\Re }^{{n}_{i}×{n}_{1}}$ and ${G}_{i}\left({x}_{i}\right)\in {\Re }^{{n}_{i}×m}$ are continuous nonlinear function matrices. Denote $C\left(t\right)={\left({c}_{ij}\left(t\right)\right)}_{N×N}$ as the outer coupling configuration matrix (OCCM) representing the coupling strength and the topological structure of the network (1) at time t, in which ${c}_{ij}\left(t\right)e 0$ if there is a connection from node i to node j $\left(je i\right)$ , otherwise ${c}_{ij}\left(t\right)=0\left(je i\right)$ , and all the coupling coefficients ${c}_{ij}\left(t\right)\left(i,j=1,2,.\cdots ,N\right)$ are bounded, that is, there exists a positive constant $\delta$ such that $\|{c}_{ij}\left(t\right)\|\le \delta$ , (2) where $\delta$ is called as the coupling coefficients common bound. Assumption 1. Consider the parameters given in the time-varying network (1). Without loss of generality, the first node is taken as the reference node. There exist N state feedbacks ${u}_{i}={\alpha }_{i}\left({x}_{i}\right)+{B}_{i}\left({x}_{i}\right){v}_{i}\text{\hspace{0.17em}}\left(i=1,2,\cdots ,N\right)$ satisfying $\frac{\partial {\phi }_{i}\left({x}_{i}\right)}{\partial {x}_{i}}\left({f}_{i}\left({x}_{i}\right)+{G}_{i}\left({x}_{i}\right){\alpha }_{i}\left({x}_{i}\right)\right)={f}_{1}\left({\phi }_{i}\left ({x}_{i}\right)\right)+{G}_{1}\left({\phi }_{i}\left({x}_{i}\right)\right){\alpha }_{1}\left({\phi }_{i}\left({x}_{i}\right)\right)$(3) $\frac{\partial {\phi }_{i}\left({x}_{i}\right)}{\partial {x}_{i}}{G}_{i}\left({x}_{i}\right){B}_{i}\left({x}_{i}\right)={G}_{1}\left({\phi }_{i}\left({x}_{i}\right)\right){B}_{1}\left({\phi }_{i}\ left({x}_{i}\right)\right)$ , (4) $\frac{\partial {\phi }_{i}\left({x}_{i}\right)}{\partial {x}_{i}}={\left(\frac{\partial {\phi }_{ik}\left({x}_{i}\right)}{\partial {x}_{ij}}\right)}_{{n}_{1}×{n}_{i}}\left(j=1,2,\cdots ,{n}_{i};k= 1,2,\cdots ,{n}_{1}\right)$ is the Jacobi matrix with dimension ${n}_{1}×{n}_{i}$ , ${\alpha }_{i}\left({x}_{i}\right)$ are m-dimensional smooth nonlinear mappings and ${B}_{i}\left({x}_{i}\right)$ are $m×m$ invertible smooth nonlinear function matrices. Remark 1. It is not difficult to find the state-feedback controllers ${u}_{i}={\alpha }_{i}\left({x}_{i}\right)+{B}_{i}\left({x}_{i}\right){v}_{i}$ satisfying (3) and (4). Firstly, for the given ${\ phi }_{i}\left({x}_{i}\right)$ and ${G}_{i}\left({x}_{i}\right)$ , and the invertibility of ${B}_{i}\left({x}_{i}\right)$ and ${B}_{1}\left({\phi }_{i}\left({x}_{i}\right)\right)$ , we can find the compatible ${B}_{i}\left({x}_{i}\right)$ ; ${B}_{1}\left({\phi }_{i}\left({x}_{i}\right)\right)$ and ${G}_{i}\left({\phi }_{i}\left({x}_{i}\right)\right)$ satisfying (3). Then, substituting ${G}_{i}\ left({\phi }_{i}\left({x}_{i}\right)\right)$ into (4), the suitable ${\alpha }_{i}\left({x}_{i}\right)$ and ${\alpha }_{i}\left({\phi }_{i}\left({x}_{i}\right)\right)$ can be obtained. In general, many controllers ${u}_{i}$ can satisfy (3) and (4). 5.2. Exponential Synchronization for the Time-Varying Network In real-world networks, although it is often difficult to obtain the exact information about coupling coefficients, it is easy to get the coupling coefficients common bound. Hence, the objective of this section is to synthesize appropriate decentralized state-feedback controllers ${u}_{i}\left(i=1,2,\cdots ,N\right)$ for network (1), which guarantee the network (1) can realize exponential synchronization [7] . For simplicity, denote ${y}_{i}={\phi }_{i}\left({x}_{i}\right)$ and the synchronous errors between node i and node j be as follows, ${e}_{ij}={\phi }_{i}\left({x}_{i}\right)-{\phi }_{j}\left({x}_{j}\right)={y}_{i}-{y}_{j}\text{\hspace{0.17em}}\left(i,j=1,2,\cdots ,N\right)$ . (5) Assumption 1. The coupling coefficients common bound is available, that is, the parameter $\delta$ in (2) is known. The time-varying complex dynamical network (1) can achieve global exponential synchronization with the following decentralized state-feedback controllers, ${u}_{i}={\alpha }_{i}\left({x}_{i}\right)+{B}_{i}\left({x}_{i}\right){v}_{i}$ , (6) ${v}_{i}={\left({B}_{1}\left({\phi }_{i}\left({x}_{i}\right)\right)\right)}^{-1}\left[-{\alpha }_{1}\left({\phi }_{i}\left({x}_{i}\right)\right)-\left(2k\delta +\eta \right){\left({G}_{1R}\left({\phi }_{i}\left({x}_{i}\right)\right)\right)}^{-1}{\phi }_{i}\left({x}_{i}\right)\right]$ . (7) where $i=1,2,\cdots ,N$ , $k,\eta \in \Re$ are both designing positive constants. Proof. For simplicity, notice that ${e}_{ii}={\phi }_{i}\left({x}_{i}\right)-{\phi }_{i}\left({x}_{i}\right)=0$ . We can get the error dynamics for the networks (1) with controllers (5) and (7) as $\begin{array}{c}{\stackrel{˙}{e}}_{ij}=\frac{\partial {\phi }_{i}\left({x}_{i}\right)}{\partial {x}_{i}}{\stackrel{˙}{x}}_{i}-\frac{\partial {\phi }_{j}\left({x}_{j}\right)}{\partial {x}_{j}}{\ stackrel{˙}{x}}_{j}\\ =\frac{\partial {\phi }_{i}\left({x}_{i}\right)}{\partial {x}_{i}}\left[{f}_{i}\left({x}_{i}\right)+{G}_{i}\left({x}_{i}\right){\alpha }_{i}\left({x}_{i}\right)+{G}_{i}\left({x} _{i}\right){B}_{i}\left({x}_{i}\right){v}_{i}+\delta \underset{j=1}{\overset{N}{\sum }}{c}_{ij}{H}_{i}\left(x{}_{i}\right){\phi }_{j}\left({x}_{j}\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace {0.17em}}-\frac{\partial {\phi }_{j}\left({x}_{j}\right)}{\partial {x}_{j}}\left({f}_{1}\left(s\right)+\delta \underset{j=1}{\overset{N}{\sum }}{c}_{ij}{\stackrel{˜}{H}}_{j}\left({x}_{j}\right)s+{G}_ {j}\left({x}_{j}\right){\alpha }_{i}\left({x}_{i}\right)+{G}_{i}\left({x}_{i}\right){B}_{i}\left({x}_{i}\right){v}_{i}\right)\\ ={f}_{1}\left({\phi }_{i}\left({x}_{i}\right)\right)+{g}_{1}\left({\phi }_{i}\left({x}_{i}\right)\right)\left({\alpha }_{1}\left({\phi }_{i}\left({x}_{i}\right)\right)+{\beta }_{1}\left({\phi }_{i}\left({x}_{i}\right)\right)v{}_{i}\right)\\ \text{\hspace{0.17em}}\text{\ hspace{0.17em}}-{f}_{1}\left(s\right)+\delta \underset{j=1}{\overset{N}{\sum }}{c}_{ij}{\stackrel{˜}{H}}_{i}\left({x}_{i}\right)\left({\phi }_{j}\left({x}_{j}\right)-s\right)\\ ={f}_{1}\left({\phi }_ {i}\left({x}_{i}\right)\right)-{f}_{1}\left(s\right)+{d}_{i}{g}_{1}\left({\phi }_{i}\left({x}_{i}\right)\right){g}_{1}^{\text{T}}\left({\phi }_{i}\left({x}_{i}\right)\right){e}_{i}+\delta \underset{j =1}{\overset{N}{\sum }}{c}_{ij}{\stackrel{˜}{H}}_{i}\left({x}_{i}\right){e}_{j}.\end{array}$(8) where $i,j=1,2,\cdots ,N$ . Choose the Lyapunov functional candidate as, $V=\underset{i=1}{\overset{N}{\sum }}\left(\frac{{e}_{i}^{\text{T}}{e}_{i}}{2}+\frac{{\left({d}_{i}+d\right)}^{2}}{2{k}_{i}}\right)$ . (9) Then, with Equation (8), the time derivative of $V\left(t\right)$ along the trajectories of error dynamics (9) is $\begin{array}{c}\stackrel{˙}{V}=\underset{i=1}{\overset{N}{\sum }}{e}_{i}^{\text{T}}{\stackrel{˙}{e}}_{i}+\underset{i=1}{\overset{N}{\sum }}\frac{\left({d}_{i}+d\right)}{{k}_{i}}{\stackrel{˙}{d}}_ {i}\\ \le \mu \underset{i=1}{\overset{N}{\sum }}{e}_{i}^{\text{T}}{e}_{i}-d\underset{i=1}{\overset{N}{\sum }}{e}_{i}^{\text{T}}{g}_{1}\left({\phi }_{i}\left({x}_{i}\right)\right){g}_{1}^{\text{T}}\ left({\phi }_{i}\left({x}_{i}\right)\right){e}_{i}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\delta \underset{i=1}{\overset{N}{\sum }}\underset{j=1}{\overset{N}{\sum }}\frac{{e}_{i}^{\text{T}}{\ stackrel{˜}{H}}_{i}\left({x}_{i}\right){\stackrel{˜}{H}}_{i}^{\text{T}}\left({x}_{i}\right){e}_{i}+{c}_{ij}^{2}{e}_{j}^{\text{T}}{e}_{j}}{2}\\ \le \underset{i=1}{\overset{N}{\sum }}{e}_{i}^{\text{T}} \left[\left(\mu +\frac{\delta \stackrel{˜}{c}}{2}\right){I}_{n1}-d{g}_{1}\left({\phi }_{i}\left({x}_{i}\right)\right){g}_{1}^{\text{T}}\left({\phi }_{i}\left({x}_{i}\right)\right)+\frac{\delta N}{2} According to Equation (5) and Equation (10), it can be deduced that $\stackrel{˙}{V}<0$ . 5.3. Numerical Example The Hindmarsh-Rose (HR) dynamical system is a well-known model of neuronal activity [8] , which can exhibit rich firing behaviors [7] . In what follows, we consider a network with N = 10 different-dimensional HR neurons, where the dynamics of each uncontrolled isolated node is as follows, ${\stackrel{˙}{x}}_{1}={f}_{1}\left({x}_{1}\right)=\left[\begin{array}{c}a\left({x}_{12}-{x}_{11}\right)\\ -{x}_{11}{x}_{13}+{c}_{1}{x}_{12}\\ {x}_{11}{x}_{12}-b{x}_{13}\end{array}\right]$ , ${\ stackrel{˙}{x}}_{i}={f}_{i}\left({x}_{i}\right)=\left[\begin{array}{c}a\left({x}_{i2}-{x}_{i1}\right)+{x}_{i4}\\ -{x}_{i1}{x}_{i3}+{c}_{2}{x}_{i2}\\ {x}_{i1}{x}_{i2}-b{x}_{i3}\\ {x}_{i1}{x}_{i3}+d{x} _{i4}\\ \mathrm{arctan}{x}_{i5}\\ ⋮\\ \mathrm{arctan}{x}_{i,i+2}\end{array}\right]$ . where a, b, c, d are the constant parameters that govern the dynamics of the neural system. The variable ${x}_{i1}$ is a voltage associated to the membrane potential, variable ${x}_{i2}$ although in principle associated to a recovery current of factions has dimensions of voltage, variable ${x}_{i3}$ is a slow adaptation current associated to slow ions, variable ${x}_{i4}$ represents an even slower process than ${x}_{i3}$ variable, respectively. In order to estimate costs reasonably, we consider hired employees as inflow of human capital in the dynamic network. Then, we allocate the recruitment of personnel according to the original proportion of the various positions. The churn rate of ICM is 18%. Owing to the time-varying complex dynamical network model and exponential synchronization model, the organization’s recruitment rate is 8.6%. Combined with the organization of the ICM position configuration status, we can get the recruitment of staff situation as follows. Combined with Table 1 and Table 5, we can calculate ICM recruitment and training budgets of 34.22σ. 6. Dynamic Simulation Model of the Staff Turnover 6.1. Assumption and Reasonable Explanation of the Model ・ There will be job vacancy due to the employee turnover. Under the circumstances, ICM prefers to recruit new staff whose professional ability is similar with the ones leaving the company from the outside in order to keep the service behavior of work group in ICM as steady as possible. Staff turnover of every enterprise is inevitable and the loss of staff is bound to bring job vacancies. There are usually two methods to fill the job vacancy, external recruitment and promotion of internal employees. The external recruitment has the advantage of being able to hire experienced employees in given position, so that the new employees can involve themselves in this position Meanwhile, the department’s schedule can also operate normally according to Table 5. The number of staff recruitment. Chart source: organize by the author. the previous work pace. But we will take some risks if we promote internal cadres. Because the internal promoted employees are usually inexperienced in a new position and whether he can reach the work efficiency of previous employee or not is unknown. Only from work efficiency’s point of view, it will reduce the work efficiency of the departments and increase the risk of turnover to replace the vacant position with internal promoted employees. Based on the assumption mentioned above, the integrity of the staff is only related to staff turnover rate named $u$ and hiring rate named $\mu$ . 6.2. The Specific of Full Status for Positions On the basis of human capital’s dynamic complex network model, we use the visualization tools of Pajek to make analogue simulation on selected research object’s human capital flow condition [9] . 6.2.1. The Simulation Result When the Changing Rate Is 25% We study whether the research object can maintain position integrity of 80% when the churn rate of the employees reaches 25%. We let the changing churn rate be 25%. Owing to the time-varying complex dynamical network model, we can know the recruitment rate $\mu =10.31%$ at this moment. Next, we let the position’s integrity degree of 80% be a screened threshold value $\delta$ , and we can get the emulation results as Figure 5 shows. In Figure 5, the green points represent the departments that can pass the threshold value. In other words, these parts can maintain the degree of position integrity above 80% on the condition that churn rate is 25%. The red points represent the departments that can’t pass the threshold value. That is to say, these Figure 5. The overall situation of department when staff turnover rate is 25%. Chart source: organize by the author. parts can’t maintain the degree of position integrity above 80% when the churn rate is 25%. From Figure 5, we can reach the research object’s condition of position integrity when the churn rate is 25%, which is shown as Table 6. 6.2.2. The Simulation Result When the Changing Rate Is 35% We study whether the research object can maintain position integrity of 80% when the changing churn rate of the employees is 35%. When the changing churn rate is 35%, according to the time-varying complex dynamical network model, we can know the recruitment rate $\mu =14.66%$ . Besides, we let threshold value $\delta$ be 80%. By the visualization tools of Pajek, we get the emulation results as Figure 6 shows. In a similar way, we can reach the research object’s condition of position integrity when the churn rate is 35%, which is showed as Table 7. Compared with Table 6, the research object can almost not maintain its position integrity of 80% when the churn rate is 35%. We can see that the degree of position integrity is lower with the higher churn rate of human capital. At such a high churn rate of employees of 35%, most of the departments of ICM can’t guarantee its degree of position integrity above 80%, in which case we may think the human resource condition of ICM is very unhealthy at this moment. 6.2.3. The Bad Influence of High Staff Changing Churn Rate on Enterprise From above analysis, the high staff churn rate is of great harm to an enterprise and has a direct effect on the following points: 1) Affecting the staff morale of ICM. The departure of an employee will affect Figure 6. The overall situation of department when staff turnover rate is 35%. Chart source: organize by the author. Table 6. The overall situation of department when staff turnover rate is 25%. Chart source: organize by the author. Table 7. The overall situation of department when staff turnover rate is 35%. Chart source: organize by the author. the morale of several employees because the departed one tends to make negative remarks about the enterprise in public. 2) Increasing the operating costs of ICM. With the departure of the former employees, there must be some new employees needed, which will invisibly increase the recruitment costs and training costs. At the same time the efficiency and job performance of newcomers are low, which makes the operating costs high all the time. 3) Influencing the team efficiency of ICM. Human resource shortages caused by employee churn will directly affect the team efficiency of ICM. 4) Affecting the long-term development of ICM. High rate of staff churn will certainly lead to a decline in the overall level of enterprise’s talent, and consequently lead to the decline of enterprises’ competitiveness. And the decline of the enterprise’s competitiveness is bound to affect the long-term development of the enterprise. 6.3. The Direct and Indirect Costs of High Turnover Rate The high churn rate of staff will certainly bring a high rate of staff turnover and the high turnover rate is bound to increase the cost of human resources. So we can roughly divide this costs into direct costs and indirect costs: 6.3.1. Direct Costs of High Turnover Rate 1) Recruitment costs mainly include: the preparation, screening resumes, interviewing costs, the cost of preparation for hiring and handling the recruitment procedures etc. 2) Training costs mainly include: pre-post training preparation, training materials, training management costs etc. 3) Costs that internal staff to fill job vacancies mainly include: the cost of internal staff to fill job vacancies, additional overtime working costs, the costs that executives coordinate to complete the work etc. 4) Costs during the new employees adapting to the position. When employees come to a new position, there must be an adjustment period. And in this period, the enterprise still needs to pay wages, which undoubtedly increases the enterprise’s costs. 6.3.2. Indirect Costs of High Turnover Rate 1) Costs caused by the flagging morale. The departure of one staff will make the other staff to leave the company like domino. What’s more there will be a process of consideration before they leave the job. During the process, employees will inevitably look for colleagues to discuss and naturally affect the emotion of other employees, which will reduce the efficiency of team cooperation. 2) Costs caused by the lack of reserve forces of enterprise. The frequent staff churn makes it difficult to select the middle managers because of the lack of talents, thereby making the enterprise unable to fill the middle-level position vacancies inside if they select senior personnel from the mid-level position, which will cause a shortage of talent. Eventually it will influence the echelon building of enterprise. 3) Costs caused by the leaks of important secrets. These important secrets are about technology, loss of customer resources, the re-use of management methods etc. 7. The Simulation Model of ICM’s Staff Flow 7.1. Assumption and Reasonable Explanation of the Model ・ Due to the employee turnover, ICM fill the job vacancy only by promoting internal employees without external recruitment. For this assumption, we try to consider ICM as a completely closed system and there is the output but no input of the human capital in this system. So we can better research and analyze the flow condition of internal human resources in ICM. ・ The internal promotion of ICM is in strict accordance with rigid structure of position, that is to say, the position of the upper hierarchy can only be replaced by the position of lower hierarchy just adjacent to the upper one, and there is no Great promotion. As we all know, the condition of the Great promotion is relatively rare in reality. There are certain obstacles among hierarchical structures of enterprise. The obstacles can be divided into ability obstacle, experience obstacle and communication. The bigger the hierarchical span is, the more obviously the obstacles perform, which is the objective law existed in enterprise management. Therefore the assumption is relatively close to reality. For example, the vacant position, senior manager can only be replaced by the lower position or experienced supervisor. In addition, when the employee turnover occurs, the vacant position will be soon filled with personnel qualified to promote. This process doesn’t cost any time and it continues until no second-string employees can be promoted. 7.2. The Simulation Model of Integrity Degree for Junior Managers 7.2.1. The Status Quo of ICM ICM is now facing a serious problem. ICM recognizes that middle managers (junior managers, experienced supervisors, inexperienced supervisors) often feel stuck in their jobs with little opportunity to advance, causing them to leave the company when they find a comparable or better job. These mid-level positions are critical ones that unfortunately suffer high turn-over (twice the average rate of the rest of the company) and seem to need filling all the time. This is the main cause of high churn rate and lower recruitment rate of middle managers at present. Hence we select 5 junior managers and 15 experienced supervisors as object to research. 7.2.2. The Display of ICM Staff’s Flow Simulation Results On the basis of human capital’s dynamic complex network model, we use the visualization tools of Pajek to make analogue simulation of the next two years’ flow condition on 5 junior managers and 15 experienced supervisors of the research objects. In this way, we make simulation of changing condition during the two years on 5 junior managers of research objects. First, we set the churn rate as 30%. Based on the assumptions above, we can know that the present recruitment rate is 0. Second, we set the threshold value (the integrity rate of the bottom employees) as 85%, 70% and 50%， respectively. And then through the simulation by Pajek, we get the results as Figure 7 shows. In Figure 7, the red point means that the position is vacant in the i’th year. If we only consider the employee promotion, then there is no one qualified to fill the position. The green point means that there is someone in the position of the i’th year. 7.3. The Simulation Model of Integrity Degree for Experienced Supervisors In a similar way, we make simulation of changing condition during the two years on 15 experienced supervisors of research objects. First, we set the churn rate as 30%. On the base of the assumptions above, we can know that the present recruitment rate is 0. Second, we set the threshold value as 85%, 70% and 50%, respectively. And then through the simulation of the visualization tools of Pajek, we get the results as Figure 8 shows. 7.4. The Summary of These Two Simulation Models We sort out the results of Figure 7 and Figure 8, and consequently get the Table 8 and Table 9. From the Figure 7, we can obtain the position’s integrity condition of research object named junior managers in different position’s integrity degree, which is shown in Table 8. From the Figure 7, in a similar way, the position’s integrity condition of research object named experienced supervisor has been got. We can see it in Table 9. Under normal conditions, ICM usually has 85% of its 370 positions filled at any time. So we may think it is healthy when the position’s integrity degree of Figure 7. The full status for positions of junior manager. Chart source: organize by the author. Figure 8. The full status for positions of experienced supervisors. Chart source: organize by the author. Table 8. The full status for positions of junior manager. Chart source: organize by the author. Table 9. The full status for positions of experienced supervisor. Chart source: organize by the author. research object is above 85%. However, from the results we can see that when the churn rate reaches 30%, the selected research object is difficult to reach healthy condition even the integrity rate of the bottom employees is merely kept at the lever of 50%. At last, we draw the conclusion that at the churn rate of 30%, the human resource organization is in an unhealthy condition of constant employee turnover without recruitment from the outside. 8. Testing the Model According to the calculation formula of human capital, the value of human capital can be expressed as the following form, $x=\frac{\left(1-\delta \right)\beta +{n}_{ij}×\left(m×\frac{As}{12}+Mc+At\right)}{\delta }$ . Aiming at each factor of the formula, we study the effect on human capital value when these factors change in the same ratio while the other factors keep unchanged. For easy figures, we set each factor’s changing ratio as the same values of 10%. And the each factor’s raw value of the formula is set as follows, $\delta =80%,\beta =70%,{n}_{ij}=4,m=6,As=4\sigma ,Mc=0.7\sigma ,At=0.6\sigma$ . 8.1. The Sensitivity Index of Churn Rate of the Employees to the Human Capital Value The changing churn rate of the employees is ${\delta }^{\prime }=\delta ×\left(1+10%\right)$ . The changing human capital value is ${x}^{\prime }=\frac{\left(1-{\delta }^{\prime }\right)\beta +{n}_{ij}×\left(m×\frac{As}{12}+Mc+At\right)}{{\delta }^{\prime }}$ . The changing ratio of human capital value is ${\lambda }_{1}=\frac{{x}^{\prime }-x}{x}$ . So, the sensitivity index of churn rate of the employees to the human capital value is ${\eta }_{1}=\frac{{\lambda }_{1}}{10%}$ . We calculate and simplify the equation, drawing that ${\eta }_{1}=\frac{-\left[\beta +{n}_{ij}×\left(m×\frac{As}{12}+Mc+At\right)\right]}{1.1\beta ×\left(1-\delta \right)+1.1{n}_{ij}×\left(m×\frac{As}{12}+Mc+At\right)}$ Substituting the given data into calculation, we can draw that ${\eta }_{1}=-\frac{7+132\sigma }{1.54+145.2\sigma }×100%$ . 8.2. The Sensitivity Index of the Satisfaction Rate The changing satisfaction rate of the company is ${\beta }^{\prime }=\beta ×\left(1+10%\right)$ . The changing human capital value is ${x}^{\prime }=\frac{\left(1-\delta \right){\beta }^{\prime }+{n}_{ij}×\left(m×\frac{As}{12}+Mc+At\right)}{\delta }$ . The changing ratio of human capital value is ${\lambda }_{2}=\frac{{x}^{\prime }-x}{x}$ . So, the sensitivity index of the satisfaction rate value is ${\eta }_{2}=\frac{{\lambda }_{2}}{10%}$ . We calculate and simplify the equation, drawing that ${\eta }_{2}=\frac{1.4}{1.4+132\sigma }×100%$ . 8.3. The Sensitivity Index of the Other Five Factors In a similar way, we can obtain that the sensitivity index of the amount of employees of each hierarchy to the human capital value ${\eta }_{3}=\frac{4488\sigma }{1.4+132\sigma }×100%$ . The sensitivity index of median time to recruit (months) to the human capital value is ${\eta }_{4}=\frac{80\sigma }{1.4+132\sigma }×100%$ . The sensitivity index of employees’ average annual salary rate for given lever to the human capital value is ${\eta }_{5}=\frac{80\sigma }{1.4+132\sigma }×100%$ . The sensitivity index of median cost of recruitment to the human capital value is ${\eta }_{6}=\frac{28\sigma }{1.4+132\sigma }×100%$ . The sensitivity index of average annual training cost to the human capital value is ${\eta }_{7}=\frac{24\sigma }{1.4+132\sigma }×100%$ 8.4. The Comparison of the Sensitivity Index’s Absolute Value Comparing their absolute value, we can obtain that ${\eta }_{3}\gg \|{\eta }_{1}\|>{\eta }_{4}={\eta }_{5}>{\eta }_{6}>{\eta }_{7}\gg {\eta }_{2}$ . That is to say, the amount of employees of each hierarchy has greatest influence on human capital value, while the satisfaction rate of the company has the least influence. Besides, the median time to recruit has the same influence as employees’ average annual salary rate on human capital value. It’s clear that the sensitivity index of churn rate of the employees is a negative number. But the other factors’ sensitivity indexes are all positive number. It suggests that the increase of employees’ churn rate will reduce the human capital value. However, the other factors’ influence on the human capital is positive. In combination with the results above, we can draw the conclusion that: to maximize the human capital value, we should mainly focus on distribution of every lever’s quantity of employees carefully. What’s more, we should also try to reduce company’s churn rate of the employees, allocate the recruiting time and employees’ average annual salary rate reasonably, and plan recruitment costs as well as employees’ average annual salary rate properly. To achieve the required goal of human capital value, we can make the same change on the recruitment time or the employees’ average annual salary 9. Final Conclusion Through setting up the dynamic complex network model of human capital, the author draw the conclusion that the next two years’ budget of recruiting and training is 34.22σ when the annual churn rate goes to 18%. Through establishing the dynamic simulation model of the staff turnover, the author reached the position’s integrity condition of ICM when the job churn rate is 25% and 35%, Then the author explained the costs caused by high turnover rates and the indirect effects of high churn rates. The author simulated the change of position’s integrity degree of junior managers and experienced supervisors in the next two years. The author concluded that the HR health of the organization is below the expectation. In addition, the author made sensitivity analysis on the 7 factors of human capital value and consequently drew that the amount of employees of each hierarchy has the greatest influence on human capital value. 10. Limitations and Further Discussion of the Model How to connect our Human Capital network with other layers of organizational network such as information flow, trust, influence and friendship? We have an idea. We know that different employees have different relationships in ICM. This kind of relationship can be formatted, on the base of several levels such as information flow, trust, influence and friendship etc. As the human resources department of ICM, we have the ability to collect the interpersonal networks at different levels and we can let each employee provide their personal networks at different levels, whose form is shown in Figure 9. Figure 9 shows the relationships of an employee in ICM at different levels such as information flow, trust, influence and friendship [4] . By analyzing the interpersonal relationship plate at different levels provided by each employee, we can aggregate the relationships among all the employees in ICM so that we get the interpersonal networks at every level of the whole ICM, as is shown in Figure 10. In Figure 10, different colors of board represent the interpersonal networks at different levels of all employees [4] . At each level such as friendship, we assume that the employee 1 has contact with the employee 2 so we link this two with a solid line. At different levels such as the friendship and trust, we assume that the employee 1 and the employee 2 have a relation in the friendship. In the trust level, Figure 9. The personal networks at four main levels. Chart source: organize by the author. Figure 10. The personal networks at four main levels. Chart source: organize by the author the employee 1 still has connection with the employee 2. Under such circumstances, we will link the employee 1 and the employee 2 at different levels with a dotted line. Thus we can get a three-dimensional interpersonal relationship network of all employees in ICM at different levels. By doing this, not only can we clearly see the interpersonal relationships among employees at each level, but we can also observe the interpersonal relationships of all the staff in ICM at different levels from the three-dimensional point of view. But the model the author built is lack of practicality to some extent especially when facing different companies and organizations. The author didn’t have enough consideration for organizational However, the author can define that the solid line represents strong relation while the dotted line link represents weak relation. When the staff turnover occurs, it is bound to have an impact on interpersonal relationship network related to him. How it affects the others? The effect is strong or weak? We can all analyze these problems by the network.	{"url":"https://scirp.org/journal/paperinformation?paperid=90134","timestamp":"2024-11-08T09:05:35Z","content_type":"application/xhtml+xml","content_length":"262387","record_id":"<urn:uuid:6e1e8637-96d3-4977-8c8e-8a0d517f96ef>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00404.warc.gz"}
Ignore Significance. Embrace Relevance! Listen to the Audio Version here “Although significance testing only dates back to the 1930s, when Fisher propagated it, and although it is logically flawed and clearly retards the growth of knowledge, social science researchers are enslaved by this very bad idea. Even now, with the American Statistical Association's explicit repudiation of the concept, researchers cling to it desperately.” Ed Rigdon, professor at Georgia State University, in a recent LinkedIN post. Wow. That's a big statement. Can it be true? Market researchers and data scientists cling desperately to the concept of "significance"! When looking at market research and analysis results in business, two questions often arise: Is it representative and is it significant? What people want to know is "does it apply to everyone" and "is the effect large"? If we ignore the question of representativeness for a moment and focus on significance, the usual answer - in both science and business - is to quote the p-value. This is also known as significance level or probability of error. At 0.01, the p-value is really good, and in practice, even 0.1 is sometimes accepted as sufficient. Significance is used to distinguish between right and wrong. “Significance is used to judge between right and wrong.“ Let's take a look at the official statement from the American Statistical Association on the p-value: • Principle 1: P-values can indicate whether data are inconsistent with a particular statistical model. • Principle 2: P-values do not measure the probability that the hypothesis being tested is true or the probability that the data occurred by chance alone. • Principle 3: Scientific conclusions or policy decisions should not be based solely on whether a p-value exceeds a certain threshold. • Principle 4: Proper conclusions require full reporting and transparency. • Principle 5: A p-value or statistical significance is not a measure of the magnitude of an effect or the importance of a finding. • Principle 6: A p-value alone is not a good measure of the power of a model or hypothesis. There now seems to be a scientific consensus that significance values are not very useful, if not dangerous. Thanks for reading 10x Insights! Subscribe for free to receive new posts and support my work. Significance can be created at will Language is a poor guide. We all know sentences like "Mr. Y had a significant impact on X". What is meant here is "great impact" or "important impact". But just because something is statistically significant does not mean that it must be important. On the contrary, something that is statistically significant can be very small and irrelevant. In market research jargon, significant means "proven to be true". In science, the term "p-hacking" has emerged. Models and hypotheses are changed and data selected until the P-value is below the desired threshold. P-hacking is common in science. Guess how it is in market research! What do you think? Significance has nothing to do with relevance In practice, almost any correlation becomes significant if the sample is large enough. Significance does not measure the strength of a correlation, but whether it can be assumed to be true or present, assuming models IMPLICIT assumption are true. A strong correlation usually requires a smaller sample to become significant. This phenomenon often leads to the misconception that significance also measures relevance. This is not the case. Every Bavarian is a German, but not every German is a Bavarian. Significance has nothing to do with relevance. The misunderstanding arises from the fact that significant, important effects often have a better p-value than effects that are only supposed to be significant. On the other hand, any supposed effect only needs a sample large enough to be significant. Thus, a minimal effect can be significant. But all this is clear to many market researchers. The real problem lies elsewhere: Significance in itself is not meaningful We have all seen the outrageous examples of correlation, such as that between the "age of the Miss America of a given year" and the "number of murders with steam or other hot objects". With N=8 years, this statistic already has a P-value of 0.001. There is no clearer example of how inappropriate the P-value is for testing a correlation for truth. But why is it still used to judge right and wrong? The answer is varied. Some will say, "The client wants it that way". But the more relevant question is: How do I know if a correlation is trustworthy? To answer that, we should clarify what the p-value is trying to judge: the p-value is trying to judge a) differences between facts OR b) the existence of a causal relationship. Questions about differences are: "Are more customers buying product X or Y?" The result of a survey shows two outcomes that are then compared. A comparison using a significance test is limited when 1. the representativeness is limited. If a smartphone brand surveys only young people, the result will not reflect the population as a whole because older people have different needs. If the sample does not represent the population well, the results will not be accurate. However, only the characteristics of the people that have an impact (see correlation) on the measurement result are relevant. This point is often overlooked. It is common to quote only on the basis of age and gender without checking whether these are the relevant factors for representativeness. 2. the measurement is distorted. I can ask consumers, "Would you buy this phone?" But whether the answer is true (i.e., undistorted) is another matter. A central focus of marketing research in recent years has been the development of valid scales. More recently, implicit measures have been added. The art of questionnaire design is one of them. On the other hand, questions about relationships are: "Do customers in the target group buy my product more than other target groups?" The assumption is that the target group characteristic is the cause of the purchase. A relationship is assumed between consumer characteristics and willingness to buy. It is no longer just a matter of showing the difference between target groups, because that would be the same as a correlation analysis, which, as the example above shows, is a poor form of analysis for relationships. In my opinion, the question of relationships is not discussed enough, although it is of particular importance. The Evidence Score The vague term "relationship" refers to something very specific: a causal relationship. All business decisions are based on it. They are based on assumptions about causality. "If I do X, Y will happen". Discovering, exploring, and validating these "relationships" is what most market research is about (consciously or unconsciously). But whether we can trust a statement about a relationship is determined by the product of the following three criteria: 1. Completeness (C for complete): How many other possible causes and conditions are there that also affect the target variable but have not yet been considered in the analysis? This can be expressed as a subjective probability (a priori probability in the Bayesian sense): 0.8 for "reasonably complete", more likely 0.2 for "actually most are missing", or 0.5 "the most important are included". But why is completeness so important? One example: Shoe size has some predictive power for career success because, for various reasons, men tend to climb the corporate ladder higher on average and have larger feet. If gender is not included in the analysis, there is a high risk of falling for spurious effects. In causal research, this is known as the "confounder problem. Confounders are unaccounted for variables that affect both cause and effect. Even today, most driver models are calculated with "only a handful" of variables, which greatly increases the risk of spurious results. The issue of representativeness is logically related to completeness. Either you ensure a representative sample (which is more or less impossible) and control for confounding factors, or you measure the factors that influence the relationships to be measured (demographics, types of buyers, etc.) and integrate them into the multivariate analysis of the relationships. 1. Correct direction of effect (D - Directed correctly): How certain can we be that A is the cause of B and not the other way around? In many cases, you can rely on prior knowledge, or you may have longitudinal data. Otherwise, statistical methods of "d-separation" (e.g., PC algorithm) must be applied. So again, the question is how subjective is the probability: more like 0.9 for "well, that's well documented" or 0.5 for "well, that could be either way"? 2. Predictive power (P - Prognostic): How much variance in the affected variable is explained by the cause? Effect size measures the absolute amount of variance explained by a variable. In his research, Nobel Laureate G. Granger once stated: In a complete (C), correctly specified (D) model, the explanatory power of a variable proves its direct causal influence. The predictive power should be assessed by a Causal AI algorithm, because only those are optimized to look for causal drivers instead of just indicators. If one of the three variables C, D, or P is missing, the evidence for the relationship is very weak. This is because all three are interdependent. Predictive power without completeness or direction is worthless. Mathematically, all three values can be combined multiplicatively. If one is small, the product is very small: Evidence = C x D x P This evidence is a proven tool in Bayesian information theory and at the same time a practical and useful value for judging a relationship. Moving beyond black-and-white thinking If the Evidence is high, then, yes, you can ask again: what is the significance of the relationship? But this is not about reaching a threshold, because the P-value cannot be the only criterion for accepting a result. The P-value, as a continuous measure, is more informative and tells us how stable the statement is. Nothing more. I'm back on LinkedIN and I read another post by Ed Rigdon presenting his paper on the subject. He writes: “How about just treating P-values as the continuous quantities they are. Don't let an arbitrary threshold turn your P-value into something else, and don't misinterpret it. And while you're at it, remember that your statistical analysis probably overlooked a lot of uncertainties, which means your confidence intervals are probably much too narrow.” The next time a customer asks me what the P-value is, I'm going to tell them, "I don't recommend looking at the P-value as a confidence check. We measure the evidence score these days. Ours is 0.5, and the last model you used to make a high significance decision was only 0.2". The concept of evidence is certainly not as convenient as significance. But I didn't say that truth was easy to come by. Imagine if your entire organization started to wake up from the "significance illusion," started to think more holistically. It would simply make better decisions that have greater impact. That is how you 10x your impact. [1] “How improper dichotomization and the misrepresentation of uncertainty undermine social science research”, Edgar Rigdon, Journal of Business Research, Volume 165, October 2023, 114086 [2] “The American Statistical Association statement on P-values explained”, Lakshmi Narayana Yaddanapudi, www.ncbi.nlm.nih.gov/pmc/articles/PMC5187603/	{"url":"https://10xinsights.substack.com/p/ignore-significance-embrace-relevance-b12","timestamp":"2024-11-08T02:53:06Z","content_type":"text/html","content_length":"165006","record_id":"<urn:uuid:e949eae8-7e85-4f56-8d1f-c54ebd870edb>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00677.warc.gz"}
R programming Archives - Michael's Bioinformatics Blog Genomics is the study of an organism’s complete set of genetic material, including its DNA sequence, genes, and regulation of gene expression. Other “omics” techniques, such as proteomics and metabolomics, focus on the study of proteins and metabolites, respectively. By analyzing these different types of data together, researchers can generate new insights into the inner workings of an organism and how it responds to its environment. For example, by combining genomics data with proteomics and metabolomics data, researchers can gain a more complete understanding of an organism’s gene expression, protein production, and metabolic processes, and how these processes work together to create health, or dysfunction to create disease. This knowledge can provide valuable insights for a wide range of applications, including drug development, disease diagnosis, and environmental monitoring. Finding correlations between related datasets means looking for patterns or relationships between different sets of data. This can provide valuable insights into the underlying biological processes and functions of an organism. For example, if two datasets show a strong positive correlation, it suggests that they are related in some way and that changes in one dataset may be associated with changes in the other. By identifying these correlations, researchers can better understand the mechanisms behind biological processes and how they are regulated. This can be useful for a variety of applications, such as predicting the effects of potential drugs or identifying potential targets for medical intervention. I have surveyed the literature for tools to integrate multiple -omics datasets together. As is the case for any task in bioinformatics, there are dozens of options. However, when I considered criteria such as ease of installation, documentation quality, robust user community, user support and published analyses, I believe that the “mixOmics” package (available for download and installation from Bioconductor) is one of the best tools out there for doing this type of integration analysis. The mixOmics approach The mixOmics package encompasses many different versions of multivariate algorithms for integrating multiple datasets. Multivariate analysis is well-suited to this problem space where there are far more features than samples. By reducing the dimension of the data, the analysis makes it easier for a human analyst to see patterns and interpret correlations. One of the most common types of algorithm in mixOmics for doing this is called “partial least squares.” Fig 1. An overview of the mixOmics package. The methods can handle single ‘omics, multiple ‘omics on the same samples (N-integration), and the same ‘omics on multiple sets of samples (P-integration) to find correlations in the data. Some examples of the graphic outputs are shown in the bottom two panels of the figure. The partial least squares (PLS) method is a mathematical technique used to analyze relationships between two or more datasets. It works by identifying the underlying patterns and correlations in the data, and then using this information to construct a set of “composite” variables that capture the most important features of the data (this is analogous to PCA analysis, but differs by focusing on maximizing correlation/covariance among latent variables). These composite (latent) variables can then be used to make predictions or draw conclusions about the relationships between the datasets. For example, if two datasets are known to be related in some way, the PLS method can be used to identify the specific features of each dataset that are most strongly correlated with the other, and then construct composite variables based on these features. PLS is more robust than PCA to highly correlated features and can be used to make predictions between the dependent and independent variables. mixOmics takes the PLS method a step further by integrating a ‘feature-selection’ option called “sparse PLS” or just “sPLS” that uses “lasso” penalization to reduce unnecessary features from the final model to aid interpretation and also computational time. Lasso regression works by adding a regularization term to the ordinary least squares regression model, which is a measure of the complexity of the model. This regularization term, called the “lasso,” forces the coefficients of the model to be zero for the less important predictors, effectively eliminating them from the model. This results in a simpler and more interpretable model that is better able to make accurate predictions. Lasso regression is particularly useful for datasets with a large number of predictors, as it can help to identify the most important predictors and reduce the risk of overfitting the model. In future posts, I will describe in more detail what can be done with MixOmics and show some results from our own studies that have produced stunningly detailed and intricate correlation networks. If you are interested in this kind of work, I would encourage you to check out mixOmics as a possible avenue for analysis. There are other packages, and many of them are excellent, but the learning curve of mixOmics is quite shallow and it is well supported with a dynamic and active user community. It is also very flexible to different experimental scenarios, so that you can analyze your data several different ways while using the same package and R script. Perturbation analysis of spatial single cell RNA-seq with ‘augur’ Spatial single cell RNA-seq data are essentially regular single-cell RNA-seq data that have spatial coordinates associated through localization on a special capture slide. I had previously used so-called “perturbation” analysis successfully with 10X single-cell data and I wanted to apply the technique to spatial single cell to understand how a treatment affects the spatially-resolved Here, I want to briefly describe the steps I went through to perform ‘augur’ perturbation analysis of 10X Visium Spatial single cell RNA-seq data. augur works as follows: Augur is an R package to prioritize cell types involved in the response to an experimental perturbation within high-dimensional single-cell data. The intuition underlying Augur is that cells undergoing a profound response to a given experimental stimulus become more separable, in the space of molecular measurements, than cells that remain unaffected by the stimulus. Augur quantifies this separability by asking how readily the experimental sample labels associated with each cell (e.g., treatment vs. control) can be predicted from molecular measurements alone. This is achieved by training a machine-learning model specific to each cell type, to predict the experimental condition from which each individual cell originated. The accuracy of each cell type-specific classifier is evaluated in cross-validation, providing a quantitative basis for cell type prioritization. I followed both the Seurat 10X Visium vignette as well as a dataset integration protocol to combine two treatment (a gene knockout, in this case) and control samples (S1 and S2). Normalization was performed by “SCTransform” as recommended for spatial RNA-seq data prior to integration. PCA, K-nearest neighbors, clustering, and uMAP were calculated as described in the Seurat vignette using default values. Cell types were assigned in collaboration with the experimentalists. With the integrated, clustered and, assigned dataset in hand, I was ready to enter the “augur” workflow as described in the paper, with some minor tweaks. First, because this is spatial and not regular scRNA-seq, there is no “RNA” default assay to set after integration. I chose to set “SCT” as the assay instead, because this represents the normalized and scaled dataset which is what you want for input to an ML model. ```{r, celltype_priority} DefaultAssay(s1s2.int) <- "SCT" augur <- Augur::calculate_auc(s1s2.int, label_col = "orig.ident", cell_type_col = "cell_type", n_threads = 6, rf_params = list(trees = 15, mtry = 2, min_n = NULL, importance = "accuracy"), n_subsamples = 25, Above, you can see the actual call to augur “calculate_auc” method. I found that by specifying ‘rf_params’ and reducing the number of trees, I got better separation between cell types in the AUC readout. The calculation takes about 20 minutes to run on a 2018 MacBook Pro 13 inch laptop. When the algorithm completes, you can visualize your results. Using the vignette for regular scRNA-seq you can do this: p1 <- plot_umap(augur, s1s2.int, mode = "default", palette = "Spectral") p1 <- p1 + geom_point(size=0.1) + ggtitle("Augur Perturbation by Type (Red = Most)") p2 <- DimPlot(s1s2.int, reduction = "umap", group.by = "cell_type") + ggtitle("S1/S2 Integrated Cell Types") p1 + p2 The resulting plot looks like this: Augur perturbation analysis by AUC (red is more perturbed; left) and UMAP plot of cell types (right). This is great and helpful, but it doesn’t take advantage of the spatially resolved nature of the data. To do that, you have to modify the integrated seurat object with the augur results: ### Make a dataframe of AUC results auc_tab <- augur$AUC auc_tab$rank <- c(1:9) ### Grab the cells by type and barcode tib <- s1s2.int$cell_type %>% as_tibble(rownames = "Barcode") %>% rename(cell_type=value) ### Join the AUC information to the barcode on cell_type tib <- tib %>% left_join(., auc_tab) ### Sanity check assertthat::are_equal(colnames(s1s2.int), tib$Barcode) ### Update the seurat object with new augur metadata s1s2.int$AUC <- round(tib$auc, 3) s1s2.int$RANK <- tib$rank Here, I am simply pulling out the AUC results into a table by cell type. Then I get the cell type information from the seurat object and merge the AUC information into it. I just set new metadata on the seurat object to transfer information about AUC and Rank for each barcode (i.e., cell). I do a sanity check to make sure the barcodes match (they do, as expected). Now you can plot the spatially resolved AUC information: SpatialDimPlot(s1s2.int, group.by = "AUC", cols = rev(c("#D73027", "#F46D43", "#FDAE61", "#FEE090", "#FFFFBF", "#E0F3F8", "#ABD9E9", "#74ADD1", "#4575B4"))) This takes advantage of the “group.by” flag in the Spatial Dim Plots command to use the AUC metadata. I’m also using a custom color scheme from ColorBrewer that shades the cell types from low to high AUC along a rainbow for ease of viewing. The plot looks like this: Spatially-resolved perturbation (AUC) of cell clusters in the WT (left) and knockout (right) samples. Video tutorial: Bioconductor and NCBI GEO data access From our IIHG Bioinformatics Workshop series: Calculate % mitochondrial for mouse scRNA-seq Seurat is a popular R/Bioconductor package for working with single-cell RNA-seq data. As part of the very first steps of filtering and quality-controlling scRNA-seq data in Seurat, you calculate the % mitochondrial gene expression in each cell, and filter out cells above a threshold. The tutorial provides the following code for doing this in human cells: mito.genes = grep(pattern = "^MT-", x = rownames(x = pbmc@data), value = TRUE) percent.mito = Matrix::colSums(pbmc@raw.data[mito.genes, ])/Matrix::colSums(pbmc@raw.data) pbmc = AddMetaData(object = pbmc, metadata = percent.mito, col.name = "percent.mito") VlnPlot(object = pbmc, features.plot = c("nGene", "nUMI", "percent.mito"), nCol = 3) Creating a catalog of mitochondrial genes by searching with ‘grep’ for any gene names that start with “MT-” works just fine for the human reference transcriptome. Unfortunately, it doesn’t work for mouse (at least for mm10, which is the reference assembly I’m working with). There are two workarounds for this, in my opinion. The easiest is to change the regular expression in the “grep” command from “^MT-” to “^mt-” since a search through the mm10 reference (version 3.0.0) in the cellranger reference files reveals that for whatever reason, the MT genes are labeled with lowercase ‘mt’ instead. A second, and perhaps more thorough, approach is to take advantage of the Broad Institute’s “Mouse Mitocarta 2.0” encyclopedia of mitochondrial genes (note that you could do this same procedure for human MT genes too). By creating a list of the top 100-200 genes with the strongest evidence for MT expression, it seems likely that you more accurately capture true mitochondrial gene expression. Below is some code to use the “MitoCarta 2.0” (downloaded as a CSV file) for this procedure. You will need to import “tidyverse” to work with tibbles: mouse_mito = as.tibble(read.csv("Mouse.MitoCarta2.0_page2.csv", header = TRUE)) mouse_mito = mouse_mito %>% select(c(Symbol, MCARTA2.0_score)) %>% slice(1:100) mito.genes = as.character(mouse_mito$Symbol) mito.genes = mito.genes[mito.genes %in% rownames(sample2@raw.data)] percent.mito = Matrix::colSums(sample2@raw.data[mito.genes,]) / Matrix::colSums(sample2@raw.data) Gene expression boxplots with ggplot2 The ubiquitous RNAseq analysis package, DESeq2, is a very useful and convenient way to conduct DE gene analyses. However, it lacks some useful plotting tools. For example, there is no convenience function in the library for making nice-looking boxplots from normalized gene expression data. There are other packages one can rely on, for example ‘pcaExplorer’, but I like a simple approach sometimes to plot just a couple of genes. So below I show you how to quickly plot your favorite gene using only ggplot2 (there is no “one weird trick” though…): traf1_counts <- counts(ddsTxi['ENSG00000056558',], normalized = TRUE) m <- list(counts = as.numeric(traf1_counts), group = as.factor(samples$group)) m <- as.tibble(m) q <- ggplot(m, aes(group, counts)) + geom_boxplot() + geom_jitter(width = 0.1) q <- q + labs(x = "Experimental Group", y = "Normalized Counts ", title = "Normalized Expression of TRAF1") q <- q + scale_x_discrete(labels=c("00hn" = "PMN, 0hrs", "06hn" = "PMN, 6hrs", "06hy" = "PMN+Hp, 6hrs", "24hn" = "PMN, 24hrs", "24hy" = "PMN+Hp, 24hrs")) As you can see above, first we must grab the normalized counts at the row corresponding with the Traf1 Ensembl ID using the ‘counts‘ function that operates on the ‘ddsTxi’ DESeqDataSet object. In order to create a dataframe (well, a tibble to be specific) for plotting, we first create a list (‘m’) that combines the counts (as a numeric vector) and metadata group. These two vectors will form the columns of the tibble for plotting, and we must give them names (i.e., “counts” and “group”) so the tibble conversion doesn’t complain. The list, m, is then converted to a tibble with ‘as.tibble‘ and plotted with ggplot2, using an ‘aes(group,counts)‘ aesthetic plus a boxplot aesthetic. The rest of the code is just modifying axis labels and tickmarks. The final product looks like this: Boxplot of normalized Traf1 expression in 5 different conditions (3 replicates each).	{"url":"https://www.michaelchimenti.com/category/r/","timestamp":"2024-11-05T07:25:16Z","content_type":"text/html","content_length":"114217","record_id":"<urn:uuid:3d416a50-782e-49c6-a49b-4f07c0592966>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00845.warc.gz"}
Number of digits in prime power reciprocals Reciprocals of prime powers Let p be a prime number. This post explores a relationship between the number of digits in the reciprocal of p and in the reciprocal of powers of p. By the number of digits in a fraction we mean the period of the decimal representation as a repeating sequence. So, for example, we say there are 6 digits in 1/7 because 1/7 = 0.142857 142857 … We will assume our prime p is not 2 or 5 so that 1/p is a repeating decimal. If 1/p has r digits, is there a way to say how many digits 1/p^a has? Indeed there is, and usually the answer is r p^a−1. So, for example, we would expect 1/7² to have 6×7 digits, and 1/7³ to have 6×7² digits, which is the case. As another example, consider 1/11 = 0.09 09 09 … Since 1/11 has 2 digits, we’d expect 1/121 to have 22 digits, and it does. You may be worried about the word “usually” above. When does the theorem not hold? For primes p less than 1000, the only exceptions are p = 3 and p = 487. In general, how do you know whether a given prime satisfies the theorem? I don’t know. I just ran across this, and my source [1] doesn’t cite any references. I haven’t thought about it much, but I suspect you could get to a proof starting from the theorem given here. What if we’re not working in base 10? We’ll do a quick example in duodecimal using bc. $ bc -lq Here we fire up the Unix calculator bc and tell it to set the output base to 12. In base 12, the representation of 1/5 repeats after 4 figures: 0.2497 2497 …. We expect 1/5² to repeat after 4×5 = 20 places, so let’s set the scale to 40 and see if that’s the case. scale = 40 OK, it looks like it repeats, but we didn’t get 40 figures, only 38. Let’s try setting the scale larger so we can make sure the full cycle of figures repeats. That gives us 40 figures, and indeed the first 20 repeat in the second 20. But why did we have to set the scale to 44 to get 40 figures? Because the scale sets the precision in base 10. Setting the scale to 40 does give us 40 decimal places, but fewer duodecimal figures. If we solve the equation 10^x = 12^40 we get x = 43.16… and so we round x up to 44. That tells us 44 decimal places will give us 40 duodecimal places. Related posts [1] Recreations in the Theory of Numbers by Alfred H. Beiler. 3 thoughts on “Reciprocals of prime powers” 1. 1/7 “has” 6 digits because the first power-of-ten-less-one that 7 divides is 999999. Similarly 11 divides 99 and that is why 1/11 has only 2 digits, et simile. But it is not obvious from this why 1/49 should have 42 digits, i.e. why the lowest (10^^n)-1 that 49 divides is (10^^42)-1 . What’s going on there? 2. The first thing that came to mind was Euler’s theorem (which is related to Fermat’s Little Theorem in the linked post about periods of fractions). phi(7) = 6, which is why 10^6 – 1 divides 7; Euler’s theorem says 10^phi(7) = 1 (mod 7). And generally phi(p^k) = p^(k-1) * (p-1), so phi(7^2) = 7^1 * 6 = 42, so 10^42 = 1 (mod 49). The formula for the totient of prime powers seems to give us the recurrence we want. But it’s not the whole story, because phi(11) = 10, and technically 1/11 is periodic of 10 digits, but the minimum period is 2, not 10. So we need some kind of “reduced” totient with some factors taken out… but not Carmichael’s, something else. The case of 11 might make you think you can just take out any 2s or 5s you see, but that’s not it. Looking into one of the oddball numbers in the main post, phi(487) is of course 486. 486 = 2 * 3^5. The period of 1/487 is 27 = 3^3, less by a factor of 18. Gonna leave it for someone to pick up from there. 3. Kenneth A. Ross. Repeating Decimals: A Period Piece. Mathematics Magazine. February 2010. The theorem: If p is an odd prime that is relatively prime to the base b, and the period of 1/p has ℓ digits, then there exists a positive integer n such that the period of every fraction in the series 1/p, 1/p², …, 1/p^n has ℓ digits, and for every integer a > n, the period of 1/p^a has ℓ p^(a−n) digits. The “usual” primes in the base b are those for which n = 1. (The prime 2 is a special case and must be treated differently.) For every base b, the “unusual” primes are rare. For every prime p and positive integer n, there exist bases in which the period of every fraction 1/p, 1/p², …, 1/p^n is the same. See the paper: Wilfrid Keller and Jörg Richstein. Solutions of the Congruence a^(p−1)≡1 mod p^r. Mathematics and Computation. June 8, 2004.	{"url":"https://www.johndcook.com/blog/2021/09/28/reciprocals-of-prime-powers/","timestamp":"2024-11-13T21:07:38Z","content_type":"text/html","content_length":"57955","record_id":"<urn:uuid:5413c543-5af0-464d-9b95-e7e26bf4a868>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00183.warc.gz"}
In this chapter, we covered ways of representing and traversing a graph and looked at shortest path algorithms. Graphs are also an optimal data structure for some problems we haven't mentioned yet. This section aims to introduce some of them. A minimum spanning tree of a graph is a subset of the set of edges E of a connected graph that connects all vertices together, without any cycles and with the minimum total edge weight. It is a tree because every two vertices in it are connected by exactly one path. In order to understand the applicability of minimum spanning trees, consider the problem of a telecommunications company that is moving into a new neighborhood. The company wants to connect all the houses, but also wants to minimize the length of cable that it uses in order to cut costs. One way to solve the problem is by computing the minimum spanning tree of a graph whose vertices are the houses of the neighborhood, and the edges between houses...	{"url":"https://subscription.packtpub.com/book/programming/9781789537178/6/ch06lvl1sec33/summary","timestamp":"2024-11-08T23:26:07Z","content_type":"text/html","content_length":"166801","record_id":"<urn:uuid:4fc287f4-3510-4dbc-8a91-a33ec01b85b1>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00449.warc.gz"}
Noether-Lefschetz theory and class groups (John Brevik, Cal State Long Beach) - Claremont Center for the Mathematical Sciences Noether-Lefschetz theory and class groups (John Brevik, Cal State Long Beach) November 5 @ 12:15 pm - 1:10 pm The classical Noether-Lefschetz Theorem states that a suitably general algebraic surface S of degree d ≥ 4 in complex projective 3-space P3 contains no curves besides complete intersections, that is, curves of the form S ∩ T where T is another surface. After discussing briefly Noether’s non-proof of this theorem and hinting at the idea behind Lefschetz’s proof, I will sketch some of our recent progress in generalizing this theorem and its implications for global and local divisor class groups. We explore the question of what class groups are possible for local rings on surfaces in a particular analytic isomorphism class and show the ubiquitousness of unique factorization domains among such rings. Joint work with Scott Nollet (Texas Christian University). Related Events	{"url":"https://colleges.claremont.edu/ccms/event/antc-talk-john-brevik-cal-state-long-beach/","timestamp":"2024-11-10T22:31:51Z","content_type":"text/html","content_length":"206838","record_id":"<urn:uuid:cbb7d2fe-81da-4c16-84ce-c06da8496421>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00518.warc.gz"}
[Solved] If the annual fixed costs are 54,000 dina \| SolutionInn Answered step by step Verified Expert Solution If the annual fixed costs are 54,000 dinars, the occupation expense represents 20%, the contribution margin is 25%, and the unit selling price is 40 If the annual fixed costs are 54,000 dinars, the occupation expense represents 20%, the contribution margin is 25%, and the unit selling price is 40 dinars. Required: Calculate the closing point of the factory (in quantities) - 2 If it is Subsidiary costs 42,000 dinars, the target profit before tax is 8000 dinars, the unit selling price is 20 dinars, and the contribution margin is 8 dinars. Required: Calculate the sales value 3- If the tax rate is 25% and the target wind after tax is 12000 dinars, the annual fixed costs are 22,000 dinars, and the contribution margin is 8 dinars for the unit required to be calculated Amount of sales If the fixed costs are 60000 dinars in the unit price is 20 dinars and the variable cost is 64 dinars per unit, and the administration decides to increase the annual settlement employees’ salaries by 6000 dinars and reduce the sales commission by 2 dinars per unit. 6000 units, fixed costs 36000 dinars, realized wind 12000 dinars, selling price 18 dinars, variable cost 10 dinars, and the management decided to increase the price, which leads to a decrease in sales by 20% or more. To maintain the same level of profit required: Calculate the new selling price • • • More O (! Possible solution to the question is necessary necessary necessary There are 3 Steps involved in it Step: 1 Get Instant Access to Expert-Tailored Solutions See step-by-step solutions with expert insights and AI powered tools for academic success Ace Your Homework with AI Get the answers you need in no time with our AI-driven, step-by-step assistance Get Started Recommended Textbook for Authors: William G. Sullivan, Elin M. Wicks, C. Patrick Koelling 16th edition 133439275, 133439274, 9780133819014 , 978-0133439274 More Books Students also viewed these Accounting questions View Answer in SolutionInn App	{"url":"https://www.solutioninn.com/study-help/questions/if-the-annual-fixed-costs-are-54000-dinars-the-occupation-239538","timestamp":"2024-11-12T13:00:28Z","content_type":"text/html","content_length":"114408","record_id":"<urn:uuid:66766f48-25af-4e3c-84ac-da6afcfa2cec>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00504.warc.gz"}
Theory of Computing Report Clément Canonne asked to pass along the following registration info for ISAAC 2024, which will be in Sydney in early December. I was lucky to visit Sydney in July 2017 for ICML, and it was one of the most fun conference trips I’ve had! (Painfully long flight from Boston aside…) Definitely attend if you can! … Continue reading "Guest Post: ISAAC’24 in Sydney: Registration deadline soon!" Clément Canonne asked to pass along the following registration info for ISAAC 2024, which will be in Sydney in early December. I was lucky to visit Sydney in July 2017 for ICML, and it was one of the most fun conference trips I’ve had! (Painfully long flight from Boston aside…) Definitely attend if you can! The registration for the 35th International Symposium on Algorithms and Computation (ISAAC 2024), to be held in Sydney (Australia) on December 8-11, is open for only a few more weeks! Besides the conference itself, the registration includes lunches as well as a ticket to the conference banquet: https://www.trybooking.com/events/landing/1249687 Registrations are open until November 29, AoE: Student: AUD 480 (~USD 325) Regular: AUD 960 (~USD 650) (Note: There will be no on-site registration.) For more details, see https://sites.google.com/view/isaac2024/registration?authuser=0 Looking forward to seeing you at ISAAC in Sydney!	{"url":"https://theory.report/","timestamp":"2024-11-11T10:34:38Z","content_type":"text/html","content_length":"233832","record_id":"<urn:uuid:6ac7c835-406a-4c0c-b420-168074064b94>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00654.warc.gz"}
The cost of an air cooler has increased from Rs.$2250$ to $2500$. What is the percentage change? Hint: We are given the initial cost price and the final cost price of the air cooler. First, find the difference in the cost by subtracting the initial cost from the final cost. This will tell us the increase in the cost price. Now use the formula- $ \Rightarrow $ Percentage change=$\dfrac{{{\text{Increase in cost}}}}{{{\text{Initial cost}}}} \times 100$ Put the given values in the formula and solve to find the percentage change. Complete step-by-step answer: Given, the initial cost of the air cooler =Rs. $2250$ The final cost of the air cooler =Rs. $2500$ Now, we have to find the percentage change. For this we need to find the increase in the cost. First, we will find the difference between the initial cost price and the final cost of the air cooler. This difference tells us the increase in the cost of the air cooler. So we can write- $ \Rightarrow $ Increase in the cost=Final cost-initial cost=$2500 - 2250$ On solving, we get- $ \Rightarrow $ Increase in the cost=Rs.$250$ Now to find the percentage change we will use the formula- $ \Rightarrow $ Percentage change=$\dfrac{{{\text{Increase in cost}}}}{{{\text{Initial cost}}}} \times 100$ On putting the given values in the formula, we get- $ \Rightarrow $ Percentage change=$\dfrac{{{\text{250}}}}{{{\text{2250}}}} \times 100$ On simplifying, we get- $ \Rightarrow $ Percentage change=$\dfrac{{25}}{{225}} \times 10$ On dividing numerator and denominator, we get- $ \Rightarrow $ Percentage change=$\dfrac{{\text{5}}}{{45}} \times 100$ On again dividing the numerator and denominator, we get- $ \Rightarrow $ Percentage change=$\dfrac{1}{{\text{9}}} \times 100$ On again dividing the numerator and denominator, we get- $ \Rightarrow $ Percentage change=$\dfrac{{100}}{9} = 11.11\% $ Hence the correct answer is $11.11\% $ Note: Here you can also directly use the formula- Percentage change=$\dfrac{{{\text{Final cost - Initial cost}}}}{{{\text{Initial cost}}}} \times 100$ On putting the given values, we get- Percentage change=$\dfrac{{2500 - 2250}}{{2250}} \times 100$ On solving, we get- Percentage change=$\dfrac{{250}}{{2250}} \times 100$ On solving, we get- Percentage change=$\dfrac{5}{{45}} \times 100 = \dfrac{1}{9} \times 100$ On solving, we get- Percentage change=$11.11\% $ So we will get the same answer.	{"url":"https://www.vedantu.com/question-answer/the-cost-of-an-air-cooler-has-increased-from-class-7-maths-cbse-5f8a8e70ae44175ee5e5bf97","timestamp":"2024-11-09T10:13:00Z","content_type":"text/html","content_length":"152500","record_id":"<urn:uuid:ad8c5cd8-5941-4dc5-b776-eb049b3957d0>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00694.warc.gz"}
Nonlinear SciencesTangent Space Causal Inference: Leveraging Vector Fields for Causal Discovery in Dynamical SystemsA kneading map of chaotic switching oscillations in a Kerr cavity with two interacting light fieldsSolving the Kinetic Ising Model with Non-ReciprocitySpatiotemporal pattern formation of membranes induced by surface molecular binding/unbindingPhysics of collective transport and traffic phenomena in biology: progress in 20 yearsNon-linear sigma models for non-Hermitian random matrices in symmetry classes AI$^{\dagger}$ and AII$^{\dagger}$A Martingale-Free Introduction to Conditional Gaussian Nonlinear SystemsOscillons from Q-balls through RenormalizationExact calculation of the large deviation function for $k$-nary coalescence2D magnetic stabilityPredictive Non-linear Dynamics via Neural Networks and Recurrence Plots https://papers.cool/arxiv/nlin 2024-11-01T00:00:00+00:00 python-feedgen Cool Papers - Immersive Paper Discovery https://papers.cool/arxiv/2410.23499 2024-11-01T00:00:00+00:00 Kurt Butler Daniel Waxman Petar M. Djurić Causal discovery with time series data remains a challenging yet increasingly important task across many scientific domains. Convergent cross mapping (CCM) and related methods have been proposed to study time series that are generated by dynamical systems, where traditional approaches like Granger causality are unreliable. However, CCM often yields inaccurate results depending upon the quality of the data. We propose the Tangent Space Causal Inference (TSCI) method for detecting causalities in dynamical systems. TSCI works by considering vector fields as explicit representations of the systems' dynamics and checks for the degree of synchronization between the learned vector fields. The TSCI approach is model-agnostic and can be used as a drop-in replacement for CCM and its generalizations. We first present a basic version of the TSCI algorithm, which is shown to be more effective than the basic CCM algorithm with very little additional computation. We additionally present augmented versions of TSCI that leverage the expressive power of latent variable models and deep learning. We validate our theory on standard systems, and we demonstrate improved causal inference performance across a number of benchmark tasks. https://papers.cool/arxiv/2410.23588 2024-11-01T00:00:00+00:00 Rodrigues D. Dikandé Bitha Andrus Giraldo Neil G. R. Broderick Bernd Krauskopf Optical systems that combine nonlinearity with coupling between various subsystems offer a flexible platform for observing a diverse range of nonlinear dynamics. Furthermore, engineering tolerances are such that the subsystems can be identical to within a fraction of the wavelength of light; hence, such coupled systems inherently have a natural symmetry that can lead to either delocalization or symmetry breaking. We consider here an optical Kerr cavity that supports two interacting electric fields, generated by two symmetric input beams. Mathematically, this system is modeled by a four-dimensional $\mathbb{Z}_2$-equivariant vector field with the strength and detuning of the input light as control parameters. Previous research has shown that complex switching dynamics are observed both experimentally and numerically across a wide range of parameter values. Here, we show that particular switching patterns are created at specific global bifurcations through either delocalization or symmetry breaking of a chaotic attractor. We find that the system exhibits infinitely many of these global bifurcations, which are organized by $\mathbb{Z}_2$-equivariant codimension-two Belyakov transitions. We investigate these switching dynamics by means of the continuation of global bifurcations in combination with the computation of kneading invariants and Lyapunov exponents. In this way, we provide a comprehensive picture of the interplay between different switching patterns of periodic orbits and chaotic attractors. https://papers.cool/arxiv/ 2410.23615 2024-11-01T00:00:00+00:00 Gabriel Weiderpass Mayur Sharma Savdeep Sethi Non-reciprocal interactions are a generic feature of non-equilibrium systems. We define a non-reciprocal generalization of the kinetic Ising model in one spatial dimension. We solve the model exactly using two different approaches for infinite, semi-infinite and finite systems with either periodic or open boundary conditions. The exact solution allows us to explore a range of novel phenomena tied to non-reciprocity like non-reciprocity induced frustration and wave phenomena with interesting parity-dependence for finite systems of size $N$. We study dynamical questions like the approach to equilibrium with various boundary conditions. We find new regimes, separated by $N^{th}$-order exceptional points, which can be classified as overdamped, underdamped and critically damped phases. Despite these new regimes, long-time order is only present at zero temperature. Additionally, we explore the low-energy behavior of the system in various limits, including the ageing and spatio-temporal Porod regimes, demonstrating that non-reciprocity induces unique scaling behavior at zero temperature. Lastly, we present general results for systems where spins interact with no more than two spins, outlining the conditions under which long-time order may exist. https://papers.cool/arxiv /2410.23635 2024-11-01T00:00:00+00:00 Hiroshi Noguchi Nonequilibrium membrane pattern formation is studied using meshless membrane simulation. We consider that molecules bind to either surface of a bilayer membrane and move to the opposite leaflet by flip-flop. When binding does not modify the membrane properties and the transfer rates among the three states are cyclically symmetric, the membrane exhibits spiral-wave and homogeneous-cycling modes at high and low binding rates, respectively, as in an off-lattice cyclic Potts model. When binding changes the membrane spontaneous curvature, these spatiotemporal dynamics are coupled with microphase separation. When two symmetric membrane surfaces are in thermal equilibrium, the membrane domains form 4.8.8 tiling patterns in addition to stripe and spot patterns. In nonequilibrium conditions, moving biphasic domains and time-irreversible fluctuating patterns appear. The domains move ballistically or diffusively depending on the conditions. https://papers.cool/arxiv/2410.23735 2024-11-01T00:00:00+00:00 Debashish Chowdhury Andreas Schadschneider Katsuhiro Nishinari Enormous progress have been made in the last 20 years since the publication of our review \cite{csk05polrev} in this journal on transport and traffic phenomena in biology. In this brief article we present a glimpse of the major advances during this period. First, we present similarities and differences between collective intracellular transport of a single micron-size cargo by multiple molecular motors and that of a cargo particle by a team of ants on the basis of the common principle of load-sharing. Second, we sketch several models all of which are biologically motivated extensions of the Asymmetric Simple Exclusion Process (ASEP); some of these models represent the traffic of molecular machines, like RNA polymerase (RNAP) and ribosome, that catalyze template-directed polymerization of RNA and proteins, respectively, whereas few other models capture the key features of the traffic of ants on trails. More specifically, using the ASEP-based models we demonstrate the effects of traffic of RNAPs and ribosomes on random and `programmed' errors in gene expression as well as on some other subcellular processes. We recall a puzzling empirical result on the single-lane traffic of predatory ants {\it Leptogenys processionalis} as well as recent attempts to account for this puzzle. We also mention some surprising effects of lane-changing rules observed in a ASEP-based model for 3-lane traffic of army ants. Finally, we explain the conceptual similarities between the pheromone-mediated indirect communication, called stigmergy, between ants on a trail and the floor-field-mediated interaction between humans in a pedestrian traffic. For the floor-field model of human pedestrian traffic we present a major theoretical result that is relevant from the perspective of all types of traffic phenomena. https://papers.cool/arxiv/2410.24043 2024-11-01T00:00:00+00:00 Anish Kulkarni Kohei Kawabata Shinsei Ryu Symmetry of non-Hermitian matrices underpins many physical phenomena. In particular, chaotic open quantum systems exhibit universal bulk spectral correlations classified on the basis of time-reversal symmetry$^{\dagger}$ (TRS$^{\dagger}$), coinciding with those of non-Hermitian random matrices in the same symmetry class. Here, we analytically study the spectral correlations of non-Hermitian random matrices in the presence of TRS$^{\dagger}$ with signs $+1$ and $-1$, corresponding to symmetry classes AI$^{\dagger}$ and AII$^{\dagger}$, respectively. Using the fermionic replica non-linear sigma model approach, we derive $n$-fold integral expressions for the $n$th moment of the one-point and two-point characteristic polynomials. Performing the replica limit $n\to 0$, we qualitatively reproduce the density of states and level-level correlations of non-Hermitian random matrices with TRS$^{\dagger}$. https://papers.cool/arxiv/2410.24056 2024-11-01T00:00:00+00:00 Marios Andreou Nan Chen The Conditional Gaussian Nonlinear System (CGNS) is a broad class of nonlinear stochastic dynamical systems. Given the trajectories for a subset of state variables, the remaining follow a Gaussian distribution. Despite the conditionally linear structure, the CGNS exhibits strong nonlinearity, thus capturing many non-Gaussian characteristics observed in nature through its joint and marginal distributions. Desirably, it enjoys closed analytic formulae for the time evolution of its conditional Gaussian statistics, which facilitate the study of data assimilation and other related topics. In this paper, we develop a martingale-free approach to improve the understanding of CGNSs. This methodology provides a tractable approach to proving the time evolution of the conditional statistics by deriving results through time discretization schemes, with the continuous-time regime obtained via a formal limiting process as the discretization time-step vanishes. This discretized approach further allows for developing analytic formulae for optimal posterior sampling of unobserved state variables with correlated noise. These tools are particularly valuable for studying extreme events and intermittency and apply to high-dimensional systems. Moreover, the approach improves the understanding of different sampling methods in characterizing uncertainty. The effectiveness of the framework is demonstrated through a physics-constrained, triad-interaction climate model with cubic nonlinearity and state-dependent cross-interacting noise. https://papers.cool/arxiv/2410.24109 2024-11-01T00:00:00+00:00 F. Blaschke T. Romańczukiewicz K. Sławińska A. Wereszczyński Using a renormalization-inspired perturbation expansion we show that oscillons in a generic field theory in (1+1)-dimensions arise as dressed Q-balls of a universal (up to the leading nonlinear order) complex field theory. This theory reveals a close similarity to the integrable complex sine-Gordon model which possesses exact multi-$Q$-balls. We show that excited oscillons, with characteristic modulations of their amplitude, are two-oscillons bound states generated from a two $Q$-ball solution. https://papers.cool/arxiv/2410.24118 2024-11-01T00:00:00+00:00 R. Rajesh V. Subashri Oleg Zaboronski We study probabilities of rare events in the general coalescence process, $kA\rightarrow \ell A$, where $k>\ell$. For arbitrary $k, \ell$, by rewriting these probabilities in terms of an effective action, we derive the large deviation function describing the probability of finding $N$ particles at time $t$, when starting with $M$ particles initially. Additionally, the most probable trajectory corresponding to a fixed rare event is derived. https://papers.cool/arxiv/2410.24156 2024-11-01T00:00:00+00:00 Douglas Lundholm This article is a contribution to the proceedings of the 33rd/35th International Colloquium on Group Theoretical Methods in Physics (ICGTMP, Group33/35) held in Cotonou, Benin, July 15-19, 2024. The stability of matter is an old and mathematically difficult problem, relying both on the uncertainty principle of quantum mechanics and on the exclusion principle of quantum statistics. We consider here the stability of the self-interacting almost-bosonic anyon gas, generalizing the Gross-Pitaevskii / nonlinear Schrödinger energy functionals to include magnetic self interactions. We show that there is a type of supersymmetry in the model which holds only for higher values of the magnetic coupling but is broken for lower values, and that in the former case supersymmetric ground states exist precisely at even-integer quantized values of the coupling. These states constitute a manifold of explicit solitonic vortex solutions whose densities solve a generalized Liouville equation, and can be regarded as nonlinear generalizations of Landau levels. The reported work is joint with Alireza Ataei and Dinh-Thi Nguyen and makes an earlier analysis of self-dual abelian Chern-Simons-Higgs theory by Jackiw and Pi, Hagen, and others, mathematically rigorous. https://papers.cool/arxiv/2410.23408 2024-11-01T00:00:00+00:00 L. Lober M. S. Palmero F. A. Rodrigues Predicting and characterizing diverse non-linear behaviors in dynamical systems is a complex challenge, especially due to the inherently presence of chaotic dynamics. Current forecasting methods are reliant on system-specific knowledge or heavily parameterized models, which can be associated with a variety of drawbacks including critical model assumptions, uncertainties in their estimated input hyperparameters, and also being computationally intensive. Moreover, even when combined with recurrence analyses, these approaches are typically constrained to chaos identification, rather than parameter inference. In this work, we address these challenges by proposing a methodology that uses recurrence plots to train convolutional neural networks with the task of estimating the defining control parameters of two distinct non-linear systems: (i) the Logistic map and (ii) the Standard map. By focusing on the neural networks' ability to recognize patterns within recurrence plots, we demonstrate accurate parameter recovery, achieving fairly confident levels of prediction for both systems. This method not only provides a robust approach to predicting diverse non-linear dynamics but also opens up new possibilities for the automated characterization of similar non-linear dynamical systems.	{"url":"https://papers.cool/arxiv/nlin/feed","timestamp":"2024-11-03T20:32:39Z","content_type":"application/atom+xml","content_length":"18481","record_id":"<urn:uuid:a8333791-a11c-4dce-9566-350398fc5093>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00720.warc.gz"}
Santa math check. I found these figures on some internet site and again in the paper this morning. Being dopers, I’m sure you’d like to check the accuracy of the math and perhaps add some figures of your own: 75 million — Approximate miles Santa must travel 650 miles per second — Approximate speed Santa must travel to serve each household with a Christian child 822.6 — Approximate number of homes Santa must visit each second on Christmas Eve 2,950.7 tons — Approximate weight Santa would gain if he consumed 1 cookie and an 8-ounce glass of milk at each home 321,000 tons— Estimated weight of Santa’s sleigh, assuming 2 pounds of toys per child 214,200— Reindeer Santa would need to pull that load (assuming aerial reindeer can pull 10 times a normal reindeer load) 17,500 — G forces inflicted on Santa as his sled accelerates to 650 miles per second 14.3 quintillion — Estimated energy, in joules, absorbed by the lead reindeer on Santa’s sleigh 0.0042 second — Estimated time it takes Santa and sleigh to vaporize from friction during acceleration My contribution: 1,285,200 feet (or 243.409 miles) - Length of 214,200 reindeer placed end to end, assuming a six foot average length per reindeer. Wouldn’t it be half that? Every image I’ve seen of Santa’s sleigh has the reindeer hitched in a 2 x 2 fashion. ETA: you should check out Norad Santa, I think it has some more details. (Or it used to, I can’t find the comparisons or technical data up there this year.) Question for you, how do you determain this? I mean some households have more than one Child. When I was a child (in England) we used to leave a glass of sherry and a mince pie. Can you calculate Santa’s blood-alcohol level? And don’t forget the carrots left out for the Reindeer (and, this year, a dog biscuit for “Olive, the Other Reindeer” since we watched that last night)! The Reindeer will be gaining weight as they go, to make up for the toys left behind. I’m wondering what the roof loading would be like for that many reindeer plus the sleigh/toys weight. I’m sorry, but you are all wrong. Every true believer knows Santa’s sled is pulled by 8 tiny reindeer. But does he ever really land? I mean, they can fly. I’ve always assumed that since they don’t have wings, they must be able to hover. Those are impressive numbers. Unfortunately, most of them are meaningless, given the fact that Santa does not actually visit every house in the world in a single night. He doesn’t have to, because Santa Claus is a Time Lord. And the fine/length of jail time Santa will receive when he’s arrested for driving under the influence. Year that I first read this - 1990 Number of websites where this appears - 362,000 Threadkiller - 1 You’re welcome. For the last few years I’ve been thinking that Santa’s reindeer are going to absolutely hate Christmas Eve when we finally colonize Mars. That’ll add, what? 30-250 million miles, depending on our respective orbits? Sure, it sounds like Santa’s got heat shielding figured out what with traveling 650 m/s, but what about radiation shielding? Reindeer-shaped pressurized suits? Freeze-and-vacuum-proofed toys? The upside is that once they’ve broken free of Earth’s gravitational field they’ll be able to coast a good portion of the way and rest up for the braking burn, but I just don’t think Ole Nick will consider it worth the trouble for a few dozen lousy souls and will secretly switch them to the naughty list. Something to consider for all you prospective spacefarers. Here’s a thought: go shit in someone else’s thread. Why carrots? There aren’t any carrots growing on the tundra. Kids should leave out things that reindeer like to eat, not carrots. Anyone remember the Sci-Fi short story involving the space station with the crew putting on a Christmas celebration for the amusement/edification of the local BEMs? It was a lot of work, but they managed it, and the natives had a blast. Then it turned out that the natives’ concept of a “year” was about one Earth week, and they were going to expect annual Only Christian childs? How bigoted. Especially considering that Santa Claus has a pagan origin. Needed for accurate calculations: time taken at each stop, amount of friction and delay generated by a fat man sliding down a thin chimney, time taken retracing steps for houses he forgot, refueling time (cookies, milk and reindeer biscuits)… Isaac Asimov’s “Christmas on Ganymede”.	{"url":"https://boards.straightdope.com/t/santa-math-check/478587","timestamp":"2024-11-13T01:02:24Z","content_type":"text/html","content_length":"53506","record_id":"<urn:uuid:f0b6001f-5031-4edf-a129-f53c1053545c>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00495.warc.gz"}
2004 AMC 10B Problems/Problem 24 In triangle $ABC$ we have $AB=7$, $AC=8$, $BC=9$. Point $D$ is on the circumscribed circle of the triangle so that $AD$ bisects angle $BAC$. What is the value of $\frac{AD}{CD}$? $\text{(A) } \dfrac{9}{8} \qquad \text{(B) } \dfrac{5}{3} \qquad \text{(C) } 2 \qquad \text{(D) } \dfrac{17}{7} \qquad \text{(E) } \dfrac{5}{2}$ Solution 1 - (Ptolemy's Theorem) Set $\overline{BD}$'s length as $x$. $\overline{CD}$'s length must also be $x$ since $\angle BAD$ and $\angle DAC$ intercept arcs of equal length (because $\angle BAD=\angle DAC$). Using Ptolemy's Theorem, $7x+8x=9(AD)$. The ratio is $\frac{5}{3}\implies\boxed{\text{(B)}}$ Solution 2 - Similarity Proportion $[asy] import graph; import geometry; import markers; unitsize(0.5 cm); pair A, B, C, D, E, I; A = (11/3,8sqrt(5)/3); B = (0,0); C = (9,0); I = incenter(A,B,C); D = intersectionpoint(I--(I + 2(I - A)), circumcircle(A,B,C)); E = extension(A,D,B,C); draw(A--B--C--cycle); draw(circumcircle(A,B,C)); draw(D--A); draw(D--B); draw(D--C); label("A", A, N); label("B", B, SW); label("C", C, SE); label ("D", D, S); label("E", E, NE); markangle(radius = 20,B, A, C, marker(markinterval(2,stickframe(1,2mm),true))); markangle(radius = 20,B, C, D, marker(markinterval(1,stickframe(1,2mm),true))); markangle(radius = 20,D, B, C, marker(markinterval(1,stickframe(1,2mm),true))); markangle(radius = 20,C, B, A, marker(markinterval(1,stickframe(2,2mm),true))); markangle(radius = 20,C, D, A, marker (markinterval(1,stickframe(2,2mm),true))); [/asy]$ Let $E = \overline{BC}\cap \overline{AD}$. Observe that $\angle ABC \cong \angle ADC$ because they both subtend arc $\overarc{AC}.$ Furthermore, $\angle BAE \cong \angle EAC$ because $\overline{AE}$ is an angle bisector, so $\triangle ABE \sim \triangle ADC$ by $\text{AA}$ similarity. Then $\dfrac{AD}{AB} = \dfrac{CD}{BE}$. By the Angle Bisector Theorem, $\dfrac{7}{BE} = \dfrac{8}{CE}$, so $\dfrac{7}{BE} = \dfrac{8}{9-BE}$. This in turn gives $BE = \frac{21}{5}$. Plugging this into the similarity proportion gives: $\dfrac {AD}{7} = \dfrac{CD}{\tfrac{21}{5}} \implies \dfrac{AD}{CD} = {\dfrac{5}{3}} = \boxed{\text{(B)}}$. Solution 3 - Angle Bisector Theorem We know that $\overline{AD}$ bisects $\angle BAC$, so $\angle BAD = \angle CAD$. Additionally, $\angle BAD$ and $\angle BCD$ subtend the same arc, giving $\angle BAD = \angle BCD$. Similarly, $\angle CAD = \angle CBD$ and $\angle ABC = \angle ADC$. These angle relationships tell us that $\triangle ABE\sim \triangle ADC$ by AA Similarity, so $AD/CD = AB/BE$. By the angle bisector theorem, $AB/BE = AC/CE$. Hence, $\[\frac{AB}{BE} = \frac{AC}{CE} = \frac{AB + AC}{BE + CE} = \frac{AB + AC}{BC} = \frac{7 + 8}{9} = \frac{15}{9} = \boxed{\frac{5}{3}}.\]$ (Where did E come from?????) See Also The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.	{"url":"https://artofproblemsolving.com/wiki/index.php/2004_AMC_10B_Problems/Problem_24","timestamp":"2024-11-12T21:52:57Z","content_type":"text/html","content_length":"50440","record_id":"<urn:uuid:2a285de3-583e-42a9-8f5a-ee680d2a2270>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00749.warc.gz"}
This HTML automatically generated with rman for NEMO Table of Contents king - tabulate Osipkov-Merritt generalization of King model king out=out_file [parameter=value] ... king tabulates an Osipkov-Merritt-King model. The output file, written in binary format, may be used with mkommod(1NEMO) . The following parameters are recognized: Write a binary table OsipkovMerrittModel to out_file. No default. w0=dimensionless central potential This parameter defines the starting point for the integration of the corresponding differential equations (see King (1966) or Binney & Tremaine (1987)). Default: -1.0. mass=total mass of system Used for calculating the real values from the previously computed unscaled model. Default: 1.0. rc=core radius The core radius is the scaling parameter for the radius. It is defined as radius at which the surface brightness falls down to half of its initial value. Default: 1.0 nsteps=number of steps Number of steps to tabulate. Default: 101 epsilon=accuracy for integration Epsilon is used to define the precision of the numerical integration Default: 1e-6 tab=table filename If a filename given, the calculated values will also be tabulated in an ASCII-file. If no filename is given, no table is output. Default: no table file. g=gravitational constant to create models with physical quantities, set this constant. Default: not used. (as it should be) the source code may have been lost mkommod(1NEMO) , king(1NEMO) , plummer(1NEMO) , tablst(1NEMO) , mkking(1falcON) Patrick A. Frisch ~/usr/pat king.c xx-mar-91 V0.9 Initial coding and testing PAF 27-jun-97 V1.0 more scaling options, code cleaning for NEMO V2 PJT/JJM	{"url":"https://teuben.github.io/nemo/man_html/king.1.html","timestamp":"2024-11-07T18:51:25Z","content_type":"text/html","content_length":"4542","record_id":"<urn:uuid:a9524579-c2df-4b29-bf55-1dd29f0bcbf0>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00681.warc.gz"}
The difference between the upper and the lower class limits is ... \| Filo The difference between the upper and the lower class limits is called Not the question you're searching for? + Ask your question The difference between the upper and the lower class limits is called the class size. Was this solution helpful? Found 7 tutors discussing this question Discuss this question LIVE 6 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 from Statistics View more Practice questions on similar concepts asked by Filo students View more Stuck on the question or explanation? Connect with our Mathematics tutors online and get step by step solution of this question. 231 students are taking LIVE classes Question Text The difference between the upper and the lower class limits is called Topic Statistics Subject Mathematics Class Class 9 Answer Type Text solution:1 Upvotes 84	{"url":"https://askfilo.com/math-question-answers/the-difference-between-the-upper-and-the-lower-class-limits-is-called","timestamp":"2024-11-04T21:44:00Z","content_type":"text/html","content_length":"125850","record_id":"<urn:uuid:e8d842eb-e5c7-48f5-9f5e-94b177da776a>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00389.warc.gz"}
Meter Candle to Lumen/Square Meter Converter (m·c to lm/m2) \| Kody Tools 1 Meter Candle = 1 Lumen/Square Meter One Meter Candle is Equal to How Many Lumen/Square Meter? The answer is one Meter Candle is equal to 1 Lumen/Square Meter and that means we can also write it as 1 Meter Candle = 1 Lumen/Square Meter. Feel free to use our online unit conversion calculator to convert the unit from Meter Candle to Lumen/Square Meter. Just simply enter value 1 in Meter Candle and see the result in Lumen/Square Meter. Manually converting Meter Candle to Lumen/Square Meter can be time-consuming,especially when you don’t have enough knowledge about Illuminance units conversion. Since there is a lot of complexity and some sort of learning curve is involved, most of the users end up using an online Meter Candle to Lumen/Square Meter converter tool to get the job done as soon as possible. We have so many online tools available to convert Meter Candle to Lumen/Square Meter, but not every online tool gives an accurate result and that is why we have created this online Meter Candle to Lumen/Square Meter converter tool. It is a very simple and easy-to-use tool. Most important thing is that it is beginner-friendly. How to Convert Meter Candle to Lumen/Square Meter (m·c to lm/m2) By using our Meter Candle to Lumen/Square Meter conversion tool, you know that one Meter Candle is equivalent to 1 Lumen/Square Meter. Hence, to convert Meter Candle to Lumen/Square Meter, we just need to multiply the number by 1. We are going to use very simple Meter Candle to Lumen/Square Meter conversion formula for that. Pleas see the calculation example given below. $\text{1 Meter Candle} = 1 \times 1 = \text{1 Lumen/Square Meter}$ What Unit of Measure is Meter Candle? Meter candle is a unit of measurement for illuminance. It can be defined as illuminance of a one candela source illuminating a surface one meter away from the light source. What is the Symbol of Meter Candle? The symbol of Meter Candle is m·c. This means you can also write one Meter Candle as 1 m·c. What Unit of Measure is Lumen/Square Meter? Lumen per square meter is a unit of measurement for illuminance. It can be defined as illuminance of a one lumen source illuminating a surface of one square meter from the light source. What is the Symbol of Lumen/Square Meter? The symbol of Lumen/Square Meter is lm/m2. This means you can also write one Lumen/Square Meter as 1 lm/m2. How to Use Meter Candle to Lumen/Square Meter Converter Tool • As you can see, we have 2 input fields and 2 dropdowns. • From the first dropdown, select Meter Candle and in the first input field, enter a value. • From the second dropdown, select Lumen/Square Meter. • Instantly, the tool will convert the value from Meter Candle to Lumen/Square Meter and display the result in the second input field. Example of Meter Candle to Lumen/Square Meter Converter Tool Meter Candle to Lumen/Square Meter Conversion Table Meter Candle [m·c] Lumen/Square Meter [lm/m2] Description 1 Meter Candle 1 Lumen/Square Meter 1 Meter Candle = 1 Lumen/Square Meter 2 Meter Candle 2 Lumen/Square Meter 2 Meter Candle = 2 Lumen/Square Meter 3 Meter Candle 3 Lumen/Square Meter 3 Meter Candle = 3 Lumen/Square Meter 4 Meter Candle 4 Lumen/Square Meter 4 Meter Candle = 4 Lumen/Square Meter 5 Meter Candle 5 Lumen/Square Meter 5 Meter Candle = 5 Lumen/Square Meter 6 Meter Candle 6 Lumen/Square Meter 6 Meter Candle = 6 Lumen/Square Meter 7 Meter Candle 7 Lumen/Square Meter 7 Meter Candle = 7 Lumen/Square Meter 8 Meter Candle 8 Lumen/Square Meter 8 Meter Candle = 8 Lumen/Square Meter 9 Meter Candle 9 Lumen/Square Meter 9 Meter Candle = 9 Lumen/Square Meter 10 Meter Candle 10 Lumen/Square Meter 10 Meter Candle = 10 Lumen/Square Meter 100 Meter Candle 100 Lumen/Square Meter 100 Meter Candle = 100 Lumen/Square Meter 1000 Meter Candle 1000 Lumen/Square Meter 1000 Meter Candle = 1000 Lumen/Square Meter Meter Candle to Other Units Conversion Table Conversion Description 1 Meter Candle = 1 Lux 1 Meter Candle in Lux is equal to 1 1 Meter Candle = 0.0001 Centimeter Candle 1 Meter Candle in Centimeter Candle is equal to 0.0001 1 Meter Candle = 0.092903039997495 Foot Candle 1 Meter Candle in Foot Candle is equal to 0.092903039997495 1 Meter Candle = 0.0001 Phot 1 Meter Candle in Phot is equal to 0.0001 1 Meter Candle = 1000 Nox 1 Meter Candle in Nox is equal to 1000 1 Meter Candle = 1 Candela Steradian/Square Meter 1 Meter Candle in Candela Steradian/Square Meter is equal to 1 1 Meter Candle = 1 Lumen/Square Meter 1 Meter Candle in Lumen/Square Meter is equal to 1 1 Meter Candle = 0.0001 Lumen/Square Centimeter 1 Meter Candle in Lumen/Square Centimeter is equal to 0.0001 1 Meter Candle = 0.092903039997495 Lumen/Square Foot 1 Meter Candle in Lumen/Square Foot is equal to 0.092903039997495 1 Meter Candle = 0.023225759999374 Flame 1 Meter Candle in Flame is equal to 0.023225759999374 1 Meter Candle = 1.4641288433382e-7 Watt/Square Centimeter 1 Meter Candle in Watt/Square Centimeter is equal to 1.4641288433382e-7	{"url":"https://www.kodytools.com/units/illuminance/from/mcd/to/lmpm2","timestamp":"2024-11-13T22:40:50Z","content_type":"text/html","content_length":"78620","record_id":"<urn:uuid:45baae27-b057-48b4-9074-e4ffa222dd6b>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00495.warc.gz"}
We study the conditions under which an algebraic curve can be modelled by a Laurent polynomial that is nondegenerate with respect to its Newton polytope. Such nondegenerate polynomials have become popular objects in explicit algebraic geometry, owing to their connection with toric geometry; however, despite their ubiquity, the intrinsic property of nondegeneracy has not seen much detailed study. We prove that every curve of genus $g \geq 4$ over an algebraically closed field is nondegenerate in the above sense. More generally, let $\mathcal{M}_g^{\textup{nd}}$ be the locus of nondegenerate curves inside the moduli space of curves of genus $g \geq 2$. Then we show that $\dim \mathcal{M}_g^{\textup{nd}} = \min(2g+1,3g-3)$, except for $g=7$ where $\dim \mathcal{M}_7^{\textup {nd}} = 16$; thus, a generic curve of genus $g$ is nondegenerate if and only if $g \geq 4$	{"url":"https://www4.math.duke.edu/media/videos.php?cat=566&sort=most_viewed&time=last_week&page=1&seo_cat_name=All","timestamp":"2024-11-13T03:13:34Z","content_type":"text/html","content_length":"109896","record_id":"<urn:uuid:5b54fd45-1a1c-4d30-8902-15c414de0bb5>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00410.warc.gz"}
Solving Quadratic Equations by Completing the Square \| sofatutor.com Solving Quadratic Equations by Completing the Square Basics on the topic Solving Quadratic Equations by Completing the Square One of the most effective methods of solving a quadratic equation is by completing the square. It is also the method that makes it easiest to graph the equation as well. By completing the square, we can find, not only the roots of the equation, but also the vertex of the graph of the equation. This is because we will pass from the standard form, ax^2 + bx + c = 0, to the vertex form, a (x - h)^2 + k = 0, with (h,k) as the vertex of the graph. The steps of this method must be taken carefully, though. The standard form must be gradually solved to arrive at the vertex form: 1. Finding the value that needs to be added on both sides of the equation to complete the square; 2. Finding the square roots of both sides of the equation; 3. Finding the roots of the equation by finally solving for x using the positive and negative values of the square root of the right side. In this video, the method of completing the square to solve an equation and plotting its graph is explained. Analyze Functions Using Different Representations. Transcript Solving Quadratic Equations by Completing the Square Kata Ana is always late for ninja school. Luckily, she’s had a lot of practice finding the quickest way to her school. Kata needs to know how to solve quadratic equations by completing the square in order to avoid some of the dangerous obstacles on her way to school. Kata has to use her grappling hook to swing over to the lantern over there. She throws her grappling hook onto the school’s roof. Kata knows that she has to avoid the school's guard Komodo dragon. She uses a technique passed down for generations called completing the square. Kata knows that if she does this, she can find the zeros of any parabola as well as the parabola’s vertex by putting the standard form of the quadratic formula, ax² + bx + c = 0 into vertex form, a(x - h)² + k = 0. Completing the Square certainly is a powerful tool. Vertex Form The first equation Kata has is one-eighth x² plus five over two 'x' plus nine over two equals 0. Kata looks at this problem. She tries to list the factors of negative nine over two: 1, negative nine over two and -1, nine over two. But neither of these options sum to five over two. Kata needs a more powerful tool. To complete the square, Kata needs to get the equation in vertex form. First, she moves the nine over two to the right hand side using opposite operations. This leaves her with one-eighth x² plus five over two 'x' equals negative nine over two. Next, she removes one-eighth from the equation by dividing each coefficient by one-eighth, leaving her with one-eighth times the quantity 'x' squared plus twenty 'x' equals negative nine over two. Now she takes half of her ‘b’ term, 20, giving her 10. She then squares the 10 and needs to add the result on both sides of the equation. However, since this part of the equation is inside the parentheses, and the contents of the parentheses is multiplied by one-eighth, we have to add 100 times one-eighth on the right side of the equation. We should simplify the fractions on the right side of the equation as much as possible. Kata notices that the left side of the equation is now a perfect square trinomial, which she knows how to factor. All Kata has to do now is move the 8 to the left side of the equation and compare her result to the standard vertex form of the equation to figure out the signs of the vertex. One-eighth times (x + 10)² minus 8= 0.Since the signs inside the parentheses don't match, we have to change 10 to negative 10. The signs are also different for the constant, so we have to change 8 to negative 8. The vertex has revealed itself! The vertex is (-10, -8)! Finding the Zeros by Isolating x Now, to get the zeros of the parabola! Let's start by getting rid of the one-eighth on the left side of the equation. We can do this by multiplying both sides of the equation by 8, giving us 'x' plus 10 quantity squared equals 64. Now we can take the square root of both sides. We're left with 'x' plus 10 equals plus/minus 8. For the last step, we isolate the 'x' and get our solutions, negative 2 and negative 18! Completing the Square really is a powerful tool. Kata Ana can clearly see why her ancestors have been using it for generations! Second Example Kata now needs to be able to clear that wall without hitting the roof so she can land safely on the ledge of her school. The equation for this part is negative one-tenth x² + eight-fifths 'x' plus 18-fifths equals zero. For this equation, Kata Ana repeats the process she used to solve the first equation. She first moves the constant to the right-hand side. Then, she divides the left side of the equation by negative one-tenth. Again, she takes half of her ‘b’ term, -16, giving her -8, which she then squares and adds to both sides of the equation. She remembers that, since the negative one-tenth is outside the parentheses, she has to multiply 64 by one-tenth before adding it to the right side of the equation. The left side of the equation is a perfect square trinomial! Completing the Square never fails! Using opposite operations, Kata rewrites the left side of the equation as negative one-tenth times the quantity 'x' minus 8, squared plus 10. Great, now the equation is in standard form. Once again, we compare the signs. Can you tell what the vertex is? That's right! it's at (8, 10)! Once again, to get the zeros of the parabola! Taking the square root on both sides of the equation leaves 'x' minus eight equal to plus/minus 10. And finally, isolating the variable gives us our solutions. Thanks to her knowledge of completing the square and some quick thinking, Kata is able to land safely on the ledge of her school. What’s this?! What's happened to her classmates?! Kata Ana is Shell Shocked! Solving Quadratic Equations by Completing the Square exercise Would you like to apply the knowledge you’ve learned? You can review and practice it with the tasks for the video Solving Quadratic Equations by Completing the Square. • Determine the vertex of the parabola. What is the format of a perfect square trinomial? You will need to add a term to the expression inside the parentheses on the left hand side of the equation in order to turn it into a perfect square trinomial. For an expression in the form $ax^2 + bx$, the third term that must be added to make it a perfect square trinomial is of the form: $(\frac{b}{2})^2$. Remember that the vertex form of the equation is $a (x-h)^2 + k = 0$. Pay close attention to the signs when you are determining the values of $h$ and $k$. Kata Ana knows that if she rearranges the function into vertex form, she will be able to easily identify the coordinates of the vertex. Vertex form is $a(x-h)^2+k = 0$, and gives the vertex as $ If the given equation was in standard form, she would begin by moving the constant term to the right hand side of the equation. Then she would factor out the coefficient on the $x^2$ term to give an expression in parentheses on the left hand side. She notices that this has already been done to the given equation, $\frac{1}{8}(x^2+20x)=-\frac{9}{2}$. She knows that she now needs to get the left hand side of the given function into a form that includes a perfect square trinomial. She knows that this trinomial is the sum of three terms: a term with $x^2$, another with $x$, and a constant. The expression on the left hand side of the given function is missing the constant term. Kata Ana knows that she must determine what constant term should be added to result in an expression that is a perfect square. She can find this constant by taking the coefficient of the $x$ term, $20$, dividing it by two, and then squaring it. She finds $(\frac{20}{2})^2 = 10^2$. She wants to add this term to the expression that is inside the parentheses on the left hand side. However, she knows that this entire expression in parentheses is being multiplied by $\frac{1} {8}$. So she really needs to add $\frac{1}{8}\times10^2$ to both sides, in order to add $10^2$ inside the parentheses on the left hand side. $\begin{array}{rclcl} \frac{1}{8}(x^2+20x+10^2) & = & -\frac{9}{2}+\frac{10^2}{8} & \| & \text{simplifying} \\ \frac{1}{8}(x^2+20x+10^2) & = & 8 \\ \end{array}$ Now she has a perfect square trionomial on the left hand side, $(x^2+20x+10^2)$. She can factor this trinomial to result in the following function: $\frac{1}{8}(x+10)^2 = 8$. Now, she can use opposite operations to get the function into vertex form: $\frac{1}{8}(x+10)^2 -8 = 0$. This shows her that $h = -10$, and $k=-8$. Therefore the vertex coordinates are $(-10,-8)$. • Find the zeroes of the quadratic equation. Remember that a positive or negative number can result from taking the square root of a given number. You must pay close attention to signs. Kata Ana knows that she must find the zeroes, or roots, of the quadratic function. To do this, she must isolate the variable $x$. She starts by observing that the given equation is in vertex form, which is $a (x-h)^2 + k = 0$. Next she uses opposite operations to move $k$ over to the the right hand side, and then divides both sides by $a$. $ \begin{array}{rclcl} -\frac{1}{10}(x-8)^2 + 10 &=& 0&\|&\text{subtract 10 from both sides} \\ -\frac{1}{10}(x-8)^2 &=&-10 &\|&\text{divide by -1/10} \\ (x-8)^2 &=&100&\|&\text{divide by -1/10} \\ \end{array} $ To continue to isolate $x$, she must take the square root of both sides of the equation. When taking the square root of a number, it is important to remember that the answer can be positive or negative. For example $\sqrt9 = \pm 3$ because $9=3^3=(-3)^2$. In this case: $x-8=\pm 10$. She then finishes isolating $x$, and arrives at: $x = 8 \pm 10$. She can now find the zeroes of the function to be: $8 - 10 = -2$ $8 + 10 = 18$. • Solve for the zeroes of each quadratic equation. You can find the zeroes by isolating $x$. The given functions are in the form $a (x-h)^2 + k = 0$. How can you move $k$ and $a$ to the right hand side? Remember that when you take the square root of a number, there are two possible answers. For example $\sqrt 81 = \pm9$. These functions have already been set equal to zero. The pair of roots for each function can therefore be found by isolating $x$. For the first function: We move the constant to the right hand side: Then we can divide through by the term that is outside the parentheses on the left hand side: Then we can take the square root of each side: Remember that the square root can be a positive or negative value: $x-5=\pm 2$. Finally, we can move the constant to the right hand side, and determine the roots: $x=5 \pm 2$ $\begin{array}{rcl} x=5+2 & or & x=5-2 \\ x=7 & or & x=3 \end{array}$ So we find that the roots are $3$ and $7$. Similarly, for the other equations, we find the roots: $\begin{array}{rcl} -3(x+2)^2+27=0 & \| & \text{roots are -5 and 1} \\ 4(x-2)^2-64=0 & \| & \text{roots are -2 and 6} \\ -3(x-9)^2+12=0 & \| & \text{roots are 7 and 11} \\ \end{array}$ • Calculate the vertex and zeroes of the quadratic equation. It will be helpful if you always reduce rational numbers to their simplest form during your solution. At every step of the way, every term in this problem (both coefficients and constants) can be simplified to an integer. If you have a rational number that you can't immediately simplify to an integer in your solution, you should double check your calculations. Remember, you can convert the standard form equation to vertex form by completing the square. The vertex will then be at the coordinates $(h,k)$. Once you have the function in vertex form, you can solve for the zeroes by setting the function equal to zero, and isolating $x$. There are two roots for this function. We need to find the vertex, and the roots of this function. We can find the vertex by converting the function into vertex form, $a (x-h)^2 + k = 0$. We have the standard form function: We start by moving the constant term to the right hand side: Next, we factor out the coefficient on the $x^2$ term from the left hand side: Then we take the coefficient on the $x$ term, divide it by two, and square the result: $(\frac{12}{2})^2 = 6^2 = 36$. Adding this constant to the expression inside the parentheses on the left hand side of the equation will give us a perfect square trinomial, which we know how to factor. In order to add the constant inside the parentheses on the left hand side, we have to add the constant times negative two on the right hand side. This is because all terms inside the parentheses on the left hand side are multiplied by negative two. So really, we add $36\times-2$ to both sides. $ \begin{array}{rclcl} -2(x^2-12x+36) & = & 64+(36\times-2) &\|&\text{Simplifying} \\ -2(x^2-12x+36) & = & 64-72 \\ -2(x^2-12x+36) & = & -8 \end{array} $ Now we can factor the perfect square trinomial on the left hand side, to get: $-2(x-6)^2 = -8$. And finally, we can move the constant to the left hand side, to get the equation in vertex form: This tells us that the vertex is at $(6,8)$. Now, we can isolate for $x$ to find the roots. The first step is to move the constant to the right hand side, which we have already done in an earlier step: $-2(x-6)^2 = -8$. Then we can divide through by the term that is outside the parentheses on the left hand side: $ \begin{array}{rclcl} (x-6)^2 & = & \frac{-8}{-2} &\|&\text{simplifying} \\ (x-6)^2 & = & 4 \end{array} $ Taking the sqaure root of both sides is the next step to isolate $x$: $x-6 = \sqrt4$. This gives us two possible solutions, a positive one and a negative one: $x-6 = \pm 2$. We can then find the two roots of the function: $\begin{array}{rcl} x=6+2 & or & x=6-2 \\ x=8 & or & x=4 \end{array}$ • Identify the vertex form of a quadratic function. Remember that the standard form is the most simplified form. This is not the vertex form. Remember that the vertex takes its name from the fact that it includes the coordinates of the parabola's vertex. The vertex form gives the coordinates of the vertex as $(h,k)$. The standard form of a quadratic functon is $ax^2+bx+c=0$. This is the most simplified format, and is a trinomial. The vertex form is $a (x-h)^2 + k = 0$. It reveals the coordinates of the vertex, which are $(h,k)$. The factored form of a quadratic function is $a(x-p)(x-q)=0$. It shows the intercepts, which are $p$ and $q$. $ax^2+bx=-c$ is a rearranged version of the standard form. It shows the first step in converting a standard form function into the vertex form. But it is not yet in the vertex form. • Identify the graphs of the functions. The equations are given in vertex form. Pay close attention to the signs. How can you determine the coordinates of the vertex from the values $h$ and $k$ in a vertex form equation? It may help to start by finding the correct vertices of each given equation. Then you can compare those vertices to those on the graph. We are given the vertices of three parabolas, and many quadratic functions. The quadratic functions are in vertex form. We know that in general, the vertex form is: $a (x-h)^2 + k = 0$, Where the vertex is $(h,k)$. Let's find the values of $h$ and $k$ that correspond to the vertex that is given for each parabola. For the purple parabola, the vertex is given as $(2,-4)$. Therefore $h = 2$ and $k = -4$. Substituting these values into the vertex form of a quadratic function, we get: $a (x-2)^2 -4 = 0$. We don't yet know the value of $a$. The only function that is listed that matches the values that we found for $h$ and $k$ is: Therefore this is the function that describes the purple parabola. Similarly, for the blue parabola, we know the vertex is at $(4,12)$, which means that $h = 4$ and $k = -12$. The only given function that matches is: For the gold parabola, we know the vertex is at $(-1,-4)$, which means that $h = -1$ and $k = -4$. The only given function that matches is $(x+1)^2-4=0$.	{"url":"https://us.sofatutor.com/math/videos/solving-quadratic-equations-by-completing-the-square","timestamp":"2024-11-11T08:03:00Z","content_type":"text/html","content_length":"165561","record_id":"<urn:uuid:62e6e776-31d7-4e3a-8b25-dffc36396799>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00259.warc.gz"}
Math worksheets percents word problems pdf This generator makes worksheets where the student calculates a percentage of a number, finds the percentage when the number and the part are given, or finds the number when the percentage and the part are given. Each printable comes with a place for the student to select what. Students will solve realworld problems involving money, stock prices percent change, discounts, etc. Some of the worksheets displayed are word problem practice workbook, 501 math word problems, solving proportion word problems, percent word problems, word problems with integers, all decimal operations with word problems, two step word problems, fractions word problems. Once you have cookies enabled, you can click here to go to the math antics home page. You can make the worksheets either in pdf or html formats both are easy to print. Click on popout icon or print icon to worksheet to print or download. 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Free games and worksheets pdf sixth grade math worksheets with answers and games online. Worksheets are percent word problems, grade 6 math word problems with percents, handouts on percents 2 percent word, percent word problems, percent proportion word problems, percents, word.	{"url":"https://viechaemepdo.web.app/684.html","timestamp":"2024-11-11T19:51:37Z","content_type":"text/html","content_length":"20008","record_id":"<urn:uuid:bf65cd7b-5a8f-4714-8c1d-15a551d26ad0>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00085.warc.gz"}
Big-O Notation: Beginners Guide In the world of programming write code as today is your last day on earth is not the main thing in the art of programming. Beyond understand what's the problem that our code will solve, we have other variables that we need to take care when we are writing code. For example: • Complexity • Maintainability • Design Patterns • Algorithms • and beyond... One of the most important things and the main topic of this article is the performance of our application when we are implementing an algorithm. The performance of our algorithm helps us to make more user-friendly our application, letting the user invest only the necessary time to do the tasks and focus on other things before the go bored and close the window (browser or anything else). No matter how fast and proficient was the device from where the user is using our app. In today's world, the time is the most precious thing that we have. Are many variables to take care when we talk about performance like the use of memory, disk space, execution time and another long beyond. One of the most important tools to determinate the performance of our algorithm is the Big-O Notation. Big O Notation The Big-O Notation is a tool that let us determinate the complexity of an algorithm, letting us take a measure of the performance of a specific algorithm. This tool measures the worst case scenario where the algorithm goes to the highest point of demand. The main Big-O terms are these: • O(1) -> Constant • O(n) -> Linear • O(log n) -> Logarithmic • O(n ^ 2) -> Quadratic • O(2 ^ n) -> Exponential O(1): Constant Complexity The constant complexity is the easier, all the algorithms who, no matter the size of the input or output, the execution time and the resources used will be always the same, have this complexity. No matter how many times or where the algorithm is executed, this one will have exactly the same behavior all the time. For example: As we see, no matter how big is the size of the array given as an argument, the behavior will always be the same. The only thing that might change will be the output, this because the array might not have the same data stored all the time. O(n): Linear Complexity We say that an algorithm has linear complexity when his execution time and/or resources used are directly proportional (grows up in a linear way) to the input size. For example: To get an easier way to understand this complexity, we can compare this to an activity that we do every day (or mostly), for example, read a book or watch a movie. The time that we spend reading a book or watching a movie, depends on the number of pages of a book and the duration of the movie. For example, if a movie has a duration of two hours, you'll spend two hours watching a movie. If the book has one hundred pages and you read fifty pages in one hour, you'll spend two hours to read all the book. O(log n): Logarithmic Complexity This complexity is present where the impact of an algorithm is directly proportional to the logarithmic result of the input size. In other words, when we have an input size of 10 and we need 1 second to accomplish the task with our algorithm, we need to do exactly the same task with an input size of 100 in 2 seconds, with an input size of 1000 in 3 seconds and so on. One interesting example is the binary search. In this algorithm, we divide the array (previously ordered) into two parts. We take the middle index as a reference to get the value at the middle of the array. If the number in that spot is equal to the number we are looking for, we return the middle index adding up the prefix that we use to find the spot where the number we are looking for is stored. Thanks to that, if the number is bigger than the number stored at the middle of the array, we look for the number in the right side of the array, instead, if is lower, we look in the left side of the array. After that, we use the prefix to use as the new prefix in the next iteration as a recursive way. We repeat this loop until get the given number. O( n^2 ): Quadratic Complexity The quadratic complexity is present when the impact of the algorithm is directly proportional to the square of the input size. This means, if we have an array as input with a length of 4 spots and we want to compare to look if the array has repeated items, we need to do 16 comparations to accomplish our task. This complexity is common in sorting algorithms like bubble sort, insertion sort, and the selection sort. In this function, the complexity can grow up if we add more for loops. This could make us go to an O(n * n) complexity. O( 2^n ): Exponential Complexity When an algorithm has exponential complexity, this means that the complexity will double with each additional item in the input. If 10 items take 100 seconds, 11 would take 200 seconds, 13 would take 400 seconds and so on. This doesn't mean that only exists O( 2^n ), this is just for explanation purpose. Can grow to O( 3^n ), O( 4^n ) and so on. Wrapping Up I hope this helps you to understand more the Big-O Notation, it's a great help to measure the complexity of an algorithm, exists many more tools to do this but this is one of the more common. If you have questions or want to correct some mistakes, feel free to leave your comment below. I'm leaving below a little list of resources about Big-O Notation. More resources Top comments (24) Itachi Uchiha • I love this post. Very useful. Thanks Carlos! Carlos Fuentes • • Edited on • Edited Thanks for your words. I love to help! Greetings :)! Rishi Giri • Wonderfully written! One suggestion - Next time when you describe something which involves writing code codes, don't use the screenshots, written code is more useful than the screenshots. Thank You! Carlos Fuentes • True, sorry about that. Thanks for your suggestion, I'll do it in the next article! 😁 Mykal • Awesome Post! Really helped my general understanding of Big O. that said, I did have a question regarding the O(n^2) definition. If you have a function that has 3 nested for loops, as opposed to 2 (as you have in your example) does the complexity then become O(n^3)? I can't seem to see anything online that would say so... Carlos Fuentes • Hi! Thanks, I really appreciate it! Yes, absolutely. If you make your for-loops grows with the same input, you can have something like an array of 4 spots and go through him three, four or five times growing the complexity in something like O(4 ^3, ^4, ^5 ^...n ). If you're going in a recursive way, the complexity it doesn't is O(n ^n ) and becomes exponential. Mykal • That makes a tonne of sense! Thanks for the reply 😊 Carlos Fuentes • You're welcome. I'm glad to helped you 🙏 benali djamel • awesome ! Joseph Galindo • Good post :) Would the exponential example not be 100, 200, (...400), 800 though? Carlos Fuentes • Yes, sure! Sorry, my bad. I'll fix it, thanks! ;) Joseph Galindo • • Edited on • Edited No problem...did want to post back though since I saw the article was updated. In the example you laid out, I think an algorithm running in O(2^n) time would be more like this: 10 items - 100s 11 items - 200s (12 items - not listed - 400s) 13 items - 800s Just wanted to point it out, since the typo makes the algorithm sound more performant than it really is. Hammed Oyedele • Thanks, this post really helped me alot! Carlos Fuentes • You're welcome. It's good to help! kiecodes • Very well put! Noorullah Ahmadzai • Thanks a lot Carlos! Fantastic Topic Mark Nicol • What a super clear description. Carlos Fuentes • Thank's. I appreciate :) Hammed Oyedele • Thanks, this post really helped me a lot! For further actions, you may consider blocking this person and/or reporting abuse	{"url":"https://dev.to/metcoder/big-o-notation-beginners-guide-1h38","timestamp":"2024-11-10T09:54:13Z","content_type":"text/html","content_length":"285128","record_id":"<urn:uuid:fd4485f2-2585-4221-a20f-2d1a54348d54>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00454.warc.gz"}
Star-convex polygons This example currently only runs on the develop branch In the grey area between convex and non-convex sets, there is a geometry which is called star-convex (star-domain, star-shaped, radially convex). As the name reveals, a classical drawing of a star is a special case. Mathematically, the definition of a (origin-centered) star-convex set is that all points between the origin and any point in the set is in the set. Compare this to the definition of a convex set where all points on a line between any two points are in the set. Loosely speaking, it is convex w.r.t a particular point (here the origin). The origin can be changed to some other so called vantage point by translating the whole set and defining star-convexity w.r.t the translated origin. Here, we will play around with star-convex polygons, modelling them both manually and by using built-in support. Note that you need a sos2 capable solver such as GUROBI or CPLEX. Star-convex polygons Define some data representing the vertices of a star (remember, a star-convex set does not have to look like a star, it is just a name) n = 6; th = linspace(-pi, pi, 2n+1); xi = (1+rem((1:2n+1),2)).cos(th); yi = (1+rem((1:2n+1),2)).sin(th); hold on; grid on plot( xi, yi, 'b-' ); axis equal; Note that we have generated the coordinates so that they represent a closed curve, i.e. the first and last values are the same. This is important to remember when you create these sets using the methods described here. The feasible set inside the star can be represented as the union of 7 polytopes. Hence, a general approach to representing this set is to describe these polytopes, and then use logic programming and binary variables to represent the union as illustrated in the big-M tutorial. However, star-convex polygons can be represented much more conveniently using sos2 constructs. To derive a sos2 representation, we first focus on representing the border of the polygon. With given vertices $v_i = (x_i,y_i)$, every point on the border of the star can be written as a linear combination of two adjacent vertices. This can be written as $V\lambda$ where where $V$ are the vertices stacked in columns and $\lambda$ is a non-negative vector with at most two adjacent non-zero elements summing up to $ 1$. This is the classical application of sos2, and precisely the same model YALMIP uses for linear interpolation in interp1 which essentially is what we are doing Assuming we work with coordinates $x$ and $y$ as decision variables in our model, and we want to say that $(x,y)$ is on the border, we arrive at the model sdpvar x y lambda = sdpvar(length(xi),1) Model = [sos2(lambda), lambda>=0, sum(lambda)==1, x == xilambda, y == yilambda]; That’s all! To see that this really models what we want, we could plot the set. When doing so, you will note that it is not drawing the star, but the convex hull of the star. This is not due to an incorrect model, but simply a limitation of the plot command. When you plot a set, it performs ray-shooting to find points on the boundary, and then draws the convex hull of these points (the only way to draw non-convex sets in YALMIP is to explicitly model mixed-integer models, as YALMIP then will perform enumeration of all combinatorial combinations of binry variables and draw each set individually to depict the union) To make sure we actually have modelled the border of the star, let us solve a simple problem where we find the point closest to a point outside. optimize(Model, (x-1.5)^2 + (y-1)^2) Scaling and translating So how can we include the interior? This is where star-convexity comes into play. Since the interior corresponds to all scaled border points, it means we can scale the interpolating $\lambda$ -variables with an arbitrary scale $0 \leq t \leq 1$. It also means we can take the adjacent interpolated vertices and scale them individually first, and then interpolate between them. Effectively, this means we can generate the model with sdpvar x y lambda = sdpvar(length(xi),1) Model = [sos2(lambda), lambda>=0,sum(lambda)<=1, x == xilambda, y == yilambda]; This model can be extended further by allowing an arbitrary scaling of the set by using any upper bound on the sum, even a decision variable as the bound enters affinely. What about the translations and the more general case of star-convexity w.r.t other vantage points than the origin? Let us start by defining a star centered outside the origin, so that star-convexity w.r.t the origin is violated. n = 6; th = linspace(-pi, pi, 2n+1); xi = 1+(1+rem((1:2n+1),2)).cos(th); yi = 2+(1+rem((1:2n+1),2)).sin(th); hold on; grid on plot( xi, yi, 'b-' ); axis equal; If we define the set using the same code as before, the case which only includes the border will still be valid, but the generalization to include the interior is flawed. As the interpolating $\ lambda$ is allowed to be zero, the origin will be included as a feasible point. The problem is that the set is not star-convex w.r.t the origin and the set we would create using our old code is the union of all stars scaled towards the origin. No problems though, we shift the origin and define the set as a translated star-convex set. Draw its convex hull as a sanity check. In this particular case, we can shift the origin to a vantage point defined as the mean of the coordinates. Note that the last element is a repetition of the first, so the mean is computed on the unique coordinates. xc = mean(xi(1:end-1)); yc = mean(yi(1:end-1)); lambda = sdpvar(length(xi),1) Model = [sos2(lambda), lambda>=0,sum(lambda)<=1, x == xc + (xi-xc)lambda, y == yc + (yi-yc)lambda]; Note that the using the mean of the coordinates as the vantage point is definitely not something which works in all cases. For highly symmetric objects it does, but in general problem specific insight is needed. As an example, the following set (blue) is star-convex (slide it to the left and it is star-convex wr.t. the origin) but after shifting all coordinates using the mean (red) we see that the set is not star-convex w.r.t the origin. xi = [1 2 2 3 3 4 4 1 1]; yi = [1 1 5 5 1 1 0 0 1]; hold on; grid on plot( xi, yi, 'b-' ); plot( xi-mean(xc), yi-mean(yc), 'r--' ); axis equal; Built-in support As we have seen, by using sos2 it is straightforward to derive a model, but YALMIP has built-in support for creating these sets even more conveniently. By default it only takes the coordinates and the element intended to be in the set and assumes star-convexity w.r.t to the origin. A fourth argument can be used to translate the set, and there is a fifth option to allow for scaling. With a sixth options, you can ask for YALMIP to automatically shift the model to derive a star-convexity model around a particular point, the mean, median or center of bounding box. Model = starpolygon(xi,yi,z); % Define model z in star-convex polygon defined by (xi,yi). Origin assumed to be vantage point Model = starpolygon(xi,yi,z,c); % Translate polygon by c (optional) Model = starpolygon(xi,yi,z,c,t); % Scale polygon by t (relative vantage point) (optional) Model = starpolygon(xi,yi,z,c,t,p); % Vantage-point (optional, either a point, or 'mean', median' or 'box') Construct data for a weird set which is star-convex around (e.g.) $ (1,2) $. Plot the star-convex set and its convex hull (shaded pink), the convex hull of the scaled set (blue), the convex hull of the translated set (yellow), and a scaled version with an alternative vantage point (red). n = 6; th = linspace(-pi, pi, 2n+1); xi = 1+(1+rem((1:2n+1),3)).cos(th).^3; yi = 2+(1+rem((1:2n+1),2)).sin(th); hold on; grid on plot( xi, yi, 'b-' ); axis equal; sdpvar x y Model = starpolygon(xi,yi,[x;y],[],[],[1;2]); Model = starpolygon(xi,yi,[x;y],[],.25,[1;2]); Model = starpolygon(xi,yi,[x;y],[],0.25,[1;1]); Model = starpolygon(xi,yi,[x;y],[-4;0],[],[1;2]); Optimizing over star-convex polygons Let us use our new set and solve some silly optimization problems. As a first example, we are given a point-cloud (almost looking like a star) and our task is to find the smallest possible star containing all points. We can formulate this as saying that all points in the point-cloud should be in a scaled and translated star, and the goal is to find the optimal translation and scaling. % Define the template star n = 6; th = linspace(-pi, pi, 2n+1); xs = (1+rem((1:2n+1),2)).cos(th); ys = (1+rem((1:2n+1),2)).sin(th); hold on; grid on plot( xs, ys, 'b-' ); axis equal; % Create the point-cloud thn = linspace(-pi, pi, 23n+1); xn = 3+interp1(th,xs,thn)+.1randn(1,23n+1) yn = 2+interp1(th,ys,thn)+.1randn(1,23n+1) plot( xn, yn, 'r' ); % Now say that each point in point-cloud is in scaled and translated polygon sdpvar t c = sdpvar(2,1); Model = []; for i = 1:length(xn) Model = [Model, starpolygon(xs,ys,[xn(i);yn(i)],c,t)]; % Minimize scale factor % ...and plot the scaled and translated star plot( value(t)xs+value(c(1)), value(t)ys+value(c(2)), 'k-' );	{"url":"https://yalmip.github.io/example/starshaped/","timestamp":"2024-11-09T10:01:19Z","content_type":"text/html","content_length":"51463","record_id":"<urn:uuid:8ed10715-a1b1-41e7-8b50-3f6ceb890371>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00741.warc.gz"}
This number is a composite. (85^11 - 85)/11 ± 1 are twin primes. [Luhn] Between 641 and 114689 (the second smallest Fermat divisor) there exist 85 prime numbers that could conceivably be Fermat divisors. There are 85 five-digit primes that begin with 85. [Bopardikar] The only composite number resulting from the sum of three double-digit consecutive emirps, 17, 31, 37. Note that the ordered concatenation of these emirps, i.e., 173137 is prime. [Loungrides] The only known Smith number n such that the aliquot sum of n = π(n). [Gupta] (There are 2 curios for this number that have not yet been approved by an editor.) Printed from the PrimePages <t5k.org> © G. L. Honaker and Chris K. Caldwell	{"url":"https://t5k.org/curios/page.php?number_id=646","timestamp":"2024-11-12T12:46:56Z","content_type":"text/html","content_length":"10283","record_id":"<urn:uuid:0f703380-4f5c-4467-9619-2fbb999500dd>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00081.warc.gz"}
Basic routines to handle periodic boundary conditions with SYCL. Artem Zhmurov zhmur.nosp@m.ov@g.nosp@m.mail..nosp@m.com static void pbcDxAiucSycl (const PbcAiuc &pbcAiuc, const rvec &r1, const rvec &r2, rvec dr) Computes the vector between two points taking PBC into account. More... static void pbcDxAiucSycl ( const PbcAiuc & pbcAiuc, const rvec & r1, const rvec & r2, inlinestatic rvec dr Computes the vector between two points taking PBC into account. Computes the vector dr between points r2 and r1, taking into account the periodic boundary conditions, described in pbcAiuc object. Note that this routine always does the PBC arithmetic for all directions, multiplying the displacements by zeroes if the corresponding direction is not periodic. For triclinic boxes only distances up to half the smallest box diagonal element are guaranteed to be the shortest. This means that distances from 0.5/sqrt(2) times a box vector length (e.g. for a rhombic dodecahedron) can use a more distant periodic image. This routine operates on rvec types and uses PbcAiuc to define periodic box, but essentially does the same thing as SIMD and GPU version. These will have to be unified in future to avoid code duplication. See Issue #2863: https://gitlab.com/gromacs/gromacs/-/issues/2863 [in] pbcAiuc PBC object. [in] r1 Coordinates of the first point. [in] r2 Coordinates of the second point. [out] dr Resulting distance.	{"url":"https://manual.gromacs.org/2022.2/doxygen/html-lib/pbc__aiuc__sycl_8h.xhtml","timestamp":"2024-11-03T09:54:49Z","content_type":"application/xhtml+xml","content_length":"11856","record_id":"<urn:uuid:58cfe5ce-7449-4b4b-8e20-6b02e7e69666>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00663.warc.gz"}
Recession Model Forecast: 01-01-2016 The following article updates the diffusion index, recession slack index, aggregate recession model, and aggregate peak-trough model through December 2015. Over the past 12 months, I added a number of new economic and market-based variables with very strong explanatory power to the recession model (including one at the end of December). This allowed me to cull three of the original independent variables with the weakest historical performance and most questionable cause and effect recessionary influence. The current 20-variable model has a diverse set of explanatory variables and is quite Each of the explanatory variables has predictive power individually; when combined together, the group of indicators is able to identify early recession warnings from a wide range of diverse market-based, fundamental, technical, and economic sources. After the latest additions and deletions, the total number of explanatory recession model variables is now 20. The current and historical data in this report reflect the current model configuration with all 20 variables. Diffusion Index The Trader Edge diffusion index equals the percentage of independent variables indicating a recession. With the recent changes, there are now a total of 20 explanatory variables, each with a unique look-back period and recession threshold. The resulting diffusion index and changes in the diffusion index are used to calculate the probit, logit, and neural network model forecasts. The graph of the diffusion index from 1/1/2006 to 01/1/2016 is presented in Figure 1 below (in red - left axis). The gray shaded regions in Figure 1 below represent U.S. recessions as defined (after the fact) by the National Bureau of Economic Research (NBER). The value of the S&P 500 index is also included (in blue - right axis). In December 2014, for the first time since December 2012, one of the 20 explanatory variables indicated a recessionary environment. The number of variables indicating a recession varied between zero and one from December 2014 through May 2015 and between one and two from June 2015 through November 2015. The number of variables indicating a recession jumped from two in November to five in December, which is the highest diffusion index value since the end of 2011. The diffusion index remained positive throughout most of 2015, which was troubling. The large spike to five in December is much more serious. Unless the market stages a significant rally in the last week of January, one or more market-based explanatory variables would move into recession territory next month. If that happens (and no other variables fall below their recession thresholds), it would constitute the highest diffusion index reading (6/20) since the Great Recession. Going back to 1959, diffusion index readings of six or more have always been associated with U.S. recessions. In other words, the diffusion index has only reached a level of six shortly before, during, or after NBER recessions. In non-recessionary environments, weakness typically persists for a few months and then dissipates. However, if the weakness becomes more widespread or lingers for many months, that can be more problematic. The weakness persisted throughout much of 2015 at a relatively modest level, but has now jumped alarmingly. We are nearing a tipping point. Please note that past estimates and index values will change whenever the historical data is revised. All current and past forecasts and index calculations are based on the latest revised data from the current data set. Recession Slack Index The Trader Edge recession slack index equals the median standardized deviation of the current value of the explanatory variables from their respective recession thresholds. The resulting value signifies the amount of slack or cushion relative to the recession threshold, expressed in terms of the number of standard deviations. The gray shaded regions in Figure 2 below represent U.S. recessions as defined (after the fact) by the NBER. The median recession slack index is depicted in purple and is plotted against the right axis, which is expressed as the number of standard deviations above the recession threshold. The dark-red, horizontal line at 0.50 standard deviations denotes a possible warning threshold for the recession slack index. Many of the past recessions began when the recession slack index crossed below 0.50. Similarly, many of the past recessions ended when the recession slack index crossed back above 0.0. In mid-2014, the revised median recession slack index peaked at 1.21, far above the warning level of 0.50. The revised values of the recession slack index declined alarmingly to 0.68 in March 2015, perilously close to the early warning level of 0.50. The value of 0.68 matched the lowest value recorded since the end of the Great Recession. The median recession slack index remained between 0.97 and 0.94 from April through July 2015, and between 0.80 and 0.82 from August through November 2015. In December 2015, the slack index dropped back to 0.68, again matching its lowest level since the end of the Great Recession. The cushion above the warning level has shrunk considerably since mid-2014 and the slack index is now only marginally above its warning threshold. The ability to track small variations and trend changes over time illustrates the advantage of monitoring the continuous recession slack index in addition to the diffusion index above, which moves in discrete steps. While it is useful to track the actual recession slack index values directly, the values are also used to generate the more intuitive probit and logit probability forecasts. Aggregate Recession Probability Estimate The Trader Edge aggregate recession model is the average of four models: the probit and logit models based on the diffusion index and the probit and logit models based on the recession slack index. The aggregate recession model estimates from 1/1/2006 to 01/01/2016 are depicted in Figure 3 below (red line - left vertical axis). The gray shaded regions represent NBER recessions and the blue line reflects the value of the S&P 500 index (right vertical axis). I suggest using a warning threshold of between 30-40% for the aggregate recession model (green horizontal line). The aggregate recession model probability estimate for 01/01/2016 increased from 0.1% in November to 3.6% in December 2015. According to the model, the probability that the U.S. is currently in a recession continues to be remote. Aggregate Peak-Trough Probability Estimate The peak-trough model forecasts are different from the recession model. The peak-trough models estimate the probability of the S&P 500 being between the peak and trough associated with an NBER recession. The S&P 500 typically peaks before recessions begin and bottoms out before recessions end. As a result, it is far more difficult for the peak-trough model to fit this data and the model forecasts have larger errors than the recession model. The Trader Edge aggregate peak-trough model equals the weighted-average of nine different models: the probit and logit models based on the diffusion index, the probit and logit models based on the recession slack index, and five neural network models. The aggregate peak-trough model estimates from 1/1/2006 to 1/01/2016 are depicted in Figure 4 below, which uses the same format as Figure 3, except that the shaded regions represent the periods between the peaks and troughs associated with NBER recessions. The aggregate peak-trough model probability estimate for 01/01/2013 was 29.2%, which is up sharply (+19.9%) from the revised value of 9.3% at the end of November. With the exception of one month in late 2011, the value of 29.2% was the highest reading since the end of the Great Recession. If current market levels hold, it is likely that this probability will increase again at the end of January December 2015 marked the most significant increase in U.S. recession risk since late 2011. The diffusion index increased to 5/20 in December, its highest value since late 2011. To make mattes worse, at least one additional explanatory variable is currently above its recession threshold (month-to-date January). This is particularly important, given that the diffusion index has never reached 6/20 outside of a recessionary environment. The use of several market-based indicators makes the Trader Edge recession model more responsive. Relative to traditional economic variables, market-based data have important advantages: they are highly predictive, they are never restated, and there is no lag in receiving the data. The five diffusion index variables above their recession thresholds at the end of December include market-based and non-market-based data. In March 2015, the recession slack index matched its lowest level since the end of the Great Recession (0.68). It rebounded briefly in subsequent months, but has again dropped back to 0.68. The peak-trough recession probability estimate increased sharply in December, ending the month at 29.2%. All of the recession forecasts remain inside their respective warning thresholds, but we could be approaching a tipping point. Given the close proximity of a few diffusion index variables relative to their respective recession thresholds, the recession model forecasts over the next few months will continue to be very Print and Kindle Versions of Brian Johnson's 2nd Book are Available on Amazon (75% 5-Star Reviews) Exploiting Earnings Volatility: An Innovative New Approach to Evaluating, Optimizing, and Trading Option Strategies to Profit from Earnings Announcements. Print and Kindle Versions of Brian Johnson's 1st Book are Available on Amazon (79% 5-Star Reviews) Option Strategy Risk / Return Ratios: A Revolutionary New Approach to Optimizing, Adjusting, and Trading Any Option Income Strategy Trader Edge Strategy E-Subscription Now Available: 20% ROR The Trader Edge Asset Allocation Rotational (AAR) Strategy is a conservative, long-only, asset allocation strategy that rotates monthly among five large asset classes. The AAR strategy has generated annual returns of approximately 20% over the combined back and forward test period. Please use the above link to learn more about the AAR strategy. Your comments, feedback, and questions are always welcome and appreciated. Please use the comment section at the bottom of this page or send me an email. If you found the information on www.TraderEdge.Net helpful, please pass along the link to your friends and colleagues or share the link with your social or professional networks. The "Share / Save" button below contains links to all major social and professional networks. If you do not see your network listed, use the down-arrow to access the entire list of networking sites. Thank you for your support. Brian Johnson Copyright 2016 - Trading Insights, LLC - All Rights Reserved. This entry was posted in Economic Indicators, Fundamental Analysis, Market Commentary, Market Timing, Recession Forecasting Model, Risk Management and tagged aggregate peak-trough model, aggregate recession model, diffusion index, logit model, probit model, recession forecast, recession forecast December 2015, Recession Slack Index, Trader Edge. Bookmark the permalink. One Response to Recession Model Forecast: 01-01-2016	{"url":"https://traderedge.net/2016/01/24/recession-model-forecast-01-01-2016/","timestamp":"2024-11-13T12:41:00Z","content_type":"text/html","content_length":"75190","record_id":"<urn:uuid:cb08c99a-a2e6-49f8-a3f8-d949c3f44c9d>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00638.warc.gz"}
(B) Establishment of a flow cytometry gate for live/dead staining with PI (B) Establishment of a flow cytometry gate for live/dead staining with PI (B) Establishment of a flow cytometry gate for live/dead staining with PI. of genotypes WT/WT, WT/L262P and WT/L262P (kDNA0). The data is represented by filled dots. The coloured lines represent median fits of the model; the shaded regions indicate 95% predictive intervals, where 95% of future data would be predicted to lie according to the model and the data already observed. (A, C, E, G) The mathematical model used involves SIF-dependent and SIF-independent differentiation terms. (B, D, F, H) The mathematical model only includes a SIF-dependent differentiation term.(TIF) ppat.1007195.s003.tif (2.1M) GUID:?521F2938-21E8-466D-A436-9A8658DD148A S4 Fig: Fit of the model including only a SIF dependent term for differentiation. (A) Standardised residuals (blue circles) of parasite density and slender fraction, by time, of the model fits with SIF-dependent differentiation only to all mice. Under a true model standardised residuals have an approximately standard normal distribution (i.e., zero mean and unit standard deviation (SD)). Inadequate fit of a model is indicated by its residuals deviating from a standard normal distribution (such as Leucovorin Calcium residuals further than ~3 SD from zero, represented by the lightest grey shading, Leucovorin Calcium or a set of residuals consistently above or below zero. The red line shows the average, across all mice, of the residuals at a particular time point. (B) Assessment of the quality of fit of the two alternative models to infection data from MacGregor et al., 2011, using the Akaike information criterion (AIC). The AIC measures the quality of a fit of mathematical model to a set of data, taking into account the goodness of fit and the number of parameters estimated in the model. As increasing the number of parameters improves the Leucovorin Calcium goodness of fit, AIC penalizes models with more estimated parameters to discourage overfitting. Hence the model with the lowest AIC, i.e. the model with the lowest number of parameters to prevent overfitting, is preferred.(TIF) ppat.1007195.s004.tif (3.2M) GUID:?232A56E2-36AC-44D1-89F9-1DEDC4DD7F3A S5 Fig: Physiological analysis of cell lines. (A) Cell cycle analysis with Hoechst 33342 dye and flow cytometry to assess slender form (SL) contamination. Stumpy forms (ST) are cell cycle arrested in G1 phase. The absence of G2 peaks (except in the SL control) suggests that slender contamination was minimal. (B) Establishment of a flow cytometry gate for live/dead staining with PI. 1×106 cells were analysed. Stumpy cells killed by heat treatment (red), live cells (orange) and a mix of live and dead cells (green) were analysed. (C) Measurement of m in WT/WT stumpy cells maintained in the presence and absence of azide. Cells were incubated in HMI-9 medium for 0, 24 or 48 h, +/- 0.5 mM sodium azide. At each time point, 1×106 cells were stained with TMRE and analysed by flow cytometry. The black line shows the no m gate which is dictated IFNGR1 by the TMRE fluorescence of cells treated with uncoupler FCCP (20 M; grey population in the background in all panels; note that the grey population is difficult to discern as it almost completely overlaps with the azide-treated populations). The average % cells that retain m in the absence of azide treatment is indicated. Left panel: dark green, plus azide; apricot, no azide. Middle panel: magenta, plus azide; yellow, no azide. Right panel: light green, plus azide; purple, no azide. (D) Cells were harvested from mice at maximum parasitaemia, with approximately 90% stumpy forms, and placed in Creeks minimal medium, supplemented as indicated. GlcNAc, N-acetyl glucosamine. The percentage of live cells after 24 hrs was assessed by PI staining and flow cytometry; n = 3 for Leucovorin Calcium each cell line.(TIF) ppat.1007195.s005.tif (4.1M) GUID:?8CB3BBB6-67BA-4641-BA36-DCAB374A5AC1 Data Availability StatementAll relevant data are within the paper and its Supporting Information files. Abstract The sleeping sickness parasite has a complex life cycle, alternating between a mammalian host and the tsetse fly vector. A tightly controlled developmental programme ensures parasite transmission between hosts as well as survival within them and involves strict regulation of mitochondrial activities. In the glucose-rich bloodstream, the replicative slender stage is Leucovorin Calcium thought to produce ATP exclusively via glycolysis and uses the mitochondrial F1FO-ATP synthase as an ATP hydrolysis-driven proton pump to generate the mitochondrial membrane potential (m). The procyclic stage in the glucose-poor tsetse midgut depends on mitochondrial catabolism of amino acids for energy production, which involves oxidative phosphorylation with ATP production via the F1FO-ATP synthase. Both modes of the F1FO enzyme critically depend on FO subunit and in mice, which significantly.	{"url":"http://www.chiflatironsofficial.com/2022/02/08/%EF%BB%BFb-establishment-of-a-flow-cytometry-gate-for-live-dead-staining-with-pi/","timestamp":"2024-11-09T13:33:30Z","content_type":"text/html","content_length":"37259","record_id":"<urn:uuid:3c8be82a-b14a-4fe2-8116-02fefc266851>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00416.warc.gz"}
Introduction to Introduction to waves Introduction to waves: A wave is a transfer of energy between two points without any transfer of matter. Mechanical: waves transferred through a material by particle vibration. For example, sound waves and primary seismic waves. Electromagnetic: waves transferred by the concurrent oscillations of electric and magnetic fields. For example, light waves, radio waves, X-rays… etc. Longitudinal: An oscillation where the displacement of particles from the mean position is parallel to the direction of energy propagation. For example, sound waves and p-seismic waves. Transverse: An oscillation where the displacement of particles from the mean position is perpendicular to the direction of energy propagation. For example, all EM waves and secondary seismic waves. Key Terms: • Wavelength (Represented by the Greek symbol, Lambda, λ): The distance between a point on one wave and the identical point on the next wave. I.e. Crest to crest or trough to trough. This can be linked with wave speed and frequency by the equation, v (velocity/wave speed) = λ (wavelength) x f (frequency) • Displacement: The displacement of a vibrating particle is the distance and direction from its equilibrium position, i.e the particle’s displacement from its mean position. • Amplitude, a: The amplitude of a wave is the maximum displacement of a vibrating particle. For a transverse wave, this is the height of a wave crest or the depth of a wave trough from its mean (equilibrium) position. • Period, T: The time for one complete wave to pass a fixed point. The period of a wave is equal to 1/f (frequency). • Frequency, f: This is the number of waves which pass a fixed point in one second. The unit is Hertz (Hz). Using the wavelength equation: QUESTION- What is the frequency of orange light with a wavelength of 600 nm? ANSWER- Use the equation v= λf, rearrange it to get f= v/λ to find the frequency. Since you are finding the frequency of a light wave, you must know the speed of light, 3×10⁸. Substitute in your values: f= 3×10⁸/ 600= 500,000 You can put this value into standard form. Therefore your final answer is 5 x 10⁵	{"url":"https://backnotes.com/as-a-level/physics-9702/introduction-to-waves/","timestamp":"2024-11-13T12:02:18Z","content_type":"text/html","content_length":"65402","record_id":"<urn:uuid:450cc575-53ea-4398-a5da-34c1888208e5>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00653.warc.gz"}
(PDF) Reducing the Effects of Inaccurate Fault Estimation in Spacecraft Stabilization J. Aerosp. Technol. Manag., São José dos Campos, Ahead of Print, 2017 ABSTRACT: Reference Governor is an important component of Active Fault Tolerant Control. One of the main reasons for using Reference Governor is to adjust/modify the reference trajectories to maintain the stability of the post-fault system, especially when a series of actuator faults occur and the faulty system can not retain the pre-fault performance. Fault estimation error and delay are important properties of Fault Detection and Diagnosis and have destructive effects on the performance of the Active Fault Tolerant Control. It is shown that, if the fault estimation provided by the Fault Detection and Diagnosis (initial “fault estimation”) is assumed to be precise (an ideal assumption), the controller may not show an acceptable performance. Then, it is shown that, if the worst “fault estimation” is considered, it will be possible to reduce the effects of fault estimation error and delay and to preserve the performance of the controller. To reduce the effects of this conservative assumption (worst “fault estimation”), a quadratic cost function is dened and optimized. One of the advantages of this method is that it gives the designer an option to select a less sophisticated Fault Detection and Diagnosis for the mission. The angular velocity stabilization of a spacecraft subjected to multiple actuator faults is considered as a case KEYWORDS: Active Fault Tolerant Control, Fault estimation error and delay, Reference Governor, Angular velocity Reducing the Effects of Inaccurate Fault Estimation in Spacecraft Stabilization Rouzbeh Moradi1, Alireza Alikhani1, Mohsen Fathi Jegarkandi2 Active Fault Tolerant Control (AFTC) is an important eld in automatic control that has attracted a large amount of attention. e main responsibility of an AFTC is to tolerate component malfunctions while maintaining desirable performance and stability properties of the faulty system (Zhang and Jiang 2008). Latterly, a review paper published recent developments of the spacecra AFTC system (Yin et al. 2016). One of the main components of any AFTC is the Fault Detection and Diagnosis (FDD) module. ere are several challenges that FDD designs have in common (Zhang and Jiang 2008). Among them, fault estimation error and delay are considered in this paper. ese challenges have destructive eects on the stability and performance (Zhang and Jiang 2008). Reference Governor (RG) is one of the components of the general AFTC structure (Zhang and Jiang 2008). The terms Command Governor (CG) and Reference Trajectory Management (RTM) have been also used in the literature. e main responsibility of RG is to adjust/modify the reference trajectories, so the post-fault model of the system remains stable, even aer the occurrence of multiple actuator faults (Garone et al. 2016). ere are several papers in the literature that have studied the eects of RG on the performance and stability of the post-fault model (Boussaid et al. 2010; Boussaid et al. 2011; Boussaid et al. 2014; Almeida 2011). According to these papers, RG has been able to deal with the actuator faults/ failures eciently. To the authors’ best knowledge, reducing the eects of fault estimation error and delay using the concept of RG still remains an open problem. This is the main subject that is pursued in this paper. It is shown that, as long as the estimated fault doi: 10.5028/jatm.v9i4.826 1.Ministry of Science, Research and Technology – Aerospace Research Institute – Astronautics Department – Tehran/Tehran – Iran. 2.Sharif University of Technology – Engineering College – Department of Aerospace Engineering – Tehran/Tehran – Iran. Author for correspondence: Alireza Alikhani \| Ministry of Science, Research and Technology – Aerospace Research Institute – Astronautics Department \| PO box: 14665-834 – Tehran/Tehran – Iran \| Email: aalikhani@ari.ac.ir Received: Oct. 29, 2016 \| Accepted: Mar. 25, 2017 J. Aerosp. Technol. Manag., São José dos Campos, Vol.9, No 4, pp.453-460, Oct.-Dec., 2017 J. Aerosp. Technol. Manag., São José dos Campos, Vol.9, No 4, pp.453-460, Oct.-Dec., 2017 454 Moradi R, Alikhani A, Fathi Jegarkandi M reported by the FDD (initial “fault estimation”) is assumed to be precise (an ideal assumption), the controller may not show an acceptable performance. However, if the maximum fault estimation error is considered (worst “fault estimation”), RG can be used to reduce the eects of FDD errors and preserve the performance of the closed-loop system. To reduce the e ects of this conservative assumption (considering maximum fault estimation error), a quadratic cost function is de ned and optimized. In order to validate the results, the angular velocity stabilization of a spacecra subjected to multiple actuator faults is considered. It is shown that, if the initial “fault estimation” (the fault estimation reported by the FDD) is considered accurate, the response will not converge to the origin. However, if RG is designed based on the worst “fault estimation”, AFTC will be able to asymptotically stabilize the faulty spacecra in a wide range of actuator fault and despite FDD errors. is paper consists of the following sections: rstly, the modeling of the proposed RG is described. en, the spacecra dynamics and controller are shown. Finally, results obtained and the discussions are presented. e structure of the considered AFTC is shown in Fig. 1. It is assumed that the FDD block provides “an estimation of” the post-fault model of the system. e RG block uses the proposed methodology to nd the most suitable reference trajectories for the post-fault model, despite the presence of fault estimation error and delay. e signals ω and ωd are the plant output (angular velocity) and the desired reference trajectory vectors, respectively. It is assumed that the actuator fault/failure occurs at t = tfault and the FDD determines ˆ tfault (estimated tfault) with a fault estimation delay equal to: Figure 1. Structure of the AFTC. In this paper, the mission of the controller is to make the origin an asymptotically stable equilibrium for the post-fault system, i.e. ω → 0 as t → tf ( nal time). which is a positive value, since ˆ tfault is always bigger than tfault. Fault estimation error is another property of the considered FDD block. e control inputs are bounded according to the following saturation function: where umax is the maximum torque that can be produced by the actuators. e reduction in the actuator region is considered as the actuator fault and is modeled according to Eq. 3 (Miksch and Gambier 2011): The subscript p-f shows the post-fault condition. The relation between pre- and post-fault actuator region is given according to: where a is the actuator e ec tiveness coe cient (Sobhani-Tehrani and Khosravi 2009), a real value between 0 and 1; umax is the pre-fault actuator region. FDD determines the estimated value of a (shown by â). It is assumed that the FDD provides â with an estimation error given by: where δa/â is a value between 0 and 1. e larger/smaller values of δa/â show better/worse fault estimation, respectively. According to the considered mission, the goal of RG is to determine ωd such that the faulty model of the system remains asymptotically stable, even a er the occ urrence of multiple actuator faults and in the presence of fault estimation error and delay J. Aerosp. Technol. Manag., São José dos Campos, Vol.9, No 4, pp.453-460, Oct.-Dec., 2017 Reducing the Effects of Inaccurate Fault Estimation in Spacecraft Stabilization in the FDD module. e RG owchart is presented in Fig. 2. e consecutive steps are explained in the following paragraphs. According to Fig. 3, ωd (t1) ... ωd (tn) are initialized by the solver, which is the Genetic Algorithm (GA), as will be explained in the results section. Note 1: although the GA is used to solve the problem, other numerical solvers can be also employed. However, the main concern of this paper is to nd a method to decrease the consequences of fault estimation error and delay. erefore, any numerical solver (possibly faster than GA) that solve the problem can be considered as well. Note 2: as will be seen in the simulation section, GA can nd a solution within a reasonable time. When these points are determined, a cubic spline is passed through them, similarly to Fig. 4. A detailed analysis about cubic spline interpolation can be found in de Boor (1978). One of the main advantages of cubic splines is their smoothness (they are twice continuously di erentiable). is will prevent the controller inputs from being discontinuous (refer to Eqs. 25 – 27). According to the FDD information, an estimation of the post-fault model of the system is known. e faulty closed-loop system is simulated from tfault to tf . is simulation is a part of the owchart shown in Fig. 2 and several simulations may be needed to obtain ωd. A er simulation, the value of ω (tf ) is checked to see whether the following equality is satis ed or not: Figure 3. Initializing ωd (t1) ... ωd (tn). Figure 2. RG owchart. ωd(t1) ... ωd(tn) are initialized Determine ωd via cubic interpolating splines Simulate the closed loop system from tfault to tf Equation 34 is satisfied Ye s t1 = tfault t1 = tfault Figure 4. ωd produced by cubic spline. Such a nal state constraint is well-known in the literature and is introduced to ensure asymptotic stability (Fontes 2001). Since this equality will never hold numerically, Eq. 34 will be considered in simulations. Note 3: to ensure that ωd approaches the origin before t = tf, its value is set to 0 as t passes ts (settling time). In other To give the solver more exibility, another variable (ks) is introduced, satisfying Eq. 8: In addition to ωd (t1) ... ωd (tn), ks is another variable that should be found by the solver. The rigid body spacecraft rotational dynamics in the principal coordinate system is described by the following equations (Sidi 2000): J. Aerosp. Technol. Manag., São José dos Campos, Vol.9, No 4, pp.453-460, Oct.-Dec., 2017 456 Moradi R, Alikhani A, Fathi Jegarkandi M where ω1, ω2, ω3 are the angular velocities; u ´ 1, u ´ 2, u ´ 3 are the normalized control inputs; J1, J2, J3 are the principal moments of inertia of the rigid body. e relation between control torques and inputs are given by Eqs. 12 – 14: and the following form of control inputs where u1, u2, u3 are the control moments acting on the spacecra . e error signal is de ned as: where ωd and ωe are the desired and error angular velocity vectors, respectively. Inserting the scalar form of Eq. 15 into Eqs. 9 – 11 and eliminating ω, one has: Canceling the non-linear terms using feedback lineari- zation, the closed-loop system will change into the following simple linear time invariant form: will lead to the exponential stabilization of ωe to 0; consequen- tl y, ω will converge to ωd exponentially. e numerical values of k1, k2 and k3 determine the exponential convergence rate of ωe to 0. erefore, larger values of k1, k2 and k3 mean a faster response and vice-versa. Considering Eqs. 16 – 18 and Eqs. 22 – 24, the following relations will be obtained: For feedback purposes, it is better to rewrite u ´ 1, u ´ 2 and u ´ 3 as a function of the original variables: According to Eqs. 28 – 30, for the control inputs to be continuous, the desired reference trajectory (ωd) should be continuously differentiable. As stated previously, this is one of the main reasons for using cubic spline interpolation to find ωd. These are the desired control inputs that will lead to the exponential convergence of ω to ωd. If ωd = 0, the equations of closed-loop system will be: J. Aerosp. Technol. Manag., São José dos Campos, Vol.9, No 4, pp.453-460, Oct.-Dec., 2017 Reducing the Effects of Inaccurate Fault Estimation in Spacecraft Stabilization Clearly, as long as there is no saturation and the actuators can produce the required control inputs, will remain globally exponentially stable (GES). However, a er the occurrence of severe actuator faults, GES will not be guaranteed. e system/controller parameters and initial conditions are given in Table 1. e values chosen for the moments of inertia are taken from Wang et al. (2013), and the range of variables is presented in Table 2. respectively. e direction of the arrows shows the direction of the forces produced by the thrusters (Fig. 5). erefore, the relation between control torques (u1, u2, u3) and T1 – T6 can be obtained according to the following equations: Optimization variable Range ωd[–100 100] deg/s ks[0.5 0.9] Table 1. System/controller parameters and initial conditions Initial conditions Moments of inertia (kgm2) k1 = 0.1 ω1 (0) = 10 J1 = 449.5 k2 = 0.1 ω2 (0) = –10 J2 = 449.5 k3 = 0.1 ω3 (0) = 5 J3 = 449.5 Table 2. Range of variables. In order to satisfy the nal state constraint given by Eq. 6, the following inequality is de ned: As already mentioned, to determine ωd , GA (Goldberg 989) is used as the solver; [ω1d (t1) ... ω1d (tn)], [ω2d (t1) ... ω2d (tn)] and [ω3d (t1) ... ω3d (tn)] are initialized every 10 s ( ∆t = 10 s or equivalently, n = 10) from the beginning of the fault time (tfault). erefore, considering ks, the total number of decision variables will be 31. e considered parameters for GA are presented in Table 3. Other GA parameters are the default values considered in MATLAB® (MathWorks® 2011). The actuation system consists of 6 thrusters (without considering hardware redundancy), that are placed in opposite directions, and each thruster can produce maximum 50 N variable thrust. e e ective moment arm of all thrusters is 1 m along the principal body axis. However, the con guration of the thrusters is such that (T1 − T2), (T3 − T4) and (T5 − T6) produce net moments about the rst, second and third principal axes, where the superscripts + and – show the positive and negative control torques, respectively. Note 4: it seems that the thrusters T3, T4, T5 and T6 pass through the center of gravity. However, as indicated before, they have a moment arm of 1 m along the rst body axis. ree important concepts are introduced: • Initial “fault estimation”: the fault estimation reported by the FDD. • Worst “fault estimation”: the biggest error of the FDD in providing the fault information. Its value is determined from the initial “fault estimation”, according to the experience or the FDD speci cations. • Real fault: the fault that happens in reality (unknown). e fault scenario that FDD reports is: Figure 5. Thruster con guration. Parameter Value Cross-over fraction 0.8 Elite count 2 Population size 5 × number of decision variables = 5 × 31 = 155 Initial population ωd,initial = 0 , ks,initial = 0 Table 3. GA parameters. J. Aerosp. Technol. Manag., São José dos Campos, Vol.9, No 4, pp.453-460, Oct.-Dec., 2017 458 Moradi R, Alikhani A, Fathi Jegarkandi M • Initial “fault estimation”: T5 and T6 have lost 99% of their e ectiveness (â5 = â6 = 0.01) and the remaining thrusters are at a good health (â1 = â2 = â3 = â4 = 1). e fault occurs at ˆ tfault = 10 s. • Worst “fault estimation”: based on the experience or the FDD specifications; in the worst case, the following parameters are given: δtfault = 5 s and δa/â = 0.01. erefore, it can be concluded that, in the worst case, a5 = a6 = 0.0001, i.e. T5 and T6 can produce a maximum 0.05 N thrust and the fault occurrence time is tfault = 5 s . Note 5: it is assumed that the real fault is less severe than the one reported by the worst “fault estimation”. In this case, the controller will show an acceptable performance for less severe, and therefore, a wide range of faults. Qualitatively, it is assumed that the severity of the faults satis es the following inequalities: where S is a quality that represents the severity of the fault; the subscripts w.f.e, r.f and i.f.e stand for worst “fault estimation”, real fault and initial “fault estimation”, respectively. According to the previous discussion, the proposed method is very conservative, because it considers the worst “fault estimation”. To reduce the adverse e ects of this assumption, the following quadratic cost function is introduced: Minimizing this cost function will decrease the adverse e ects of considering the worst fault estimation. e consi- dered sample time for integration is 0.1 s. e problem consists of 2 phases: first, GA tries to satisfy the constraint given by Eq. 34. Then, the result is used as an initial solution to optimize Eq. 39. e following penalty on cost function is It was verified that 1,000 s elapsed time is considered as the stopping criterion for the second phase — Intel(R) Core™ 2 CPU, T7200@2.00 GHz; MATLAB® (MathWorks® 2011). To observe the consequences of employing the proposed method, 2 different cases are considered and summarized in Table 4. Case Fault estimation 1 Considering the initial “fault estimation” 2 Considering the worst “fault estimation” Table 4. Cases consi dered. CASE 1 If the initial “fault estimation” is considered (FDD is assumed to report the precise fault information), the results shown in Figs. 6 and 7 will be obtained. Figure 6. Angular velocities, initial “fault estimation” (case 1). Figure 7. Control inputs, initial “fault estimation” (case 1). Time [s] Time [s] Time [s] Time [s] Time [s] Time [s] ω3 [deg/s] ω2 [deg/s] ω1 [deg/s] Eq. 34 is not sa t isfie d Eq. 34 is satis ed Eq. 34 is not satis ed J. Aerosp. Technol. Manag., São José dos Campos, Vol.9, No 4, pp.453-460, Oct.-Dec., 2017 Reducing the Effects of Inaccurate Fault Estimation in Spacecraft Stabilization Figure 6 shows that RG can not make the closed-loop system asymptotically stable, because it assumes the fault scenario reported by the FDD (initial “fault estimation”), which is precise. However, since the real fault is worse than the fault reported by the FDD (initial “fault estimation”), does not converge to the origin. is simulation shows the consequences of considering the initial “fault estimation”. e main conclusion of this simulation is: if the FDD is assumed to report the precise fault information, the response of the controller may not be acceptable. CASE 2 e result of considering the worst “fault estimation” is illustrated in Fig. 8. e control inputs are illustrated in Fig. 9. According to Fig. 8, RG can asymptotically stabilize the closed-loop system, when the worst “fault estimation” is considered. A comparison of Figs. 6 and 8 shows the consequences of considering the worst “fault estimation” in the RG design. Clearly, considering the initial “fault estimation” (case 1) can lead to the poor performance of the controller and even to a non-convergent response. On the other hand, if RG is designed for the worst “fault estimation” (case 2), it can cover less severe faults and stabilize the faulty system for a wide range of faults (Note 5). Since the assumption of worst “fault estimation” is conservative, the response is optimized via minimizing the cost function (Eq. 39). e GA performance is illustrated in Fig. 10. As stated previously, the quadratic cost function has been introduced to reduce the adverse consequences of considering the worst “fault estimation” (maximum fault estimation error). According to Fig. 10, after 14 generations (1,000 s elapsed time), the cost function is reduced from 8,758 to 5,944 Time [s] T5 [N] Time [s] T3 [N] Time [s] T1 [N] Time [s] T6 [N] Time [s] T4 [N] Time [s] T2 [N] ω3 [deg/s] ω2 [deg/s] ω1 [deg/s] (about 32%). is reduction in the cost function decreases the adverse consequences of considering the worst fault estimation. Figure 8. Angular velocities, worst “fault estimation” (case 2). Figure 10. Cost function versus generations (1,000 s elapsed Figure 9. Control inputs, worst “fault estimation” (case 2). Fault estimation error and delay are important characteristics of FDD schemes. RG is a method to adjust/ modify the reference trajectories to handle actuator fault/ failure. It was shown that, if the initial “fault estimation” was assumed to be precise (an ideal assumption), the controller might not be able to show an acceptable performance. On the other hand, if the worst “fault estimation” was considered, it J. Aerosp. Technol. Manag., São José dos Campos, Vol.9, No 4, pp.453-460, Oct.-Dec., 2017 460 Moradi R, Alikhani A, Fathi Jegarkandi M would be possible to reduce the destructive effects of fault estimation error. A quadratic cost function was defined to reduce the adverse consequences of this conservative assumption (assuming maximum fault estimation error). Therefore, a less sophisticated FDD can be used to satisfy the mission objectives. Conceptualization, Moradi R; Methodology, Moradi R, Alikhani A, and Fathi Jegarkandi M; Writing – Original Dra, Moradi R and Alikhani A; Writing – Review & Editing, Moradi R, Alikhani A, and Fathi Jegarkandi M. Almeida FA (2011) Reference management for fault-tolerant model predictive control. J Guid Control Dynam 34(1):44-56. doi: Boussaid B, Aubrun C, Abdelkrim MN (2010) Fault adaptation based on reference governor. Proceedings of the Conference on Control and Fault-Tolerant Systems; Nice, France. Boussaid B, Aubrun C, Abdelkrim MN (2011) Two-level active fault tolerant control approach. Proceedings of the 8th International Multi- Conference on Systems, Signals and Devices; Sousse, Tunisia. Boussaid B, Aubrun C, Jiang J, Abdelkrim MN (2014) FTC approach with actuator saturation avoidance based on reference management. International Journal of Robust and Nonlinear Control 24(17):2724- 2740. doi: 10.1002/mc.3020 De Boor C (1978) A practical guide to splines. Berlin: Springer. Fontes FACC (2001) A general framework to design stabilizing nonlinear model predictive controllers. Systems and Control Letters 42(2):127-143. doi: 10.1016/S0167-6911(00)00084-0 Garone E, Di Cairano S, Kolmanovsky IV (2016) Reference and command governors for systems with constraints: A survey on theory and applications. Automatica 75:306-328. doi: 10.1016/j. Goldberg DE (1989) Genetic algorithms in search, optimization & machine learning. Reading: Addison-Wesley. MathWorks® (2011) MATLAB® and SIMULINK®. Natick: Miksch T, Gambier A (2011) Fault-tolerant control by using lexicographic multi-objective optimization. Proceedings of the 8th Asian control conference (ASCC); Kaohsiung, Taiwan. Sidi MJ (2000) Spacecraft dynamics and control: a practical engineering approach. Cambridge: Cambridge University Press. Sobhani-Tehrani E, Khosravi KH (2009) Fault diagnosis of nonlinear systems using a hybrid approach. Lecture Notes in Control and Information Sciences. Dordrecht; New York: Springer. Wang D, Jia Y, Jin L, Xu S (2013) Control analysis of an underactuated spacecraft under disturbance. Acta Astronautica 83:44-53. doi: Yin S, Xiao B, Ding S, Zhou D (2016) A review on recent development of spacecraft attitude fault tolerant control system. IEEE Trans Ind Electron 63(5):3311-3320. doi: 10.1109/TIE.2016.2530789 Zhang Y, Jiang J (2008) Bibliographical review on recongurable fault-tolerant control systems. Ann Rev Contr 32(2):229-252. doi:	{"url":"https://www.researchgate.net/publication/317013467_Reducing_the_Effects_of_Inaccurate_Fault_Estimation_in_Spacecraft_Stabilization","timestamp":"2024-11-04T18:54:10Z","content_type":"text/html","content_length":"540549","record_id":"<urn:uuid:d54a067c-1ee3-4b4c-a129-1d063c1023b0>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00717.warc.gz"}
Chapter 7: A Deep Dive on Cryptographic Algorithms in Bitcoin Chapter 7: A Deep Dive on Cryptographic Algorithms in Bitcoin# In this chapter, we’ll explore the key cryptographic algorithms used in Bitcoin: SHA-256, ECDSA, and RIPEMD-160. We’ll also revisit the general concepts of hashing and why Bitcoin chose these particular algorithms. The Power of Hashing# Before we dive into specific algorithms, let’s understand the concept of hashing and why it’s so crucial in cryptography and Bitcoin. Imagine you have a magic box. You can put anything into this box - a letter, a book, or even a digital file. No matter what you put in, the box always gives you back a fixed-size string of characters. This string is unique to what you put in, like a fingerprint. If you change even a tiny bit of your input, the output string changes completely. This is essentially what a hash function Key properties of hash functions: 1. One-way function: It’s easy to calculate the hash from the input, but practically impossible to recreate the input from the hash. 2. Deterministic: The same input always produces the same hash. 3. Avalanche effect: A small change in the input results in a significantly different hash. 4. Collision resistance: It’s extremely unlikely to find two different inputs that produce the same hash. These properties make hash functions incredibly useful in cryptography and, by extension, in Bitcoin. SHA-256: The Workhorse of Bitcoin# SHA-256 stands for Secure Hash Algorithm 256-bit. It’s part of the SHA-2 family of cryptographic hash functions, designed by the U.S. National Security Agency (NSA). How SHA-256 Works (Simplified)# 1. The input message is padded and divided into blocks. 2. Each block goes through 64 rounds of operations involving bitwise functions, modular addition, and bit rotations. 3. The result is a 256-bit (32-byte) hash, regardless of the input size. Why Bitcoin Chose SHA-256# Bitcoin uses SHA-256 for two main purposes: 1. In the proof-of-work system 2. As part of the process of creating Bitcoin addresses Bitcoin chose SHA-256 for several reasons: • Strong security: As of 2024, SHA-256 remains unbroken and is considered highly secure. • Fixed output size: The 256-bit output provides a good balance between security and efficiency. • Resistance to collisions: It’s extremely difficult to find two different inputs that produce the same hash. • Relatively fast computation: While not the fastest hash function, it’s efficient enough for Bitcoin’s needs. # Import necessary libraries import hashlib import ecdsa import binascii print("Cryptographic Algorithms in Bitcoin: Code Examples") # SHA-256 Example print("\n1. SHA-256 Example") def sha256_hash(message): return hashlib.sha256(message.encode()).hexdigest() message = "Hello, Bitcoin!" hashed_message = sha256_hash(message) print(f"Original message: {message}") print(f"SHA-256 hash: {hashed_message}") # Demonstrate avalanche effect message2 = "Hello, Bitcoin?" # Changed last character hashed_message2 = sha256_hash(message2) print(f"\nSlightly changed message: {message2}") print(f"New SHA-256 hash: {hashed_message2}") Cryptographic Algorithms in Bitcoin: Code Examples 1. SHA-256 Example Original message: Hello, Bitcoin! SHA-256 hash: 8a208c3f523f64f8a52434688d9ca442483cd3007a108fd79325a0fab9b71376 Slightly changed message: Hello, Bitcoin? New SHA-256 hash: 5968a0a38635aa884d2418d7aa739e24747e44f1389c32880b311fcb9772b0db ECDSA: Signing and Verifying Transactions# ECDSA stands for Elliptic Curve Digital Signature Algorithm. It’s used in Bitcoin for creating and verifying digital signatures in transactions. How ECDSA Works (Simplified)# 1. A user generates a private key (a random number) and a corresponding public key (a point on the elliptic curve). 2. To sign a transaction: □ The transaction data is hashed. □ The hash is combined with the private key to create a signature. 3. To verify a transaction: □ The verifier uses the signer’s public key, the signature, and the transaction data. □ If the verification equation holds, the signature is valid. Why Bitcoin Chose ECDSA# Bitcoin chose ECDSA for several reasons: • Strong security: ECDSA provides a high level of security with relatively small key sizes. • Efficiency: Compared to other digital signature algorithms like RSA, ECDSA is faster and requires less computational power. • Smaller signatures: ECDSA signatures are compact, which is beneficial for a blockchain where space is at a premium. # ECDSA Example print("\n2. ECDSA Example") def generate_ecdsa_keys(): private_key = ecdsa.SigningKey.generate(curve=ecdsa.SECP256k1) public_key = private_key.get_verifying_key() return private_key, public_key def sign_message(private_key, message): return private_key.sign(message.encode()) def verify_signature(public_key, message, signature): return public_key.verify(signature, message.encode()) return False private_key, public_key = generate_ecdsa_keys() message = "Send 1 BTC to Alice" signature = sign_message(private_key, message) print(f"Message: {message}") print(f"Signature: {binascii.hexlify(signature)}") print(f"Signature valid: {verify_signature(public_key, message, signature)}") # Try to verify with a different message fake_message = "Send 1 BTC to Eve" print(f"\nTrying to verify with a fake message: {fake_message}") print(f"Signature valid: {verify_signature(public_key, fake_message, signature)}") 2. ECDSA Example Message: Send 1 BTC to Alice Signature: b'e6c432179da059c60bd7a5fa4842a99d4a33665dbaeba4d3823733806af5661f5cde6fc521e498c077dda48f960d020d7dd9a9c034f60586a5fe93bccb8bb5ad' Signature valid: True Trying to verify with a fake message: Send 1 BTC to Eve Signature valid: False RIPEMD-160: Creating Shorter Addresses# RIPEMD-160 stands for RACE Integrity Primitives Evaluation Message Digest 160-bit. In Bitcoin, it’s used in combination with SHA-256 to create Bitcoin addresses. How RIPEMD-160 is Used in Bitcoin# 1. First, a public key is hashed with SHA-256. 2. The resulting hash is then hashed again with RIPEMD-160. 3. This double-hashed result is used as part of the Bitcoin address. Why Bitcoin Chose RIPEMD-160# Bitcoin uses RIPEMD-160 for a specific reason: • Shorter output: RIPEMD-160 produces a 160-bit (20-byte) hash, which helps create shorter Bitcoin addresses while still maintaining a high level of security. By using both SHA-256 and RIPEMD-160, Bitcoin adds an extra layer of hashing security while keeping addresses reasonably short. # RIPEMD-160 Example print("\n3. RIPEMD-160 Example") def ripemd160_hash(message): hash_object = hashlib.new('ripemd160') return hash_object.hexdigest() message = "Hello, Bitcoin!" hashed_message = ripemd160_hash(message) print(f"Original message: {message}") print(f"RIPEMD-160 hash: {hashed_message}") # Bitcoin Address Generation (simplified) print("\n4. Simplified Bitcoin Address Generation") def generate_bitcoin_address(public_key): # Step 1: SHA-256 hash of the public key sha256_hash = hashlib.sha256(public_key.to_string()).digest() # Step 2: RIPEMD-160 hash of the result ripemd160_hash = hashlib.new('ripemd160') return ripemd160_hash.hexdigest() # Generate a new key pair private_key, public_key = generate_ecdsa_keys() # Generate a simplified Bitcoin address address = generate_bitcoin_address(public_key) print(f"Simplified Bitcoin address: {address}") print("\nNote: This is a simplified demonstration. Actual Bitcoin addresses include additional steps like version bytes and checksums.") 3. RIPEMD-160 Example Original message: Hello, Bitcoin! RIPEMD-160 hash: 772ed84249d4fa71819ed1035673d6377fb6944c 4. Simplified Bitcoin Address Generation Simplified Bitcoin address: accd5a8408bcfb2ed3ed9aebc2d54ce3d61a3f58 Note: This is a simplified demonstration. Actual Bitcoin addresses include additional steps like version bytes and checksums. Base58Check: Making Addresses User-Friendly# While not strictly a cryptographic algorithm, Base58Check encoding plays a crucial role in Bitcoin by making addresses and keys more user-friendly and error-resistant. What is Base58Check?# Base58Check is a way of encoding Bitcoin addresses and private keys that makes them easier for humans to read, write, and transcribe without errors. It’s called Base58 because it uses 58 characters (hence, base 58) from the standard ASCII character set. How Base58Check Works# 1. Start with binary data (like a hash of a public key). 2. Add a version byte at the beginning (to indicate what type of address it is). 3. Calculate a checksum by hashing the result twice with SHA-256 and taking the first 4 bytes. 4. Append the checksum to the end. 5. Convert the result to Base58 encoding. Why Bitcoin Chose Base58Check# Bitcoin uses Base58Check for several reasons: • Human-friendly: It avoids using characters that might be mistaken for one another, like ‘0’ (zero), ‘O’ (capital o), ‘I’ (capital i), and ‘l’ (lowercase L). • Compact: It’s more compact than standard Base64 encoding. • Error-checking: The checksum allows detection of typos or transcription errors. • Versatility: It can encode different types of data (addresses, private keys) by using different version bytes. Base58Check encoding is what gives Bitcoin addresses their familiar appearance, typically starting with ‘1’ or ‘3’ for standard addresses, or ‘bc1’ for newer SegWit addresses. # Base58Check Example print("\n5. Base58Check Encoding Example") import hashlib # Base58 character set BASE58_ALPHABET = '123456789ABCDEFGHJKLMNPQRSTUVWXYZabcdefghijkmnopqrstuvwxyz' def sha256(data): return hashlib.sha256(data).digest() def ripemd160(data): h = hashlib.new('ripemd160') return h.digest() def base58_encode(data): # Convert data to integer n = int.from_bytes(data, 'big') # Convert to base58 res = [] while n > 0: n, r = divmod(n, 58) res = ''.join(res[::-1]) # Add '1' characters for leading zero bytes pad = 0 for b in data: if b == 0: pad += 1 return BASE58_ALPHABET[0] * pad + res def base58check_encode(version, payload): # Add version byte data = bytes([version]) + payload # Double SHA-256 for checksum checksum = sha256(sha256(data))[:4] # Combine data and checksum final_data = data + checksum # Encode with Base58 return base58_encode(final_data) # Example: Create a simplified Bitcoin address public_key_hash = ripemd160(sha256(b'example_public_key')) version = 0x00 # Mainnet public key hash version bitcoin_address = base58check_encode(version, public_key_hash) print(f"Simplified Bitcoin address: {bitcoin_address}") print("\nNote: This is a simplified demonstration. Real Bitcoin address generation involves additional steps.") 5. Base58Check Encoding Example Simplified Bitcoin address: 151f3GWNePpmL8aegYpotp6H8f4NUyZiJV Note: This is a simplified demonstration. Real Bitcoin address generation involves additional steps. What We Learned# In this chapter, we explored the key cryptographic algorithms used in Bitcoin: 1. Hashing Concepts: We learned about the fundamental properties of hash functions and why they’re crucial for Bitcoin’s security. 2. SHA-256: We discovered how this strong, collision-resistant hash function is used in Bitcoin’s proof-of-work system and address creation. 3. ECDSA: We explored how this efficient digital signature algorithm enables secure transaction signing and verification in Bitcoin. 4. RIPEMD-160: We learned how this algorithm is used in combination with SHA-256 to create shorter, yet secure, Bitcoin addresses. 5. Base58Check Encoding: We understood how this encoding scheme makes Bitcoin addresses more user-friendly and error-resistant. By understanding these algorithms, we gain insight into the cryptographic foundation that makes Bitcoin secure, efficient, and user-friendly. Quick Check# Test your understanding with these questions: 1. What are the four key properties of hash functions discussed in this chapter? 2. Why did Bitcoin choose SHA-256 for its proof-of-work system? 3. What are the main advantages of using ECDSA for digital signatures in Bitcoin? 4. In the Bitcoin address creation process, why is RIPEMD-160 used after SHA-256? 5. How does Base58Check encoding help prevent errors when users are working with Bitcoin addresses? 6. Can you describe the general process of how a Bitcoin transaction is signed using ECDSA? 7. What would happen if you changed one character in a message before hashing it with SHA-256? 8. Why is it important that hash functions like SHA-256 and RIPEMD-160 are one-way functions? (Answers to these questions can be found by reviewing the relevant sections in the chapter.)	{"url":"https://sorukumar.github.io/TheCryptographyExpress/content/Chapter7_cryptographyexpress.html","timestamp":"2024-11-12T05:43:56Z","content_type":"text/html","content_length":"56228","record_id":"<urn:uuid:74dccbe3-6451-4ee2-8f60-900066fa97af>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00779.warc.gz"}
Y Intercept - Meaning, Examples \| Y Intercept Formula - [[company name]] [[target location]], [[stateabr]] Y-Intercept - Explanation, Examples As a learner, you are always working to keep up in class to avert getting swamped by subjects. As guardians, you are continually searching for ways how to motivate your children to be successful in academics and furthermore. It’s particularly essential to keep the pace in math because the concepts always build on themselves. If you don’t comprehend a particular topic, it may hurt you for months to come. Comprehending y-intercepts is a perfect example of topics that you will revisit in mathematics over and over again Let’s look at the fundamentals regarding the y-intercept and let us take you through some tips and tricks for working with it. Whether you're a math wizard or beginner, this preface will equip you with all the knowledge and instruments you must possess to tackle linear equations. Let's dive right in! What Is the Y-intercept? To entirely comprehend the y-intercept, let's imagine a coordinate plane. In a coordinate plane, two straight lines intersect at a junction to be stated as the origin. This junction is where the x-axis and y-axis meet. This means that the y value is 0, and the x value is 0. The coordinates are noted like this: (0,0). The x-axis is the horizontal line traveling through, and the y-axis is the vertical line traveling up and down. Each axis is counted so that we can locate points on the plane. The vales on the x-axis increase as we drive to the right of the origin, and the values on the y-axis grow as we drive up along the origin. Now that we have gone over the coordinate plane, we can specify the y-intercept. Meaning of the Y-Intercept The y-intercept can be taken into account as the initial point in a linear equation. It is the y-coordinate at which the graph of that equation overlaps the y-axis. Simply put, it portrays the value that y takes once x equals zero. Next, we will illustrate a real-world example. Example of the Y-Intercept Let's imagine you are driving on a long stretch of road with a single path runnin in respective direction. If you begin at point 0, where you are sitting in your vehicle this instance, therefore your y-intercept would be similar to 0 – considering you haven't moved yet! As you begin you are going the track and started gaining speed, your y-intercept will increase until it archives some greater value when you reach at a destination or stop to induce a turn. Consequently, once the y-intercept may not look particularly applicable at first sight, it can give insight into how things transform over a period of time and space as we shift through our world. So,— if you're ever puzzled trying to understand this theory, bear in mind that nearly everything starts somewhere—even your journey through that long stretch of road! How to Find the y-intercept of a Line Let's think about how we can discover this number. To support you with the process, we will make a synopsis of handful of steps to do so. Then, we will give you some examples to show you the process. Steps to Discover the y-intercept The steps to discover a line that intersects the y-axis are as follows: 1. Locate the equation of the line in slope-intercept form (We will expand on this further ahead), which should look as same as this: y = mx + b 2. Plug in 0 for x 3. Calculate the value of y Now once we have gone through the steps, let's see how this procedure will function with an example equation. Example 1 Locate the y-intercept of the line portrayed by the equation: y = 2x + 3 In this instance, we can substitute in 0 for x and solve for y to discover that the y-intercept is equal to 3. Thus, we can say that the line goes through the y-axis at the point (0,3). Example 2 As another example, let's consider the equation y = -5x + 2. In such a case, if we place in 0 for x one more time and solve for y, we discover that the y-intercept is equal to 2. Consequently, the line crosses the y-axis at the point (0,2). What Is the Slope-Intercept Form? The slope-intercept form is a method of representing linear equations. It is the cost common form utilized to represent a straight line in mathematical and scientific uses. The slope-intercept formula of a line is y = mx + b. In this function, m is the slope of the line, and b is the y-intercept. As we went through in the previous portion, the y-intercept is the coordinate where the line goes through the y-axis. The slope is a measure of how steep the line is. It is the unit of deviation in y regarding x, or how much y changes for each unit that x moves. Since we have revised the slope-intercept form, let's check out how we can employ it to locate the y-intercept of a line or a graph. Discover the y-intercept of the line signified by the equation: y = -2x + 5 In this case, we can observe that m = -2 and b = 5. Thus, the y-intercept is equal to 5. Consequently, we can conclude that the line intersects the y-axis at the point (0,5). We could take it a step further to explain the slope of the line. Founded on the equation, we know the slope is -2. Replace 1 for x and calculate: y = (-2*1) + 5 y = 3 The answer tells us that the next point on the line is (1,3). When x changed by 1 unit, y replaced by -2 units. Grade Potential Can Help You with the y-intercept You will review the XY axis repeatedly throughout your math and science studies. Ideas will get further complicated as you progress from solving a linear equation to a quadratic function. The time to peak your understanding of y-intercepts is now before you lag behind. Grade Potential provides expert instructors that will support you practice finding the y-intercept. Their customized explanations and work out problems will make a positive distinction in the results of your test scores. Whenever you believe you’re stuck or lost, Grade Potential is here to support!	{"url":"https://www.clearwaterinhometutors.com/blog/y-intercept-meaning-examples-y-intercept-formula","timestamp":"2024-11-02T15:22:59Z","content_type":"text/html","content_length":"76388","record_id":"<urn:uuid:876aff34-ce77-4ba5-b38f-b3e0179021cc>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00613.warc.gz"}
Where's the winner? Max-finding and sorting with metric costs Traditionally, a fundamental assumption in evaluating the performance of algorithms for sorting and selection has been that comparing any two elements costs one unit (of time, work, etc.); the goal of an algorithm is to minimize the total cost incurred. However, a body of recent work has attempted to find ways to weaken this assumption - in particular, new algorithms have been given for these basic problems of searching, sorting and selection, when comparisons between different pairs of elements have different associated costs. In this paper, we further these investigations, and address the questions of max-finding and sorting when the comparison costs form a metric; i.e., the comparison costs c[uv] respect the triangle inequality c[vw] + c[uw] ≥ c[uw] for all input elements u, v and w. We give the first results for these problems - specifically, we present - An O(log n)-competitive algorithm for max-finding on general metrics, and we improve on this result to obtain an O(1) -competitive algorithm for the max-finding problem in constant dimensional spaces. - An O(log^2 n)-competitive algorithm for sorting in general metric spaces. Our main technique for max-finding is to run two copies of a simple natural online algorithm (that costs too much when run by itself) in parallel. By judiciously exchanging information between the two copies, we can bound the cost incurred by the algorithm; we believe that this technique may have other applications to online algorithms. ASJC Scopus subject areas • Theoretical Computer Science • General Computer Science Dive into the research topics of 'Where's the winner? Max-finding and sorting with metric costs'. Together they form a unique fingerprint.	{"url":"https://nyuscholars.nyu.edu/en/publications/wheres-the-winner-max-finding-and-sorting-with-metric-costs","timestamp":"2024-11-14T02:26:03Z","content_type":"text/html","content_length":"54912","record_id":"<urn:uuid:9df8b00b-6ea0-4925-a4e4-85b4d55bec9a>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00449.warc.gz"}
Properties of number 1682 1682 has 6 divisors (see below), whose sum is σ = 2613. Its totient is φ = 812. The previous prime is 1669. The next prime is 1693. The reversal of 1682 is 2861. It is a Cunningham number, because it is equal to 41^2+1. 1682 is an esthetic number in base 5, because in such base its adjacent digits differ by 1. It can be written as a sum of positive squares in 2 ways, for example, as 1681 + 1 = 41^2 + 1^2 . It is an ABA number since it can be written as A⋅B^A, here for A=2, B=29. It is an Ulam number. It is a Duffinian number. It is a plaindrome in base 9 and base 11. It is a nialpdrome in base 8, base 12, base 14 and base 15. It is a self number, because there is not a number n which added to its sum of digits gives 1682. It is an unprimeable number. 1682 is an untouchable number, because it is not equal to the sum of proper divisors of any number. It is a pernicious number, because its binary representation contains a prime number (5) of ones. It is a polite number, since it can be written in 2 ways as a sum of consecutive naturals, for example, 44 + ... + 72. 2^1682 is an apocalyptic number. 1682 is a deficient number, since it is larger than the sum of its proper divisors (931). 1682 is an equidigital number, since it uses as much as digits as its factorization. With its successor (1683) it forms a Ruth-Aaron pair, since the sum of their distinct prime factors is the same (31). 1682 is an odious number, because the sum of its binary digits is odd. The sum of its prime factors is 60 (or 31 counting only the distinct ones). The product of its digits is 96, while the sum is 17. The square root of 1682 is about 41.0121933088. The cubic root of 1682 is about 11.8925594332. The spelling of 1682 in words is "one thousand, six hundred eighty-two".	{"url":"https://www.numbersaplenty.com/1682","timestamp":"2024-11-14T03:59:53Z","content_type":"text/html","content_length":"8848","record_id":"<urn:uuid:3395469f-9efe-4f51-92c5-44127da5bd54>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00315.warc.gz"}
divide(x1: array, x2: array, /) array¶ Calculates the division of each element x1_i of the input array x1 with the respective element x2_i of the input array x2. If one or both of the input arrays have integer data types, the result is implementation-dependent, as type promotion between data type “kinds” (e.g., integer versus floating-point) is Specification-compliant libraries may choose to raise an error or return an array containing the element-wise results. If an array is returned, the array must have a real-valued floating-point data type. ☆ x1 (array) – dividend input array. Should have a numeric data type. ☆ x2 (array) – divisor input array. Must be compatible with x1 (see Broadcasting). Should have a numeric data type. out (array) – an array containing the element-wise results. The returned array must have a floating-point data type determined by Type Promotion Rules. Special cases For real-valued floating-point operands, □ If either x1_i or x2_i is NaN, the result is NaN. □ If x1_i is either +infinity or -infinity and x2_i is either +infinity or -infinity, the result is NaN. □ If x1_i is either +0 or -0 and x2_i is either +0 or -0, the result is NaN. □ If x1_i is +0 and x2_i is greater than 0, the result is +0. □ If x1_i is -0 and x2_i is greater than 0, the result is -0. □ If x1_i is +0 and x2_i is less than 0, the result is -0. □ If x1_i is -0 and x2_i is less than 0, the result is +0. □ If x1_i is greater than 0 and x2_i is +0, the result is +infinity. □ If x1_i is greater than 0 and x2_i is -0, the result is -infinity. □ If x1_i is less than 0 and x2_i is +0, the result is -infinity. □ If x1_i is less than 0 and x2_i is -0, the result is +infinity. □ If x1_i is +infinity and x2_i is a positive (i.e., greater than 0) finite number, the result is +infinity. □ If x1_i is +infinity and x2_i is a negative (i.e., less than 0) finite number, the result is -infinity. □ If x1_i is -infinity and x2_i is a positive (i.e., greater than 0) finite number, the result is -infinity. □ If x1_i is -infinity and x2_i is a negative (i.e., less than 0) finite number, the result is +infinity. □ If x1_i is a positive (i.e., greater than 0) finite number and x2_i is +infinity, the result is +0. □ If x1_i is a positive (i.e., greater than 0) finite number and x2_i is -infinity, the result is -0. □ If x1_i is a negative (i.e., less than 0) finite number and x2_i is +infinity, the result is -0. □ If x1_i is a negative (i.e., less than 0) finite number and x2_i is -infinity, the result is +0. □ If x1_i and x2_i have the same mathematical sign and are both nonzero finite numbers, the result has a positive mathematical sign. □ If x1_i and x2_i have different mathematical signs and are both nonzero finite numbers, the result has a negative mathematical sign. □ In the remaining cases, where neither -infinity, +0, -0, nor NaN is involved, the quotient must be computed and rounded to the nearest representable value according to IEEE 754-2019 and a supported rounding mode. If the magnitude is too large to represent, the operation overflows and the result is an infinity of appropriate mathematical sign. If the magnitude is too small to represent, the operation underflows and the result is a zero of appropriate mathematical sign. For complex floating-point operands, division is defined according to the following table. For real components a and c and imaginary components b and d, c dj c + dj a a / c -(a/d)j special rules bj (b/c)j b/d special rules a + bj (a/c) + (b/c)j b/d - (a/d)j special rules In general, for complex floating-point operands, real-valued floating-point special cases must independently apply to the real and imaginary component operations involving real numbers as described in the above table. When a, b, c, or d are all finite numbers (i.e., a value other than NaN, +infinity, or -infinity), division of complex floating-point operands should be computed as if calculated according to the textbook formula for complex number division \[\frac{a + bj}{c + dj} = \frac{(ac + bd) + (bc - ad)j}{c^2 + d^2}\] When at least one of a, b, c, or d is NaN, +infinity, or -infinity, □ If a, b, c, and d are all NaN, the result is NaN + NaN j. □ In the remaining cases, the result is implementation dependent. For complex floating-point operands, the results of special cases may be implementation dependent depending on how an implementation chooses to model complex numbers and complex infinity (e.g., complex plane versus Riemann sphere). For those implementations following C99 and its one-infinity model, when at least one component is infinite, even if the other component is NaN, the complex value is infinite, and the usual arithmetic rules do not apply to complex-complex division. In the interest of performance, other implementations may want to avoid the complex branching logic necessary to implement the one-infinity model and choose to implement all complex-complex division according to the textbook formula. Accordingly, special case behavior is unlikely to be consistent across implementations. Changed in version 2022.12: Added complex data type support.	{"url":"https://data-apis.org/array-api/latest/API_specification/generated/array_api.divide.html","timestamp":"2024-11-06T17:19:10Z","content_type":"text/html","content_length":"36669","record_id":"<urn:uuid:c9559a40-5d56-45c8-b466-5353d33a957c>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00415.warc.gz"}
Online multivariable graphing calculator online multivariable graphing calculator Related topics: free worksheets elementary plotting points on cartesian coordinate plane hyperbola help maths exercise year 11 www,solving math problems free 9th grade algebra formulas free matrices, vectors and systems of linear equations graph quadrant numbers factoring trinomials when a is not 1 worksheet Show Balancing Equation Worksheets And Answers simplify cubed roots trinomial cubes factoring ti-83 calculaters that i can download from my computer for free pre-calculus algebra,2 solving linear equations in one variable Author Message Nanani Posted: Tuesday 22nd of Jun 07:31 I am in a real bad state of mind. Somebody save me please. I experience a lot of problems with equation properties, rational equations and unlike denominators and especially with online multivariable graphing calculator. I have to show some speedy change in my math. I heard there are plenty of Applications available online which can assist you in algebra. I can shell out some moolah too for an effective and inexpensive software which helps me with my studies. Any reference is greatly appreciated. Thanks. From: Norway Back to top IlbendF Posted: Wednesday 23rd of Jun 16:48 There are umpteen inside the extensive subject area of online multivariable graphing calculator, for example, linear algebra, rational inequalities and quadratic equations. I am acquainted with several individuals who rejected the expensive choices for aid also . Notwithstanding , don't panic because I learned of an alternative result that is low-priced , easy to utilize and to a greater extent more useful than I would have ever supposed . Subsequent to trials with demonstrative math computer products and nearly hanging it up , I found Algebrator. This software has correctly furnished answers to every math problem I have provided to the software program . Just as monumental , Algebrator as well leaves all Registered: of the working strides needed to derive the final solution . Even though one could apply the program just to dispose of worksheets , I am uncertain about a user should be granted 11.03.2004 permission to exercise the program for exams . From: Netherlands Back to top Matdhejs Posted: Friday 25th of Jun 07:19 Algebrator is a splendid software. All I had to do with my difficulties with graphing lines, side-angle-side similarity and syntehtic division was to basically type in the problems; hit the ‘solve’ and presto, the answer just popped out step-by-step in an effortless manner. I have done this to problems in Intermediate algebra, Remedial Algebra and Algebra 1. I would for sure say that this is just the answer for you. From: The Back to top Dolknankey Posted: Friday 25th of Jun 15:34 I remember having problems with roots, slope and simplifying fractions. Algebrator is a really great piece of math software. I have used it through several algebra classes - Basic Math, Basic Math and Basic Math. I would simply type in the problem and by clicking on Solve, step by step solution would appear. The program is highly recommended. From: Where the trout streams flow and the air is nice Back to top	{"url":"https://softmath.com/algebra-software/exponential-equations/online-multivariable-graphing.html","timestamp":"2024-11-08T22:22:27Z","content_type":"text/html","content_length":"40639","record_id":"<urn:uuid:038090dd-ac2d-48ea-8624-11dc8c861b06>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00064.warc.gz"}
3-Dimensional Geometry - SAT II Math I All SAT II Math I Resources Example Questions Example Question #1 : 3 Dimensional Geometry A circular swimming pool has diameter Correct answer: The pool can be seen as a cylinder with diameter Example Question #1 : Volume The above depicts a rectangular swimming pool for an apartment. 60% of the pool is six feet deep, and the remaining part of the pool is four feet deep. How many cubic feet of water does the pool Possible Answers: None of the other choices gives the correct answer. Correct answer: None of the other choices gives the correct answer. The cross-section of the pool is the area of its surface, which is the product of its length and its width: Since 60% of the pool is six feet deep, this portion of the pool holds Since the remainder of the pool - 40% - is four feet deep, this portion of the pool holds Add them together: the pool holds This answer is not among the choices. Example Question #1 : Volume Find the volume of a cube in inches with a side of Correct answer: Convert the side dimension to inches first before finding the volume. Write the volume for a cube and substitute the new side to obtain the volume in inches. Example Question #1 : 3 Dimensional Geometry Figure not drawn to scale. What is the volume of the cylinder above? Correct answer: 56.55 in^3 In order to find the volume of a cylinder, you find the area of the circular top and multiply it by the height. The volume of the cylinder is 56.55 in^3 Example Question #1 : 3 Dimensional Geometry Figure not drawn to scale If the volume of the cone above is 47.12 ft^3, what is the radius of the base? Because we have been given the volume of the cone and have been asked to find the radius of the base of the cone, we must work backwards using the volume formula. The radius of the base of the cone is 3 ft. Example Question #2 : 3 Dimensional Geometry Figure not drawn to scale. What is the volume of the above image? You can find the volume of a box by following the equation below: The surface area of the box is 42 yd^3 (remember that volume measurements are cubic units NOT square units) Example Question #1 : How To Find The Volume Of A Tetrahedron Correct answer: A regular tetrahedron is composed of four equilateral triangles. The formula for the volume of a regular tetrahedron is: Plugging in our values we get: Example Question #1 : How To Find The Volume Of A Tetrahedron Find the volume of a tetrahedron with an edge of Correct answer: Write the formula for the volume of a tetrahedron. Substitute in the length of the edge provided in the problem. Rationalize the denominator. Example Question #1 : How To Find The Volume Of A Tetrahedron Find the volume of a tetrahedron with an edge of Correct answer: Write the formula for the volume of a tetrahedron. Substitute in the length of the edge provided in the problem: Cancel out the A square root is being raised to the power of two in the numerator; these two operations cancel each other out. After canceling those operations, reduce the remaining fraction to arrive at the correct answer: Example Question #1 : How To Find The Volume Of A Tetrahedron Find the volume of a tetrahedron with an edge of Correct answer: Write the formula for finding the volume of a tetrahedron. Substitute in the edge length provided in the problem. Cancel out the Expand, rationalize the denominator, and reduce to arrive at the correct answer: Certified Tutor University of Pittsburgh-Pittsburgh Campus, Bachelor of Engineering, Industrial Engineering. Certified Tutor Massachusetts Institute of Technology, Bachelor of Science, Mathematics. Certified Tutor Kennesaw State University, Bachelor of Science, Mathematics. All SAT II Math I Resources	{"url":"https://cdn.varsitytutors.com/sat_ii_math_i-help/geometry/3-dimensional-geometry","timestamp":"2024-11-08T14:44:38Z","content_type":"application/xhtml+xml","content_length":"174456","record_id":"<urn:uuid:6ce4c958-da93-42a3-ad64-c333eb96e68e>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00458.warc.gz"}
A response to van Benthem - COGENCY \| Journal of Reasoning and COGENCY Vol. 4, N0. 2 (115-134), Summer 2012 ISSN 0718-8285 Whose Toulmin, and Which Logic? A response to van Benthem ¿Cuál Toulmin y qué lógica? Una respuesta a van Benthem Yun Xie Institute of Logic and Cognition, Sun Yat-sen University, Guangzhou, China Minghui Xiong Institute of Logic and Cognition, Sun Yat-sen University, Guangzhou, China Received: 24-06-2012 Accepted: 20-11-2012 Abstract: In a recent paper, “One Logician’s Perspective on Argumentation”, van Benthem expressed his reservations on Toulmin’s diagnosis and abandonment of formal logic, and argued that Toulmin was wrong for leading the study of argumentation apart from formal approach. In this paper we will try to reveal two serious misunderstandings of Toulmin’s ideas in his discussions, and thereby make an apology for Keywords: Argumentation Theory, Johan van Benthem, Informal Logic, Logic, Stephen Toulmin. Resumen: En un artículo reciente, “La perspectiva de un lógico sobre la argumentación”, van Benthem expresó sus dudas sobre el diagnóstico de Toulmin y el abandono de la lógica formal, y argumentó que Toulmin estuvo equivocado en llevar el estudio de la argumentación aparte de un acercamiento formal. En este trabajo trataremos de revelar dos serias incomprensiones de las ideas de Toulmin en su discusión, y de este modo desarrollamos una defensa de Toulmin. Palabras clave: Johan van Benthem, Lógica, Lógica Informal, toería de la argumentación, Stephen Toulmin. COGENCY Vol. 4, N0. 2, Summer 2012 1. Introduction It is still easy today to recall that, historically, when Toulmin first put forward his theory of argument (1958), his book was either poorly brushed aside, or fiercely criticized as a hostile “anti-logic book”, by logicians who endorsed the view of modern formal logic. Ever since that unfortunate encounter, for the following fifty years, modern logic has continued to achieve its own glorious accomplishments, unaffected by any of what Toulmin has said. Toulmin’s theory of argument, in parallel, has also been well taken and appreciated by scholars who study ordinary argumentation and communication. The hasty divorce not only exiles Toulmin’s idea outside logic and philosophy, but also decreases the successive efforts from modern logicians to investigate how much truth there is in Toulmin’s accusations. However, nowadays many logicians may think that it could be a better occasion to pick up Toulmin’s charges again, and to scrutinize his ideas in light of a semi-century’s outstanding new developments of modern logic. In a recent paper (2009), Johan van Benthem, currently one of the most prestigious formal logicians in the world, makes this interesting and rare occasion possible, for us to see how Toulmin’s ideas could be seriously taken up again by formal logicians of the present age. In that paper, van Benthem offers for us many insightful commentaries on Toulmin’s theory of argument “in the light of modern developments in logic”, as well as some serious reservations on Toulmin’s diagnosis and abandonment of formal logic. In particular, he argues that mathematical formalism is still the powerful methodology in logic, and that Toulmin is wrong for leading the study of argumentation away from the formal logical approach. This paper is a response to van Benthem’s discussion on Toulmin. We think that he has misinterpreted, in a quite important way, Toulmin’s accusations against formal logic, and his defence thereby seems to be misleading, and his arguments against Toulmin not so persuasive. In sections 2 and 3 of what follows, we first examine van Benthem’s reading of and his critiques of Toulmin. In sections 4 and 5, two responses are made by revealing, respectively, two main misunderstandings of Toulmin’s ideas in van Benthem’s discussions. In section 6, we provide a different reading of the historical developments of logic, which will help us on a just re-evaluation of Toulmin’s ideas. Taken together, we hope they can make an apology Whose Toulmin, and Which Logic? A response to van Benthem / Y. Xie and M. Xiong for Toulmin and his rejection of mathematical formalism. In the last section, we offer some concluding remarks. 2. Van Benthem on Toulmin: An Overview Van Benthem begins his paper with an appreciation of the richness of the Toulmin model of argument (which he terms the Toulmin Scheme). He agrees that “Toulmin’s scheme for the structure of argument is more natural and evident”, and that it captures “many aspects of ordinary reasoning”, when compared with the traditional simple binary scheme of inference, i.e. the bare transition from premise to conclusion (van Benthem 2009, pp. 15-16). However, van Benthem continues, if seen from the perspective of current developments of modern logical theories, the idea that arguments should be understood in a rich way to get its “broader maps” is no longer controversial today. Accordingly, the contrast between Toulmin’s views and that of mathematical logic is not as sharp as it was at the time of Toulmin’s book (1958). And this is due to the fact that “things have changed” (van Benthem 2009, p. 16): Toulmin’s ideas that inference comes in different forces, and is task-relative, have already been widely taken and well managed in the development of modern formal logic. Specifically, these changes are fairly manifest in those new branches of logic developed since 1980s, such as semantics of natural language and the studies of commonsense reasoning, as well as the new systems of nonmonotonic logics and belief-revision logics. Grounded on this, van Benthem concludes his first commentary point on Toulmin by claiming that logic, through its practical turn, “has absorbed similar ideas to Toulmin’s, largely through meetings with computer science and artificial intelligence” (van Benthem 2009, p. 17). Moreover, Toulmin’s view that “standards of inference are task-relative, and logic with its universal claims must be rejected” is not only no longer as threatening as before, but it has also been disproved by the historical fact that “but the opposite has happened” (van Benthem 2009, p. 17). That is, the discipline of logic has indeed “been enriched” and been given a “much greater scope”, it is obviously not as unviable or unpromising as it was alleged to be by Toulmin. Besides, to better support his point, van Benthem has also provided for COGENCY Vol. 4, N0. 2, Summer 2012 us two historical analogies. The first one is Bernard Bolzano, who also “saw the task of logic as charting different natural styles of reasoning in different settings”, and “predates Toulmin in not placing the central emphasis on logical forms” (van Benthem 2009, p. 17). The second one is the HempelOppenheim scheme of explanation, which “predates the Toulmin schema by a decade” and shows that “philosophers of science had long noted that standard logical consequence will not do for the variety of reasoning found in science” (van Benthem 2009, p. 18). However, these two “parallel observations on a practice that clearly goes beyond standard logic” reveal otherwise a spirit which is just opposite to that of Toulmin. That is, as it happened in both cases, “logical form has not been dismissed altogether”, and “logic is not abandoned, but enriched” (van Benthem 2009, pp. 1718, italics original). Therefore, van Benthem raises his main criticism that “Toulmin’s decision to break away [from logic] was fateful, and I wonder how justified it really was” (van Benthem 2009, p. 19). The second part of van Benthem’s discussion of Toulmin concerns the opposition of ‘logical/mathematical form’ and ‘procedural/juridical formalities’. According to van Benthem, by highlighting “the procedure by which we draw inference and the importance of procedure in argumentation generally”, Toulmin is “right on the mark” and “does point at a major theme that has been neglected in modern logic” (van Benthem 2009, p.19). In a nutshell, “reasoning is an activity”, and it is performed “interactively with others”, thus logic should study not only some products of such acts, but also “those acts themselves” (van Benthem 2009, p. 19, italics original). However, once again, in the light of modern development of logic, these ideas have already been long pursued and properly realized by “current systems of dynamic epistemic logic”, which “incorporate [into modern logic] a wide variety of dynamic viewpoints, without giving up its classical methodological standards” (van Benthem 2009, p. 19). But on the contrary, as van Benthem has criticized, Toulmin seems to have overemphasized the opposition between ‘form’ and ‘formalities’, thereby hastily judging that the mathematical methodology is incapable of offering effective tools in dealing with those procedural concerns. Whereas, van Benthem argues, Toulmin’s position is untenable, since virtually “formalities have form, there is no opposition,” and “formalities have procedural structure, and that structure can be studied by bringing out the major operations creat- Whose Toulmin, and Which Logic? A response to van Benthem / Y. Xie and M. Xiong ing it in a mathematical formalism” (van Benthem 2009, p. 21). This could not only be easily confirmed by recent developments of Logical Dynamics, but has indeed also been well exemplified in the 1950s by works on dialogue games of Paul Lorenzen. Consequently, in conclusion, van Benthem claims that “years before Toulmin’s book, logic had already started developing tools for some of the very things he was asking for: formalities, and task dependence” (van Benthem 2009, p. 20). Moreover, he even suggests that we should realize that actually “one should go further than Toulmin’s own scheme - and one very effective tool for doing that is using not less but more logic!” (van Benthem 2009, p. 20, italics original). Here again, his criticism on Toulmin is nicely echoed. The immature disbelief in the power and capacity of mathematical formalism, the unlucky ignorance of the coeval logical ideas and developments at his time, make Toulmin’s judgment on logic feeble, and his decision to “leave the party” not truly justified. The last part of the paper is a discussion about the joint concerns for studies of logic and argumentation theory. Here van Benthem shares with us his optimism, as well as his insightful proposals, for the future interaction between these two fields. He indicates to us that by taking argumentation more adequately, logical systems can be better enriched. In particular, “it seems promising to merge dynamic logics of information flow with concrete models of argumentation” (van Benthem 2009, p. 22). In the end, as his final verdict, van Benthem concludes, Toulmin “was right in many of his major observations, but I would say that he was wrong in his decision to leave the party [of logic]. Working together, argumentation theory and logic can advance along Toulmin’s lines extending both practical coverage and theoretical insight” (van Benthem 2009, p. 22). 3. Van Benthem on Toulmin: A Reconstruction As a whole, by his paper van Benthem offers us a critical scrutiny of Toulmin’s ideas, which is, needless to say, quite thoughtful and subtle. His reading of Toulmin’s work is impressively careful, and his reservations and criticisms elaborate and well-advised. To better capture the gist, here we will try to give an illustrative reconstruction of van Benthem’s reading of Toulmin, and of his corresponding criticisms as well. COGENCY Vol. 4, N0. 2, Summer 2012 In the first place, the line of argument that Toulmin has followed to make his case, as it is refined in van Benthem’s paper, is as follows: Firstly, Toulmin makes the case that (T1) “comparing with what has been delivered in traditional simple binary scheme, ordinary reasoning has more aspects and elements, as revealed by the Toulmin Scheme.” Given this, on the one hand, it leads Toulmin to find that (T2) “inference comes in different forces, and is task-relative”, which directly leads to the claim that (T3) “logic with its universal claims must be rejected”. On the other hand, it also helps him to emphasize (T4) “the importance of procedure in argumentation”, thus to put forward “a procedural view of argument, which replaces mathematical form by juridical formalities”. In the meantime, Toulmin also holds that (T5) there is “an opposition of logical form and procedural formalities,” and he believes that (T6) “logic with a mathematical formalism is incapable of developing effective tools for formalities.” From this he has been led to (T7) “a dismissal of form altogether”. Ultimately, given (T3) and (T7), which have indicated respectively the incompetence of logic in dealing with task dependence and formalities, Toulmin has urged (T8) “the abandonment of logic, which breaks away the connection between logic and argumentation theory”. Accordingly, van Benthem tries to put forward his criticisms into a three-fold strategy. First and foremost, it is obvious that (B1) history, by the development of modern logic, simply proves that “the opposite has happened”, that is, “logic has not been abandoned, but enriched”. It is supported by the fact that (B2) “logic has experienced its practical turn since 1980s, thus has absorbed similar ideas to Toulmin’s through meetings with computer science and artificial intelligence”. This could be further exemplified by (B3) “the development of study of semantics of natural language and commonsense reasoning, as well as the new systems of nonmonotonic logics and belief-revision logics.” Secondly, it could also be argued that (B4) “Toulmin’s abandonment of logic is hasty and not truly justified”, because (B5) “his disbelief in the power and capacity of logic (with mathematical formalism) is ill-founded”, and (B6) “his negative judgment of logic is immature and radical, because of his ignorance of the coeval logical ideas and developments at his time”. (B6) can be simply verified by indicating (B7) “three historical analogies: Bolzano’s ideas of logic, the Hempel-Oppenheim scheme of explanation, and Whose Toulmin, and Which Logic? A response to van Benthem / Y. Xie and M. Xiong dialogue games of Lorenzen”. The first two indicates, quite differently from Toulmin, a due respect for the role of form in logic, and the latter offers the evidence that logic can and indeed has already started, years before Toulmin’s book, to develop tools for dealing with formalities. Meanwhile, (B5) could be well supported when pointing out (B8) “Toulmin’s opposition between form and formalities is illusive”. Because, not only (B9) “formalities have form, i.e. procedural structure, thus can be studied by bringing out the major operations creating it in a mathematical formalism,” but also that (B10) “those procedural formalities embodied in Toulmin’s own scheme can be even better and effectively handled by using more logic”, once you see (B11) “the connection between formalities and logical dynamics”. And thirdly, it has also been hinted that (B12) “the breaking away of logic and argumentation is actually unfortunate and fateful”. Since, first, (B13) “these two fields have joint concerns, thus a lot of works which try to bridge and merge them will be promising”, and second, (B14) “current argumentation theory might need to bring in not less but more logic”. That is because, for example, not only (B10) “Toulmin’s own scheme can be even better handled by using more logic”, but also (B15) “some of current informal paradigms might acquire some more dynamics with a dose of logical insights from the last decades”. And third, (B16) “argumentation theory cannot just stay on its own”, otherwise (B17) “it is not sure where argumentation theory is heading.” To sum up, for one thing, Toulmin’s rejection of logic based on ideas like the variable force of reasoning and task dependence [(T3)] has been easily falsified by the historical developments [(B1)]. For another, his disbelief of logic based on formalities [(T7)] is hasty, and relies indeed on an illusory opposition, thus his abandonment of logic unjustified [(B4)]. Moreover, Toulmin’s decision to break away from logic is proved to be fateful, when considering the promising possibilities, and even the necessity, of the interaction between these two fields [(B12)]. Therefore, taking those three counter-arguments together, it is easy to conclude that (C) “Toulmin was wrong in his decision to leave the party of logic.” Obviously, van Benthem’s criticisms are well-constructed, and appear to be quite compelling. They provide us a particular and insightful way for our understanding and appraisal of Toulmin. However, as we will argue in the following two sections, there also exist some possible flaws in his COGENCY Vol. 4, N0. 2, Summer 2012 arguments, which turn out to be based on some fateful misinterpretations of Toulmin. The first one is concerned with Toulmin’s hasty rejection of mathematical formalism, while the second one is related to Toulmin’s unjustified abandonment of logic. 4. Formality, Formalism and Practical Discontinuity The trouble does not lie within the formal systems themselves: it would be pointless to argue that one could not have formal mathematical calculi concerned with relations between propositions, since everyone knows what elaborate and sophisticated propositional calculi have in fact been built up in recent years. The objections turn rather on the question, what application these calculi can have to the practical assessment of arguments - whether the relations so elegantly formalized in these systems are, in fact, the ones which concern us when we ask in practice about the cogency, force and acceptability of arguments. (Toulmin 1958, pp. 179-80, italics original) As one of his substantial critiques, van Benthem accuses Toulmin of being blind to the power and capacity of mathematical formalism, and thereby proposing a hasty and unjustified abandonment of logic. According to him, Toulmin has misconceived the contrast between form and formalities, and has over-reacted on this illusory opposition to dismiss logical form altogether. Furthermore, he has also wrongly predicted that logic with formalism cannot handle issues like field-dependence, thus misjudging the fate of formal logic. However, as van Benthem has attempted to demonstrate with great efforts, this is not the case at all. When you have figured out that formalities have form, and procedural structure can all be studied by bringing out the major operations creating it in a mathematical formalism, or, when you have a look at the works of dialogue games of Lorenzen, which has already hinted at the effective formal tools in dealing with procedures, or, when you have witnessed the developments of new nonmonotonic and epistemic logics and systems, all of which make it manifest that reasoning forces or field dependence can all be represented within formal logical systems, then you should conclude forthwith that Toulmin is obviously at fault for his insufficient judgment of the power of mathematical formalism. As a result, he is apparently also wrong in his negative view on logic. Whose Toulmin, and Which Logic? A response to van Benthem / Y. Xie and M. Xiong Properly speaking, if Toulmin did reject mathematical formalism, and formal logic as well, just simply based on his judgment that issues like procedure, reasoning forces or field-dependence cannot possibly be handled in a manner of mathematical formalism, then van Benthem is certainly advancing a very strong and persuasive critique, and Toulmin is definitely guilty of this unjustified ignorance of both the power of formalism and the development of logic. However, despite the apparent strength of this argument, if that is not what Toulmin really had in mind, we are probably facing a straw person. There is no doubt that Toulmin did criticize mathematical formalism in his works, and as a result, rejected it in his theory of argument. But his criticisms and his corresponding ideas of logic, in our reading, indeed go in a quite different way. Toulmin starts at the very beginning with the central question of “how we are to set out and analyze arguments in order that our assessments shall be logically candid?”(Toulmin 1958, p. 9). That is, how we are to show the validity or invalidity of an argument clearly, and to make explicit the grounds it relies on and the bearing of the conclusion, thereby to allow us to do justice to all different issues in the establishment of conclusions when subjecting our arguments to rational assessment (Toulmin 1958, p.95). With this task and principle in mind, Toulmin did propose some sort of opposition between mathematical form and jurisprudential formalities, in such a way of treating and comparing them as two available rival models, “one mathematical, the other jurisprudential” (Toulmin 1958, p. 95). Then he took pains to argue that the former is less candid than it was customarily presumed to be. Because, as he has recognized, the “establishment of conclusions” in the normal run of life turns out to be more complex than a simple “logical demonstration”. A number of issues of different sorts will emerge in our analysis and criticism of the former kind, but they either have no room, or have to be distorted, in our customary theories of the latter kind. For one thing, statements used in our ordinary arguments have functional differences, they give birth to subtle distinctions (such as data, warrant, backing, etc.), but most of them have been overlooked or covered up when we try to lay out our arguments in a tidy and simple geometrical form (Toulmin 1958, pp. 97-107). Meanwhile, some categories and distinctions (for example, major premise, deductive/inductive) produced by formal lo- COGENCY Vol. 4, N0. 2, Summer 2012 gicians endorsing the geometrical model are ambiguous, thus turn out to be a misinterpretation of reality (Toulmin 1958, pp. 107-118). Moreover, some functional differences, due to its field-dependence and contextual complexity, “cannot be explained away by formal devices: e.g., by inventing separate formal systems of alethic, deontic, or epistemic logic for every purpose and field” (Toulmin 1976, p. 277). For another, by choosing the geometrical model, formal logicians have also been led to over-value what Toulmin calls “analytical arguments”. Accordingly, they try to sublime it; to make it out to be the only good and ideal type of argument, and to demand that all the other classes of argument should conform to its standards regardless. Consequently, “they have built up their systems of formal logic entirely on this foundation; and they have felt free to apply to arguments in other fields the categories so constructed” (Toulmin 1958, p. 166). However, in reality, analytical argument is far from a representative type, since there exist a lot of substantial arguments “whose cogency cannot be displayed in a purely formal way, even validity is something entirely out of reach and unobtainable” (Toulmin 1958, p. 154). Therefore, Toulmin points out, “the differences between the criteria we employ in different fields can be circumvented in this way only at the price of robbing our logical systems of all serious application to substantial arguments” (Toulmin 1958, p. 167). With all those reasons provided, Toulmin has informed us that “unfortunately an idealized logic, such as the mathematical model leads us to, cannot keep in serious contact with its practical application” (Toulmin 1958, p. 147). As he has observed, there in fact has already grown up a “Great Divide between the formal logician and the practical arguer” (Toulmin 1958, p. 163), viz. “a systematic divergence between two sets of categories: those we find employed in the practical business of argumentation, and the corresponding analyses of them set in books of formal logic” (Toulmin 1958, p. 147). At last, it is due to this practical discontinuity that Toulmin has raised his rejection of mathematical formalism in the study of argument: “so far as formal logicians claim to say anything of relevance to arguments of other than analytic sorts, judgment must therefore be pronounced against them” (Toulmin 1958, p. 147). If our reading of Toulmin is reasonable, it can be seen that he is probably not guilty of hastily dismissing mathematical formalism based on Whose Toulmin, and Which Logic? A response to van Benthem / Y. Xie and M. Xiong some immature disbelief of the power of formalism. By contrasting mathematical form and jurisprudential formalities, he indeed has no intention to argue that procedural issues cannot possibly be handled in a manner of mathematical formalism. Rather, what he is trying to show is that if we do it exactly in that way, we are ending with a logical theory losing it original practical application. Accordingly, the real disbelief underlying Toulmin’s rejection of mathematical formalism has never been concerning with the possible development of formal tools or systems for representing or modeling those procedural aspects of arguments. It is, otherwise, related to the legitimacy and applicability of formalization with respect to actual practice of argument analysis and assessment. Therefore, understood properly, Toulmin’s criticisms of mathematical formalism stem from his suspicion that “can one cast into a timeless mathematical mould the relations upon which the soundness and acceptability of our arguments depend, without distorting them beyond recognition?” (Toulmin 1958, p. 182) And the real forces of his rejection rest indeed upon the fact that “practical critic of arguments, as of morals, is in no position to adopt the mathematician’s Olympian posture…strength, cogency, evidential support and the like…resist idealization as much as our utterances themselves” (Toulmin 1958, p. 183). Moreover, even Toulmin himself has made it quite clear that he is not against formal logical systems themselves which glorify the methodology of mathematical formalism. He clarifies that what he is really advocating is not that “the elaborate mathematical systems which constitute ‘symbolic logic’ must now be thrown away; but only that people with intellectual capital invested in them should retain no illusions about the extent of their relevance to practical arguments” (Toulmin 1958, p. 185). As a matter of fact, his real and exact appeal is that “in order to get a logic which is lifelike and applicable…we shall have to replace mathematically-idealized logical relations…by relations which in practical fact are no more timeless than the statements they relate” (Toulmin 1958, p. 185, italics added). To sum up, the contrast between form and formality, as well as the issues like field-dependence or variability of force of reasoning articulated by Toulmin, were not brought forward intending to reveal the incompetence of formalism. Rather, they have been advanced to unveil the practical discontinuity of formalization with our real lives. Consequently, in defence of mathematical formalism against Toulmin’s accusation by arguing that COGENCY Vol. 4, N0. 2, Summer 2012 there could be no difficulties in treating them successfully in a formalized way, will probably be beside the point. Mathematical formalism is certainly a methodology with great power, as the modern developments of logic have well illustrated. But that is not the issue Toulmin has addressed and decried. 5. Taking Logic Seriously The question still needs to be pressed, whether this branch of mathematics is entitled to the name of ‘logical theory’…this branch of mathematics does not form the theoretical part of logic in anything like the way that the physicist’s mathematical theories form the theoretical part of physics. (Toulmin 1958, p. 186) The real problem of rational assessment - telling sound argument from untrustworthy ones, rather than consistent from inconsistent ones - requires experience, insight and judgment, and mathematical calculations can never be more than one tool among others of use in this task. (Toulmin 1958, p. 188) As another substantial critique, also his final verdict, van Benthem charges Toulmin with the fateful breaking away and unjustified abandonment of logic. As he has argued, not only both fields of argumentation and logic could, and indeed should, interact more with mutual benefits, but also Toulmin’s own scheme could be better dealt with by using more logical tools. Besides, Toulmin should have shown enough respect to logic with mathematical formalism, rather than “turning to the Law as the major paradigm of reasoning” (van Benthem 2009, p. 15), since “historically, logic probably had its origins in dialectical and legal practice, but it was the combination of these origins with mathematical methodology that produced its great strength and staying power” (van Benthem 2009, pp. 20-21, italics original). Furthermore, as was nicely proved by recent developments of logic in modern times, logic has been prosperously enriched in this mathematically formal approach. However, as we read his works, Toulmin has nowhere urged such a breaking away or abandonment of logic per se, notwithstanding his serious resistance against mathematical formalism in the study of argument. Quite Whose Toulmin, and Which Logic? A response to van Benthem / Y. Xie and M. Xiong the contrary, surprisingly, within all his discussions, he has always been caring so much about logic itself, and been talking quite a lot about logic as a science. At the very beginning of his book (1958), he has already informed us that he actually intends to discuss the “problems about logic, but not problems in logic” (Toulmin 1958, p. 1, italics original). These are specific problems “which arise with special force not within the science of logic, but only when one withdraws oneself for a moment from the technical refinements of the subject, and inquires what bearing the science and its discoveries have on anything outside itself—how they apply in practice, and what connections they have with the canons and methods we use when, in everyday life, we actually assess the soundness, strength and conclusiveness of argument” (Toulmin 1958, p. 1). With this in mind, virtually his whole discussion and comparison between geometrical and jurisprudential model is just an attempt to demonstrate those problems about logic, namely, the then-current logical theory “has indeed lost touch with its application” (Toulmin 1958, p. 9). And then, great efforts are made by Toulmin, with a whole chapter, to explain the origins of this phenomenon, viz. the divergences existing between idealized logic and working logic. He reveals that “the science of logic has throughout its history tended to develop in a direction leading it away…from practical questions about the manner in which we have occasion to handle and criticize arguments in different fields” (Toulmin 1958, p. 2). He strives to trace its historical origins back to the Aristotelian idea of logic as a formal science comparable to geometry, and then reveals that within the developments in this direction some influential prejudices are implicitly endorsed and reinforced. In particular, the mathematical model is treated as the ideal model of knowledge, and logical studies are regarded as, and are expected to be, a kind of successful episteme, like geometry, studying the universal, timeless and formal relations among different sorts of propositions (Toulmin 1958, pp. 178-179). As he has observed, “logicians have evidently cherished this hope, and…have been trying to free theoretical logic of the field-dependence which marks all logical practice” (Toulmin 1958, p. 167), that is, making logic “a formal science comparable to geometry”, and “casting the whole logical theory into a mathematical form” (Toulmin 1958, p. 9). However, as he has argued, those prejudices will, COGENCY Vol. 4, N0. 2, Summer 2012 regretfully, result in some fatal consequences in epistemology and philosophy (Toulmin 1958, pp. 211-252), as well as an unbalanced account of reason and rationality (Toulmin 2001, pp. 14-28). Given all these diagnoses, Toulmin then expresses his doubt about “how far logic can hope to be a formal science, and yet retain the possibility of being applied in the critical assessment of actual arguments” (Toulmin 1958, p. 3). When he found that “ever since Frege and Russell, philosophers have discussed logic, not as ‘the art and theory of rational criticism’, but as a field which is purely formal” (Toulmin 1976, p. 270), he has no choice but to exert all his powers, on the one hand, to warn us that “we must be careful before we allow any formal calculus to assume the title of ‘logic’” (Toulmin 1958, p. 209), and on the other hand, to advance an alternative proposal to revive the idea that “logic is concerned with the soundness of the claims we make - with the solidity of the grounds we produce to support them, the firmness of the backing we provide for them” (Toulmin 1958, p. 7). Metaphorically, he urges a re-conceptualization of logic as generalized jurisprudence, expecting that “but of one thing I am confident: that by treating logic as generalized jurisprudence and testing our ideas against actual practice of argument assessment rather than against a philosopher’s ideal, we shall eventually build up a picture very different from the traditional one” (Toulmin 1958, p. 10). Once again, if our reading of Toulmin is plausible, he cannot be accused of abandoning logic. On the contrary, logic in fact has never been expelled from his considerations. Probably we can even say that no one has taken logic more seriously than Toulmin. Obviously his discussions have been centered on logical theory and logical practice, with a goal of saving the science of logic from both the theoretical dangers and its practical predicaments. With this said, if there is any negativism within Toulmin’s attitude towards logic, it is only his reluctance of allowing formal logic (with mathematical methodology) to assume the whole title of “logic”. Therefore, the only possible way of accusing Toulmin of abandoning logic has to rely on the assumption that “formal logic (with mathematical methodology) is the only legitimate subject of logic”. Interestingly, as was expected, it is just the view of logic that van Benthem has endorsed, “logic may be a normative mathematical study of valid inference patterns” (van Benthem 2009, p.14). However, even though this could be granted as a tenable view of log- Whose Toulmin, and Which Logic? A response to van Benthem / Y. Xie and M. Xiong ic, which we seriously doubt, suffice it to point out that criticizing Toulmin in this way van Benthem is simply talking past Toulmin, and his argument committing a fallacy known as circular reasoning, or definitional dodge. 6. History Once Again: A Voice Unheard Still, the real issue remains, what kind of judgment can we pass on Toulmin in light of a semi-century’s new developments of logic, if the history does help in our re-evaluation of Toulmin’s ideas about logic? Note that the recent developments of modern formal logic do help quite a lot in van Benthem’s discussions and criticisms of Toulmin: they have not only made Toulmin’s expectation subverted and his predication on logic falsified, and they have also made the power and progress of mathematical formalism fully exhibited. However, history could always be complex with more than one facet. Let’s provide some other historical facts which can also reveal that the discipline of logic has been developed and enriched, but in a very different way from what van Benthem has sketched. Looking back over the past 50 years, we have also witnessed the emergence of Informal Logic and its development as a sub-field of logical studies. To be brief, in the early 1970s, in response to both practical frustration and pedagogical critiques of formal logic, informal logic first emerged as an attempt to reform the teaching of introductory-level logic course. In the very beginning, informal logicians promoted a set of important shifts in the textbooks of the day, trying to replace those artificial and trivial examples abounding in the traditional logical textbooks, and to provide relevant and useful logical tools for the analysis and criticism of real life arguments (Kahane, 1971; Scriven, 1976, Johnson & Blair, 1977). Later it quickly became clear that a lot of theoretical problems and topics need to be fully explored in order to get a comprehensive understanding of arguments analysis and critique (Johnson & Blair, 1980). Accordingly, informal logicians started to develop wide-ranging theories of argument by probing into various issues concerning the nature of argument, the elements of argument, missing premises, the principle of charity, argument structure, argument typology, argument schemes, norms and criteria of cogent argument, and theory of fallacy, and so forth. As a result, informal logic gradually conceived of itself COGENCY Vol. 4, N0. 2, Summer 2012 as a theoretical discipline “whose task is to develop non-formal standards, criteria, procedures for the analysis, interpretation, evaluation, criticism and construction of argumentation in everyday discourse” (Johnson & Blair, 1987, p.148; 2002, p. 358). Over the last thirty years, it has already achieved a lot of notable successes on both the pedagogical front and the theoretical level (Blair, 2006, Johnson, 2009). A proliferation of studies and achievements in recent years has made this discipline better established and well recognized, and nowadays works in informal logic community have also made considerable impacts on studies in speech communication, linguistics, rhetoric, communication, psychology, computer science, artificial intelligence. Obviously, rather than heading obstinately to the direction of mathematical formalization, informal logic tries to enrich the discipline of logic by “adding a new domain which is more attentive to the needs of argumentation analysis and critique, and reminding logicians of the art and craft of logic – views which have their roots in Aristotle” (Johnson & Blair, 2002, p. 351). Indeed, from a historical point of view, we do have inherited from Aristotle a balanced view on logic which conceives logic as having both analytical/formal and topical/substantial parts (Toulmin, 1976, pp. 266-267). Even in the late 19th century we still see that logic was mainly understood as both a theoretical and formal science and a practical and informal art (Hyslop, 1892; Jevons, 1883). It is until the beginning of the 20th century that philosophers and logicians, following Frege and Russell, started to draw much narrower boundaries around the subject of logic, thereby making it recognized as only a purely formal field. Hence, insofar as we have in mind this general picture of the history of logic, we could reasonably regard the development of informal logic in the last decades as displaying a course of new development within the discipline of logic itself, though a course of development very different from that of modern formal logics. That is, by taking informal logic seriously, we can see logic itself has actually been developed by reviving a part which has long belonged to this enterprise, and by redressing a balanced understanding of this discipline which has traditionally set up its agenda in a proper manner. Likewise, and more importantly, if philosophers and logicians of the present age could really drop some of their ingrained prejudices, and then give a fair hearing of the voice from the informal logic community, they can Whose Toulmin, and Which Logic? A response to van Benthem / Y. Xie and M. Xiong easily find that Toulmin’s ideas of logic have already been nicely echoed by works of informal logicians, and his innovative proposal has also been substantially developed within the enterprise of informal logic. To be more specific, in line with Toulmin, informal logicians have demonstrated that ordinary arguments are quite different from inference and implication in a number of respects, and probed into substantial issues in its analysis and evaluation. Also, they have offered reasonable challenges to deductivism and abusive formalizations, and developed alternative methods, tools, and criteria for our effective criticism of real argumentation (Govier; 1987, Johnson 2000, 2011; Johnson & Blair; 2000). In brief, informal logicians have sensed the same problems about logic as Toulmin has diagnosed, and they have also tried to reclaim this discipline from losing its practical application in ordinary practice in the same way as Toulmin has urged. 1 With those new facts being added, now it is as though the historical development of logic in the last semi-century could also provide us with another version of the claim that “things have changed and logic has been enriched”: despite the fact that many new branches of formal logic has been well developed, with a great deal of new logics and formal systems abound in, great efforts have also been devoted in line with what Toulmin has envisaged, and steady progress has been made for a reconfiguration of logic in the direction of making it lifelike and applicable. Therefore, to be fair, it could also be the case that history does not really go against Toulmin. Here we believe it’s an appropriate occasion to address a specific criticism we occasionally encountered when presenting the earlier versions of this paper. That is, many readers would find that we didn’t take any concern to further address the issue of “whether those new developments of logic portrayed by van Benthem really can or cannot provide adequate tools or methods that can be of practical use in ordinary argument analysis and evaluation”. Hence this paper may have a weakness of failing to conclusively disprove van Benthem’s claim and his criticisms on Toulmin. But readers should be reminded that van Benthem’s paper didn’t do that either, it gives no example or arguments to show how those advances in formal methods could actually help in our understanding and in dealing with the issues in practical arguments. So the real question is “who has the burden of proof to do that”? However, we believe it is exactly by the development of informal logic and the works of informal logicians that a presumption has already been established in favour of Toulmin’s diagnosis that formal mathematical methodology does lose its practicality in real argument analysis and evaluation. Therefore, we don’t think this is a shortcoming of our paper. COGENCY Vol. 4, N0. 2, Summer 2012 7. Conclusion Toulmin confessed more than once that, after the publication of The Uses of Argument, he suffered the pain of being misunderstood as having written an “anti-logic” book. However, more than fifty years later, it seems that he once again has to face, unfortunately, some sort of similar confrontation: unjustly abandoning logic. Strangely enough, today those ideas underlying his analysis of force of reasons, warrant, rebuttal and qualifier etc. have already attracted more and more attention from logicians enthusiastic with formalization, and all of them turn to be the profitable sources for some new modeling and formal systems. But his more genuine appeal, that “a radical re-ordering of logical theory is needed in order to bring it more nearly into line with critical practice” (Toulmin, 1958, p. 253), has never been taken seriously by formal logicians. This is, in fact, a situation that happens commonly to many great philosophers throughout the history of ideas. Offering an explanation for the case of Toulmin could be a good topic in studies of sociology of knowledge, but suffice it to say here that even though it is a way of what indeed has happened, it is probably not the way of what should have been expected. Finally, let’s also end with a remark on the joint concern between argumentation and formal logic. We share the same optimism with van Benthem for the promising interaction between these two fields, in particular, in exploring the “argumentation perspective in modern logic”. Actually we have already witnessed a lot of successful efforts in bringing into modern formal logic those theoretical insights afforded in contemporary argumentation theory. However, at the same time, we are also in sympathy with his claim that “some of current informal paradigms [in argumentation theory] might acquire some more dynamics with a dose of logical insights from the last decades” (van Benthem 2009, p. 17), even though we are still, remaining as before, awaiting some valuable achievements. Because we believe it is only by realizing this possibility, i.e. mathematical formalism without practical discontinuity, that Toulmin’s ideas could really be surpassed. Acknowledgements: An earlier version of this paper was presented in the 9th OSSA conference held at the University of Windsor in May 2011, Whose Toulmin, and Which Logic? A response to van Benthem / Y. Xie and M. Xiong the authors are very grateful to Hans Hansen, Ralph Johnson, Trudy Govier and Harvey Siegel, for their valuable comments and criticisms. And the work in this paper is supported by the Chinese MOE Project of Key Research Institute of Humanities and Social Sciences at Universities (12JJD720006), the Chinese MOE Project of Humanities and Social Sciences (10YJC72040003), and by the “Fundamental Research Funds for the Central Universities”. Works cited van Benthem, Johan. “One Logician’s Perspective on Argumentation.” Cogency 1 (2) (2009): 13-26. Blair, J. Anthony. “Informal Logic’s Influence on Philosophy Instruction.” Informal Logic 26(3) (2006): 259-286. Govier, Trudy. Problems in Argument Analysis and Evaluation, Dordrecht: Foris, 1987. Hyslop, James. H. The Elements of Logic: Theoretical and Practical. New York: C. Scribner’s Sons, 1892. Jevons, William. The Elements of Logic. New York: American Book Company, Johnson, Ralph. Manifest Rationality: A Pragmatic Study of Argument, Mahwah, NJ: Lawrence Erlbaum Associates, 2000. Johnson, Ralph. “On the Alleged Failure of Informal Logic.” Cogency 1 (1) (2009): 59-88. Johnson, Ralph. “Informal Logic and Deductivism.” Studies in Logic 4 (1) (2011): 17-37. Johnson, R. and Blair, J. A. Logical Self-Defense. Toronto: McGraw-Hill Ryerson, 1977. Johnson, R. H. and Blair, J. A. “The Recent Development of Informal Logic.” In J. Anthony Blair and Ralph H. Johnson (eds.). Informal Logic: The First International Symposium (pp. 3-28). Inverness, CA: Edgepress, 1980. Johnson, R. H. and Blair, J. A. “The current state of informal logic.” Informal Logic 9(2) (1987): 147-151. Johnson, R. H. and Blair, J. A. “Informal Logic: An Overview.” Informal Logic 20(2) (2000): 93-99. Johnson, R. H. and Blair, J. A. “Informal logic and the reconfiguration of logic.” In Gabbay, D., Johnson, R. H., Ohlbach, Hans-Jurgen, and Woods, J. (eds.). Handbook of the Logic of Argument and Inference: The turn towards the practical (pp.339-396). Amsterdam: Kluwer, 2002. COGENCY Vol. 4, N0. 2, Summer 2012 Kahane, Howard. Logic and Contemporary Rhetoric. Belmont, CA: Wadsworth, 1971. Scriven, Michael. Reasoning. New York: McGraw-Hill, 1976. Toulmin, Stephen. The Uses of Argument. Cambridge: Cambridge University Press, 1958. Toulmin, Stephen. “Logic and the Criticism of Arguments.” In Golden, J., Berquist, G., and Coleman, W. (eds.). The Rhetoric of Western Thought (pp. 265-277). Dubuque, IA: Kendall-Hunt, 1976. Toulmin, Sephen. Return to Reason. Cambridge, MA: Harvard University Press, 2001.	{"url":"https://studylib.net/doc/8394029/a-response-to-van-benthem---cogency---journal-of-reasonin...","timestamp":"2024-11-11T03:28:33Z","content_type":"text/html","content_length":"103678","record_id":"<urn:uuid:c07abed6-b566-4dd0-bda0-85671c7b19da>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00347.warc.gz"}
Methods of Proof — Diagonalization , 10 min read Methods of Proof — Diagonalization Very clear presentation on the uncountability of the real numbers, and the halting problem. Further keywords: Cantor, natural numbers, real numbers, diagonalization, bijection, Turing halting problem, proof by contradiction. -- -- -- Quote -- -- -- A while back we featured a post about why learning mathematics can be hard for programmers, and I claimed a major issue was not understanding the basic methods of proof (the lingua franca between intuition and rigorous mathematics). I boiled these down to the "basic four," direct implication, contrapositive, contradiction, and induction. But in mathematics there is an ever growing supply of proof methods. There are books written about the "probabilistic method," and I recently went to a lecture where the "linear algebra method" was displayed. There has been recent talk of a "quantum method" for proving theorems unrelated to quantum mechanics, and many more. So in continuing our series of methods of proof, we'll move up to some of the more advanced methods of proof. And in keeping with the spirit of the series, we'll spend most of our time discussing the structural form of the proofs. This time, diagonalization. Perhaps one of the most famous methods of proof after the basic four is proof by diagonalization. Why do they call it diagonalization? Because the idea behind diagonalization is to write out a table that describes how a collection of objects behaves, and then to manipulate the "diagonal" of that table to get a new object that you can prove isn't in the table. The simplest and most famous example of this is the proof that there is no bijection between the natural numbers and the real numbers. We defined injections, and surjections and bijections, in two earlier posts in this series, but for new readers a bijection is just a one-to-one mapping between two collections of things. For example, one can construct a bijection between all positive integers and all even positive integers by mapping $n$ to $2n$. If there is a bijection between two (perhaps infinite) sets, then we say they have the same size or cardinality. And so to say there is no bijection between the natural numbers and the real numbers is to say that one of these two sets (the real numbers) is somehow "larger" than the other, despite both being infinite in size. It's deep, it used to be very controversial, and it made the method of diagonalization famous. Let's see how it works. Theorem: There is no bijection from the natural numbers $\mathbb{N}$ to the real numbers $\mathbb{R}$. Proof. Suppose to the contrary (i.e., we're about to do proof by contradiction) that there is a bijection $f: \mathbb{N} \to \mathbb{R}$. That is, you give me a positive integer $k$ and I will spit out $f(k)$, with the property that different $k$ give different $f(k)$, and every real number is hit by some natural number $k$ (this is just what it means to be a one-to-one mapping). First let me just do some setup. I claim that all we need to do is show that there is no bijection between $\mathbb{N}$ and the real numbers between 0 and 1. In particular, I claim there is a bijection from $(0,1)$ to all real numbers, so if there is a bijection from $\mathbb{N} \to (0,1)$ then we could combine the two bijections. To show there is a bijection from $(0,1) to \mathbb{R}$, I can first make a bijection from the open interval $(0,1)$ to the interval $(-\infty, 0) \cup (1, \infty)$ by mapping $x$ to $1/x$. With a little bit of extra work (read, messy details) you can extend this to all real numbers. Here's a sketch: make a bijection from $(0,1)$ to $(0,2)$ by doubling; then make a bijection from $(0,2)$ to all real numbers by using the $(0,1)$ part to get $(-\infty, 0) \cup (1, \infty)$, and use the $[1,2)$ part to get $[0,1]$ by subtracting 1 (almost! To be super rigorous you also have to argue that the missing number 1 doesn't change the cardinality, or else write down a more complicated bijection; still, the idea should be clear). Okay, setup is done. We just have to show there is no bijection between $(0,1)$ and the natural numbers. The reason I did all that setup is so that I can use the fact that every real number in $(0,1)$ has an infinite binary decimal expansion whose only nonzero digits are after the decimal point. And so I'll write down the expansion of $f(1)$ as a row in a table (an infinite row), and below it I'll write down the expansion of $f(2)$, below that $f(3)$, and so on, and the decimal points will line up. The table looks like this. The $d$'s above are either 0 or 1. I need to be a bit more detailed in my table, so I'll index the digits of $f(1)$ by $b_{1,1}, b_{1,2}, b_{1,3}, dots$, the digits of $f(2)$ by $b_{2,1}, b_{2,2}, b_ {2,3}, dots$, and so on. This makes the table look like this It's a bit harder to read, but trust me the notation is helpful. Now by the assumption that $f$ is a bijection, I'm assuming that every real number shows up as a number in this table, and no real number shows up twice. So if I could construct a number that I can prove is not in the table, I will arrive at a contradiction: the table couldn't have had all real numbers to begin with! And that will prove there is no bijection between the natural numbers and the real numbers. Here's how I'll come up with such a number $N$ (this is the diagonalization part). It starts with 0., and it's first digit after the decimal is $1-b_{1,1}$. That is, we flip the bit $b_{1,1}$ to get the first digit of $N$. The second digit is $1-b_{2,2}$, the third is $1-b_{3,3}$, and so on. In general, digit $i$ is $1-b_{i,i}$. Now we show that $N$ isn't in the table. If it were, then it would have to be $N = f(m)$ for some $m$, i.e. be the $m$-th row in the table. Moreover, by the way we built the table, the $m$-th digit of $N$ would be $b_{m,m}$. But we defined $N$ so that it's $m$-th digit was actually $1-b_{m,m}$. This is very embarrassing for $N$ (it's a contradiction!). So $N$ isn't in the table. $\square$ It's the kind of proof that blows your mind the first time you see it, because it says that there is more than one kind of infinity. Not something you think about every day, right? The Halting Problem The second example we'll show of a proof by diagonalization is the Halting Theorem, proved originally by Alan Turing, which says that there are some problems that computers can't solve, even if given unbounded space and time to perform their computations. The formal mathematical model is called a Turing machine, but for simplicity you can think of "Turing machines" and "algorithms described in words" as the same thing. Or if you want it can be "programs written in programming language X." So we'll use the three words "Turing machine," "algorithm," and "program" interchangeably. The proof works by actually defining a problem and proving it can't be solved. The problem is called the halting problem, and it is the problem of deciding: given a program $P$ and an input $x$ to that program, will $P$ ever stop running when given $x$ as input? What I mean by "decide" is that any program that claims to solve the halting problem is itself required to halt for every possible input with the correct answer. A "halting problem solver" can't loop infinitely! So first we'll give the standard proof that the halting problem can't be solved, and then we'll inspect the form of the proof more closely to see why it's considered a diagonalization argument. Theorem: The halting program cannot be solved by Turing machines. Proof. Suppose to the contrary that $T$ is a program that solves the halting problem. We'll use $T$ as a black box to come up with a new program I'll call meta-$T$, defined in pseudo-python as def metaT(P): run T on (P,P) if T says that P halts: loop infinitely halt and output "success!" In words, meta-$T$ accepts as input the source code of a program $P$, and then uses $T$ to tell if $P$ halts (when given its own source code as input). Based on the result, it behaves the opposite of $P$; if $P$ halts then meta-$T$ loops infinitely and vice versa. It's a little meta, right? Now let's do something crazy: let's run meta-$T$ on itself! That is, run So meta. The question is what is the output of this call? The meta-$T$ program uses $T$ to determine whether meta-$T$ halts when given itself as input. So let's say that the answer to this question is "yes, it does halt." Then by the definition of meta-$T$, the program proceeds to loop forever. But this is a problem, because it means that metaT(metaT) (which is the original thing we ran) actually does not halt, contradicting $T$'s answer! Likewise, if $T$ says that metaT(metaT) should loop infinitely, that will cause meta-$T$ to halt, a contradiction. So $T$ cannot be correct, and the halting problem can't be solved. $\square$ This theorem is deep because it says that you can't possibly write a program to which can always detect bugs in other programs. Infinite loops are just one special kind of bug. But let's take a closer look and see why this is a proof by diagonalization. The first thing we need to convince ourselves is that the set of all programs is countable (that is, there is a bijection from $\mathbb{N}$ to the set of all programs). This shouldn't be so hard to see: you can list all programs in lexicographic order, since the set of all strings is countable, and then throw out any that are not syntactically valid programs. Likewise, the set of all inputs, really just all strings, is countable. The second thing we need to convince ourselves of is that a problem corresponds to an infinite binary string. To do this, we'll restrict our attention to problems with yes/no answers, that is where the goal of the program is to output a single bit corresponding to yes or no for a given input. Then if we list all possible inputs in increasing lexicographic order, a problem can be represented by the infinite list of bits that are the correct outputs to each input. For example, if the problem is to determine whether a given binary input string corresponds to an even number, the representation might look like this: Of course this all depends on the details of how one encodes inputs, but the point is that if you wanted to you could nail all this down precisely. More importantly for us we can represent the halting problem as an infinite table of bits. If the columns of the table are all programs (in lex order), and the rows of the table correspond to inputs (in lex order), then the table would have at entry $(x,P)$ a 1 if $P(x)$ halts and a 0 otherwise. here $b_{i,j}$ is 1 if $P_j(x_i)$ halts and 0 otherwise. The table encodes the answers to the halting problem for all possible inputs. Now we assume for contradiction sake that some program solves the halting problem, i.e. that every entry of the table is computable. Now we'll construct the answers output by meta-$T$ by flipping each bit of the diagonal of the table. The point is that meta-$T$ corresponds to some row of the table, because there is some input string that is interpreted as the source code of meta-$T$. Then we argue that the entry of the table for $(textup{meta-}T, textup{meta-}T)$ contradicts its definition, and we're done! So these are two of the most high-profile uses of the method of diagonalization. It's a great tool for your proving repertoire. Until next time!	{"url":"https://eklausmeier.mywire.org/blog/2015/06-13-methods-of-proof-diagonalization","timestamp":"2024-11-03T12:53:13Z","content_type":"text/html","content_length":"49486","record_id":"<urn:uuid:38637f34-64e6-4d03-a49b-fd140ad20f7f>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00318.warc.gz"}
Sander's Roboblog Since my previous post on AHRS types, I have made some additional changes to the three systems. Specifically, I have added the magnetometer data as an update to the yaw in each system. My technique for adding the mag in each system was derived from this by Sebastian Madgwick. The idea is that the Earth's magnetic field can be described as having a components in a single horizontal axis and a single vertical axis. By rotating these components inversely, by our quaternion, we can directly compare our results to the measured magnetic field vector. This was especially helpful because it simplified the equation for R in the EKF and UKF, because I could directly use square of the rms-noise of the magnetometer sensors as the diagonals of the matrix. The toughest part of this problem was deriving each of the equations for Madgwick, the EKF and the UKF. I would post it my work, but it is extremely tedious. However, the results are very good and you can see that it clearly improved the yaw from the previous post in all three cases. This was an extremely exciting result for me to see, because I worked very hard on the solutions. The one area that I need to work on is when to provide updates to the EKF and UKF. Currently, I use an accelerometer threshold, just to make sure there is no movement. However, this causes the system to be very sensitive to slight movements. Preferably, I would like to make a more robust system for performing updates. That's all for now! In recent weeks, I have spent some time brushing up on as many types of Attitude and Heading Reference Systems (AHRS) as I can. I wanted to code them, and compare them in a meaningful way and eventually implement them on an Arduino. My initial goal has been to work with three types: The Madgwick Filter, an Extended Kalman Filter and an Unscented Kalman Filter. As of right now, I have each of them working and am able to play back a few different types of csv datasets. All of my code is on my github. I should note that I collected data from my phone, which is a Google Pixel, that holds a BMI160 IMU sensor with a 3 axis accelerometer, 3 axis gyroscope, 3 axis magnetometer, a barometer and a temperature sensor. I am sampling the data using an app called HyperIMU (available on the Google Play Store). As of right now, I have only integrated the gyroscope and the accelerometer and am working on the magnetometer updates. All of my implementations are very simple models, that do not yet introduce the gyroscope bias as parameter in the filter. This will be added later, because it complicates the model. My states are the quaternion values: w, x, y, z and all operations are done in quaternion space in order to avoid the singularity when the pitch is at 90 degrees. The Madgwick Filter The Madgwick Filter is based on this paper by Sebastian Madgwick. Remarkably, it is a very new algorithm, but has been widely used across many systems. The idea of this filter is to incorporate updates to the classic gyroscope integration via an optimization assumption. The initial update is to correct for drift in the pitch and roll directions by taking advantage of the direction of gravity from the accelerometers. Essentially, the algorithm forms an objective function between the gravitational vector rotated into the frame of the sensor and the acceleration vector in the frame of the sensor. The idea is that at all times, the acceleration is an approximation of the gravity, even though there may be some acceleration due to movement and noise. The optimization is solved with a gradient descent solution and is therefore, always attempting to correct any drift originating from the gyroscope in the gravity related directions. Here is an image of the results of the Madgwick filter when applied to my phone spinning along the three axes. This is particular run is using the recommended beta gain value from the paper, however, I have found that setting it to between 0.04 and 0.2, allows it to converge faster and more accurately. As you can see in the image, the prediction of roll, pitch and yaw works well. In the roll and pitch directions, you can see that the filter is slowly converging back to 0 degrees. If I increase the beta value, I can speed up that convergence, but it comes at the cost of factoring in any acceleration that is not due to gravity. To see the divergence, I decided to compare residual between the estimate and the measurement. What is interesting to see is what happens when we have actual movement of the phone and how it causes divergence in the filter values. The Extended Kalman Filter The EKF is the standard equation for most estimation problems and it fits well for the AHRS, as well. Essentially, the EKF is a typical Kalman filter that linearizes the prediction and update equations in order to estimate the uncertainty of each of the states. The uncertainty is used to weight measurement updates in order to shrink the overall error of the system. When the sensor is moving with extra acceleration, the gravity updates are far more damaging than they are in the Madgwick filter. In order to mitigate this problem, I decided to only apply updates when the change in acceleration along all three axes is less than a threshold. This way, we know that the phone is stationary during this period. In the future, I will work on a more robust way to find allowable update Here is the results of the EKF. The Euler plot shows fast convergence back to 0 degrees in the pitch and roll after rotations. We can see a bit more noise in the solution than Madgwick Filter, but faster convergance. This is probably because the linearized function does not approximate the uncertainty distribution as well. I have only plotted the residuals when I have done updates. As you can see, the residual is zero mean and has a nice error distribution after the updates. This means that the filter is doing its job. The Unscented Kalman Filter The UKF was a curious addition to this batch of algorithms. Typically, a UKF is used if there is an unclear distribution function. It works by creating a distribution from a few "Sigma Points", which are projections of the system states with a fraction of the noise added back in. This creates a pseudo space that approximates the distribution of the uncertainty of each state. This image was very helpful in my understanding of the UKF. Basically, a proportion of the standard deviation of the uncertainty is added to each state and then either projected forward in time by the state transition matrix or rotated to the measurement frame. Here are the results of the UKF. Again, we are seeing good results in terms of convergence back to zero after large movements in pitch and roll. We also see that we have somewhat Gaussian Error in the residual. Here are the histograms of the Roll and Pitch error. Both are very Gaussian and has almost exactly the same amount of mean and error as the EKF. Now that I have this done, I have a few other things that I want to do to improve my results. These things include: • Yaw correction via Magnetometer • Gyroscope bias correction, also with the Magnetometer and possibly the Temperature Sensor • Process and Measurement noise improvement via adaptive EKF • Implementation in C for realtime estimation on Arduino This is a Recurrent Neural Network diagram from here Sporadically, I have been working on this little project to both learn more about recurrent neural networks and build something useful to predict future cryptocurrency prices. As I talked about before, I have been looking into ways of predicting the price on a rolling basis. As of right now, I am predicting the next day's price from a history of 6 days before. Let's take a look at what I Recurrent Neural Networks are a good choice for this type of timeseries because they can incorporate new values and keep track of history in order to make new predictions. I am using Keras to create the network and here is how I built it: model = Sequential() batch_size = 1 model.add(LSTM(4, input_shape=(lookback, 1))) model.compile(loss='mse', optimizer='rmsprop', metrics=['mae']) As you can see, I used 4 Long-Short Term Memory blocks and a lookback of 6 days. I used "rmsprop" as my optimizer because it is essentially a more advanced gradient descent method which is usually fine for regression tasks. The Loss Metric chosen was Mean Square Error, which is the classic loss function for regression problems. I am also keeping track of Mean Absolute Error, just to confirm the results. The data in this example consists of BTC/USD daily closes from January 2016 to February 2018. This is the plot of that data. Before training, I scale the data between 0 and 0.9 to account for higher prices in the future, with a Min-Max Scaler from Sci-kit Learn. In the future, I may try dividing by 1 million instead, to better account for future prices (I don't see it hitting 1 million any time soon, but it could in the future). Then I split the data into training and testing datasets with a 67% training split. During the train, I also check a 20% validation set, just to watch how each iteration of the model performs. I have plotted these values during the train. This allows me to see at what point the model begins to over-train. We can see this by looking at the point at which the validation loss (MSE) significantly diverges from the training loss. This is an image of that plot, with the validation loss filtered to discard noise: In this example, I have trained to 1000 iterations. It is kind of tough to see the divergence, but it happens around 125 iterations. I am curious if I were to leave it training for 10,000 iterations, whether there might be a more clear divergence point. Anyway, if we train to about 125 iterations, we get a result that looks like the one below. The green line is the prediction of trained data and the red line is the prediction of the untrained portion of the data. Although the result is clearly worse, I am pretty happy with how well it did. The results are as follows: - On Trained data the RMSE is 44.67 - On Test data the RMSE is 1342.08 The question is, how can I improve this result? My initial thoughts are to experiment with different look-back values, and possibly more LSTM blocks. However, I suspect that the most practical way to improve the result is to also add in open's, high's and low's as features as well. This may vastly improve the model because it will be able to see momentum and other patterns at each timestep. This where I will focus next. Check out my code here: https://github.com/Sanderi44/crypto-analysis Over the past few weeks, I have been working on a few different projects to broaden my skills and learn about some new technologies. One area that I have been jumping into is computer vision. Recently, I have been working my way through the book "OpenCV with Python Blueprints". Some of the projects I have done so far include building a Cartoonizer and some Feature Matching. Let me show This is me just after finishing the Cartoonizer. As you can see, the program cartoonizes live video and shows the result in real-time. The process involves: - Applying a bilateral filter in order to smooth flat areas while keeping edges sharp - Converting the original to grayscale - Applying a median blur to reduce image noise - Using an adaptive threshold to detect edges - Combining the color image with the edge mask The result is really great! Another project I just completed is the Feature Matching project. The idea here is to find robust features of an image, match them in another image and find the transform that converts between the two images. Here is an examples of what that looks like in practice. On the left, is the still image that I took from my computer and on the right is a live image of the scene. The red lines show where the feature in the first frame is located in the second frame. This seemed to work pretty well for very similar scenes, but had some trouble when I changed the scene significantly. However, it is not unexpected that it would fail on different scenes, because the features are not very similar at all. Here is how I did it: - First I used SURF (Speeded-Up Robust Features) to find some distinctive keypoints. - Then, I used FLANN (Fast Library for Approximate Nearest Neighbors) to check whether the other frame has similar keypoints and then match them. - Then, using outlier rejection, I was able to pare down the number of features to only the good ones. - Finally, using a perspective transform, I was able to warp the second image so that it matched the first (not shown here). I am currently in the middle of the 3D scene reconstruction project. This something I have been meaning to do for a long time and I am currently really enjoying working on it. It's been a while since I last posted, because I was working hard over at Navisens. After about 3 years, I am now back on the market looking for a new position. In the meantime, I have started working on another project that has fascinated me for a while. For about a year now, I have been looking into and trading cryptocurrencies. I find the whole market exciting to follow and very lucrative (if you do it right). This is where my new project comes in. I am building a tool for querying historical cryptocurrency price data in order to analyze and use it for making future price predictions. My current progress is located on my Github. To get started, I have built a simple api that uses the phenomenal ccxt api to query from tons of exchanges to build up data. Then, once I have a significant amount of data, I will test some machine learning algorithms on that data. Here are some questions that I am starting to think about: - How is one cryptocurrency related to another? Can I use the data from one crypto to train a classifier/regressor for predicting a different crypto? - What type of Machine Learning algorithms will work best on this time-series data? Neural Networks? Recurrent Neural Networks? Decision Trees? Bayesian Estimators? - What features should I use as inputs to the ML algorithms? Do I need scaling? (probably) How many features will be sufficient? - What should I predict? A new price (regressor)? Whether it will go up or down (classifier)? As I start to look at these problems more carefully, I will continue to write about the conclusions that I come to. If you have questions or thoughts, I would love to hear them! Feel free to comment on this post or send me an email!	{"url":"https://www.sanderbot.com/2018/","timestamp":"2024-11-12T07:38:35Z","content_type":"text/html","content_length":"101067","record_id":"<urn:uuid:b375e46b-2fc0-4752-b4b7-e52e27255e1d>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00229.warc.gz"}
Find the Maximum Clique Finds the $O(\|V\|/(log\|V\|)^2)$ apx of maximum clique/independent set in the worst case. GNetworkX graph Undirected graph The apx-maximum clique of the graph If the graph is directed or is a multigraph. A clique in an undirected graph G = (V, E) is a subset of the vertex set C subseteq V such that for every two vertices in C there exists an edge connecting the two. This is equivalent to saying that the subgraph induced by C is complete (in some cases, the term clique may also refer to the subgraph). A maximum clique is a clique of the largest possible size in a given graph. The clique number omega(G) of a graph G is the number of vertices in a maximum clique in G. The intersection number of G is the smallest number of cliques that together cover all edges of G. Boppana, R., & Halldórsson, M. M. (1992). Approximating maximum independent sets by excluding subgraphs. BIT Numerical Mathematics, 32(2), 180–196. Springer. doi:10.1007/BF01994876 >>> G = nx.path_graph(10) >>> nx.approximation.max_clique(G) {8, 9}	{"url":"https://networkx.org/documentation/latest/reference/algorithms/generated/networkx.algorithms.approximation.clique.max_clique.html","timestamp":"2024-11-06T20:35:22Z","content_type":"text/html","content_length":"32870","record_id":"<urn:uuid:22cc11a4-0300-48bf-8a1e-33f3d221b3e1>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00664.warc.gz"}
Haplotyping for Disease Association: A Combinatorial Approach We consider a combinatorial problem derived from haplotyping a population with respect to a genetic disease, either recessive or dominant. Given a set of individuals, partitioned into healthy and diseased, and the corresponding sets of genotypes, we want to infer “bad” and “good” haplotypes to account for these genotypes and for the disease. Assume, for example, that the disease is recessive. Then, the resolving haplotypes must consist of bad and good haplotypes so that 1) each genotype belonging to a diseased individual is explained by a pair of bad haplotypes and 2) each genotype belonging to a healthy individual is explained by a pair of haplotypes of which at least one is good. We prove that the associated decision problem is NP-complete. However, we also prove that there is a simple solution, provided that the data satisfy a very weak requirement.	{"url":"https://kilthub.cmu.edu/articles/journal_contribution/Haplotyping_for_Disease_Association_A_Combinatorial_Approach/6705722","timestamp":"2024-11-07T06:17:08Z","content_type":"text/html","content_length":"123201","record_id":"<urn:uuid:3732811a-a770-4b02-96f9-b4d983b71b45>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00444.warc.gz"}
DOI Number We have introduced the paranormed sequence spaces cF (f; Λ; ∆m; p), cF0 (f; Λ; ∆m; p) and l1 F (f; Λ; ∆m; p) of fuzzy numbers associated with the multiplier sequence Λ = (λk) determined by a sequence of moduli f = (fk). Some of their properties like solidity, symmetricity, completeness etc. and inclusion relations are studied. fuzzy numbers, paranorm, modulus function, difference sequence Fuzzy numbers, paranorm, modulus function, difference sequence Y.Altin and M.Et, Generalized difference sequence spaces defined by a modulus function in a locally convex space, Scoo. J. Math., 31(2005), no. 2, 233-243. S.Debnath, D. Bhattacharya and J.Debnath, On Some Sequence Spaces of IFNs, Bull.Kerala Math. Asso. 12 (2015), no. 2, 125-133. S.Debnath and J.Debnath, Some generalized statistical convergent sequence spaces of fuzzy numbers via ideals, Math. Sci. Lett., 2(2013), no. 2, 151-154. S.Debnath and J.Debnath, Some ideal convergent sequence spaces of fuzzy real numbers, Palestine J. Mathematics, 3 (2014), no. 1, 27-32. S.Debnath and J.Debnath, On I-statistically convergent sequence spaces defined by sequences of Orlicz functions using matrix transformation, Proecciones J. Math. 33(2014),no. 3, 277-285. S.Debnath and S.Saha, On some I-convergent generalized difference sequence spaces associated with multiplier sequence defined by a sequence of modulli, Proecciones J. Math., (2015), no. 2, 137-145. M. Et and R. Colak, On some generalized difference sequence spaces, Soochow J. Math.,21(1995), 377-386. G. Goes and S. Goes, Sequences of bounded variation and sequences of Fourier coefficients,Math. Zeift., 118(1970), 93-102. P. K. Kamthan, Bases in certain class of Frechet spaces, Tamking J. Math, 7(1976), 41-49. H. Kizmaz, On certain sequence spaces, Canad. Math. Bull, 24(1981), no. 2, 169-176. I. J. Maddox, Spaces of strongly summable sequences, Quart. Jour. Math (2nd Ser.)18(1967), 345-355. H. Nakano, Concave modulars, J Math Soc. Japan, 5(1953), 29-49. K.Raj and S.K.Sharma, Some strongly convergent difference sequence spaces defined by a modulus functions, NOVISAD J.Math, 42(2012), no.2, 61-73. S. Simons, The sequences spaces l(Pv) and m(pv), Proc. London Math. Soc.,15(1965), no.3, 422-436. B. C. Tripathy and A. Baruah, New type of difference sequence spaces of fuzzy real numbers;Math. Model. Anal, 14(2009), no. 3, 391-397. B. C. Tripathy and P. Chandra, On some generalized difference paranormed sequence spaces associated with multiplier sequence defined by modulus function, Anal. Theory Appl, 27 (2011), no. 1, 21-27. B. C. Tripathy and S. Debnath, On generalized difference sequence spaces of fuzzy numbers,Acta Sinet. Tech., 35(2013), no. 1, 117-121 B. C. Tripathy and S. Mahanta, On a class of vector valued sequences associated with multiplier sequences; Acta Math. Appl. Sin. (Eng. Ser.), 20(2004), no. 3, 487-494. L. A. Zadeh, Fuzzy sets, Inform. and control, 8(1965), no. 1, 338-353 • There are currently no refbacks. © University of Niš \| Created on November, 2013 ISSN 0352-9665 (Print) ISSN 2406-047X (Online)	{"url":"https://casopisi.junis.ni.ac.rs/index.php/FUMathInf/article/view/2215","timestamp":"2024-11-10T08:21:13Z","content_type":"application/xhtml+xml","content_length":"24070","record_id":"<urn:uuid:3ffda906-a9a1-4e78-a41b-2eb8d7cfa98e>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00242.warc.gz"}
Simultaneous Numerical Solution of Differential-Algebraic Equations A unified method for handling the mixed differential and algebraic equations of the type that commonly occur in the transient analysis of large networks or in continuous system simulation is discussed. The first part of the paper is a brief review of existing techniques of handling initial value problems for stiff ordinary differential equations written in the standard form y′ f(y, t). In the second part one of these techniques is applied to the problem F(y, y′, t)=0. This may be either a differential or an algebraic equation as ∂F/∂y′ is nonzero or zero. It will represent a mixed system when vectors F and y represent components of a system. The method lends itself to the use of sparse matrix techniques when the problem is sparse. All Science Journal Classification (ASJC) codes Dive into the research topics of 'Simultaneous Numerical Solution of Differential-Algebraic Equations'. Together they form a unique fingerprint.	{"url":"https://collaborate.princeton.edu/en/publications/simultaneous-numerical-solution-of-differential-algebraic-equatio","timestamp":"2024-11-06T18:00:07Z","content_type":"text/html","content_length":"48119","record_id":"<urn:uuid:0d119f28-6ee7-4bf1-858b-4ec702a6e38b>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00663.warc.gz"}
Lesson 1 Find the Largest Product Lesson Purpose The purpose of this lesson is for students to recognize and explain place value patterns as they find products using the standard algorithm they learned in Unit 4. Lesson Narrative In previous units, students learned the standard algorithm for multiplying whole numbers. In this lesson, students compare and contrast different products that can be made with the same set of 3 or 4 digits. They look for patterns in the arrangement of the digits and explain those patterns in terms of place value. For example, students might notice that the product of 2 two-digit numbers results in a greater product than that of a three-digit number and 1 one-digit number. If students need additional support with the concepts in this lesson, refer back to Unit 4, Section A in the curriculum Activity 1: Talk About it Learning Goals Teacher Facing • Fluently multiply multi-digit whole numbers using the standard algorithm. Student Facing • Let’s look for patterns when we multiply multi-digit numbers. Lesson Timeline Warm-up 10 min Activity 1 15 min Activity 2 20 min Lesson Synthesis 10 min Cool-down 5 min Teacher Reflection Questions What did you learn about students' understanding of the standard algorithm for multiplication during the lesson today? How can you use what you learned to support students in tomorrow's lesson? Additional Resources Google Slides For access, consult one of our IM Certified Partners. PowerPoint Slides For access, consult one of our IM Certified Partners.	{"url":"https://curriculum.illustrativemathematics.org/k5/teachers/grade-5/unit-8/lesson-1/preparation.html","timestamp":"2024-11-14T12:24:41Z","content_type":"text/html","content_length":"70108","record_id":"<urn:uuid:97487056-889a-4227-aaab-921e8bf043d0>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00644.warc.gz"}
Python Arithmetic Operators - TestingDocs.com Python Arithmetic Operators Python Arithmetic operators are used to perform arithmetical operations like addition, subtraction, multiplication, division, etc, in Python programs. We use these operators in scientific calculations and mathematical expressions. Python Arithmetic Operators Python arithmetic operators are as follows: Python Operator Name Description + Addition Operator The + operator adds two operands. The – operator subtracts the right operand from the left operand. z = x – y – Subtraction Operator For example, the value of y is subtracted from the x and then assigned to the variable z. * Multiplication Operator The * operator multiplies two operands. / True Division Operator The / operator divides the left operand by the right one and returns the result into a float. % Modulus Operator The % operator returns the remainder of the division of the left operand by the right. // Floor Division Operator The // operator divides the left operand by the right one and returns results into the whole number adjusted to the left in the number line. Exponential Operator The operator returns the value of the left operand raised to the power of the right. The below program illustrates the use of the arithmetic operators: # Python Arithmetic Operators program # Python Tutorials - www.TestingDocs.com number1 = input('Enter a number:') number1 = int(number1) number2 = input('Enter a number:') number2 = int(number2) result = number1 + number2 print('Addition =', result) diff = number1 - number2 print('Subtraction =', diff) product = number1 * number2 print('Multiplication =', product) truediv = number1 / number2 print('True Division =', truediv) floordiv = number1 // number2 print('Floor Division =', floordiv) Sample output of the program: Enter a number:25 Enter a number:7 Addition = 32 Subtraction = 18 Multiplication = 175 True Division = 3.5714285714285716 Floor Division = 3 Python Tutorials Python Tutorial on this website can be found at: More information on Python is available at the official website:	{"url":"https://www.testingdocs.com/python-arithmetic-operators/?amp=1","timestamp":"2024-11-13T01:11:02Z","content_type":"text/html","content_length":"68107","record_id":"<urn:uuid:89294fe3-f152-4b41-9380-c241e9ec4e8f>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00083.warc.gz"}
Multiplying Two Digit Numbers Worksheet Pdf Multiplying Two Digit Numbers Worksheet Pdf work as fundamental tools in the realm of maths, supplying a structured yet versatile platform for learners to discover and grasp mathematical ideas. These worksheets supply a structured method to understanding numbers, supporting a strong structure upon which mathematical efficiency flourishes. From the most basic counting workouts to the intricacies of sophisticated calculations, Multiplying Two Digit Numbers Worksheet Pdf satisfy learners of varied ages and skill degrees. 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discrete object I added a hatnote. Okay, good point. I changed the name of discrete space to discrete object such that it is now consistent with codiscrete object. In discrete object, I saw two mentions of the diagonal map “$X \times X \to X$”, so I made them both $X \to X \times X$. The first paragraph under Discrete Geometric Spaces puzzled me, where it says, “the converse holds if $X$ satisfies the $T_0$ separation axiom” (i.e., if the diagonal map is open, then $X$ is discrete provided we assume $T_0$). I don’t understand why we need that assumption. Suppose $X \to X \times X$ is an open map. In particular the image of the diagonal map is an open set in $X \times X$, i.e., for each $(x, x)$ there is a basic open $U \times V$ containing $(x, x)$ that is entirely contained in the diagonal. Thus the subset $\{x\} \times V$ of $U \times V$ would also be entirely contained in the diagonal, i.e., $(x, y) \in \{x\} \times V$ implies $x = y$, for any $y \in V$. So the open $V$ is the singleton $\{x\}$. (By similar reasoning, $U$ is also the singleton $\{x\}$.) So $\{x\}$ is open, for every $x \in X$. No separation axiom needed. Am I missing A topological space has open diagonal if and only if it is discrete, indeed. I prefer this argument: for every $x \in X$, the intersection of the diagonal and $\{ x \} \times X$ is the singleton $\{ (x, x) \}$, hence $\{ x \}$ is open in $X$. Oh, I see: the inverse image of the open $\Delta$ along $y \mapsto (x, y)$, for any given $x \in X$. I went ahead and edited that point in.	{"url":"https://nforum.ncatlab.org/discussion/3613/discrete-object/?Focus=29659","timestamp":"2024-11-07T07:27:49Z","content_type":"application/xhtml+xml","content_length":"51767","record_id":"<urn:uuid:4dbdce17-869b-4a5d-814e-c3c2bf72f097>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00417.warc.gz"}
Physics - Uncertainty What are the two types of uncertainty? • Absolute uncertainty • Percentage uncertainty What is absolute uncertainty? • A number that gives the range of values for an experiment. • E.g. $7.3mm \pm 0.2mm$ What is percentage uncertainty equal to for a single measurement? The resolution of the equipment. What is percentage uncertainty equal to for multiple measurements? The range of values divided by two. What is the percentage uncertainty for $0.65mm, 0.58mm, 0.62mm$? How do you calculate percentage uncertainty? \[\frac{\text{Absolute uncertainty}}{\text{Actual value}} \times 100\%\] How does multiplication by a constant affect percentage uncertainties? How does multiplication by a constant affect absolute uncertainties? They are multiplied by the constant. How does addition or subtraction of quantities affect absolute uncertainties? The uncertainties are added together. How does multiplication or division of quantities affect percentage uncertainties? The uncertainties are added together. How does raising a value to a power affect percentage uncertainties? The percentage uncertainty is multiplied by the power. How do you deal with absolute uncertainties after using an operation like $\sin(x)$ or $\ln(x)$? • Calculate the difference between the two min and max values after being put into the operation. How can you calculate the $\text{percentage uncertainty}$ from the $\text{range}$ and the $\text{mean}$? \[\frac{\text{range}}{2\text{percentage uncertainty}}\] Related posts	{"url":"https://ollybritton.com/notes/a-level/physics/topics/uncertainty/","timestamp":"2024-11-07T09:49:07Z","content_type":"text/html","content_length":"505709","record_id":"<urn:uuid:24423ba9-f16e-4110-830b-d3f66706067d>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00449.warc.gz"}
Sandwich sudoku -- Grid Puzzle How to play Sandwich sudoku Sandwich Sudoku Rules • Fill the numbers 1-9 into each row, column, and 3x3 subgrid of a 9x9 grid. • Each number can only appear once in each row, column, and 3x3 subgrid. • Fill in the blank cells with the numbers 1-9 so that each row, column, and 3x3 subgrid contains the numbers 1-9. • contains extra numbers outside the grid. These extra clues indicate the sum of digits between 1 and 9 in the corresponding row or column. Sandwich Sudoku is a type of Sudoku puzzle that uses a non-standard grid. The grid is divided into several smaller grids, and the numbers in each smaller grid must be different. In addition, there are also numbers outside of the smaller grids that provide clues to the solution. The outer borders of the grid contain the sum of the middle numbers in the corresponding row or column. For example, in the row with the numbers 2, 4, and 6, the outer border would contain the number 12. These numbers can be used to help solve the puzzle by providing clues to the possible values of the missing numbers. Sandwich Sudoku puzzles can be more challenging than standard Sudoku puzzles because they require players to think more creatively and use more problem-solving skills. For example, players may need to use the clues outside of the smaller grids to help them solve the puzzle, and they may also need to use logic and deduction to narrow down the possible solutions.	{"url":"https://en.gridpuzzle.com/sandwich-sudoku","timestamp":"2024-11-10T03:19:19Z","content_type":"text/html","content_length":"95792","record_id":"<urn:uuid:969011d0-ab12-4f83-930f-d7783bd296e9>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00578.warc.gz"}
Effect of density and Poisson’s ratio on thermal induced vibration of parallelogram plate The present paper provides mathematical model for the study of natural (free) vibration of non homogeneous tapered parallelogram plate on clamped boundary condition. Here non homogeneity (in material) of the plate’s means that the density and Poisson’s ratio varies circularly and exponentially respectively. For tapered, we assumed that thickness of the plate varies linearly in one direction. Bi parabolic temperature (parabolic in $\zeta$-direction and parabolic in $\psi$-direction) variation on the plate is being viewed. Rayleigh Ritz method is used to solve the model (governing differential equation of motion). 1. Introduction The study of vibration of tapered (non uniform) plates with non homogeneity in the material (non homogeneous plate) is the vast area of research due to its utility in various engineering applications like marine engineering, ocean engineering, optical instruments and mechanical engineering. Non homogeneous tapered plate plays significant role in engineering structures because of high tensile strength, durability and elastic behavior. All the engineering structure worked under great influence of temperature which causes non homogeneity. Therefore, with-out consideration of temperature the study of vibration means nothing. A significant work has been reported in these directions. An excellent work on vibration of plates with various shapes has been described by Chakraverty [1]. Chen et al. [2] discussed the free vibration of cantilevered symmetrically laminated thick trapezoidal plates. Gupta and Mamta [3] studied non linear thickness variation of non homogeneous rectangle plate using spline technique. Free vibration has been discussed by Gupta and Sharma [4] on trapezoidal plate with thickness variation under temperature effect. Rotary inertial effect in isotropic plates (uniform and tapered thickness) has been carried out in two companion papers by Kalita et al. [5, 6]. The study of vibration of non uniform and non homogeneous rectangular plate with temperature effect has been studied by Khanna and Kaur [7-9] with exponential variation in non homogeneity. Leissa [10] provided vibration of plates (of different shapes) on different combination of boundary (clamped, simply supported and free) conditions in his excellent monograph. Leissa et. al. [11, 12] studied approximate analysis of the forced vibration response of plates and vibration of completely free triangular plates. Transverse vibration and instability of an eccentric rotating circular plate is studied by Ratko [13]. Sharma et al. [14-16] discussed natural vibration on orthotropic non homogeneous of rectangular plate, non homogeneous square plate (with circular variation in density) and non homogeneous trapezoidal plate with temperature effect. The literature shows that the significant work has been done on vibration of non uniform (tapered) and non homogeneous plates with thermal gradient. Literature also emphasis on that, for non homogeneity either density or Poisson’s ratio varies linearly, parabolic and exponentially. But none of the researcher focused on other variation. This aspect provides good motivation to us to study the effect of circular variation in density as a new interesting aspect to frequency modes. Author also studies the effect of exponential variation in Poisson’s ratio as another parameter of non homogeneity (i.e., simultaneous variation in density and Poisson’s ratio) with the help of this model. The model presented in this paper computes the vibrational frequency modes (first two modes) of non uniform and non homogeneous clamped parallelogram plate. This model also provides effect of other aspects such as effect of temperature and thickness to frequency modes. 2. Analysis 2.1. Description of model A non uniform and non homogeneous parallelogram (thin) plate having skew angle $\theta$ is shown in Fig. 1. The skew coordinates for the parallelogram plate are: $\zeta =x-y\mathrm{t}\mathrm{a}\mathrm{n}\theta ,\mathrm{}\mathrm{}\psi =y\mathrm{s}\mathrm{e}\mathrm{c}\theta .$ The boundaries of the plate in skew coordinates are: $\zeta =0,a\mathrm{}\mathrm{}\text{and}\mathrm{}\mathrm{}\psi =0,b.$ For natural (free) vibration of plate, deflection (displacement) is assumed as [8]: $\varphi \left(\zeta ,\psi ,t\right)=\mathrm{\Phi }\left(\zeta ,\psi \right)\mathrm{}T\left(t\right),$ where $\mathrm{\Phi }\left(\zeta ,\psi \right)$ and $T\left(t\right)$ are known as maximum deflection (displacement) at time $t$ and time function respectively. The differential equation of motion (kinetic energy ${T}_{s}$ and strain energy ${V}_{s}$) for natural frequency of non uniform parallelogram plate is given by [10]: ${T}_{s}=\frac{1}{2}{p}^{2}\mathrm{c}\mathrm{o}\mathrm{s}\theta \iint \rho g{\mathrm{\Phi }}^{2}d\zeta d\psi ,$ ${V}_{s}=\frac{1}{2\mathrm{c}\mathrm{o}{\mathrm{s}}^{3}\theta }\iint D\left[\begin{array}{l}{\left(\frac{{\partial }^{2}\mathrm{\Phi }}{\partial {\zeta }^{2}}\right)}^{2}-4\mathrm{s}\mathrm{i}\mathrm {n}\theta \mathrm{}\left(\frac{{\partial }^{2}\mathrm{\Phi }}{\partial {\zeta }^{2}}\right)\left(\frac{{\partial }^{2}\mathrm{\Phi }}{\partial \zeta \mathrm{}\partial \psi }\right)\\ +2\left(\mathrm {s}\mathrm{i}{\mathrm{n}}^{2}\theta +u \mathrm{c}\mathrm{o}{\mathrm{s}}^{2}\theta \right)\left(\frac{{\partial }^{2}\mathrm{\Phi }}{\partial {\zeta }^{2}}\right)\left(\frac{{\partial }^{2}\mathrm{\ Phi }}{\partial {\psi }^{2}}\right)\\ +2\left(1+\mathrm{s}\mathrm{i}{\mathrm{n}}^{2}\theta -u \mathrm{c}\mathrm{o}{\mathrm{s}}^{2}\theta \right){\left(\frac{{\partial }^{2}\mathrm{\Phi }}{\partial \ zeta \mathrm{}\partial \psi }\right)}^{2}\\ -4\mathrm{s}\mathrm{i}\mathrm{n}\theta \mathrm{}\left(\frac{{\partial }^{2}\mathrm{\Phi }}{\partial \zeta \mathrm{}\partial \psi }\right)\left(\frac{{\ partial }^{2}\mathrm{\Phi }}{\partial {\psi }^{2}}\right)+{\left(\frac{{\partial }^{2}\mathrm{\Phi }}{\partial {\psi }^{2}}\right)}^{2}\end{array}\right]d\zeta d\psi ,$ where $\rho$, $u$ and $g$ are known as density, Poisson’s ratio and thickness of the plate. Here $D=E{g}^{3}/12\left(1-{u }^{2}\right)$ is known flexural rigidity; $E$ is Young’s modulus. 2.2. Assumptions for the model Due to the wide range and general scope of vibrations, we require little limitations in the form of assumptions in this study. 1) The thickness of the plate is assumed to be linear in $\zeta$-direction as shown in Fig. 2 as: $g={g}_{0}\left(1+\beta \frac{\zeta }{a}\right),$ where $\beta$, $\left(0\le \beta \le 1\right)$ is known as tapering parameter. Thickness of plate become constant i.e., $g={g}_{0}$ at $\zeta =0$. Fig. 1Parallelogram plate with skew angle θ Fig. 2Tapered parallelogram plate 2) For non homogeneity in plate’s material, we taken into consideration that the density and Poisson’s ratio of the plate varies circularly and exponentially in $\zeta$-direction: $\rho ={\rho }_{0}\left[1+{m}_{1}\left(1-\sqrt{1-\frac{{\zeta }^{2}}{{a}^{2}}}\right)\right],$ $u ={u }_{0}\left[{e}^{{m}_{2}\frac{\zeta }{a}}\right],$ where ${m}_{1}$, ${m}_{2}$, $\left(0\le {m}_{1},{m}_{2}\le 1\right)$ are known as non homogeneity constant corresponding to density and Poisson’s ratio respectively. 3) The variation of temperature on the plate is considered as bi parabolic i.e., parabolic in $\zeta$ and parabolic in $\psi$ direction: $\tau ={\tau }_{0}\left(1-\frac{{\zeta }^{2}}{{a}^{2}}\right)\left(1-\frac{{\psi }^{2}}{{b}^{2}}\right),$ where $\tau$ and ${\tau }_{0}$ denotes the temperature excess above the reference temperature on the plate at any point and at the origin respectively. The temperature dependence modulus of elasticity for engineering structures is given by: $E={E}_{0}\left(1-\gamma \tau \right),$ where ${E}_{0}$ is the Young’s modulus at mentioned temperature (i.e., $\tau =0$) and $\gamma$ is called slope of variation. Using Eq. (9), Eq. (10) becomes: $E={E}_{0}\left[1-\alpha \left\{1-{\frac{\zeta }{{a}^{2}}}^{2}\right\}\left\{1-{\frac{\psi }{{b}^{2}}}^{2}\right\}\right],$ where $\alpha$, $\left(0\le \alpha <1\right)$ is called temperature gradient, which is the product of temperature at origin and slope of variation i.e., $\alpha =\gamma {\tau }_{0}$. Using Eqs. (6), (8) and (11), flexural rigidity of the plate becomes: $D={E}_{0}{{g}_{0}}^{3}\left[1-\alpha \left\{1-{\frac{\zeta }{{a}^{2}}}^{2}\right\}\left\{1-{\frac{\psi }{{b}^{2}}}^{2}\right\}\right]{\left[1+\beta \frac{\zeta }{a}\right]}^{3}/12\left(1-{{u }_{0}}^ {2}{e}^{2{m}_{2}\frac{\zeta }{a}}\right).$ Also, using Eqs. (6), (7), (8) and (12), Eqs. (4) and (5) becomes: ${V}_{s}=\frac{{E}_{0}{{g}_{0}}^{3}}{24\mathrm{c}\mathrm{o}{\mathrm{s}}^{4}\theta }{\int }_{0}^{a}\mathrm{}{\int }_{0}^{b}\left\{\begin{array}{c}\left[\frac{\left[1-\alpha \left\{1-{\frac{\zeta }{{a} ^{2}}}^{2}\right\}\left\{1-{\frac{\psi }{{b}^{2}}}^{2}\right\}\right]{\left[1+\beta \frac{\zeta }{a}\right]}^{3}}{\left(1-{{u }_{0}}^{2}{e}^{2{m}_{2}\frac{\zeta }{a}}\right)}\right]\\ \left[\begin {array}{l}{\left(\frac{{\partial }^{2}\mathrm{\Phi }}{\partial {\zeta }^{2}}\right)}^{2}-4\left(\frac{a}{b}\right)\mathrm{s}\mathrm{i}\mathrm{n}\theta \mathrm{}\left(\frac{{\partial }^{2}\mathrm{\Phi }}{\partial {\zeta }^{2}}\right)\left(\frac{{\partial }^{2}\mathrm{\Phi }}{\partial \zeta \mathrm{}\partial \psi }\right)\\ +2{\left(\frac{a}{b}\right)}^{2}\left(\mathrm{s}\mathrm{i}{\mathrm{n}}^{2}\ theta +{u }_{0}\left\{{e}^{{m}_{2}\frac{\zeta }{a}}\right\}\mathrm{c}\mathrm{o}{\mathrm{s}}^{2}\theta \right)\left(\frac{{\partial }^{2}\mathrm{\Phi }}{\partial {\zeta }^{2}}\right)\left(\frac{{\ partial }^{2}\mathrm{\Phi }}{\partial {\psi }^{2}}\right)\\ +2{\left(\frac{a}{b}\right)}^{2}\left(1+\mathrm{s}\mathrm{i}{\mathrm{n}}^{2}\theta -{u }_{0}\left\{{e}^{{m}_{2}\frac{\zeta }{a}}\right\}\ mathrm{c}\mathrm{o}{\mathrm{s}}^{2}\theta \right){\left(\frac{{\partial }^{2}\mathrm{\Phi }}{\partial \zeta \mathrm{}\partial \psi }\right)}^{2}\\ -4{\left(\frac{a}{b}\right)}^{3}\mathrm{s}\mathrm{i} \mathrm{n}\theta \mathrm{}\left(\frac{{\partial }^{2}\mathrm{\Phi }}{\partial \zeta \mathrm{}\partial \psi }\right)\left(\frac{{\partial }^{2}\mathrm{\Phi }}{\partial {\psi }^{2}}\right)+{\left(\frac {a}{b}\right)}^{4}{\left(\frac{{\partial }^{2}\mathrm{\Phi }}{\partial {\psi }^{2}}\right)}^{2}\end{array}\right]\end{array}\right\}\mathrm{}d\psi d\zeta ,$ ${T}_{s}=\frac{1}{2}{p}^{2}{\rho }_{0}{g}_{0}{\int }_{0}^{a}\mathrm{}{\int }_{0}^{b}\left\{\left[1+{m}_{1}\left(1-\sqrt{1-\frac{{\zeta }^{2}}{{a}^{2}}}\right)\right]\left[1+\beta \frac{\zeta }{a}\ right]\right\}\mathrm{}{\mathrm{\Phi }}^{2}d\psi d\zeta .$ In this model, we are computing frequency on clamped (along all the four edges) condition (i.e., on C-C-C-C), therefore the boundary conditions are: $\mathrm{\Phi }\left(\zeta ,\psi \right)=\frac{\partial \mathrm{\Phi }\left(\zeta ,\psi \right)}{\partial \zeta }=0,\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\zeta =0,\mathrm{}a,$$\mathrm{\Phi }\ left(\zeta ,\psi \right)=\frac{\partial \mathrm{\Phi }\left(\zeta ,\psi \right)}{\partial \psi }=0,\mathrm{}\mathrm{}\mathrm{}\mathrm{}\psi =0,\mathrm{}b.$ Therefore, two term deflection (i.e., maximum displacement) which satisfy the Eq. (15) could be represented by: $\mathrm{\Phi }\left(\zeta ,\psi \right)={\left(\frac{\zeta }{a}\right)}^{2}{\left(\frac{\psi }{b}\right)}^{2}{\left(1-\frac{\zeta }{a}\right)}^{2}{\left(1-\frac{\psi }{b}\right)}^{2}\left[{\mathrm{\ Omega }}_{1}+{\mathrm{\Omega }}_{2}\left(\frac{\zeta }{a}\right)\left(\frac{\psi }{b}\right)\left(1-\frac{\zeta }{a}\right)\left(1-\frac{\psi }{b}\right)\right],$ where ${\mathrm{\Omega }}_{1}$ and ${\mathrm{\Omega }}_{2}$ are arbitrary constants. 3. Solution of model for frequency equation To solve the model (obtain equation of frequency and vibrational frequency), we use Rayleigh Ritz technique (i.e., maximum strain energy ${V}_{s}$ must equal to maximum kinetic energy ${T}_{s}$). Therefore, we have: $\delta \left({V}_{s}-{T}_{s}\right)=0.$ Using Eqs. (13), (14) (15) and (16), Eq. (17) becomes: $\delta \left({{V}_{s}}^{\mathrm{}}-{\lambda }^{2}{{T}_{s}}^{\mathrm{}}\right)=0,$ ${{V}_{s}}^{\mathrm{}}=\frac{1}{\mathrm{c}\mathrm{o}{\mathrm{s}}^{4}\theta }{\int }_{0}^{a}\mathrm{}{\int }_{0}^{b}\left\{\begin{array}{c}\left[\frac{\left[1-\alpha \left\{1-{\frac{\zeta }{{a}^{2}}} ^{2}\right\}\left\{1-{\frac{\psi }{{b}^{2}}}^{2}\right\}\right]{\left[1+\beta \frac{\zeta }{a}\right]}^{3}}{\left(1-{{u }_{0}}^{2}{e}^{2{m}_{2}\frac{\zeta }{a}}\right)}\right]\\ \left[\begin{array} {l}{\left(\frac{{\partial }^{2}\mathrm{\Phi }}{\partial {\zeta }^{2}}\right)}^{2}-4\left(\frac{a}{b}\right)\mathrm{s}\mathrm{i}\mathrm{n}\theta \mathrm{}\left(\frac{{\partial }^{2}\mathrm{\Phi }}{\ partial {\zeta }^{2}}\right)\left(\frac{{\partial }^{2}\mathrm{\Phi }}{\partial \zeta \mathrm{}\partial \psi }\right)\\ +2{\left(\frac{a}{b}\right)}^{2}\left(\mathrm{s}\mathrm{i}{\mathrm{n}}^{2}\ theta +{u }_{0}\left\{{e}^{{m}_{2}\frac{\zeta }{a}}\right\}\mathrm{c}\mathrm{o}{\mathrm{s}}^{2}\theta \right)\left(\frac{{\partial }^{2}\mathrm{\Phi }}{\partial {\zeta }^{2}}\right)\left(\frac{{\ partial }^{2}\mathrm{\Phi }}{\partial {\psi }^{2}}\right)\\ +2{\left(\frac{a}{b}\right)}^{2}\left(1+\mathrm{s}\mathrm{i}{\mathrm{n}}^{2}\theta -{u }_{0}\left\{{e}^{{m}_{2}\frac{\zeta }{a}}\right\}\ mathrm{c}\mathrm{o}{\mathrm{s}}^{2}\theta \right){\left(\frac{{\partial }^{2}\mathrm{\Phi }}{\partial \zeta \mathrm{}\partial \psi }\right)}^{2}\\ -4{\left(\frac{a}{b}\right)}^{3}\mathrm{s}\mathrm{i} \mathrm{n}\theta \mathrm{}\left(\frac{{\partial }^{2}\mathrm{\Phi }}{\partial \zeta \mathrm{}\partial \psi }\right)\left(\frac{{\partial }^{2}\mathrm{\Phi }}{\partial {\psi }^{2}}\right)+{\left(\frac {a}{b}\right)}^{4}{\left(\frac{{\partial }^{2}\mathrm{\Phi }}{\partial {\psi }^{2}}\right)}^{2}\end{array}\right]\end{array}\right\}\mathrm{}d\psi d\zeta ,$ ${{T}_{s}}^{\mathrm{}}={\int }_{0}^{a}\mathrm{}{\int }_{0}^{b}\left\{\left[1+{m}_{1}\left(1-\sqrt{1-\frac{{\zeta }^{2}}{{a}^{2}}}\right)\right]\left[1+\beta \frac{\zeta }{a}\right]\right\}\mathrm{} {\mathrm{\Phi }}^{2}d\psi d\zeta .$ Here ${\lambda }^{2}=12{\rho }_{0}{p}^{2}{a}^{4}/{E}_{0}{{g}_{0}}^{2}$ is known as frequency parameter. Eq. (18) consists of two unknown constants ${\mathrm{\Omega }}_{1}$ and ${\mathrm{\Omega }}_{2} $ (because of substitution of deflection function $\mathrm{\Phi }\left(\zeta ,\psi \right)$). These two unknowns could be calculated as follows: $\frac{\partial }{\partial {\mathrm{\Omega }}_{n}}\left({{V}_{s}}^{\mathrm{}}-{\lambda }^{2}{{T}_{s}}^{\mathrm{*}}\right)=0,\mathrm{}\mathrm{}n=1,2.$ After simplifying Eq. (19), we get system of homogeneous equation as: ${b}_{11}{\mathrm{\Omega }}_{1}+{b}_{12}{\mathrm{\Omega }}_{2}=0,$${b}_{21}{\mathrm{\Omega }}_{2}+{b}_{22}{\mathrm{\Omega }}_{2}=0.$ To obtain non-zero solution (frequency equation), the determinant of coefficient matrix (symmetric matrix) of Eq. (20) must zero i.e.: $\left\|\begin{array}{ll}{b}_{11}& {b}_{12}\\ {b}_{21}& {b}_{22}\end{array}\right\|=0.$ Eq. (21) is quadratic equation from which we get two modes as ${\lambda }_{1}$ (first mode) and ${\lambda }_{2}$ (second mode). 4. Results and discussions To examine the behavior of modes and effect of plate’s parameters (non homogeneity ${m}_{1}$, ${m}_{2}$, temperature gradient $\alpha$ and tapering parameter $\beta$), numerical computation for frequency $\lambda$ is carried out for different combination of plate’s parameters. The value of ${u }_{0}$ is taken 0.345. All the numerical computation is done with the help of MAPLE (high level software). All the findings are presented with the help of tables and graphs. Table 1 provides the frequency modes (first two modes) corresponding to non homogeneity constant (corresponding to Poisson’s ratio ${m}_{2}$ and keeping density parameter ${m}_{1}$ off) with fixed value of skew angle $\theta =$ 30° and aspect ratio $a/b=$ 1.5 for three different values of taper constant $\beta$ and temperature gradient. $\alpha$ i.e., $\beta =\alpha =$ 0, 0.4, 0.8. From Table 1, we conclude that frequency for both modes increases, when the non homogeneity corresponding to Poisson’s ratio increases from 0 to 1 for all the three values of taper constant and temperature gradient ($\beta =\alpha =$ 0, 0.4, 0.8). Also, when the combined value of taper constant $\beta$ and temperature gradient $\alpha$increases from 0 to 0.8, frequency modes increases. The rate of increment in case of non homogeneity is much smaller (due to exponential variation) than the rate of increment in case of combined value of taper constant $\beta$ and temperature gradient $\alpha$. Table 1Non homogeneity constant corresponding to Poisson’s ratio (m2) vs. frequency parameter (λ) for m1= 0, θ= 30° and a/b= 1.5 $\beta =\alpha =$ 0.0 $\beta =\alpha =$ 0.4 $\beta =\alpha =$ 0.8 ${\lambda }_{1}$ ${\lambda }_{2}$ ${\lambda }_{1}$ ${\lambda }_{2}$ ${\lambda }_{1}$ ${\lambda }_{2}$ 0.0 78.77 313.60 86.28 343.38 90.68 360.69 0.2 79.94 318.24 87.63 348.77 92.14 366.65 0.4 81.45 324.25 89.40 355.87 94.08 374.58 0.6 83.44 332.14 91.76 365.31 96.71 385.31 0.8 86.07 342.59 94.94 378.05 100.28 399.98 1.0 89.60 356.61 99.24 395.40 105.16 420.26 Table 2 provides the different set of data (frequency modes) corresponding to non homogeneity constant (variable value of Poisson’s ratio ${m}_{2}$ and fixed value of density parameter ${m}_{1}=$ 0.6) with fixed value of skew angle $\theta =$ 30° and aspect ratio $a/b=$ 1.5 for three different values of taper constant $\beta$ and temperature gradient. $\alpha$ i.e., $\beta =\alpha =$ 0, 0.4, 0.8. From Table 2, one can easily see that the frequency behaviour (increases corresponding to non homogeneity and corresponding to combined value of thermal gradient and taper constant) is same as in Table 1. But due to implementation of another non homogeneity parameter (circular variation in density parameter) the frequency modes are less when compared to Table 1. Table 2Non homogeneity constant corresponding to Poisson’s ratio (m2) vs. frequency parameter (λ) for m1= 0.6, θ= 30° and a/b= 1.5 $\beta =\alpha =$ 0.0 $\beta =\alpha =$ 0.4 $\beta =\alpha =$ 0.8 ${\lambda }_{1}$ ${\lambda }_{2}$ ${\lambda }_{1}$ ${\lambda }_{2}$ ${\lambda }_{1}$ ${\lambda }_{2}$ 0.0 75.41 298.54 82.51 325.99 86.65 341.73 0.2 76.53 302.96 83.79 331.11 88.04 347.37 0.4 77.98 308.68 85.49 337.85 89.90 354.88 0.6 79.88 316.18 87.75 346.81 92.41 365.04 0.8 82.40 326.14 90.79 358.90 95.83 378.93 1.0 85.78 339.48 94.91 375.36 100.50 398.13 Table 3 gives frequency modes corresponding to non homogeneity constant (corresponding to density parameter ${m}_{1}$ and keeping Poisson’s ratio ${m}_{2}$ off) with fixed value of skew angle $\theta =$ 30° and aspect ratio $a/b=$ 1.5 for three different values of taper constant $\beta$ and temperature gradient. $\alpha$ i.e., $\beta =\alpha =$ 0, 0.4, 0.8. From Table 3, we enlighten the fact that frequency for both decreases, when the non homogeneity corresponding to density parameter increases from 0 to 1. Here the frequency behave totally opposite (decreases corresponding to ${m}_{1}$) to Table 1 (increases corresponding to ${m}_{2}$). On the other hand frequency increases when the combined value of taper constant $\beta$ and temperature gradient $\alpha$ increases from 0 to 0.8 as in Table 1. Here the rate of decrement is much smaller (due to circular variation) as compared to rate of increment in Table 1. Table 4 provides the another set of data (frequency modes) corresponding to non homogeneity constant (variable value of density parameter ${m}_{1}$ and fixed value of Poisson’s ratio ${m}_{2}=$ 0.6) with fixed value of skew angle $\theta =$ 30° and aspect ratio $a/b=$ 1.5 for three different values of taper constant $\beta$ and temperature gradient. $\alpha$ i.e., $\beta =\alpha =$ 0, 0.4, 0.8. From Table 4, one can easily get that frequency behaves same as in Table 3 (in all respect). But due to the implementation of other non homogeneity constant (exponential variation in Poisson’s ratio) the frequency for both modes is higher when compared to Table 3. Here the rate of decrement is same as in Table 3 because of circular variation in density parameter. Table 3Non homogeneity constant corresponding to density (m1) vs. frequency parameter (λ) for m2= 0, θ= 30° and a/b= 1.5 $\beta =\alpha =$ 0.0 $\beta =\alpha =$ 0.4 $\beta =\alpha =$ 0.8 ${\lambda }_{1}$ ${\lambda }_{2}$ ${\lambda }_{1}$ ${\lambda }_{2}$ ${\lambda }_{1}$ ${\lambda }_{2}$ 0.0 78.77 313.60 86.28 343.38 90.68 360.69 0.2 77.60 308.33 84.97 337.27 89.28 354.02 0.4 76.48 303.32 83.71 331.49 87.94 347.71 0.6 75.41 298.54 82.51 325.99 86.65 341.73 0.8 74.38 293.99 81.35 320.77 85.42 336.05 1.0 73.40 289.64 80.25 315.79 84.25 330.65 Table 5 accommodates the frequency modes corresponding to thermal gradient with fixed value of non homogeneity constant (corresponding to Poisson’s ratio ${m}_{2}=0$), skew angle $\theta =$ 30° and aspect ratio $a/b=$ 1.5 for three different values of non homogeneity constant ${m}_{1}$ (correspondint to density) and tapering parameter $\beta$ i.e., $\beta ={m}_{1}=$ 0.2, 0.4, 0.8. From Table 5, it is interesting to note that when the temperature gradient on the plate increases from 0 to 0.8, frequency modes decreases for the all the three values of non homogeneity constant ${m}_{1}$ and tapering paramenter $\beta$. Also, when the combined values of non homogeneity corresponding to density ${m}_{1}$ and tapering parameter $\beta$ increases from 0.2 to 0.8, the frequency modes also Table 4Non homogeneity constant corresponding to density (m1) vs. frequency parameter (λ) for m2= 0.6, θ= 30° and a/b= 1.5 $\beta =\alpha =$ 0.0 $\beta =\alpha =$ 0.4 $\beta =\alpha =$ 0.8 ${\lambda }_{1}$ ${\lambda }_{2}$ ${\lambda }_{1}$ ${\lambda }_{2}$ ${\lambda }_{1}$ ${\lambda }_{2}$ 0.0 83.44 332.14 91.76 365.31 96.71 385.31 0.2 82.20 326.55 90.36 358.82 95.21 378.17 0.4 81.01 321.24 89.03 352.66 93.78 371.43 0.6 79.88 316.18 87.75 346.81 92.41 365.04 0.8 78.79 311.36 86.52 341.25 91.10 358.97 1.0 77.75 306.75 85.35 335.95 89.85 353.20 Table 5Thermal gradient (α) vs. frequency parameter (λ) for m2= 0, θ= 30° and a/b= 1.5 ${m}_{1}=\beta =$ 0.2 ${m}_{1}=\beta =$ 0.4 ${m}_{1}=\beta =$ 0.8 ${\lambda }_{1}$ ${\lambda }_{2}$ ${\lambda }_{1}$ ${\lambda }_{2}$ ${\lambda }_{1}$ ${\lambda }_{2}$ 0.0 85.47 339.42 92.17 364.85 105.38 414.25 0.2 81.58 324.07 88.05 348.57 100.80 396.14 0.4 77.49 307.96 83.71 331.49 95.98 377.17 0.6 73.14 290.97 79.11 313.48 90.87 357.20 0.8 68.49 272.92 74.19 294.39 85.42 336.05 When we look at the Table 6, it tells how frequency modes behave for the fixed value of non homogeneity constant (Poisson’s ratio ${m}_{2}=$ 0), skew angle $\theta =$ 30°, aspect ratio $a/b=$ 1.5 and variable value of non homogeneity constant (density parameter ${m}_{1}=$ 0.2, 0.4, 0.8) and temperature gradient ($\alpha =$ 0.2, 0.4, 0.8) corresponding to tapering parameter $\beta$ of the plate. From the Table 6, we enlighten the fact that when the tapering parameter of the plate increases from 0 to 1, modes of frequency increases (due to linear variation in thickness). On the other hand, when the combined value of density parameter ${m}_{1}$ and temperature $\alpha$ increases from 0.2 to 0.8, modes of frequency decreases. In order to get good understanding of results and discussions (variation of plate’s parameter), graphical representation of Tables 1-6 presented in the form of Figs. 3-9. Table 6Taper constant (β) vs. frequency parameter (λ) for m2= 0, θ= 30° and a/b= 1.5 ${m}_{1}=\alpha =$ 0.2 ${m}_{1}=\alpha =$ 0.4 ${m}_{1}=\alpha =$ 0.8 ${\lambda }_{1}$ ${\lambda }_{2}$ ${\lambda }_{1}$ ${\lambda }_{2}$ ${\lambda }_{1}$ ${\lambda }_{2}$ 0.0 74.00 294.18 69.19 274.76 59.30 235.40 0.2 81.58 324.07 76.36 302.80 65.62 259.84 0.4 89.37 354.65 83.71 331.49 72.11 284.85 0.6 97.31 385.75 91.20 360.66 78.72 310.28 0.8 105.37 417.25 98.81 390.22 85.42 336.05 1.0 113.52 449.07 106.50 420.08 92.20 362.08 Fig. 3Non homogeneity constant (m2) vs. frequency (λ) for fixed m1= 0, θ= 30° and a/b= 1.5 Fig. 4Non homogeneity constant (m2) vs. frequency (λ) for fixed m1= 0.6, θ= 30° and a/b= 1.5 Fig. 5Non homogeneity constant (m1) vs. frequency (λ) for fixed m2= 0, θ= 30° and a/b= 1.5 Fig. 6Non homogeneity constant (m1) vs. frequency (λ) for fixed m2= 0.6, θ= 30° and a/b= 1.5 Fig. 7Thermal gradient (α) vs. frequency (λ) for fixed m2= 0, θ= 30° and a/b= 1.5 Fig. 8Taper constant (β) vs. frequency (λ) for fixed m2= 0, θ= 30° and a/b= 1.5 5. Conclusions This model display the effect of plate’s parameter on vibration of non homogeneous tapered parallelogram plate (with the help of Tables 1-6 and Figs. 3-8). With the help of this model, author drags the attentions of the readers on two important aspects. Firstly, effect of simultaneous variation of Poisson’s ratio and density parameter (as non homogeneity effect) to vibrational frequency (in Table 2 and Table 4). The frequency is less in Table 2 (due to circular variation in density parameter) as compared to Table 1 (density parameter is off). The frequency is high in case of Table 4 (due to exponential variation in Poisson’s ratio) when compared to Table 3. Secondly, effect of circular variation in density to vibrational frequency (in Table 3). The frequency is decreasing corresponding to density parameter (in Table 3). But frequency is increasing corresponding to Poisson’s ratio (in Table 1). The rate of decrement/increment in frequency is less in Table 3 when compared to Table 1. The author also provides effects of temperature (in Table 5) and thickness (in Table 6) to vibrational frequency. This paper gives good appropriate numerical data of frequency modes which is helpful for researchers and scientists, making good optimal structural designs. • Chakraverty S. Vibration of Plates. CRC Press, Boca Raton, 2008. • Chen C. C., Kitipornchai S., Lim C. W., Liew K. M. Free vibration of cantilevered symmetrically laminated thick trapezoidal plates. International Journal of Mechanical Sciences, Vol. 41, 1999, p. • Gupta A. K., Mamta Non-Linear thickness variation on the thermally induced vibration of a rectangular plate: a spline technique. International Journal of Acoustics and Vibration, Vol. 19, Issue 2, 2014, p. 131-136. • Gupta A. K., Sharma P. Vibration study of non-homogeneous trapezoidal plates of variable thickness under thermal gradient. Journal of Vibration and Control, Vol. 22, Issue 5, 2016, p. 1369-1379. • Kalita K., Shivakoti I., Ghadai R. K., Haldar S. Rotary inertial effect in isotropic plates part I: uniform thickness. Romanian Journal of Acoustics and Vibration, Vol. 13, Issue 2, 2016, p. • Kalita K., Shivakoti I., Ghadai R. K., Haldar S. Rotary inertial effect in isotropic plates part II: taper thickness. Romanian Journal of Acoustics and Vibration, Vol. 13, Issue 2, 2016, p. • Khanna A., Kaur N. Theoretical study on vibration of non-homogeneous tapered visco-elastic rectangular plate. Proceedings of the National Academy of Sciences, India Section A: Physical Sciences, Vol. 86, Issue 2, 2016, p. 259-266. • Khanna A., Kaur N. Effect of thermal gradient on vibration of non-uniform visco-elastic rectangular plate. Journal of The Institution of Engineers (India): Series C, Vol. 97, Issue 2, 2016, p. • Khanna A., Kaur N. Effect of structural parameter on vibration of non homogeneous visco elastic rectangular plate. Journal of Vibration Engineering and Technology, Vol. 4, Issue 5, 2016, p. • Leissa A. W. Vibration of Plates. NASA SP-160, 1969. • Leissa A. W., Chern T. Y. Approximate analysis of the forced vibration response of plates. Journal of Vibration and Acoustics, Vol. 114, Issue 1, 1992, p. 106-111. • Leissa A. W., Jaber N. A. Vibration of completely free triangular plates. International Journal of Mechanical Sciences, Vol. 34, Issue 8, 1992, p. 605-616. • Ratko M. Transverse vibration and instability of an eccentric rotating circular plate. Journal of Sound and Vibration, Vol. 280, 2005, p. 467-478. • Sharma A., Sharma A. K., Raghav A. K., Kumar V. Effect of vibration on orthotropic visco-elastic rectangular plate with two dimensional temperature and thickness variation. Indian Journal of Science and Technology, Vol. 9, Issue 2, 2016. • Sharma A., Sharma A. K., Raghav A. K., Kumar V. Vibrational study of square plate with thermal effect and circular variation in density. Romanian Journal of Acoustics and Vibration, Vol. 13, Issue 2, 2016, p. 146-152. • Sharma A. K., Sharma A., Raghav A. K., Kumar V. Analysis of free vibration of non-homogenous trapezoidal plate with 2D varying thickness and thermal effect. Journal of Measurements in Engineering, Vol. 4, Issue 4, 2016, p. 201-208. About this article Mechanical vibrations and applications parallelogram plate thermal induced circular variation Copyright © 2018 Amit Sharma, et al. 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Printable Calendars AT A GLANCE Algebra Worksheets Grade 9 Algebra Worksheets Grade 9 - Never runs out of questions. Plus every worksheet includes a free answer key. Define functions and evaluate points in tables, graphs, and contextual situations using function notation. Web children will practice their skill at solving rational expressions and equations with this printable. Web free printable math worksheets for 9th grade. Mixed problems on writing equations of lines. Web algebra i for beginners the ultimate step by step guide to acing algebra i. Therefore, 3x is the mother’s age. Our 9th grade math worksheets cover topics from. Explore our collection of free printable resources to enhance your lessons. Topics in this unit include: Discover a vast collection of free printable math worksheets by quizizz, tailored to enhance learning and understanding. Let x represent the daughter’s age. Adding and subtracting rational numbers. Web free printable math worksheets for 9th grade. 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Probabilistic Maps and Maximum Probability Maps BrainVoyager v23.0 Probabilistic Maps and Maximum Probability Maps When performing fMRI group studies, it is useful to investigate the spatial consistency of activity patterns or regions-of-interest across subjects. One useful approach to quantify consistency is offered by the calculation of probabilistic (functional) maps. At each spatial location, such maps represent the relative number of subjects leading to significant task activity or exhibit the same region-of-interest. In a study of 20 subjects, for example, a value of 60% in a probabilistic functional map would indicate that 12 subjects activated the respective brain region. It is evident that the calculation of probabilistic functional (or other) maps depends to some extent on the chosen brain normalization method since probabilities are determined by counting how many subjects activate at the "same" spatial location. For volumetric normalization schemes, spatial coordinates (e.g. in Talairach or MNI space) address corresponding regions in the brains of different subjects. For cortex-based normalization schemes, aligned surface points (mesh vertices) are used to address corresponding brain regions. Relative to a good macro-anatomical alignment, probabilistic functional maps may reveal the spatial consistency of areas across subjects. Probabilistic Functional Maps In BrainVoyager, probabilistic maps can be calculated using volumetric maps (VMPs) or surface maps (SMPs) to quantify how many subjects surpass a statistical threshold at corresponding locations. For each subject, the same type of information must be represented in the provided maps per subject; in the context of a GLM analysis, for example, the same contrast(s) must be calculated for each subject, which can be easily performed as described in topic Creating Multi-Subject T and Beta Maps from Mult-Subject GLMs. Since volume or surface maps can also be created using several other tools (e.g. Independent Component Analysis, Granger Causality Analysis, Cortical Thickness Analysis), probabilistic maps provide a general means to evaluate the spatial consistency of effects across subjects. When subjects have been co-registered using cortex-based alignment, the surface maps of participating subjects must be transformed in group-aligned space prior to calculation of probabilistic maps. This step is not necessary when calculating GLM-based probabilistic surface maps since the cortex-based GLM performs the cortex-based alignment step implicitly, but it is necessary, e.g., for cortical thickness maps. Probabilistic Maps of Regions-Of-Interest Probabilistic maps can also be calculated based on regions-of-interests (ROIs) specified for each subject in volume (VOIs) or surface (POIs) space. When ROIs have been specified on subject's cortex (SPH) meshes, the POIs must be transformed first into group-aligned space prior to calculation of probabilistic maps. If corresponding VOIs or POIs have been defined for each subject, resulting probabilistic maps are likely more focal and easier interpretable than probabilistic maps from volume or surface maps. On the other hand, whole-brain (whole-hemisphere for cortex meshes) probabilistic maps calculated on the basis of subject's (statistical) maps may help to reveal consistent regions or extended networks across subjects at unexpected locations. Visualizing Probabilistic Maps In order to properly represent probabilistic maps, the map type "PM" has been defined, which represents at each voxel or vertex a percent value in the range from 0 - 100. This map type is recognizable by the "PM" label in the displayed color bar as shown in the snapshot below. The threshold value can be changed as usual; if one wants, for example, to see only the locations in a probabilistic map where 50 or more percent of the subjects overlap, the Min threshold must be set accordingly in the Volume Maps or Surface Maps dialog. It is also possible to use the Increase Threshold and Decrease Threshold icons to modify the percent threshold. The information how many subjects were included in calculating a probabilistic map is provided in the DF 1 field of the Volume Maps and Surface Maps dialog (the degrees-of-freedom value is not used for any statistical calculation in case of a PM map). In order to see better transition zones in probabilistic maps, special look-up tables may be used; examples for a useful look-up-tables are provided in the "MapLUTs" folder (e.g. "ProbMap_Red.olt", "ProbMap_Green.olt", "ProbMap_Blue.olt"); each volume or surface map may have its own look-up-table (it is also possible to assign an individual color gradient for each map, which is interpolated between a specified "min" and "max" color). Maximum Probability Maps and Creation of Atlases Since version 20.6 of BrainVoyager maximum probability maps (MPMs) are calculated automatically when calculating probability maps for multiple ROIs (if not turned off). MPMs assign a single area label to a voxel or vertex corresponding to the ROI probability map with the hightest (maximum) probability value. In case that two or more probability maps have the same probability value at a voxel or vertex, a decision need to be made to which map the voxel or vertex belongs. To break such tie situations, the program inspects an increasing voxel/vertex neighborhood counting how often the respective ROIs are represented. While ties are less likely for statistical maps, they may occur frequently in case of VOI/POI based probability maps since only as many non-zero values can be calculated per voxel/vertex as subjects are available (prob[i] = n[i]/N[i]*100 with n indicating the number of overlapping subjects for a specific map i and N the number of all included subjects). Breaking ties by inspecting voxel or vertex neighborhoods extents the functionality provided by the "winner" map function available in the Combine Maps dialog available for volume and surface space. The calculation of MPMs is especially useful to define boundaries between multiple overlapping probability maps (see example for VOIs and POIs in topics below). Since computed MPMs integrate all multi-subject source probability maps in one visualization of non-overlapping regions, they form the basis for building and visualizing custom brain atlases. The following sections describe in more detail how to calculate: Copyright © 2023 Rainer Goebel. 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NHETC Seminar Rutgers University Department of Physics and Astronomy Title: The CKN Bound and Particle Physics Abstract: The holographic principle implies that quantum field theory overcounts the number of independent degrees of freedom in quantum gravity. An argument due to Cohen, Kaplan, and Nelson (CKN) suggests that the number of degrees of freedom well-described by QFT is even smaller than required by holographic bounds, and CKN interpreted this result as indicative of a correlation between the UV and IR cutoffs on QFT. We consider an alternative interpretation in which the QFT degrees of freedom are depleted as a function of scale, and we use a simple recipe to estimate the impact of depleted densities of states on precision observables. Although these observables are not sensitive to the level of depletion motivated by gravitational considerations, the phenomenological exercises also provide an interesting test of QFT that is independent of underlying quantum gravity assumptions. A depleted density of states can also render the QFT vacuum energy UV-insensitive, reconciling the success of QFT in describing ordinary particle physics processes and its apparent failure in predicting the cosmological constant. For help, please contact Webmaster.	{"url":"https://www.physics.rutgers.edu/het/video/2021/10_05_21.html","timestamp":"2024-11-07T20:22:22Z","content_type":"text/html","content_length":"2381","record_id":"<urn:uuid:db854ef4-bf86-4f2e-aa6f-5572493bbe62>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00219.warc.gz"}
20 is What Percent of 52? = 38.4615% [With 2 Solutions] Determining percentages is an important mathematical skill used in many real-world situations. In this article, we will explore the question “20 is what percent of 52?” step-by-step and discuss various methods to find the percentage. Utilizing the “X is What Percent of Y Calculator”,we can compute that 20 is 38.4615% of 52. Step-by-Step Solution: 20 is What Percent of 52? Now that we know we want to find 20 is what percent of 52? , let’s look at two different step-by-step methods we can use to solve this percentage problem. Each approach arrives at the same solution, but understanding different ways to think through the math builds flexibility and deeper learning. Method 1: Convert Ratio to Percentage Fractions consist of a numerator and denominator. Fractions consist of a numerator and denominator. The numerator represents the number of pieces being considered, while the denominator is the total number of pieces that make up the whole. “Percent” means “per hundred”. So percent is another way to express a fraction where the denominator is always 100. For example, the percentage 50% indicates we have 50 out of a total of 100 pieces. 50% can be rewritten as the fraction 50/100, since we have 50 parts out of 100 total parts. In summary, to convert a fraction to a percentage, we multiply the numerator and denominator by the same value so that the denominator becomes 100. Once we have a fraction with a denominator of 100, the numerator gives us the percentage amount. This allows any fraction to be converted to an equivalent percentage based on the concept of “out of 100”. Let’s apply this understanding to calculate “20 is what percent of 52?”. Step1: Write the Ratio as a Fraction First, we can express “20 is what percent of 52?” as the fraction 20/52. 20 of 52 can be written as: 20 / 52 Step 2: Find the Number to Multiply to Get a Denominator of 100 We want the denominator to be 100, so need to multiply by a number that makes 52 into 100 To find this number, divide 100 by 52: 100 /52 = 1.923076923 Step 3: Multiply the Fraction by the Number from Step 2 Multiply both the numerator and denominator by 1.923076923: 20/52 x 1.923076923/1.923076923 = 20 x 1.923076923/52 x 1.923076923 = 38.4615/100 Step 4: The Numerator is Now the Percentage Since our new denominator is 100, the numerator 38.4615 represents the percentage This step-by-step method of converting the ratio 20 : 52 to a fraction and then a percentage with a denominator of 100 allows us to calculate that 20 is 38.4615% of 52. Method 2: Converting Fractions to Decimals and Percents You can also use division to convert a fraction to a decimal and then to a percentage. Step 1: Write the Fraction Start with the original fraction: 20 of 52 can be written as: 20 / 52 Step 2: Divide the Numerator by the Denominator Use long division to divide the numerator by the denominator: 20 / 52 = 0.384615 Step 3: Multiply the Decimal by 100 To convert the decimal to a percentage, multiply it by 100: 0.384615 x 100 = 38.4615% Therefore, by dividing 20/52 to 0.384615 and then multiplying by 100, the result is 38.4615% Following this process carefully of dividing the fraction, multiplying the decimal by 100, and interpreting the result as a percentage provides us another method to find that 20 is 38.4615% of 52. Comparing the Two Methods Let’s summarize the two methods we used to find the percentage “20 is what percent of 52?”: Method Description Steps 1. Write ratio as fraction Convert Ratio to Percentage Converts the ratio to a fraction, then multiplies to make the denominator 100 2. Multiply to make denominator 100 3. Numerator is the percentage 1. Write the fraction Convert Fraction to Decimal and Percentage Divides the fraction to a decimal, then multiplies the decimal by 100 2. Divide numerator by denominator 3. Multiply decimal by 100 4. Result is the percentage The ratio to percentage method directly converts the ratio 20 : 52 to an equivalent percentage fraction. The fraction to decimal method indirectly converts the fraction 20/52 to a decimal and then a p>Both methods follow a clear step-by-step procedure to arrive at the same result that 20 is 38.4615% of 52. Understanding multiple approaches builds flexibility in thinking through percentage Standard Percentage of 52 Calculation Examples Now that we have calculated that 20 is 38.4615% of 52, let’s explore what 5%, 10%, up to 95% of 52 would be. 2.6 is 5% of 52 5.2 is 10% of 52 7.8 is 15% of 52 10.4 is 20% of 52 13 is 25% of 52 15.6 is 30% of 52 18.2 is 35% of 52 20.8 is 40% of 52 23.4 is 45% of 52 26 is 50% of 52 28.6 is 55% of 52 31.2 is 60% of 52 33.8 is 65% of 52 36.4 is 70% of 52 39 is 75% of 52 41.6 is 80% of 52 44.2 is 85% of 52 46.8 is 90% of 52 49.4 is 95% of 52 52 is 100% of 52 Finding percentages is an essential mathematical skill used in daily life for calculations of discounts, taxes, tips, interest rates, test scores, and more. In this article, we explored the specific question “20 is what percent of 52?” and discussed two methods to solve it – converting a ratio to a percentage and converting a fraction to a decimal and percentage. Both methods carefully followed arrive at the solution that 20 is 38.4615% of 52. We also looked at calculating a range of standard percentages from 5% to 95% of the total number 52. Practicing these types of percentage calculations helps improve mathematical proficiency and comfort applying different techniques.	{"url":"https://timehackhero.com/20-is-what-percent-of-52/","timestamp":"2024-11-05T19:23:47Z","content_type":"text/html","content_length":"174260","record_id":"<urn:uuid:37c2c45c-44b8-4a2a-b37d-023c427a455f>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00861.warc.gz"}
Free online calculators. Number theory. Collection of online calculators which will help you to solve mathematical tasks. With these calculators you can find the greatest common divisor (GCD) and the least common multiple (LCM) of two numbers and perform calculations in a column between whole numbers and decimals. Add the comment	{"url":"https://onlinemschool.com/math/assistance/number_theory/","timestamp":"2024-11-05T02:44:00Z","content_type":"text/html","content_length":"19102","record_id":"<urn:uuid:59b757ad-ef98-4884-9671-00f871de593a>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00622.warc.gz"}
CTAN Update: amscls Date: December 20, 2017 3:45:17 PM CET Barbara Beeton submitted an update to the amscls package. Version: 2.20.4 2017-10-31 License: lppl1.3c Summary description: AMS document classes for LaTeX Announcement text: Definition of \dh was changed to not assume amsfonts, saving two math families. The package’s Catalogue entry can be viewed at The package’s files themselves can be inspected at Thanks for the upload. For the CTAN Team Petra Rübe-Pugliese We are supported by the TeX users groups. Please join a users group; see amscls – AMS document classes for LaTeX This bundle contains three AMS classes, amsart (for writing articles for the AMS), amsbook (for books) and amsproc (for proceedings), together with some supporting material. This material forms one branch of what was originally the AMS-LaTeX distribution. The other branch, amsmath, is now maintained and distributed separately. The user documentation can be found in the package amscls-doc. Package amscls Version 2.20.6 2020-05-29 Copyright 2004, 2010, 2014–2020 American Mathematical Society Maintainer The American Mathematical Society	{"url":"https://www.ctan.org/ctan-ann/id/mailman.3874.1513781131.5216.ctan-ann@ctan.org","timestamp":"2024-11-14T02:18:10Z","content_type":"text/html","content_length":"15148","record_id":"<urn:uuid:df15cb9a-a87f-4b12-ba23-db4e289b91dd>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00445.warc.gz"}
Sensitivity Analysis in Financial Modeling: Understanding Impact and Risk 25.3.2 Sensitivity Analysis In the world of finance and investment, sensitivity analysis is a critical technique used to understand how changes in key input variables can affect financial outcomes. This section delves into the intricacies of sensitivity analysis, highlighting its purpose, process, applications, and limitations. By the end of this chapter, you will have a comprehensive understanding of how to apply sensitivity analysis in financial modeling to assess risk and uncertainty effectively. Understanding Sensitivity Analysis Sensitivity analysis is a method used to predict the outcome of a decision given a certain range of variables. It is a way to determine how different values of an independent variable will impact a particular dependent variable under a given set of assumptions. This technique is particularly useful in financial modeling, where it helps analysts and decision-makers understand the robustness of their models and identify which variables have the most significant effect on outcomes. Purpose of Sensitivity Analysis The primary purposes of sensitivity analysis in financial modeling include: • Assessing Impact: It helps in assessing the impact of changes in input variables on the output. By understanding how sensitive a model is to changes in assumptions, analysts can better predict potential outcomes. • Identifying Key Variables: Sensitivity analysis identifies which variables have the most significant effect on outcomes, allowing analysts to focus on these critical factors. • Evaluating Robustness: It evaluates the robustness of a financial model by testing how changes in assumptions affect the model’s predictions. The Process of Sensitivity Analysis The process of conducting a sensitivity analysis involves several key steps: 1. Identify Key Variables: The first step is to determine which inputs are uncertain or have the greatest potential impact on the model’s outcomes. Common variables include sales growth rate, discount rate, and cost of goods sold. 2. Define the Range of Variations: Once the key variables are identified, the next step is to decide on the range of values for each variable. This could be a percentage change, such as ±10% or 3. Recalculate Model Outputs: Adjust one variable at a time while keeping others constant to observe changes in outputs such as Net Present Value (NPV) or Internal Rate of Return (IRR). This helps in understanding the sensitivity of the model to each variable. Example of Sensitivity Analysis Consider a company evaluating a project with uncertain future sales growth. The sensitivity analysis might focus on the sales growth rate as a key variable. By varying the sales growth rate from 3% to 7%, the company can observe how the NPV of the project changes. If the NPV varies significantly with changes in sales growth, it indicates high sensitivity, suggesting that accurate estimation of sales growth is crucial for the project’s success. Utilizing Data Tables in Excel Excel provides powerful tools for conducting sensitivity analysis through data tables: • One-Variable Data Table: This tool shows how changing one input affects one or more outputs. It is useful for understanding the impact of a single variable on the model’s results. • Two-Variable Data Table: This tool analyzes the impact of two variables simultaneously on a single output. It provides a more comprehensive view of how multiple factors interact to affect Tornado charts are an effective way to visualize the results of a sensitivity analysis. These charts depict the relative impact of variables on the output by displaying the range of outcomes for each variable. The widest bars represent variables with the most significant effect, making it easy to identify which factors are most critical to the model’s success. graph TD; A[Key Variables] --> B[Sales Growth Rate]; A --> C[Discount Rate]; A --> D[Cost of Goods Sold]; B --> E[NPV]; C --> E; D --> E; E --> F[Tornado Chart]; Applications of Sensitivity Analysis Sensitivity analysis has several important applications in financial modeling: • Risk Assessment: By identifying critical variables that may pose risks if assumptions are incorrect, sensitivity analysis helps in assessing the risk associated with a financial model. • Decision Making: It aids in decision-making by focusing management attention on variables that require accurate estimation. This ensures that resources are allocated efficiently to address the most significant uncertainties. Limitations of Sensitivity Analysis While sensitivity analysis is a powerful tool, it has certain limitations: • Single Variable Focus: Sensitivity analysis examines changes in one variable at a time, which may not capture interactions between variables. This can lead to an incomplete understanding of the model’s dynamics. • Lack of Probability Distributions: Sensitivity analysis does not account for the probability distributions of variables, which can limit its ability to predict real-world outcomes accurately. Sensitivity analysis is a vital tool for understanding the dynamics of financial models. It helps in identifying key drivers and preparing for potential variations in outcomes. By assessing the impact of changes in input variables, sensitivity analysis provides valuable insights into the robustness of financial models and aids in effective risk management and decision-making. Quiz Time! 📚✨ Quiz Time! ✨📚 ### What is the primary purpose of sensitivity analysis in financial modeling? - [x] To assess the impact of changes in input variables on the output - [ ] To create complex financial models - [ ] To eliminate all risks in financial decisions - [ ] To predict stock market trends > Explanation: Sensitivity analysis is primarily used to assess how changes in input variables affect the output of a financial model, helping to identify key variables and evaluate model robustness. ### Which tool in Excel is used to show how changing one input affects one or more outputs? - [x] One-Variable Data Table - [ ] Two-Variable Data Table - [ ] Pivot Table - [ ] VLOOKUP > Explanation: A One-Variable Data Table in Excel is used to analyze how changes in a single input variable affect one or more outputs. ### What does a tornado chart depict in sensitivity analysis? - [x] The relative impact of variables on the output - [ ] The probability distribution of outcomes - [ ] The historical performance of a stock - [ ] The correlation between two variables > Explanation: Tornado charts depict the relative impact of different variables on the output by showing the range of outcomes for each variable, with the widest bars indicating the most significant effects. ### What is a limitation of sensitivity analysis? - [x] It examines changes in one variable at a time - [ ] It provides exact predictions of future outcomes - [ ] It eliminates all uncertainties in financial models - [ ] It requires complex software to perform > Explanation: A limitation of sensitivity analysis is that it examines changes in one variable at a time, which may not capture interactions between variables. ### In the process of sensitivity analysis, what is the first step? - [x] Identify Key Variables - [ ] Recalculate Model Outputs - [ ] Define the Range of Variations - [ ] Create a Tornado Chart > Explanation: The first step in sensitivity analysis is to identify key variables that are uncertain or have the greatest potential impact on the model's outcomes. ### How does sensitivity analysis aid in decision-making? - [x] By focusing management attention on critical variables - [ ] By eliminating all financial risks - [ ] By predicting future stock prices - [ ] By providing exact financial forecasts > Explanation: Sensitivity analysis aids in decision-making by focusing management attention on critical variables that require accurate estimation, ensuring efficient resource allocation. ### What is the role of a Two-Variable Data Table in Excel? - [x] To analyze the impact of two variables simultaneously on a single output - [ ] To create a visual representation of data - [ ] To calculate the average of a dataset - [ ] To sort data alphabetically > Explanation: A Two-Variable Data Table in Excel is used to analyze how two variables interact to affect a single output, providing a more comprehensive view of potential outcomes. ### Why is sensitivity analysis important for risk assessment? - [x] It identifies critical variables that may pose risks if assumptions are incorrect - [ ] It guarantees the accuracy of financial models - [ ] It predicts economic downturns - [ ] It eliminates the need for further analysis > Explanation: Sensitivity analysis is important for risk assessment because it identifies critical variables that may pose risks if assumptions are incorrect, allowing for better risk management. ### What does sensitivity analysis not account for? - [x] Probability distributions of variables - [ ] Changes in input variables - [ ] The impact of external factors - [ ] The robustness of financial models > Explanation: Sensitivity analysis does not account for the probability distributions of variables, which can limit its ability to predict real-world outcomes accurately. ### True or False: Sensitivity analysis can capture interactions between multiple variables simultaneously. - [ ] True - [x] False > Explanation: False. Sensitivity analysis typically examines changes in one variable at a time, which may not capture interactions between multiple variables simultaneously.	{"url":"https://csccourse.ca/25/3/2/","timestamp":"2024-11-15T03:01:34Z","content_type":"text/html","content_length":"92826","record_id":"<urn:uuid:ee341aa4-707b-44b3-a3a2-7ad08ce38b28>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00472.warc.gz"}
The Hertz-Mindlin with JKR V The Hertz-Mindlin with JKR Version 2 Model Particles may adhere together in a number of ways depending on the type of bond formed. For very small particles (smaller than 100 µm), van der Waals forces become significant and particles tend to stick to each other. The Johnson-Kendal-Roberts theory of adhesion (also known as the JKR Model) is typically used for calculating the contact forces acting on elastic and adhesive particles and assumes that the attractive forces are short range. To date, a large number of studies on many different particulate systems have been reported in the literature using the JKR model. However, most of them use a simplified version of the JKR model that depends on the surface energy of the particles involved in the contact and also for cases where materials of the same type are involved too. The Hertz-Mindlin with JKR Version 2 (JKR V2) model implemented in EDEM captures the behavior of multiple materials and uses a more accurate implementation of the JKR theory. It calculates the additional work required to break the contact (adhesion) after physical detachment of particles, thus making it applicable to contacts involving very small particles. The model is suitable for elastic and adhesive systems and the force-overlap response of the JKR model is shown in the following figure. It states that when two elastic and adhesive spheres approach each other, the force acting on the spheres is zero (from A to B). The DEM contact between these two spheres is established when they physically come into contact (B), and the normal contact force immediately drops to 8/9 f[c], where f[c] is the pull-off fore due to the presence of the van der Waals attractive forces (C). Upon loading the two spheres, the normal contact force follows the trend from C to D. During the recovery stage (unloading), the stored elastic energy is released and is converted into kinetic energy which causes the spheres to move in the opposite direction. All the work done during the loading stage is recovered when the contact overlap becomes zero (C). However, at this point, the spheres remain adhered to each other and further work is required to separate the two spheres (within the area highlighted in red). In order to break the contact, a minimum force equal to the pull-off force (E) is required and the contact breaks at F as shown. In order to account for the work of adhesion, the EDEM contact radius needs to be activated and set to be greater than the physical radius of the particle as: where r is the physical particle radius and α[f] is the relative approach where the contact breaks. The range of values where the EDEM contact radius can be considered valid should be defined prior to the simulation by using Equation (3) and (5): In EDEM simulations, in order to account for work of adhesion, it is important to increase the contact radius of the particle using EDEM Creator > Bulk Material > Particle. The contact radius being greater than the physical radius allows the influence of a negative overlap in the force calculations. Figure 1:Loading and Unloading Behavior of JKR Contact Model. The normal contact force (or adhesion force) in the JKR V2 model is defined as: where E, R, Γ and a are the relative elasticity, relative radius, interfacial surface energy (also known as work of adhesion) and contact radius (as described in Thornton, 2015), respectively. This is not the same as the EDEM contact radius. In this implementation of the model, the adhesive force depends on the interfacial surface energy and the relative approach (negative) at which the contact breaks. These are defined by the following generalized equations: here γ[1] and γ[2] are the surface energies of the two spheres and γ[1,2] is the interfacial surface energy. For the special case where two spheres of the same material come into contact, the interfacial surface energy is zero γ[1,2] = 0 , and the interfacial surface energy becomes Γ = 2γ. The relative approaching distance, α, (also known as contact overlap) and the pull-up force are defined by the following equations: where a is the normal overlap between particles. For contacts between spheres of the same material, then Γ = 2γ and so equations 1 and 4 can be rewritten as: The corresponding relative approach in the JKR model, a, is independent from the contact radius prescribed by you in the EDEM GUI. The latter is only used to activate the model and the JKR force will be calculated as described and applied for both positive and negative overlaps. (c) 2023 Altair Engineering Inc. All Rights Reserved.	{"url":"https://help.altair.com/2023/EDEM/Creator/Physics/Base_Models/Hertz-Mindlin_with_JKR_V2.htm","timestamp":"2024-11-13T01:39:39Z","content_type":"application/xhtml+xml","content_length":"13592","record_id":"<urn:uuid:f456985a-7bc4-47de-b452-fa8a6527e60f>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00352.warc.gz"}
Amps to VA calculator Amps (A) to volt-amps (VA) calculator and how to calculate. Enter phase number, the current in amps, the voltage in volts and press the Calculate button, to get the apparent power in volt-amps: Single phase amps to VA calculation formula The apparent power S in volt-amps is equal to current I in amps, times the voltage V in volts: S[(VA)] = I[(A)] × V[(V)] 3 phase amps to VA calculation formula The apparent power S in kilovolt-amps is equal to square root if 3 current I in amps, times the line to line voltage V[L-L] in volts: S[(VA)] = √3 × I[(A)] × V[L-L(V)] = 3 × I[(A)] × V[L-N(V)] See also	{"url":"https://jobsvacancy.in/calc/electric/amps-to-va-calculator.html","timestamp":"2024-11-07T05:26:28Z","content_type":"text/html","content_length":"9555","record_id":"<urn:uuid:86c70994-8ce5-4b51-bf6f-1fe68a5bb108>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00398.warc.gz"}
Alessio D'Alì: On the Koszul property for quadratic Artinian Gorenstein graded rings \| KTH Alessio D'Alì: On the Koszul property for quadratic Artinian Gorenstein graded rings Time: Mon 2022-05-02 15.00 - 16.00 Location: Zoom Video link: Meeting ID: 634 5223 2191 Participating: Alessio D'Alì (Osnabrück) Quadratic Artinian Gorenstein graded rings have been the subject of several recent papers in commutative algebra: in this talk I want to focus on two topics that involve such rings. The first goal is to introduce a combinatorial construction introduced by Gondim and Zappalà and developed in later independent work by myself and Lorenzo Venturello (KTH Stockholm). Such a construction takes a pure flag simplicial complex $\Delta$ and associates with it an Artinian Gorenstein ring $A_{\Delta}$; similarly to Stanley-Reisner theory, there exists a rich two-way vocabulary between the combinatorics of $\Delta$ and the algebraic properties of $A_{\Delta}$. In particular, $A_{\Delta}$ turns out to be Koszul (respectively, to admit a Gröbner basis of quadrics) if and only if \ (\Delta\) is Cohen-Macaulay (respectively, shellable). The second goal is to introduce a problem which is - to the best of my knowledge - still open. Conca, Rossi and Valla asked in 2001 whether every quadratic Artinian Gorenstein algebra of socle degree 3 has the Koszul property. Using Nagata idealization, in 2019 Mastroeni, Schenck and Stillman answered this question in the negative and constructed non-Koszul objects with codimension 9 and higher. In 2020, McCullough and Seceleanu further exhibited a non-Koszul object in codimension 8 and proved that this is the lowest achievable codimension when working with Nagata idealization. It is currently unknown whether Koszulness needs to hold in codimensions 6 and 7; I'd like to present some of the challenges that come with investigating such a question, and possibly to have some brainstorming with the audience.	{"url":"https://www.kth.se/math/kalender/alessio-d-ali-on-the-koszul-property-for-quadratic-artinian-gorenstein-graded-rings-1.1166109?date=2022-05-02&orgdate=2022-03-01&length=1&orglength=306","timestamp":"2024-11-12T10:58:38Z","content_type":"text/html","content_length":"57212","record_id":"<urn:uuid:59bbf562-1cf5-4841-a2f6-bb50561d1b9b>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00755.warc.gz"}
Vladimir Ivanovich Smirnov Born: 10 June 1887 in St Petersburg, Russia Died: 11 Feb 1974 in Leningrad, USSR Vladimir Smirnov attended the 2nd Gymnasium, the oldest secondary school in St Petersburg, and there he won the gold medal for mathematics. From school he entered the Physics and Mathematics Faculty of St Petersburg University. Smirnov had become friends with a number of outstanding mathematicians while at the 2nd Gymnasium. He was particularly friendly with Friedmann and Tamarkin, who were in the class below him at school. Valentina Doinikova, who was a friend of Friedmann, describes how the three went around together while undergraduates at St Petersburg University:- Friedmann, Tamarkin and Smirnov often came together, and they were called 'the boys from the second Gymnasium'. They were always smart and neatly dressed, and always called each other - in public - by their first name and patronymic. In 1910 Smirnov graduated from St Petersburg and remained at the University to study for the higher degrees which would allow him to become a university teacher. At the University a circle was formed in 1911 to study mathematical analysis and mechanics. Smirnov was a very active member of this circle, for example lecturing on the theory of algebraic equations, particularly the work of Goursat and Appell. In session 1911-12 he gave nine lectures on Goursat's books. Smirnov worked jointly with his friends from the 2nd Gymnasium. He published a joint paper with Friedmann in 1913 which was published in the Journal of the Russian Physico-Chemical Society (Physics Section). He wrote the first volume of his major five volume work A Course in Higher Mathematics jointly with Tamarkin. From 1912 Smirnov taught at the St Petersburg Institute of Railway Engineering. He taught at Simferopol University in the southern Ukraine from 1919 to 1922, then he returned to St Petersburg (by now Leningrad). Smirnov was awarded his doctorate in 1936 and he became head of the Institute of Mathematics and Mechanics. He became the head of the Mathematics School at the University of Leningrad and was elected to the USSR Academy of Sciences. In 1953 Smirnov organised the Leningrad Mathematical Seminar. To some extent this Seminar also filled the gap left when the Leningrad Mathematical Society disbanded due to political pressure in the late 1920s. Smirnov had been an active member of the Leningrad Mathematical Society through the 1920s and he was a strong believer in relaunching the Society. In 1959, mainly due to the efforts of Smirnov, it became possible to restart the Leningrad Mathematical Society and Smirnov was elected the honorary president of the Society. Smirnov's mathematical activity was in both pure and applied mathematics. He wrote the five volume work A Course in Higher Mathematics referred to above, which was widely used in Russia. He worked on conjugate functions in multidimensional euclidean space and the theory of functions of a complex variable. With Sobolev he devised a method for obtaining solutions on the propagation of waves in elastic media with plane boundaries. Other applied mathematical work resulted in him developing methods for studying oscillations of elastic spheres. In [1] the authors write:- ... V I Smirnov was not only an outstanding mathematician and a famous historian of science, but also a person of exceptional nobility, benevolence and culture. All these qualities left a lasting impression even on those who seldom had occasion to meet this remarkable man in person, still more on his pupils and associates. Their love and respect for their teacher's memory were reflected in a three-day scientific conference which was held in Leningrad in June 1987 and was dedicated to the centenary of the scientist's birth. List of References (18 books/articles) Article by: J.J. O'Connor and E.F. Robertson Source: MacTutor History of Mathematics archive	{"url":"http://www.mathsoc.spb.ru/pantheon/smirnov/JOC_EFR.html","timestamp":"2024-11-02T15:33:22Z","content_type":"text/html","content_length":"5686","record_id":"<urn:uuid:8e12dd3d-6cd7-4a6d-901a-fe2184df0f04>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00894.warc.gz"}
Random Number Generator Based on Discrete Cosine Transform Based Lossy Picture Compression Malatya Turgut Ozal University EriÅŸim HakkÄ± The The widespread use of digital data makes the security of this data important. Various cryptographic systems are used to ensure the security of this data. The most important part of these systems is random numbers. In this article, a random number generator based on the discrete cosine transform, which is the basis of image compression algorithms, is proposed. In this generator, the difference between the original image and the compressed image produced using the discrete cosine transform is used. The original picture is transferred to the frequency plane using the discrete cosine transform. It is then converted back to the space plane using the inverse discrete cosine transform. These transformations cause some losses as certain coefficients are taken into account. Raw random numbers were generated using the differences between the original image and the compressed image. Then, the possible weaknesses in the random numbers generated by passing these raw data through the hash function were fixed. The SHA-512 algorithm was used as the hash function. An important advantage of the developed system is that it can be easily produced using any digital data source. It has been shown by the analysis that the generated random numbers are safe. Anahtar Kelimeler Discrete cosine transform, Image compression, Random numbers, Hash functions YAKUT, S. Random Number Generator Based on Discrete Cosine Transform Based Lossy Picture Compression. NATURENGS, 2(2), 76-85.	{"url":"https://acikerisim.ozal.edu.tr/items/43a19411-8d7c-407a-8419-653df4f66a15","timestamp":"2024-11-07T19:07:03Z","content_type":"text/html","content_length":"481100","record_id":"<urn:uuid:5c9c90b7-30f4-4bfa-b022-4b1f4b68b414>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00400.warc.gz"}
Car driving performance \| Grease Monkey Car driving performance Car driving performance table of contents Car driving performance When a car runs, multiple resistances occur. The principal among them are rolling resistance, air resistance, grade resistance, and acceleration resistance. Rolling resistance is caused by the friction between the tires and the road surface, and air resistance is caused by the friction between the tires and the air as the vehicle moves. Gradient resistance is the resistance caused by gravity when climbing a slope, and acceleration resistance is the force required to increase speed. All of these resistance forces act in a direction that impedes running, and are collectively called running resistance. In contrast to running resistance, driving force is the force that allows a vehicle to move forward. This force is generated by the torque generated by the engine, and in order for the car to move forward, this driving force must exceed the running resistance. Specifically, the power generated by the engine is transmitted to the wheels via the gearbox to move the vehicle. The magnitude and efficiency of this driving force directly affects the vehicle’s acceleration performance and performance on slopes. • Driving performance curve diagram The driving performance curve diagram in Figure 1 is a visual representation of the driving performance of a car. This diagram shows the relationship between vehicle speed, driving force, and engine rotational speed in a diagram. This diagram provides a detailed understanding of the vehicle’s performance, including its acceleration ability, hill-climbing ability, and maximum speed. For example, it can read how much driving force is generated when the engine rotational speed is constant, or the efficiency of the engine at a specific speed. This provides important information for optimizing vehicle performance. Running resistance and driving force Figure 2 shows the relationship between the vehicle speed of a car traveling in top gear on a flat road, and the corresponding running resistance and driving force. When the vehicle speed in the diagram is point A, the driving force is located at point C and the running resistance is located at point B. At this time, the driving force C exceeds the running resistance B, and the difference becomes the margin driving force. This surplus driving force acts as an accelerating force when the vehicle travels on a flat road surface. Marginal driving force = Driving force C - Running resistance B When the vehicle travels uphill, this extra driving force acts as climbing force. However, as the slope of the slope increases, the slope resistance increases, and as a result, the margin driving force decreases. As the vehicle speed continues to increase, the driving force and running resistance become equal at point D in the diagram, and no further increase in speed occurs and the speed remains constant. The speed at this time is called the maximum speed. Maximum speed = (driving force = running resistance) From the above, the relationship between driving force, running resistance, and margin driving force is expressed by the following equation. Driving force = running resistance + margin driving force This formula is the basis for numerically understanding vehicle performance, and is an important index when evaluating vehicle acceleration performance and maximum speed. running resistance Running resistance is a general term for multiple resistances that a car must overcome when moving. Specifically, it consists of the following four elements. The resistance that tires receive from the road surface when a car is running Resistance received from airflow etc. that occurs when a car is running The resistance that a car experiences when going up or down a hill, etc. Various resistances experienced when the car accelerates by stepping on the accelerator These resistances are necessary for vehicles to run efficiently, and directly affect vehicle performance such as energy efficiency, fuel efficiency, acceleration performance, and top speed. When designing a vehicle, it is necessary to minimize these running resistances to maximize performance and reduce fuel consumption. rolling resistance Rolling resistance is the resistance force that occurs when a tire rolls on a road surface, and the main cause is energy loss due to contact between the tire and the road surface. It is affected by conditions such as tire material, air pressure, road surface quality, and tire temperature. The rolling resistance increases as the running speed increases, but the rate of increase becomes relatively small. This resistance is mainly caused by the following factors. • Deformation of tire contact area The contact surface of the tire when the car is stationary is called the tire contact area. This area changes depending on the shape and air pressure of the tire, and the area and shape of the contact with the ground also change depending on the quality and shape of the road surface. As the tire rotates, the contact area continuously deforms, and this deformation creates resistance. • Friction between tires and road surface When a car is running, the tread portion of the tire comes into contact with the road surface, and frictional force is generated at this contact portion. The amount of friction changes depending on the tire material, tread pattern, road surface condition, etc., and this frictional force causes rolling resistance. Friction also occurs in the bearing, which is the rotation axis of the tire. This is due to the bearings that connect the wheels and the car body, and the amount of resistance changes depending on the structure and lubrication condition of the bearings themselves. Friction in this area also becomes part of the rolling resistance. • Impact and deformation from the road surface Because the road surface is not perfectly flat, when driving over irregular shapes such as bumps and curves, a different resistance occurs than when driving in a normal straight line. This is because tires require additional energy to accommodate the deformation of the road surface. These factors act comprehensively as tire running resistance, and affect the fuel efficiency and power performance of automobiles. Therefore, it is important to minimize these resistances in vehicle design and maintenance. Rolling resistance is influenced by the weight of the car and road surface conditions, and there are various factors involved, but rolling resistance on a flat road is generally calculated using the following formula. $$ R1 = \mu Mg $$ $ R1 $：rolling resistance $N$ = $kg$・$m/s$$^2$ = $kg$・$m$・$s$$^{-2}$ $ \mu $：rolling resistance coefficient $ M $：Total vehicle mass（$kg$） $ g $：gravitational acceleration = 9.8 $m/s$$^2$ = 9.8$ m$・$s$$^{-2}$ Effects of road surface conditions The rolling resistance coefficient is shown in Table 1 below. What is the “coefficient” of rolling resistance coefficient? When a formula with one or more variables is used to explain the regularity of a certain phenomenon, a coefficient is a constant that is applied to the formula using those variables, and the constant changes depending on the situation. For example, the rolling resistance calculation formula ($R1 = \mu Mg$) above has only two variables: the required rolling resistance R1 and the total vehicle mass $M$. Among these variables, the rolling resistance $R1$ can be calculated by multiplying three elements: the coefficient (constant) $\mu$ which varies depending on the road surface conditions shown in Table 1, the constant gravitational acceleration$g$(which is 9.8$m/s^2$),and the variable total weight of the automobile $M$. Here, when considering determining the rolling resistance of the same car under various road conditions, the total vehicle weight M needs to be a constant value, so it can be treated as a constant instead of a variable. If you can select the road surface condition and determine the value of μ, which is the rolling resistance coefficient, then all you have to do is substitute each value into the formula $R1 = \mu Mg $, and the rolling resistance will inevitably be determined. determined. Effect of vehicle speed As shown in Figure 3, the rolling resistance coefficient tends to increase rapidly when the vehicle speed exceeds a certain level. This is due to the generation of standing waves in the tires. Standing wave is a phenomenon that occurs when a tire is continuously subjected to a load. Automobile tires always maintain a deflected state due to the load of the vehicle. Tire deflection will be greater if the tire pressure is incorrect or if the vehicle is overloaded and heavy. If the vehicle continues to drive at high speeds in this state, multiple deflections will overlap inside the tire, creating a standing wave. If this condition continues, it may cause serious trouble such as tire burst. However, even if the tire pressure is appropriate, standing waves can occur when driving at high speeds. This is because tires are not perfectly round, and even when they are in proper condition, they are slightly bent. Even if the driver is not aware of it, this deflection can cause standing waves. However, tires are designed and developed in such a way that they do not cause any trouble even when driving at a certain high speed, assuming appropriate air pressure and mounting conditions. Therefore, it is important to properly maintain and regularly check your tires, which can prevent problems caused by standing waves. Effect of tire deformation amount The higher the tire air pressure, the smaller the tire deformation, and therefore the smaller the rolling resistance coefficient. The smaller the tire’s aspect ratio, the smaller the tire deformation, and therefore the smaller the rolling resistance coefficient. Tire aspect ratio Tire aspect ratio(%) = (Tire section height / Section width) * 100 Simply put, the tire’s flatness is the flatness of the tire. In other words, when the tire is viewed from the side, the lower the height between the wheel end face and the tire end face, the lower the tire’s aspect ratio. Most tires have a wide tread (tread) and are made harder than normal tires, so there is little deformation of the tires even when driving at high speeds, and they also have good running stability. On the other hand, tires with high profile ratios have a long distance from the wheel edge to the tire edge, and are made softer to absorb shock. As a result, they tend to deform and increase rolling resistance when driving at high speeds, but they provide better ride comfort than tires with a lower profile ratio. Impact of slope On a slope as shown in Figure 6, the load perpendicular to the slope road surface ($Mng$), which is a component of the total vehicle mass * gravitational acceleration ($Mg$), decreases as the slope angle $\theta$ increases. This relationship is expressed by the following equation. $$Mng = Mg\cos\theta$$ Therefore, the rolling resistance of the slope $R1$ , $$R1 = \mu Mg \cos\theta$$ $R1$：Rolling resistance on slopes $N$=$kg$$m/s$$^2$ $\mu$：rolling resistance coefficient $M$：Total vehicle mass $g$：gravitational acceleration $9.8m/s$$^2$ $\theta$：slope angle air resistance Air resistance is the resistance that occurs when a car displaces air and changes depending on factors such as the shape of the car, the smoothness of its surface, and the speed of the car. As speed increases, the force of air resistance increases in proportion to the square of speed, so the faster you drive, the greater the effect. When the vehicle is traveling straight and there is no natural wind, drag (force opposite to the direction of travel), lift (force trying to rise), and lateral force (force perpendicular to the direction of travel) act on the vehicle body. As shown in Figure 7, when natural wind blows or the car turns, in addition to the forces in the three directions (drag, lift, and lateral force) shown in this diagram, yawing moment, rolling moment, and pitching moment are generated. As a result, the maneuverability and stability of the vehicle will be greatly affected. Drag, lift, and lateral forces are the three main aerodynamic forces that affect vehicle motion. Drag force is a force that acts rearward on the vehicle body. As the car moves forward, the air received at the front of the car body creates positive pressure, while the separation of air at the rear creates negative pressure. The combination of these positive and negative pressures acts as a force that pulls the vehicle backwards, that is, a drag force. This drag increases in proportion to vehicle speed, and has a direct impact on fuel efficiency and top speed. Lift is a force that acts perpendicularly upward on the vehicle body. The difference in pressure created by the difference in speed of air flowing between the top and bottom of the car body creates a force that lifts the car body. As the lift force increases, the ground contact force of the tire with the road surface decreases, thereby reducing steering stability. Lift is often a problem, especially when driving at high speeds, and in racing cars and high-performance cars, aerodynamic design is emphasized in order to suppress this lift. Lateral force is a force that acts in the lateral direction of the vehicle body. This force is mainly caused by crosswinds and increases in proportion to changes in wind direction (yaw angle). Lateral forces can cause the vehicle to skid or change its direction of travel, leading to unexpected behavior for the driver. The effect of crosswinds is particularly large on expressways, and disturbances in steering due to lateral forces have a direct impact on driving safety. Air resistance is proportional to the front projected area of the vehicle and the square of the airspeed, and is expressed by the following formula. $$R2 = \frac{1}{2} Cd Av^2\rho$$ $R2$：air resistance $N$ = $kg$・$m/s$$^2$ $Cd$：air resistance coefficient $A$：Full projected area $m$$^2$ $v$：airspeed $m/s$ $\rho$：air density $kg/m$$^3$ Note that the air resistance coefficient varies depending on the shape of the vehicle body (Table 2) Note 1) Airspeed: The composite speed of the vehicle speed and wind speed. Note 2) Air density is standard standby condition（1 atm/15℃）, 1.225kg/m$^2$ gradient resistance Gradient resistance is the resistance that occurs when climbing a slope, and is related to the magnitude of the slope and gravity. This resistance is directly proportional to the slope angle and independent of vehicle speed. The steeper the slope, the greater the grade resistance. When a car goes up a hill, the force of the car’s weight acts in the opposite direction to the driving force. This is called gradient resistance. As shown in Figure 8, when a car with a weight ($Mg$) climbs a slope with a slope angle $\theta$, the component force of gravity ($R3$) is the slope resistance and is expressed by the following formula. $$R3 = Mg・sin\theta$$ $R3$：gradient resistance $N$ = $kg$・$m/s$$^2$ $M$：Total vehicle mass $kg$ $g$：gravitational acceleration = 9.8$m/s$$^2$ $\theta$：slope angle The slope of a road is generally expressed as $\tan\theta$, so to calculate slope resistance, it is necessary to find $sin\theta$ from $tan\theta$. However, when $\theta$ is small, there is no practical problem in approximately assuming that $sin\theta ≒ tan\theta$. For example, when calculating the slope resistance when a car with a total mass of 1225 kg goes up a 10% slope. Considering $\tan\theta ≒ \sin\theta$, Since $\tan\theta = 0.1$, $\sin\theta = 0.1$ From the gradient resistance $R3 = Mg \cdot \sin\theta$ $R3 = 1225 \, \text{kg} \times 9.8 \, \text{m/s}^2 \times 0.1 = 1200.5 \, \text{N}$ Therefore $R3 = 1200.5 \, \text{N}$ From the above, it follows that the car is being subjected to the same gradient resistance that is being pulled backwards by a force of approximately 1200N. This is because the work of lifting the mass of the vehicle to a high position is added as the vehicle travels. The slope resistance is a constant value regardless of the vehicle speed, but as the slope changes, the resistance also changes, resulting in the characteristics shown in FIG. 9. Note that in the case of a downhill slope, the slope resistance becomes negative and reduces the running resistance, so it acts as a force to assist the driving force. acceleration resistance The resistance that occurs when a car accelerates is called acceleration resistance, which is the resistance that occurs against the force that attempts to accelerate due to the law of inertia. Acceleration resistance is the additional force required when a car accelerates, and force is needed to overcome inertia in order for a car to change speed. The magnitude of this resistance is determined by the degree to which the accelerator is depressed and the acceleration of the vehicle, and is also influenced by the driver’s driving technique. The more rapid the acceleration, the greater the acceleration resistance. Therefore, this acceleration resistance acts in the direction of decelerating the accelerated automobile. Acceleration resistance is expressed by the following formula. $$R4 = M\alpha + M’\alpha = (M + M’)\alpha$$ $R4$：acceleration resistance $\text{N} = \text{kg} \cdot \text{m/s}^2 = \text{kg} \cdot \text{m} \cdot \text{s}^{-2}$ $\alpha$：acceleration $\text{m/s}^2$ $M$：Total vehicle mass $\text{kg}$ $M’$：Inertial mass equivalent to rotating part $\text{kg}$ Driving power and driving performance If the driving force can be increased, the ability to climb hills and accelerate will increase, and the maximum speed will also increase. The driving force can be increased by increasing the engine torque or the reduction ratio of the power transmission, or by decreasing the diameter of the drive wheels. However, if the driving force exceeds the frictional force between the road surface and the tires, the vehicle will spin, and power cannot be transmitted effectively. Therefore, the maximum limit of driving force can be expressed by the following formula. $$F0 = \mu Mg$$ $F0$：Maximum limit of driving force $\text{N} = \text{kg} \cdot \text{m/s}^2 = \text{kg} \cdot \text{m} \cdot \text{s}^{-2}$ $\mu$：Coefficient of friction between the road surface and tires $M$：Total mass on drive wheels $\text{kg}$ $g$：gravitational acceleration $9.8 \, \text{m/s}^2 = 9.8 \, \text{m} \cdot \text{s}^{-2}$ The coefficient of friction ($\mu$) changes depending on the road surface and tire conditions. (Table 3) There is a limit to the driving force for each drive wheel. Four-wheel drive (4WD) systems are advantageous in maximizing driving performance such as hill-climbing ability, acceleration ability, and maximum speed, but there are also ways to reduce running resistance by improving the shape and materials of the vehicle. It will be effective to minimize it. Driving performance Driving performance curve diagram FIG. 11 shows a driving performance curve diagram of an automobile having the performance shown in the specification table of Table 4. The horizontal axis is the vehicle speed ($km/h$), and the vertical axis is the driving force and running resistance ($kN$) and engine rotation speed ($\text{min}^{-1}$). It shows. The elements shown in FIG. 11 are as follows. 1. A~E and R are the driving forces at vehicle speeds from 1st to 5th shift and reverse, respectively. 2. 0~55% is running resistance at vehicle speed on uphill slopes 3. a~e and r are engine rotational speeds at vehicle speed • Relationship with engine performance curve FIG. 12 is an engine performance curve diagram of the automobile shown in the driving performance curve diagram of FIG. 11. From Figure 12, it can be seen that this engine, with a rotational speed of 1000~6500$\text{min}^{-1}$, outputs a shaft torque of approximately 90 to 120$N・m$ in this range. Generally, the following relationship exists between engine torque and vehicle driving force. $$F = \frac{Te \cdot i \cdot \eta}{r}$$ $F$：driving force $\text{N}$ $Te$：engine torque $\text{N} \cdot \text{m}$ $i$：Total reduction ratio $\eta$：Power transmission efficiency $r$：Dynamic load radius of drive wheel tires $\text{m}$ From this equation, it can be seen that the driving force of an automobile is proportional to the engine torque, total reduction ratio, and power transmission efficiency, and inversely proportional to the dynamic load radius of the drive wheel tires. Further, there is the following relationship between the engine rotation speed and the vehicle speed. The distance the car travels during one revolution of the drive wheels is $2\pi r$ [$m$], where $r$ [$m$] is the tire’s dynamic load radius. In other words, if the drive wheel rotates $Nr$ [$\text{min}^{-1}$], the distance traveled in one hour will be $Nr \cdot 60$ [$\text{h}^{-1}$] $\cdot 2\pi r$ [$m$]. Therefore, to find vehicle speed （$v$） by $km/h$ (conversion), divide by $10^3$ [$m$] $\cdot$ [$\text{km}^{-1}$]. $v = \frac{Nr \cdot 60 \cdot 2\pi r}{10^3}$ [$km/h$] Therefore $v = \frac{Nr \cdot 60 \cdot 2\pi r}{10^3}$ [$km/h$] —➀ Also, $Nr = \frac{Ne}{i}$ (engine speed) , the relationship between engine rotation speed and vehicle speed is Substituting $Nr = \frac{Ne}{i}$ for $v = \frac{Nr \cdot 60 \cdot 2\pi r}{10^3}$ [$km/h$], we get $v = \frac{Ne \cdot 60 \cdot 2\pi r}{10^3 \cdot i}$ [$km/h$] Therefore $v = \frac{Ne \cdot 60 \cdot 2\pi r}{10^3 \cdot i}$ [$km/h$] —➁ By applying the specification table in Table 4 and the engine rotation speed and torque values in FIG. 12 to these two equations ➀ and ➁, it is possible to calculate the relationship between vehicle speed, driving force, and engine rotation speed. Furthermore, when looking at the relationship between driving force, vehicle speed, and output, the following equation holds true. $$P = Fv$$ $P$：output $\text{W} = \text{kg} \cdot \text{m}^2/\text{s}^3 = \text{kg} \cdot \text{m}^{-2} \cdot \text{s}^{-3}$ $F$：driving force $\text{N} = \text{kg} \cdot \text{m/s}^2 = \text{kg} \cdot \text{m} \cdot \text{s}^{-2}$ $v$：Vehicle speed $\text{m/s} = \text{m} \cdot \text{s}^{-1}$（For km/h display, use $\frac{1}{3.6} \cdot v$ [$\text{m} \cdot \text{s}^{-1}$]） The driving force curve, as shown in A (1st speed) in Fig. 13, has a large driving force in 1st speed, but the speed range is narrow and is biased to the left. Also, the driving force curve gradually becomes lower and wider from A to E. Note that in the case of a continuously variable transmission such as a CVT, the transmission will be as shown in the X curve in the figure. In the driving performance curve diagram of FIG. 11, each line from 0 to 55% is a curve showing uphill driving performance, and % represents the magnitude of the slope. For example, 20% indicates the slope of $tan$$\theta$=0.2（$\theta$ is slope angle）, and the 0% line is a curve that indicates running resistance on a flat road surface, which is the sum of rolling resistance and air resistance. maximum speed The maximum speed of an automobile generally refers to the intersection of the driving force curve and the running resistance curve when driving on a flat road surface in the gear with the smallest transmission ratio. Therefore, it can be seen from the intersection of the 5th speed driving force curve and the 0% running resistance curve that the maximum speed of a car with the performance shown in the running performance curve diagram of FIG. 11 is approximately 165 km/h. acceleration performance In FIG. 11, considering the case of running on a flat road, the driving force is greater than the running resistance below the maximum speed, so the difference is used for acceleration as extra driving force. The extra driving force is greatest in 1st gear and decreases in the order of 2nd, 3rd, 4th, and 5th gears, so if you shift appropriately from 1st to 5th gear, you can obtain a large acceleration. The acceleration when the vehicle is traveling at a given speed can be calculated from the traveling performance curve diagram. The relationship between acceleration ($\alpha$) and acceleration force ($F$) is $F = (M + M’)\alpha \, [\text{N}]$ or $\alpha = \frac{F}{M + M’} \, [\text{m/s}^2]$ For example, in Figure 11, when driving at 70 km/h on a flat road in 3rd gear, the extra driving force is approximately 1.7 kN. From Table 4, the total mass of the car is 1225 kg, so the acceleration can be calculated as follows. car mass = $1225 \, [\text{kg}] – 55 \, [\text{kg/person}] \times 5 \, [\text{person}] = 950 \, [\text{kg}]$ （Total vehicle mass – Vehicle mass） If $M’$ is 5% of the vehicle mass, then $M’ = 950 \, [\text{kg}] \times 0.05 = 47.5 \, [\text{kg}]$ from $ \alpha = \frac{F}{M + M’} \, [\text{m/s}^2] $ acceleration $ \alpha = \frac{1.7 \times 10^3 \, [\text{kg} \cdot \text{m} \cdot \text{s}^{-2}]}{1225 \, [\text{kg}] + 47.5 \, [\text{kg}]} = \frac{1700 \, [\text{kg} \cdot \text{m} \cdot \text{s}^ {-2}]}{1272.5 \, [\text{kg}]} \approx 1.34 \, [\text{m/s}^2]$ $ \alpha = 1.34 \, [\text{m/s}^2] $ Therefore, it can be seen that in order to improve acceleration performance, it is necessary to increase the margin driving force (increase acceleration) and reduce the total mass of the vehicle (reduce weight). Climbing ability When going up a slope at a constant speed Driving force – (rolling resistance + air resistance) = $F$ This force ($F$) is all spent on gradient resistance, so $F = Mg\sin\theta$ From this formula, the slope angle $\theta$ of the slope that can be climbed can be determined. Furthermore, in the running performance curve diagram, the speed at which the vehicle can climb the gradient at each shift stage can be determined from the intersection of the running resistance curve for each gradient and the driving force curve at each shift stage.	{"url":"https://katuhito.info/en/driving_performance_en/","timestamp":"2024-11-14T23:36:20Z","content_type":"text/html","content_length":"123911","record_id":"<urn:uuid:64c5663d-2b0f-454e-9e2a-054134f8285f>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00431.warc.gz"}
Can Pi be trademarked? Intellectual property law is complex and varies from jurisdiction to jurisdiction, but, roughly speaking, creative works can be copyrighted, while inventions and processes can be patented. In each case the intention is to protect the value of the owner’s work or possession. For the most part mathematics is excluded by the Berne convention of the World Intellectual Property Organization WIPO. An unusual exception was the successful patenting of Gray codes in 1953. More usual was the carefully timed Pi Day 2012 dismissal by a US judge of a copyright infringement suit regarding Pi, since “Pi is a non-copyrightable fact.” Pi period In January 2014, the U.S. Patent and Trademark Office granted Brooklyn artist Paul Ingrisano a trademark on his design consisting of the Greek letter Pi, followed by a period. It should be noted here that there is nothing stylistic or in any way particular in Ingrisano’s trademark — it is simply a standard Greek Pi letter, followed by a period. That’s it — Pi period. No one doubts the enormous value of Apple’s partly-eaten apple or the MacDonald’s arch. But Pi.? We live in an era of aggressive patent trolling by vulture patent firms. There is vast amount at stake. Think of the current smartphone patent wars and the sometimes successful patenting of life forms. Additionally, it is often cheaper for a firm to pay than go to court. A vague patent can be a “nice little earner,” and thus even established firms such as Microsoft and Apple go patent trolling. Other firms are more willing to “open source” some of their intellectual property, such as, for example, Tesla’s announcement that it will open some of its patents to help spur the electric automobile industry. What happened next ? To underscore that he means business, Ingrisano, through his lawyer Ronald Millet, sent a letter to Zazzle.com, a Pi novelty company, to: “Immediately cease and desist” their “unlawful” usage of their Pi trademark or “any confusingly similar trademark,” and, within 14 days, 1. Provide an accounting of all sales of any products containing their trademark. 2. Provide an inventory of all relevant products. 3. Disclose any other uses, electronic or print, that have been made of the trademark. 4. Provide an account of the date when the Pi trademark was first incorporated into their products, a list of all known links to Zazzle’s webpage, and a list of third parties who offer such The letter threatened attorney’s fees and “treble money damages.” The full text of the letter is available here. Implied in the letter is the plaintiff’s position that “any confusingly similar trademark” includes the Pi symbol itself, without the period, since none of the products offered by Zazzle features a Pi followed by a period. Indeed, according to a report in Wired, Ingrisano’s attorney Millet has asserted that many items for sale by Zazzle “clearly have a pi sign and look similar enough that folks out there might confuse it with products that my client also sells.” Zazzle responded by temporarily banning all garments featuring the Pi symbol, which involved “thousands of products,” according to the Wired report. But two days later, after being flooded with complaints, Zazzle restored the products. Millet is consulting with Ingrisano as to their next step. Along this line, it is amusing to note that a Pi design is featured by the Mathematical Association of America as a finalist for its 2014 T-Shirt Contest (and the design includes one formula that one of the present authors was instrumental in discovering). Will MAA be challenged as well? The smiley face This spat is reminiscent of a dispute over the “smiley face” between litigants Wal-Mart Stores and SmileyWorld, a London-based company that registered rights to the smiley face many years ago on behalf of Franklin Loufrani. The dispute was finally settled in June 2011, under undisclosed (but likely quite expensive) terms. Unlike the Pi case, no one has argued that the smiley face has scientific significance! But the case does demonstrate that such disputes must be taken seriously. Moreover, the smiley face is a defined and recognisable image and Loufrani explicitly makes no attempt to stop the use of it in email as plain text, such as :). Pi in modern mathematics and science The Pi. trademark, and the aggressive actions taken by the trademark holder, may seem amusing, and are certainly unfortunate for Zazzle and its owners, employees and customers. But much more is at stake here. If Ingrisano and his attorney prevail in their legal actions, this would mean, in effect, that anyone who uses the Pi symbol in any context would live under the threat that they might receive a similar “cease and desist” letter, with the threats of significant financial loss. This would be an unmitigated disaster for modern mathematics and science. It is not the slightest bit an exaggeration to say that Pi is the most important irrational constant of modern mathematics. Each year, the Pi symbol appears in thousand of published books and in tens (possibly hundreds) of thousands of technical papers, not just in books and papers related to geometry but also in fields as diverse as statistics and quantum physics. In fact, the numerical value of Pi (expressed in binary digits) is contained in every smart phone ever produced, since the computations performed to process wireless signals (using the “fast Fourier transform”) inherently involve Pi. Pi appears in several guises in the equations of quantum physics, and thus is central to semiconductor electronics. Pi even arises in GPS technology, since the frequency of clock signals broadcast by GPS satellites must be adjusted according to the formulas of Einstein’s general relativity, the equations of which involve Pi. The mind reels at the thought that the authors of every mathematical, scientific or engineering paper that uses a Pi symbol must live under a cloud of worry that they too might be accused of “trademark violation” by including Pi symbols in their articles. And can we really not put Pi on our posters and Tee-shirts? Also, if Pi is placed under a cloud of trademark violation, what is next? The letter e, the base of natural logarithms, which is almost as ubiquitous as Pi? The “sigma” summation sign (another Greek letter)? The integral sign? What to do Unlikely? Perhaps. But to even approach such a path, to place even a glimmer of doubt or worry into the workings and communications of modern science would be an unmitigated disaster. No precedent even remotely approaching this scenario must be tolerated by the scientific community. Must the American Mathematical Society or the International Mathematical Union trademark all mathematical symbols — including 5! — as logos and release them under a general public license? No, the best solution is simply to rescind the Pi. trademark and to block any future attempts at trademarking mathematical symbols. There definitely are precedents for such action, including the June 2014 action by the U.S. Patent and Trademark Office to cancel the Washington Redskins’ registration of their image, which was ruled as disparaging to Native Americans. Surely the needs of the worldwide mathematical and scientific community to use standard notation free from trademark worries is an equally compelling justification.	{"url":"https://experimentalmath.info/blog/2014/06/can-pi-be-trademarked-2/","timestamp":"2024-11-07T05:35:29Z","content_type":"application/xhtml+xml","content_length":"76356","record_id":"<urn:uuid:26b2c41e-c017-412e-9075-d2fa7d648870>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00826.warc.gz"}
CDS & AFCAT 2 2024 Exam Maths SDT Boat & Stream Class 2 Preparing for competitive exams such as the Combined Defence Services (CDS) and Air Force Common Admission Test (AFCAT) involves mastering various mathematical concepts. One of the crucial areas tested in these exams is Speed, Distance, and Time. This blog will provide a comprehensive guide on this topic, covering key sub-topics such as time to pass stationary and moving objects, boat and stream problems, and races. Additionally, we’ll highlight the importance of practicing multiple-choice questions (MCQs) based on these concepts. Time to Pass Stationary and Moving Objects When calculating the time it takes for one object to pass another, different approaches are used based on whether the object is stationary or moving: 1. Passing a Stationary Object: When an object passes a stationary object, such as a train passing a pole, the time taken is based on the length of the moving object and its speed. 2. Passing a Moving Object: When an object passes another moving object, the relative speed between the two objects must be considered. The calculation differs depending on whether the objects move in the same direction or in opposite directions. Boat and Stream Problems Problems involving boats and streams are common in competitive exams. These problems typically involve calculating the effective speed of a boat moving upstream or downstream: 1. Upstream: When a boat moves against the current, the effective speed is reduced. 2. Downstream: When a boat moves with the current, the effective speed increases. Race problems involve comparing the speeds and times of different competitors over a certain distance. Key concepts include: 1. Lead or Lag: Understanding how much one competitor leads or lags behind another. 2. Catch-up Time: Calculating the time required for a slower competitor to catch up with a faster one. Practical Application through MCQs To excel in the CDS and AFCAT exams, practicing multiple-choice questions (MCQs) on these topics is crucial. Let’s explore how these concepts are applied through examples: Example 1: Time to Pass a Stationary Object Question: A train 200 meters long passes a pole in 20 seconds. What is the speed of the train? • Calculate the speed using the formula: Speed = Distance / Time. • Substitute the values: Speed = 200 meters / 20 seconds = 10 m/s. So, the speed of the train is 10 m/s. Example 2: Time to Pass a Moving Object Question: Two trains, each 150 meters long, are moving in the same direction at speeds of 40 km/h and 60 km/h. How long will it take for the faster train to pass the slower one? • Calculate the relative speed: 60 km/h – 40 km/h = 20 km/h. • Convert the relative speed to meters per second: 20 * (5/18) = 5.56 m/s. • Calculate the total distance to be covered: 150 meters + 150 meters = 300 meters. • Calculate the time: Time = Distance / Relative Speed = 300 meters / 5.56 m/s â‰ˆ 54 seconds. So, it will take approximately 54 seconds for the faster train to pass the slower one. Example 3: Boat and Stream Question: A boat’s speed in still water is 15 km/h, and the speed of the stream is 5 km/h. What is the boat’s effective speed downstream? • Calculate the effective speed downstream: Boat speed + Stream speed = 15 km/h + 5 km/h = 20 km/h. So, the boat’s effective speed downstream is 20 km/h. Example 4: Race Question: Runner A can complete a race in 30 minutes, while Runner B takes 45 minutes. If Runner A gives Runner B a head start of 5 minutes, who will win the race and by how much time? • Calculate the speed of each runner based on the distance and time. • Determine the effective time difference given the head start. By carefully analyzing these factors, you can determine the outcome of the race. Strategies for Solving Speed, Distance, and Time Problems 1. Understand the Problem: Carefully read the question to identify whether it involves calculating speed, distance, or time. 2. Identify the Units: Ensure all measurements are in consistent units. Convert units if necessary. 3. Apply the Formulas: Use the relevant formulas to find the required solution. 4. Check the Options: In MCQs, checking the provided options can help narrow down the correct answer quickly. 5. Practice Regularly: Regular practice of different types of problems will help in understanding patterns and frequently asked questions. Mastering Speed, Distance, and Time concepts, including the time to pass stationary and moving objects, boat and stream problems, and races, is crucial for success in the CDS and AFCAT exams. By understanding the fundamentals, practicing regularly, and using strategic approaches to problem-solving, you can enhance your proficiency in this area. Each practice session brings you closer to achieving your goal. Stay focused, practice diligently, and approach each problem with a clear, analytical mind. Good luck! Leave Your Comment	{"url":"https://ssbcrackexams.com/cds-afcat-2-2024-exam-maths-sdt-boat-stream-class-2/","timestamp":"2024-11-02T17:35:53Z","content_type":"text/html","content_length":"340362","record_id":"<urn:uuid:778f550a-1759-4088-835c-e130efe491eb>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00585.warc.gz"}
Visualization for Function Optimization in Python - MachineLearningMastery.comVisualization for Function Optimization in Python - MachineLearningMastery.com Function optimization involves finding the input that results in the optimal value from an objective function. Optimization algorithms navigate the search space of input variables in order to locate the optima, and both the shape of the objective function and behavior of the algorithm in the search space are opaque on real-world problems. As such, it is common to study optimization algorithms using simple low-dimensional functions that can be easily visualized directly. Additionally, the samples in the input space of these simple functions made by an optimization algorithm can be visualized with their appropriate context. Visualization of lower-dimensional functions and algorithm behavior on those functions can help to develop the intuitions that can carry over to more complex higher-dimensional function optimization problems later. In this tutorial, you will discover how to create visualizations for function optimization in Python. After completing this tutorial, you will know: • Visualization is an important tool when studying function optimization algorithms. • How to visualize one-dimensional functions and samples using line plots. • How to visualize two-dimensional functions and samples using contour and surface plots. Kick-start your project with my new book Optimization for Machine Learning, including step-by-step tutorials and the Python source code files for all examples. Let’s get started. Tutorial Overview This tutorial is divided into three parts; they are: 1. Visualization for Function Optimization 2. Visualize 1D Function Optimization 1. Test Function 2. Sample Test Function 3. Line Plot of Test Function 4. Scatter Plot of Test Function 5. Line Plot with Marked Optima 6. Line Plot with Samples 3. Visualize 2D Function Optimization 1. Test Function 2. Sample Test Function 3. Contour Plot of Test Function 4. Filled Contour Plot of Test Function 5. Filled Contour Plot of Test Function with Samples 6. Surface Plot of Test Function Visualization for Function Optimization Function optimization is a field of mathematics concerned with finding the inputs to a function that result in the optimal output for the function, typically a minimum or maximum value. Optimization may be straightforward for simple differential functions where the solution can be calculated analytically. However, most functions we’re interested in solving in applied machine learning may or may not be well behaved and may be complex, nonlinear, multivariate, and non-differentiable. As such, it is important to have an understanding of a wide range of different algorithms that can be used to address function optimization problems. An important aspect of studying function optimization is understanding the objective function that is being optimized and understanding the behavior of an optimization algorithm over time. Visualization plays an important role when getting started with function optimization. We can select simple and well-understood test functions to study optimization algorithms. These simple functions can be plotted to understand the relationship between the input to the objective function and the output of the objective function and highlighting hills, valleys, and optima. In addition, the samples selected from the search space by an optimization algorithm can also be plotted on top of plots of the objective function. These plots of algorithm behavior can provide insight and intuition into how specific optimization algorithms work and navigate a search space that can generalize to new problems in the future. Typically, one-dimensional or two-dimensional functions are chosen to study optimization algorithms as they are easy to visualize using standard plots, like line plots and surface plots. We will explore both in this tutorial. First, let’s explore how we might visualize a one-dimensional function optimization. Want to Get Started With Optimization Algorithms? Take my free 7-day email crash course now (with sample code). Click to sign-up and also get a free PDF Ebook version of the course. Visualize 1D Function Optimization A one-dimensional function takes a single input variable and outputs the evaluation of that input variable. Input variables are typically continuous, represented by a real-valued floating-point value. Often, the input domain is unconstrained, although for test problems we impose a domain of interest. Test Function In this case we will explore function visualization with a simple x^2 objective function: This has an optimal value with an input of x=0.0, which equals 0.0. The example below implements this objective function and evaluates a single input. 2 # example of a 1d objective function 3 # objective function 4 def objective(x): 5 return x2.0 6 # evaluate inputs to the objective function 7 x = 4.0 8 result = objective(x) 9 print('f(%.3f) = %.3f' % (x, result)) Running the example evaluates the value 4.0 with the objective function, which equals 16.0. Sample the Test Function The first thing we might want to do with a new function is define an input range of interest and sample the domain of interest using a uniform grid. This sample will provide the basis for generating a plot later. In this case, we will define a domain of interest around the optima of x=0.0 from x=-5.0 to x=5.0 and sample a grid of values in this range with 0.1 increments, such as -5.0, -4.9, -4.8, etc. 1 ... 2 # define range for input 3 r_min, r_max = -5.0, 5.0 4 # sample input range uniformly at 0.1 increments 5 inputs = arange(r_min, r_max, 0.1) 6 # summarize some of the input domain 7 print(inputs[:5]) We can then evaluate each of the x values in our sample. 1 ... 2 # compute targets 3 results = objective(inputs) 4 # summarize some of the results 5 print(results[:5]) Finally, we can check some of the input and their corresponding outputs. 1 ... 2 # create a mapping of some inputs to some results 3 for i in range(5): 4 print('f(%.3f) = %.3f' % (inputs[i], results[i])) Tying this together, the complete example of sampling the input space and evaluating all points in the sample is listed below. 2 # sample 1d objective function 3 from numpy import arange 4 # objective function 5 def objective(x): 6 return x2.0 7 # define range for input 8 r_min, r_max = -5.0, 5.0 9 # sample input range uniformly at 0.1 increments 10 inputs = arange(r_min, r_max, 0.1) 11 # summarize some of the input domain 12 print(inputs[:5]) 13 # compute targets 14 results = objective(inputs) 15 # summarize some of the results 16 print(results[:5]) 17 # create a mapping of some inputs to some results 18 for i in range(5): 19 print('f(%.3f) = %.3f' % (inputs[i], results[i])) Running the example first generates a uniform sample of input points as we expected. The input points are then evaluated using the objective function and finally, we can see a simple mapping of inputs to outputs of the objective function. 1 [-5. -4.9 -4.8 -4.7 -4.6] 2 [25. 24.01 23.04 22.09 21.16] 3 f(-5.000) = 25.000 4 f(-4.900) = 24.010 5 f(-4.800) = 23.040 6 f(-4.700) = 22.090 7 f(-4.600) = 21.160 Now that we have some confidence in generating a sample of inputs and evaluating them with the objective function, we can look at generating plots of the function. Line Plot of Test Function We could sample the input space randomly, but the benefit of a uniform line or grid of points is that it can be used to generate a smooth plot. It is smooth because the points in the input space are ordered from smallest to largest. This ordering is important as we expect (hope) that the output of the objective function has a similar smooth relationship between values, e.g. small changes in input result in locally consistent (smooth) changes in the output of the function. In this case, we can use the samples to generate a line plot of the objective function with the input points (x) on the x-axis of the plot and the objective function output (results) on the y-axis of the plot. 1 ... 2 # create a line plot of input vs result 3 pyplot.plot(inputs, results) 4 # show the plot 5 pyplot.show() Tying this together, the complete example is listed below. 2 # line plot of input vs result for a 1d objective function 3 from numpy import arange 4 from matplotlib import pyplot 5 # objective function 6 def objective(x): 7 return x2.0 8 # define range for input 9 r_min, r_max = -5.0, 5.0 10 # sample input range uniformly at 0.1 increments 11 inputs = arange(r_min, r_max, 0.1) 12 # compute targets 13 results = objective(inputs) 14 # create a line plot of input vs result 15 pyplot.plot(inputs, results) 16 # show the plot 17 pyplot.show() Running the example creates a line plot of the objective function. We can see that the function has a large U-shape, called a parabola. This is a common shape when studying curves, e.g. the study of calculus. Scatter Plot of Test Function The line is a construct. It is not really the function, just a smooth summary of the function. Always keep this in mind. Recall that we, in fact, generated a sample of points in the input space and corresponding evaluation of those points. As such, it would be more accurate to create a scatter plot of points; for example: 2 # scatter plot of input vs result for a 1d objective function 3 from numpy import arange 4 from matplotlib import pyplot 5 # objective function 6 def objective(x): 7 return x2.0 8 # define range for input 9 r_min, r_max = -5.0, 5.0 10 # sample input range uniformly at 0.1 increments 11 inputs = arange(r_min, r_max, 0.1) 12 # compute targets 13 results = objective(inputs) 14 # create a scatter plot of input vs result 15 pyplot.scatter(inputs, results) 16 # show the plot 17 pyplot.show() Running the example creates a scatter plot of the objective function. We can see the familiar shape of the function, but we don’t gain anything from plotting the points directly. The line and the smooth interpolation between the points it provides are more useful as we can draw other points on top of the line, such as the location of the optima or the points sampled by an optimization algorithm. Line Plot with Marked Optima Next, let’s draw the line plot again and this time draw a point where the known optima of the function is located. This can be helpful when studying an optimization algorithm as we might want to see how close an optimization algorithm can get to the optima. First, we must define the input for the optima, then evaluate that point to give the x-axis and y-axis values for plotting. 1 ... 2 # define the known function optima 3 optima_x = 0.0 4 optima_y = objective(optima_x) We can then plot this point with any shape or color we like, in this case, a red square. 1 ... 2 # draw the function optima as a red square 3 pyplot.plot([optima_x], [optima_y], 's', color='r') Tying this together, the complete example of creating a line plot of the function with the optima highlighted by a point is listed below. 2 # line plot of input vs result for a 1d objective function and show optima 3 from numpy import arange 4 from matplotlib import pyplot 5 # objective function 6 def objective(x): 7 return x2.0 8 # define range for input 9 r_min, r_max = -5.0, 5.0 10 # sample input range uniformly at 0.1 increments 11 inputs = arange(r_min, r_max, 0.1) 12 # compute targets 13 results = objective(inputs) 14 # create a line plot of input vs result 15 pyplot.plot(inputs, results) 16 # define the known function optima 17 optima_x = 0.0 18 optima_y = objective(optima_x) 19 # draw the function optima as a red square 20 pyplot.plot([optima_x], [optima_y], 's', color='r') 21 # show the plot 22 pyplot.show() Running the example creates the familiar line plot of the function, and this time, the optima of the function, e.g. the input that results in the minimum output of the function, is marked with a red This is a very simple function and the red square for the optima is easy to see. Sometimes the function might be more complex, with lots of hills and valleys, and we might want to make the optima more visible. In this case, we can draw a vertical line across the whole plot. 1 ... 2 # draw a vertical line at the optimal input 3 pyplot.axvline(x=optima_x, ls='--', color='red') Tying this together, the complete example is listed below. 2 # line plot of input vs result for a 1d objective function and show optima as line 3 from numpy import arange 4 from matplotlib import pyplot 5 # objective function 6 def objective(x): 7 return x2.0 8 # define range for input 9 r_min, r_max = -5.0, 5.0 10 # sample input range uniformly at 0.1 increments 11 inputs = arange(r_min, r_max, 0.1) 12 # compute targets 13 results = objective(inputs) 14 # create a line plot of input vs result 15 pyplot.plot(inputs, results) 16 # define the known function optima 17 optima_x = 0.0 18 # draw a vertical line at the optimal input 19 pyplot.axvline(x=optima_x, ls='--', color='red') 20 # show the plot 21 pyplot.show() Running the example creates the same plot and this time draws a red line clearly marking the point in the input space that marks the optima. Line Plot with Samples Finally, we might want to draw the samples of the input space selected by an optimization algorithm. We will simulate these samples with random points drawn from the input domain. 1 ... 2 # simulate a sample made by an optimization algorithm 3 seed(1) 4 sample = r_min + rand(10) * (r_max - r_min) 5 # evaluate the sample 6 sample_eval = objective(sample) We can then plot this sample, in this case using small black circles. 1 ... 2 # plot the sample as black circles 3 pyplot.plot(sample, sample_eval, 'o', color='black') The complete example of creating a line plot of a function with the optima marked by a red line and an algorithm sample drawn with small black dots is listed below. 2 # line plot of domain for a 1d function with optima and algorithm sample 3 from numpy import arange 4 from numpy.random import seed 5 from numpy.random import rand 6 from matplotlib import pyplot 7 # objective function 8 def objective(x): 9 return x*2.0 10 # define range for input 11 r_min, r_max = -5.0, 5.0 12 # sample input range uniformly at 0.1 increments 13 inputs = arange(r_min, r_max, 0.1) 14 # compute targets 15 results = objective(inputs) 16 # simulate a sample made by an optimization algorithm 17 seed(1) 18 sample = r_min + rand(10) (r_max - r_min) 19 # evaluate the sample 20 sample_eval = objective(sample) 21 # create a line plot of input vs result 22 pyplot.plot(inputs, results) 23 # define the known function optima 24 optima_x = 0.0 25 # draw a vertical line at the optimal input 26 pyplot.axvline(x=optima_x, ls='--', color='red') 27 # plot the sample as black circles 28 pyplot.plot(sample, sample_eval, 'o', color='black') 29 # show the plot 30 pyplot.show() Running the example creates the line plot of the domain and marks the optima with a red line as before. This time, the sample from the domain selected by an algorithm (really a random sample of points) is drawn with black dots. We can imagine that a real optimization algorithm will show points narrowing in on the domain as it searches down-hill from a starting point. Next, let’s look at how we might perform similar visualizations for the optimization of a two-dimensional function. Visualize 2D Function Optimization A two-dimensional function is a function that takes two input variables, e.g. x and y. Test Function We can use the same x^2 function and scale it up to be a two-dimensional function; for example: This has an optimal value with an input of [x=0.0, y=0.0], which equals 0.0. The example below implements this objective function and evaluates a single input. 2 # example of a 2d objective function 3 # objective function 4 def objective(x, y): 5 return x2.0 + y2.0 6 # evaluate inputs to the objective function 7 x = 4.0 8 y = 4.0 9 result = objective(x, y) 10 print('f(%.3f, %.3f) = %.3f' % (x, y, result)) Running the example evaluates the point [x=4, y=4], which equals 32. 1 f(4.000, 4.000) = 32.000 Next, we need a way to sample the domain so that we can, in turn, sample the objective function. Sample Test Function A common way for sampling a two-dimensional function is to first generate a uniform sample along each variable, x and y, then use these two uniform samples to create a grid of samples, called a mesh This is not a two-dimensional array across the input space; instead, it is two two-dimensional arrays that, when used together, define a grid across the two input variables. This is achieved by duplicating the entire x sample array for each y sample point and similarly duplicating the entire y sample array for each x sample point. This can be achieved using the meshgrid() NumPy function; for example: 1 ... 2 # define range for input 3 r_min, r_max = -5.0, 5.0 4 # sample input range uniformly at 0.1 increments 5 xaxis = arange(r_min, r_max, 0.1) 6 yaxis = arange(r_min, r_max, 0.1) 7 # create a mesh from the axis 8 x, y = meshgrid(xaxis, yaxis) 9 # summarize some of the input domain 10 print(x[:5, :5]) We can then evaluate each pair of points using our objective function. 1 ... 2 # compute targets 3 results = objective(x, y) 4 # summarize some of the results 5 print(results[:5, :5]) Finally, we can review the mapping of some of the inputs to their corresponding output values. 1 ... 2 # create a mapping of some inputs to some results 3 for i in range(5): 4 print('f(%.3f, %.3f) = %.3f' % (x[i,0], y[i,0], results[i,0])) The example below demonstrates how we can create a uniform sample grid across the two-dimensional input space and objective function. 2 # sample 2d objective function 3 from numpy import arange 4 from numpy import meshgrid 5 # objective function 6 def objective(x, y): 7 return x2.0 + y2.0 8 # define range for input 9 r_min, r_max = -5.0, 5.0 10 # sample input range uniformly at 0.1 increments 11 xaxis = arange(r_min, r_max, 0.1) 12 yaxis = arange(r_min, r_max, 0.1) 13 # create a mesh from the axis 14 x, y = meshgrid(xaxis, yaxis) 15 # summarize some of the input domain 16 print(x[:5, :5]) 17 # compute targets 18 results = objective(x, y) 19 # summarize some of the results 20 print(results[:5, :5]) 21 # create a mapping of some inputs to some results 22 for i in range(5): 23 print('f(%.3f, %.3f) = %.3f' % (x[i,0], y[i,0], results[i,0])) Running the example first summarizes some points in the mesh grid, then the objective function evaluation for some points. Finally, we enumerate coordinates in the two-dimensional input space and their corresponding function evaluation. 1 [[-5. -4.9 -4.8 -4.7 -4.6] 2 [-5. -4.9 -4.8 -4.7 -4.6] 3 [-5. -4.9 -4.8 -4.7 -4.6] 4 [-5. -4.9 -4.8 -4.7 -4.6] 5 [-5. -4.9 -4.8 -4.7 -4.6]] 6 [[50. 49.01 48.04 47.09 46.16] 7 [49.01 48.02 47.05 46.1 45.17] 8 [48.04 47.05 46.08 45.13 44.2 ] 9 [47.09 46.1 45.13 44.18 43.25] 10 [46.16 45.17 44.2 43.25 42.32]] 11 f(-5.000, -5.000) = 50.000 12 f(-5.000, -4.900) = 49.010 13 f(-5.000, -4.800) = 48.040 14 f(-5.000, -4.700) = 47.090 15 f(-5.000, -4.600) = 46.160 Now that we are familiar with how to sample the input space and evaluate points, let’s look at how we might plot the function. Contour Plot of Test Function A popular plot for two-dimensional functions is a contour plot. This plot creates a flat representation of the objective function outputs for each x and y coordinate where the color and contour lines indicate the relative value or height of the output of the objective function. This is just like a contour map of a landscape where mountains can be distinguished from valleys. This can be achieved using the contour() Matplotlib function that takes the mesh grid and the evaluation of the mesh grid as input directly. We can then specify the number of levels to draw on the contour and the color scheme to use. In this case, we will use 50 levels and a popular “jet” color scheme where low-levels use a cold color scheme (blue) and high-levels use a hot color scheme (red). 1 ... 2 # create a contour plot with 50 levels and jet color scheme 3 pyplot.contour(x, y, results, 50, alpha=1.0, cmap='jet') 4 # show the plot 5 pyplot.show() Tying this together, the complete example of creating a contour plot of the two-dimensional objective function is listed below. 3 # create a contour plot with 50 levels and jet color scheme 4 pyplot.contour(x, y, results, 50, alpha=1.0, cmap='jet') 5 # show the plot 6 pyplot.show() 7 Tying this together, the complete example of creating a contour plot of the two-dimensional objective function is listed below. 8 # contour plot for 2d objective function 9 from numpy import arange 10 from numpy import meshgrid 11 from matplotlib import pyplot 12 # objective function 13 def objective(x, y): 14 return x2.0 + y2.0 15 # define range for input 16 r_min, r_max = -5.0, 5.0 17 # sample input range uniformly at 0.1 increments 18 xaxis = arange(r_min, r_max, 0.1) 19 yaxis = arange(r_min, r_max, 0.1) 20 # create a mesh from the axis 21 x, y = meshgrid(xaxis, yaxis) 22 # compute targets 23 results = objective(x, y) 24 # create a contour plot with 50 levels and jet color scheme 25 pyplot.contour(x, y, results, 50, alpha=1.0, cmap='jet') 26 # show the plot 27 pyplot.show() Running the example creates the contour plot. We can see that the more curved parts of the surface around the edges have more contours to show the detail, and the less curved parts of the surface in the middle have fewer contours. We can see that the lowest part of the domain is the middle, as expected. Filled Contour Plot of Test Function It is also helpful to color the plot between the contours to show a more complete surface. Again, the colors are just a simple linear interpolation, not the true function evaluation. This must be kept in mind on more complex functions where fine detail will not be shown. We can fill the contour plot using the contourf() version of the function that takes the same arguments. 1 ... 2 # create a filled contour plot with 50 levels and jet color scheme 3 pyplot.contourf(x, y, results, levels=50, cmap='jet') We can also show the optima on the plot, in this case as a white star that will stand out against the blue background color of the lowest part of the plot. 1 ... 2 # define the known function optima 3 optima_x = [0.0, 0.0] 4 # draw the function optima as a white star 5 pyplot.plot([optima_x[0]], [optima_x[1]], '', color='white') Tying this together, the complete example of a filled contour plot with the optima marked is listed below. 2 # filled contour plot for 2d objective function and show the optima 3 from numpy import arange 4 from numpy import meshgrid 5 from matplotlib import pyplot 6 # objective function 7 def objective(x, y): 8 return x2.0 + y2.0 9 # define range for input 10 r_min, r_max = -5.0, 5.0 11 # sample input range uniformly at 0.1 increments 12 xaxis = arange(r_min, r_max, 0.1) 13 yaxis = arange(r_min, r_max, 0.1) 14 # create a mesh from the axis 15 x, y = meshgrid(xaxis, yaxis) 16 # compute targets 17 results = objective(x, y) 18 # create a filled contour plot with 50 levels and jet color scheme 19 pyplot.contourf(x, y, results, levels=50, cmap='jet') 20 # define the known function optima 21 optima_x = [0.0, 0.0] 22 # draw the function optima as a white star 23 pyplot.plot([optima_x[0]], [optima_x[1]], '', color='white') 24 # show the plot 25 pyplot.show() Running the example creates the filled contour plot that gives a better idea of the shape of the objective function. The optima at [x=0, y=0] is then marked clearly with a white star. Filled Contour Plot of Test Function with Samples We may want to show the progress of an optimization algorithm to get an idea of its behavior in the context of the shape of the objective function. In this case, we can simulate the points chosen by an optimization algorithm with random coordinates in the input space. 1 ... 2 # simulate a sample made by an optimization algorithm 3 seed(1) 4 sample_x = r_min + rand(10) * (r_max - r_min) 5 sample_y = r_min + rand(10) * (r_max - r_min) These points can then be plotted directly as black circles and their context color can give an idea of their relative quality. 1 ... 2 # plot the sample as black circles 3 pyplot.plot(sample_x, sample_y, 'o', color='black') Tying this together, the complete example of a filled contour plot with optimal and input sample plotted is listed below. 2 # filled contour plot for 2d objective function and show the optima and sample 3 from numpy import arange 4 from numpy import meshgrid 5 from numpy.random import seed 6 from numpy.random import rand 7 from matplotlib import pyplot 8 # objective function 9 def objective(x, y): 10 return x2.0 + y2.0 11 # define range for input 12 r_min, r_max = -5.0, 5.0 13 # sample input range uniformly at 0.1 increments 14 xaxis = arange(r_min, r_max, 0.1) 15 yaxis = arange(r_min, r_max, 0.1) 16 # create a mesh from the axis 17 x, y = meshgrid(xaxis, yaxis) 18 # compute targets 19 results = objective(x, y) 20 # simulate a sample made by an optimization algorithm 21 seed(1) 22 sample_x = r_min + rand(10) * (r_max - r_min) 23 sample_y = r_min + rand(10) * (r_max - r_min) 24 # create a filled contour plot with 50 levels and jet color scheme 25 pyplot.contourf(x, y, results, levels=50, cmap='jet') 26 # define the known function optima 27 optima_x = [0.0, 0.0] 28 # draw the function optima as a white star 29 pyplot.plot([optima_x[0]], [optima_x[1]], '', color='white') 30 # plot the sample as black circles 31 pyplot.plot(sample_x, sample_y, 'o', color='black') 32 # show the plot 33 pyplot.show() Running the example, we can see the filled contour plot as before with the optima marked. We can now see the sample drawn as black dots and their surrounding color and relative distance to the optima gives an idea of how close the algorithm (random points in this case) got to solving the Surface Plot of Test Function Finally, we may want to create a three-dimensional plot of the objective function to get a fuller idea of the curvature of the function. This can be achieved using the plot_surface() Matplotlib function, that, like the contour plot, takes the mesh grid and function evaluation directly. 1 ... 2 # create a surface plot with the jet color scheme 3 figure = pyplot.figure() 4 axis = figure.gca(projection='3d') 5 axis.plot_surface(x, y, results, cmap='jet') The complete example of creating a surface plot is listed below. 2 # surface plot for 2d objective function 3 from numpy import arange 4 from numpy import meshgrid 5 from matplotlib import pyplot 6 from mpl_toolkits.mplot3d import Axes3D 7 # objective function 8 def objective(x, y): 9 return x2.0 + y*2.0 10 # define range for input 11 r_min, r_max = -5.0, 5.0 12 # sample input range uniformly at 0.1 increments 13 xaxis = arange(r_min, r_max, 0.1) 14 yaxis = arange(r_min, r_max, 0.1) 15 # create a mesh from the axis 16 x, y = meshgrid(xaxis, yaxis) 17 # compute targets 18 results = objective(x, y) 19 # create a surface plot with the jet color scheme 20 figure = pyplot.figure() 21 axis = figure.gca(projection='3d') 22 axis.plot_surface(x, y, results, cmap='jet') 23 # show the plot 24 pyplot.show() Running the example creates a three-dimensional surface plot of the objective function. Additionally, the plot is interactive, meaning that you can use the mouse to drag the perspective on the surface around and view it from different angles. Further Reading This section provides more resources on the topic if you are looking to go deeper. In this tutorial, you discovered how to create visualizations for function optimization in Python. Specifically, you learned: • Visualization is an important tool when studying function optimization algorithms. • How to visualize one-dimensional functions and samples using line plots. • How to visualize two-dimensional functions and samples using contour and surface plots. Do you have any questions? Ask your questions in the comments below and I will do my best to answer. 5 Responses to Visualization for Function Optimization in Python 1. Tobi Adeyemi January 17, 2021 at 12:15 am # Thanks for the insightful tutorial Jason. Looking forward to your textbook on optimization. Your books are always delightful to read. □ Jason Brownlee January 17, 2021 at 6:04 am # 2. Slava February 3, 2021 at 6:05 am # Thank you for a very interesting tutorial. Easy to follow even for a beginner. Would it be possible to write a tutorial on learned optimizers? □ Jason Brownlee February 3, 2021 at 6:28 am # Thanks for the suggestion. 3. Janakraj Oza October 13, 2021 at 6:06 pm # Beautifully explained Jason the test function in details. This tutorial was worth reading and very helpful for everyone. Leave a Reply Click here to cancel reply.	{"url":"https://machinelearningmastery.com/visualization-for-function-optimization-in-python/","timestamp":"2024-11-08T04:23:08Z","content_type":"text/html","content_length":"587623","record_id":"<urn:uuid:283b9618-89ad-4c04-befa-dd2235ccd0b5>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00241.warc.gz"}
American Mathematical Society an Inductive Mean? Frank Nielsen Communicated by Notices Associate Editor Richard Levine Notions of means The notion of means 10 is central to mathematics and statistics, and plays a key role in machine learning and data analytics. The three classical Pythagorean means of two positive reals and are the arithmetic (A), geometric (G), and harmonic (H) means, given respectively by These Pythagorean means were originally geometrically studied to define proportions, and the harmonic mean led to a beautiful connection between mathematics and music. The Pythagorean means enjoy the following inequalities: with equality if and only if . These Pythagorean means belong to a broader parametric family of means, the power means defined for . We have , and in the limits: , , and . Power means are also called binomial, Minkowski, or Hölder means in the literature. There are many ways to define and axiomatize means with a rich literature 8. An important class of means are the quasi-arithmetic means induced by strictly increasing and differentiable real-valued functional generators : Quasi-arithmetic means satisfy the in-betweenness property of means: , and are called so because is the arithmetic mean on the -representation of numbers. The power means are quasi-arithmetic means, , obtained for the following continuous family of generators: Power means are the only homogeneous quasi-arithmetic means, where a mean is said to be homogeneous when for any . Quasi-arithmetic means can also be defined for -variable means (i.e., ), and more generally for calculating expected values of random variables 10: We denote by the quasi-arithmetic expected value of a random variable induced by a strictly monotone and differentiable function . For example, the geometric and harmonic expected values of are defined by and , respectively. The ordinary expectation is recovered for : . The quasi-arithmetic expected values satisfy a strong law of large numbers and a central limit theorem (10, Theorem 1): Let be independent and identically distributed (i.i.d.) with finite variance and derivative at . Then we have as , where denotes a normal distribution of expectation and variance . Inductive means An inductive mean is a mean defined as a limit of a convergence sequence of other means 15. The notion of inductive means defined as limits of sequences was pioneered independently by Lagrange and Gauss 7 who studied the following double sequence of iterations: initialized with and . We have where the homogeneous arithmetic-geometric mean (AGM) is obtained in the limit: There is no closed-form formula for the AGM in terms of elementary functions as this induced mean is related to the complete elliptic integral of the first kind 7: where is the elliptic integral. The fast quadratic convergence 11 of the AGM iterations makes it computationally attractive, and the AGM iterations have been used to numerically calculate digits of or approximate the perimeters of ellipses among others 7. Some inductive means admit closed-form formulas: For example, the arithmetic-harmonic mean obtained as the limit of the double sequence initialized with and converges to the geometric mean: In general, inductive means defined as the limits of double sequences with respect to two smooth symmetric means and : are proven to converge quadratically 11 to (order- convergence). Inductive means and matrix means We have obtained so far three ways to get the geometric scalar mean between positive reals and : As an inductive mean with the arithmetic-harmonic double sequence: , As a quasi-arithmetic mean obtained for the generator : , and As the limit of power means: . Let us now consider the geometric mean of two symmetric positive-definite (SPD) matrices and of size . SPD matrices generalize positive reals. We shall investigate the three generalizations of the above approaches of the scalar geometric mean, and show that they yield different notions of matrix geometric means when . First, the AHM iterations can be extended to SPD matrices instead of reals: where the matrix arithmetic mean is and the matrix harmonic mean is . The AHM iterations initialized with and yield in the limit , the matrix arithmetic-harmonic mean 314 (AHM): Remarkably, the matrix AHM enjoys quadratic convergence to the following SPD matrix: When and are positive reals, we recover . When , the identity matrix, we get , the positive square root of SPD matrix . Thus the matrix AHM iterations provide a fast method in practice to numerically approximate matrix square roots by bypassing the matrix eigendecomposition. When matrices and commute (i.e., ), we have . The geometric mean is proven to be the unique solution to the matrix Ricatti equation , is invariant under inversion (i.e., ), and satisfies the determinant property . Let denote the set of symmetric positive-definite matrices. The matrix geometric mean can be interpreted using a Riemannian geometry 5 of the cone : Equip with the trace metric tensor, i.e., a collection of smoothly varying inner products for defined by where and are matrices belonging to the vector space of symmetric matrices (i.e., and are geometrically vectors of the tangent plane of ). The geodesic length distance on the Riemannian manifold is where denotes the -th largest real eigenvalue of a symmetric matrix , denotes the Frobenius norm, and is the unique matrix logarithm of a SPD matrix . Interestingly, the matrix geometric mean can also be interpreted as the Riemannian center of mass of and : This Riemannian least squares mean is also called the Cartan, Kärcher, or Fréchet mean in the literature. More generally, the Riemannian geodesic between and of for is expressed using the weighted matrix geometric mean minimizing This Riemannian barycenter can be solved as with , , and , i.e., is the arc length parameterization of the constant speed geodesic . When matrices and commute, we have . We thus interpret the matrix geometric mean as the Riemannian geodesic Second, let us consider the matrix geometric mean as the limit of matrix quasi-arithmetic power means which can be defined 13 as for , with and . We get , the log-Euclidean matrix mean defined by where and denote the matrix exponential and the matrix logarithm, respectively. We have . Consider the Loewner partial order on the cone : if and only if is positive semi-definite. A mean is said operator monotone 5 if for and , we have . The log-Euclidean mean is not operator monotone but the Riemannian geometric matrix mean is operator monotone. Third, we can define matrix power means for by uniquely solving the following matrix equation 13: Let denote the unique solution of Eq. 2. This equation is the matrix analogue of the scalar equation which can be solved as , i.e., the scalar -power mean. In the limit case , this matrix power mean yields the matrix geometric/Riemannian mean 13: In general, we get the following closed-form expression 13 of this matrix power mean for : Inductive means, circumcenters, and medians of several matrices To extend these various binary matrix means of two matrices to matrix means of matrices of , we can use induction sequences 9. First, the -variable matrix geometric mean can be defined as the unique Riemannian center of mass:	{"url":"https://www.ams.org/journals/notices/202311/noti2832/noti2832.html?adat=December%202023&trk=2832&pdfissue=202311&pdffile=rnoti-p1851.pdf&cat=none&type=.html","timestamp":"2024-11-05T17:19:22Z","content_type":"text/html","content_length":"1049019","record_id":"<urn:uuid:01a30bfb-8007-4ef7-80cd-37f82c7f568b>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00373.warc.gz"}
Consider a value to be significantly low if its score is less than or equal to $-2$ or consider the value to be significantly high if its $z$ score is greater than or equal to $2 .$ Data Set 29 "Coin Weights" lists weights (grams) of quarters manufactured after 1964 . Those weights have a mean of $5.63930 \mathrm{~g}$ and a standard deviation of $0.06194 \mathrm{~g}$. Identify the weights that are significantly low or significantly high. Short Answer Expert verified Significantly low weights are <= 5.51542 g, and significantly high weights are >= 5.76318 g. Step by step solution Understand the problem Identify the given values and understand the thresholds for significantly low and high Z-scores. Given values: mean = 5.63930 g, standard deviation = 0.06194 g. Thresholds: Z <= -2 (significantly low), Z >= 2 (significantly high). Recall the Z-score formula The formula for the Z-score is given by: \[ Z = \frac{X - \mu}{\sigma} \]Where X is the data point, $ \mu $ is the mean, and $ \sigma $ is the standard deviation. Set up equations for threshold values To find the significantly low and high weights, solve for X in the Z-score formula using Z = -2 and Z = 2:\[ -2 = \frac{X - 5.63930}{0.06194} \] \[ 2 = \frac{X - 5.63930}{0.06194} \] Solve for low threshold weight (Z = -2) Multiply both sides of the equation $ -2 = \frac{X - 5.63930}{0.06194} $ by 0.06194:\[ -2 * 0.06194 = X - 5.63930 \]\[ -0.12388 = X - 5.63930 \] Add 5.63930 to both sides:\[ X = 5.63930 - 0.12388 \]\[ X = 5.51542 \text{ g} \] Solve for high threshold weight (Z = 2) Multiply both sides of the equation $ 2 = \frac{X - 5.63930}{0.06194} $ by 0.06194:\[ 2 * 0.06194 = X - 5.63930 \]\[ 0.12388 = X - 5.63930 \] Add 5.63930 to both sides:\[ X = 5.63930 + 0.12388 \]\[ X = 5.76318 \text{ g} \] Weights significantly low are <= 5.51542 g. Weights significantly high are >= 5.76318 g. Key Concepts These are the key concepts you need to understand to accurately answer the question. Normal Distribution The normal distribution is a widely used statistical concept characterized by its bell-shaped curve. This curve is symmetrical, meaning the left and right sides are mirror images of each other. Each side of the curve extends indefinitely, getting closer but never touching the horizontal axis. The highest point in the curve represents the mean. To understand it better: • The normal distribution is used to represent real-valued random variables. • Most occurrences take place near the mean, dwindling as they move away. • In practical terms, weights of manufactured quarters can follow a normal distribution to assess consistency. Grasping the normal distribution is fundamental in statistics as it forms the basis for calculating probabilities and making inferences about a population. Significance Thresholds Significance thresholds are critical in determining which data points hold statistical importance. In the context of Z-scores, a threshold helps us decide if a value is significantly high or low based on its distance from the mean. • Z-scores below -2 are significantly low; they deviate more than two standard deviations below the mean. • Z-scores above 2 are significantly high; they deviate more than two standard deviations above the mean. This threshold gives us a clear criterion for identifying unusual data points. When you calculated that weights ≤ 5.51542 g were significantly low and ≥ 5.76318 g were significantly high, you used this concept. Understanding significance thresholds helps in making informed decisions in various fields, from quality control to research hypothesis testing. Standard Deviation Standard deviation (σ) measures the amount of variation or dispersion of a set of values. A smaller standard deviation means that values tend to be close to the mean. Larger standard deviation indicates that values are more spread out. Here's how it’s relevant: • Standard deviation allows us to understand how much individual weights of quarters vary from the average weight. • It is central to calculating Z-scores, which are key to identifying outliers. The given standard deviation of 0.06194 g for the weights indicates how tightly clustered the weights are around the mean. It's crucial in understanding whether a particular quarter's weight is typical or an outlier. Mean Calculation The mean (μ) is the average of a set of numbers. Calculating the mean involves adding up all data points and then dividing by the number of points. Here’s a step-by-step for understanding: • Sum all observed weights of quarters. • Divide by the total number of weights to get the mean (e.g., given mean = 5.63930 g). Knowing the mean is essential because it provides a central value from which deviations are measured. The mean helps in calculating Z-scores, which identify how far a data point is from the average in terms of standard deviations. Understanding mean and its role in statistical analysis simplifies many complex calculations.	{"url":"https://www.vaia.com/en-us/textbooks/math/essentials-of-statistics-6-edition/chapter-3/problem-11-consider-a-value-to-be-significantly-low-if-its-s/","timestamp":"2024-11-08T11:49:43Z","content_type":"text/html","content_length":"252742","record_id":"<urn:uuid:517807be-1a94-49ef-9ecf-f73dfe09d109>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00536.warc.gz"}
Voltage Regulation of Transformer The voltage regulation is the percentage of voltage difference between no load and full load voltages of a transformer with respect to its full load voltage. The voltage regulation is the percentage of voltage difference between no load and full load voltages of a transformer with respect to its full load voltage. Explanation of Voltage Regulation of Transformer Say an electrical power transformer is open circuited, means load is not connected with secondary terminals. In this situation, the secondary terminalvoltage of the transformer will be its secondary induced emf E2. Whenever full load is connected to the secondary terminals of the transformer, ratedcurrent I2 flows through the secondary circuit and voltage drop comes into picture. At this situation, primary winding will also draw equivalent full load current from source. The voltagedrop in the secondary is I2Z2 where Z2 is the secondary impedance of transformer. Now if at this loading condition, any one measures the voltage between secondary terminals, he or she will getvoltage V2 across load terminals which is obviously less than no load secondary voltage E2 and this is because of I2Z2 voltage drop in the transformer. Expression of Voltage Regulation of Transformer, represented in percentage, is Equivalent impedance of transformer is essential to be calculated because the electrical power transformer is an electrical power system equipment for estimating different parameters of electrical power system which may be required to calculate total internal impedance of an electrical power transformer, viewing from primary side or secondary side as per requirement. This calculation requires equivalent circuit of transformer referred to primary or equivalent circuit of transformer referred to secondary sides respectively. Percentage impedance is also very essential parameter of transformer. Special attention is to be given to this parameter during installing a transformer in an existing electrical power system. Percentage impedance of different power transformers should be properly matched during parallel operation of power transformers. The percentage impedance can be derived from equivalent impedance of transformer so, it can be said that equivalent circuit of transformer is also required during calculation of % impedance. Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail Basic Electrical and electronics : Electrical Mechanics : Voltage Regulation of Transformer \|	{"url":"https://www.brainkart.com/article/Voltage-Regulation-of-Transformer_6669/","timestamp":"2024-11-15T04:33:55Z","content_type":"text/html","content_length":"31500","record_id":"<urn:uuid:0e7c8af7-03ac-4326-bd80-7318497c1ad9>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00698.warc.gz"}
Gamebook Design Well this is something I put together for another gamebook system EDIT: which I can now announce is none other than J.H.Brennan's classic Sagas of the Demonspawn series, that is to be digitally released through Tin Man Games as announced HERE. An experience that is quite surreal for me since J.H. "Herbie" Brennan is my favourite gamebook author and a childhood idol: to say it's an honour is a massive understatement! ...So having done this work, it was a "relatively" simple thing to adapt it to other gamebook systems that I know others are/have writing/written for: namely the Fighting Fantasy and Gamebook Adventures systems. (The GA system involves quite a lot more calculation though, although from a player point of view, it's only slightly more complicated than the FF system). So if you're ever writing for either of these systems, and want a tool to help you determine how hard a particular combat is, how long it will last for etc, then this is what you need... Or maybe you're just curious to know what your chances are of surviving a particular combat or series of them in XYZ gamebook. This tool will give you a pretty definitive answer... And yes, it also demonstrates points I've made before about how well balanced (or unbalanced) these gamebook systems can be. I'll spare you the detailed analysis, as you're now able to crunch out the numbers yourself. With this tool, you can set Player and Enemy stats yourself, or use Max/Min/Average/Random Player stats, set a number of other variables for the combat, and then run 10, 100, 1000 or whatever number of trials you want and get a stats summary that can also be written out to a report as well if you wish. Hopefully you can figure out how to use it (I've tried to make it a simple as possible). All you need is Excel 97 or later (maybe Open Office will work too, I haven't tried) and to enable macros... Let me know what you think (and if you'd like additions) Happy number crunching :) This article is even more esoteric than yesterday’s topic on probabilities in the Gamebook Adventures system, but it’s related and to those that are designing and writing gamebooks of their own, this may prove useful for their own designs… Or at least give them ideas. There’s a few things happening in the “gamebook world” at the moment that could benefit from this, and so having done this conversion for a new project I’m working on, I thought I’d share my work, and to kinda follow on from my article yesterday. 15240 Hits Today’s article is all about probabilities in the Gamebook Adventures system (with some limited comparison to other systems, particularly Fighting Fantasy). Statistics can be a bit of a dry topic haha, so I’ll try to reduce it to the important elements for you… Well important that is if you’re writing for such a system or are interested to know what the game odds are… Beware though, there’s a lot of tables and charts incoming :) Calculating probabilities for a system such as Fighting Fantasy is reasonably straight forward: (In a typical battle) you roll two six-sided dice for each combatant and add a given Skill score to each combatant’s roll, with the highest score dealing 2 Stamina damage to the other combatant. Use of Luck rolls makes it a little more complicated but not by much. You can “Test your Luck”; by attempting to roll two dice equal to or under your current Luck; in order to do one more Stamina damage, but failing this roll means you do one less Stamina. Skill and Luck scores vary, resulting in an exponential scale where even a few points of Skill difference make it unlikely for one side to win, even with far more Stamina. For example, if your Skill is 3 lower than your opponent’s, then your opponent is roughly five times more likely than you are, to be the one making a hit in a given combat round, not including the Luck factor. In other words, if your hero has Skill 9 Stamina 24 and your opponent Skill 12 Stamina 5, then you both have about the same chance of winning… The Gamebook Adventures system is considerably more complicated to calculate probabilities for (even though the rules themselves are of a similar level of complexity) and shares a similar but less extreme exponential scale, such that the outcome of a combat is not as much of a sure thing. Combatants roll between one and six six-sided dice to attack (as determined by their Offence rating) and between one and six six-sided dice to defend (as determined by their Defence rating). If the attacker’s highest roll is higher than the defender’s highest roll, then they do damage equal to the sum of all their dice. (And in the case of tied rolls, the two tied dice are removed and the next two highest rolls are compared, until no dice are left). In addition, the hero can make a “Fitness check” on any given combat round by rolling two dice under their current Fitness. This is similar to “Testing your Luck” in the Fighting Fantasy system, except that the advantage given is to add 1 to their highest roll, significantly increasing the chance of hitting or defending. In the extreme case of 6 dice in Offence against an opponent with 6 dice in Defence, and including the two additional dice rolled for a Fitness check, there are 6^14 possible dice combinations for any given round (that’s 78,364,164,096 combinations). And to calculate the highest roll and damage inflicted for each of these combinations, is quite a task… But I’ve done this (well kind of; I had to take some short cuts, but the end conclusions are about the same) and this is what I present below. This isn’t meant to be a rigorous statistics paper, so I’ll spare you comprehensive details of how I came up with these numbers… But basically I listed every dice combination out on a spreadsheet, so that I could be sure I was calculating correct averages etc, until the number of possible combinations became too unwieldy. I listed all combinations up to seven dice, which is 6^7 or 279,936 combinations. The numbers for the remaining eight to fourteen dice combinations I estimated using a “best fit” exponential regression equation. In other words, I took the half-finished set of values I had manually calculated and used them to apply a formula to estimate the rest… First of all, we’ll ignore the impact of Fitness checks, and just focus on the chance of success (success being an attack that hits the defender) for each Offence / Defence combination: So to interpret what these numbers mean, it’s saying that if your Offence value was 6, and your opponent had a Defence value of 1, then (not including Fitness checks) your chance of success (i.e. hitting) is 76.00%, and if their Defence was 2, your chance is about 57.77% and so on… The yellow cells in the above table are where I had to cut corners as the number of combinations was too large to individually analyse. These are the values I extrapolated based on the trend already shown. This trend (as you can see in the above graph) is an exponential decline, where the degree to which your chances drop lessens as Defence values increase. Here are some interesting conclusions from these numbers: • Even at maximum Offence (6) versus minimum Defence (1), you have at best a 76% chance of hitting. This compares to a 100% chance for a similarly extreme matchup in the Fighting Fantasy system, and to many other dice-based game systems where the best odds tend to be 95% (anything but a roll of one on a twenty-sided dice), 97% (anything but double-one or double-six on two six-sided dice) or 99% in percentile-based systems. • Conversely though, even at minimum Offence (1) versus maximum Defence (6), you still have a 7.33% chance of hitting (more once you consider Fitness checks), which typically compares to between 0% and 5% in other systems. • Typically your chance of hitting; without including Fitness checks; is lower in the Gamebook Adventures system than that for other systems such as Fighting Fantasy, Lone Wolf, Dungeons and Dragons and the Basic Role Playing system. (However when you do hit, your damage is typically higher than what occurs in these systems). • Increasing Offence or Defence has an increasingly smaller impact. For instance, you’ll see that Offence 6 is only slightly better than Offence 5, particularly against high Defence values. (Although I suspect it’s not quite a close as shown, since the values in the yellow cells were those obtained from extrapolation). Now let’s look at the average damage inflicted when you hit: It’s actually difficult to draw much from these numbers since you need to factor in the chance of hitting to say how much damage is done on average each round… The amount of damage done for any given Offence rating only increases slightly with increasing Defence (based on the fact that the higher the Defence value you’re trying to hit, the more likely that a successful hit was based on a high By multiplying the chance to hit, by the average damage done when hit, we come to this table (which is very useful from a design / game-balancing point of view): So now we can start to see exactly how hard any given combat is. For instance, if you have an Offence of 4 and a Defence of 2, and are fighting an enemy with an Offence and Defence value of 3, then you inflict an average of 6.08 damage per attack, whilst your opponent inflicts an average of 5.44 damage per attack… Pretty useful huh? 21946 Hits In case you hadn’t noticed (and you probably haven’t haha) I haven’t done nearly as many blog posts by now as I’d like to have done… So whilst I’ve have plenty more articles I mean to get around to doing, I’m prioritising the articles I do post at the moment based on where (I consider that) there is a specific need to communicate something to a given audience. Today’s blog post is light on pictures (it doesn’t have any, just walls of text), as I just want to put this information out there and then move onto the next thing… It’s a post about my take on how to write “better gamebooks”. 33399 Hits	{"url":"https://thebrewin.com/blog/categories/gamebook-design","timestamp":"2024-11-11T23:40:30Z","content_type":"text/html","content_length":"47044","record_id":"<urn:uuid:6c61370d-ebb3-4668-b27d-5d86f5accc25>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00410.warc.gz"}
Time and space lower bounds for implementations using k-cAS This paper presents lower bounds on the time and space complexity of implementations that use k-compare&swap (k-CAS) synchronization primitives. We prove that using k-CAS primitives can improve neither the time nor the space complexity of implementations of widely used concurrent objects, such as counter, stack, queue, and collect. Surprisingly, overly restrictive use of k-CAS may even increase the space complexity required by such implementations. We prove a lower bound of Ω (log[2]n) on the round complexity of implementations of a collect object using read, write, and k-CAS, for any k, where n is the number of processes in the system. There is an implementation of collect with O(log[2]n) round complexity that uses only reads and writes. Thus, our lower bound establishes that k-CAS is no stronger than read and write for collect implementation round complexity. For k-CAS operations that return the values of all the objects they access, we prove that the total step complexity of implementing key objects such as counters, stacks, and queues is Ω (n log[k]n). We also prove that k-CAS cannot improve the space complexity of implementing many objects (including counter, stack, queue, and single-writer snapshot). An implementation has to use at least n base objects even if k-CAS is allowed, and if all operations (other than read) swap exactly k base objects, then it must use at least k ̇n base objects. • Collect • Compare&swap (CAS) • Conditional synchronization primitives • Counter • K-compare&swap (k-CAS) • Queue • Round complexity • Stack ASJC Scopus subject areas • Signal Processing • Hardware and Architecture • Computational Theory and Mathematics Dive into the research topics of 'Time and space lower bounds for implementations using k-cAS'. Together they form a unique fingerprint.	{"url":"https://cris.bgu.ac.il/en/publications/time-and-space-lower-bounds-for-implementations-using-k-cas-4","timestamp":"2024-11-06T11:46:31Z","content_type":"text/html","content_length":"60692","record_id":"<urn:uuid:0b256d5c-eac6-47bd-9160-9bbc2d121f9a>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00725.warc.gz"}
How to Find the Perimeter of a TriangleHow to Find the Perimeter of a Triangle Title: How to Find the Perimeter of a Triangle Triangles are one of the fundamental shapes in geometry. Understanding their properties and being able to calculate their various attributes is essential for both students and professionals and teachers in various fields. One of the most basic calculations for a triangle is finding its perimeter, which is the sum of the lengths of all its sides. In this article, I will try to explain into the various methods of How To Find The Perimeter Of A Triangle, including different types of triangles and the different cases to consider. Listen full article: Understanding the Basics Before I start this topic,How To Find The Perimeter Of A Triangle, let's establish a few key concepts and definitions: Triangle Types: Based on sides triangles are divided in three categories: (a)Equilateral Triangle: A triangle whose all three sides are equal is called equilateral triangle. In the picture, ABC is an equilateral triangle,whose AB = BC = AC (b)Isosceles Triangle: A triangle whose any two sides are equal and third one is different is called isosceles triangle. In the picture, PQR is an isosceles triangle whose PQ = PR (c)Scalene Triangle: A triangle whose all three sides are different to one another is called a scalene triangle. In the picture, XYZ is a scalene triangle whose, XY ≠ YZ ≠ ZX Sides and Vertices: (i)The sides of a triangle are the line segments that form the boundaries of the triangle. (ii)The vertices of a triangle are the points where the sides meet. The perimeter of a triangle is the total length or sum of all its sides. Sides of a Triangle: In an equilateral triangle, all sides have the same length, which we'll denote as 'a.' In an isosceles triangle, two sides have the same length, and the third side is 'b.' In a scalene triangle, all three sides have different lengths, which we'll denote as 'a,' 'b,' and 'c.' With these basic concepts in mind, let's discuss How To Find The Perimeter Of A Triangle in various scenarios. Procedure 1: Perimeter of an Equilateral Triangle: An equilateral triangle is a special case where all sides are of equal length. To find its perimeter, we will simply multiply the length of one side ('a') by 3, as there are three equal sides. So,Perimeter of an Equilateral Triangle (P) = 3a Some problems related on equilateral triangle:- (a) The measure of each side of an equilateral triangle is 5 cm, Find its perimeter. ∵ Each side of the equilateral triangle(a) = 5cm ∴ Perimeter of the equilateral triangle = 3a = (3 × 5)cm = 15 cm (b) The perimeter of an equilateral triangle is 21cm, find its each side. ∵The perimeter of equilateral triangle = 21 cm ∴ Its each side measure = 21/3 = 7cm Procedure 2: Perimeter of an Isosceles Triangle: An isosceles triangle has two sides of equal length ('a') and one side of different length ('b'). To calculate its perimeter, we will add the lengths of the two equal sides ('a') and the different side ('b'). Perimeter of an Isosceles Triangle (P) = 2a + b Some problems related on isosceles triangle: (a) The measure of two equal sides of an isosceles triangle is 7m each and third side is 5m, find the perimeter of the isosceles triangle. ∵ The measure of each equal side of the isosceles triangle(a) = 7m The measure of third side (b) = 5m ∴ Perimeter of the isosceles triangle = 2a + b = (2 × 7 + 5)m = (14 + 5)m = 19 m (b)The perimeter of an isosceles triangle is 27m,the measure of its third side is 3m, find the measure of its each equal side. Solution:- 1st Method: ∵ Perimeter of isosceles triangle = 27m Measure of its third side(b) = 3m Let, measure of each equal side = a meter 2a + b = 27 => 2 × a + 3 = 27 => 2 × a = 27 - 3 => 2 × a = 24 => a = 24/2 => a = 12 ∴ Measure of each equal side of the isosceles triangle (a) = 12 m 2nd Method: ∵ Perimeter of isosceles triangle = 27m Measure of its third side(b) = 3m ∴Measure of each equal side(a) = (Perimeter - b) / 2 = (27 -3)/2 = 24/2 = 12 m (c)The perimeter of an isosceles triangle is 20 cm, its equal sides measure 4cm, Find its third side. Solution:- 1st Method: The perimeter of an isosceles triangle = 20cm Measure of its equal sides (a) = 4cm Let, measure of its third side = b cm 2a + b = 20 => 2 × 4 + b = 20 => 8 + b = 20 => b = 20 - 8 => b = 12 ∴ Measure of third side(b) = 12 cm 2nd Method: The perimeter of an isosceles triangle = 20cm Measure of its equal sides (a) = 4cm ∴ Measure of its third side (b) = perimeter - 2a = (20 - 2 × 4) = 20 - 8 = 12 cm Procedure 3: Perimeter of a Scalene Triangle: In a scalene triangle, all three sides have different lengths ('a,' 'b,' and 'c'). To find the perimeter, we will simply add the lengths of all three sides. ∴Perimeter of a Scalene Triangle (P) = a + b + c Some Problems on scalene triangle: (a) Three sides of a scalene triangle are 3m, 4m, and 5m respectively, Find the perimeter of the triangle. ∵ 1st side (a) = 3m 2nd side (b) = 4m and 3rd side (c) =5m ∴ Perimeter of scalene triangle (P) = (3 + 4 + 5) = 12 m (b)Perimeter of a scalene triangle 15 cm, its two sides are 6cm and 7cm; find third side. Solution:- 1st Method: Perimeter of a scalene triangle (P) = 15 cm 1st side (a) = 6 cm 2nd side (b) =7 cm ∴ Third side (c)= P - (a + b) = 15 - (6+7) =15 - 13 = 2 cm 2nd Method: Perimeter of a scalene triangle (P) = 15 cm 1st side (a) = 6 cm 2nd side (b) =7 cm Let,the third side = c cm a + b + c = P => 6 + 7 + c = 15 => 13 + c = 15 => c = 15 - 13 => c = 2 ∴ Third side (c) = 2 cm (c)The perimeter of a scalene triangle is 30m and sum of its two side is 20m,find the third side. ∵ Perimeter of scalene triangle = 30m Sum of its two sides = 20m ∴ Measure of third side = Perimeter - Sum of its two sides = (30 - 20)m = 10m Let us, experiment this above formula of finding perimeter of scalene triangle with a calculator. Scalene Triangle Perimeter Calculator:- Procedure 4: Using the Pythagorean Theorem: As we all know thatThe Pythagorean Theorem is an important and fundamental principle in geometry that relates the sides of a right triangle. In a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This theorem can also be used to find the perimeter of a right triangle. Let us suppose, we have a right triangle with sides 'a' and 'b' and a hypotenuse 'c.' To find the perimeter, we would add the lengths of all three sides: Perimeter of a Right Triangle P = a + b + c This method is especially useful when dealing with right triangles, but it can also be applied to other types of triangles. Procedure 5: Semi-Perimeter and Heron's Formula: Heron's Formula is a powerful tool for finding the area of a triangle, and it can be used to find the perimeter as well. Heron's Formula depends on the semi-perimeter of the triangle, which is half of the perimeter. First, we will calculate the semi-perimeter, and then we will use it to find the perimeter. Semi-Perimeter of a Triangle (S): S = (a + b + c) / 2 Once we have the semi-perimeter, we can use Heron's Formula to find the area of the triangle: Area of a Triangle (A): A = √(S × (S - a) × (S - b) × (S - c)) Finally, to find the perimeter, we will double the semi-perimeter. Perimeter of a Triangle (P)= 2S Procedure 6: Using Trigonometry: If we know the angles and at least one side length, we can use trigonometric functions to find the other sides and then calculate the perimeter. The most commonly used trigonometric functions for this purpose are the sine, cosine, and tangent. For example, In a right triangle if we have an angle 'A' and the side 'a,' and we want to find the side 'b,' we can use the sine function: Sine Rule: sin(A) = (b / a) Solving for 'b,' we will get: b = a × sin(A) We will repeat this process for the other sides, 'b' and 'c,' using their corresponding angles 'B' and 'C.' Once we have all three side lengths, we can find the perimeter using the formula mentioned earlier for a scalene triangle: Perimeter of a Scalene Triangle (P) = a + b + c Procedure 7: Coordinate Geometry: In some cases, we may be given the coordinates of the vertices of a triangle on a coordinate plane. To find the perimeter, we can use the distance formula to calculate the lengths of the sides. Suppose we have the coordinates of the vertices A(x₁, y₁), B(x₂, y₂), and C(x₃,y₃). we can use the distance formula to find the lengths of sides 'a,' 'b,' and 'c': Distance Formula: Length of side (a) = √((x₂ - x₁)² + (y₂ - y₁)²) Length of side (b) = √((x₃ - x₂)² + (y₃ - y₂)²) Length of side (c) = √((x₁ - x₃)² + (y₁ - y₃)²) Once we have the side lengths, wecan find the perimeter as usual: Perimeter of a Triangle (P)= a + b + c Real-Life Applications: Understanding How To Find The Perimeter Of A Triangle is not just an academic exercise. It has numerous practical applications in everyday life and various fields, including: 1.Construction: Builders and architects always use knowledge of perimeter to determine the amount of material needed for projects involving triangles, such as roofing, flooring, or fencing. 2.Engineering: In structural engineering, finding the perimeter is crucial for designing load-bearing structures, bridges, and other architectural marvels. 3.Landscaping: Landscape designers often use perimeter calculations to plan garden beds, walkways, and other outdoor features. 4.Navigation: In navigation and geolocation, the perimeter of a triangle always help determine distances between points on the Earth's surface. 5.Art and Design: Artists and designers use triangular shapes in various projects. Understanding perimeter. How To Find The Perimeter Of A Triangle is a fundamental concept in geometry and trigonometry. Depending on the type of triangle and the given information, we can use various methods to calculate the perimeter. For scalene, isosceles, and right triangles, we simply add up the side lengths. In the case of an equilateral triangle, we multiply the length of one side by 3 to find the perimeter. When we are given angles rather than side lengths, the Law of Sines and the Law of Cosines become invaluable tools for determining the perimeter. By understanding these methods and applying them to different triangle scenarios, every one will be better equipped to work with triangles in various mathematical and real-world applications. Whether anyone is solving geometry problems, measuring physical objects, or designing structures, the ability to find the perimeter of a triangle is a valuable skill that serves as a building block for more advanced mathematical.......... FREQUENTLY ASKED QUESTIONs on Perimeter of a Triangle:- How do you find the perimeter of a triangle in Class 7? Answer:- As we know three sides of an equilateral triangle are equal. ∴The perimeter of an equilateral triangle = 3 × a [Where, a is measure of each equal side] An isosceles triangle has two sides equal. ∴Perimeter of an isosceles triangle = 2a + b [ Where, a = Measure of each equal side and b = Measure of unequal third side] A scalene triangle has three different sides. ∴ Perimeter of a scalene triangle = a +b + c where, a,b and c are its three different sides. Answer:- Sum of three angles of any triangle is = 180°, if sum of all three angle grater than 180°, then it will not form a triangle. What is a unique triangle? Answer:- If the sum of any two sides is greater than third side and sum of all angles of a triangle is 180°,then it is a unique triangle. What is a formula of trapezium? Answer:- The formula of perimeter of a trapezium = Sum of all its four sides. Again, Area of a trapezium = ½ (a+b) × h where, a and b are its parallel sides and h is the perpendicular distance between parallel sides. What is a formula of a rectangle? Answer:- Perimeter of a rectangle = 2(l + b), where, l = length of the rectangle and b = breadth of the rectangle Area of the rectangle = l × b where, l = length of the rectangle and b = breadth of the rectangle Was this article helpful? Share this post:- - * Please Don't Spam Here. All the Comments are Reviewed by Admin. How to Find the Perimeter of a Triangle Champak's educational blog	{"url":"https://www.maths-group.online/2023/10/how-to-find-perimeter-of-triangle.html","timestamp":"2024-11-13T08:36:44Z","content_type":"application/xhtml+xml","content_length":"527479","record_id":"<urn:uuid:1c4b4be3-752c-4904-ab76-4667be6c4a2d>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00878.warc.gz"}
The world of Algebra is a world of unknown, abstract things. Just as a detective carries out various investigations to track an unknown criminal, so do we use various algebraic operations to reach an unknown solution to a mathematical problem Algebra is taught in almost all schools across the globe. So, here’s presenting to you an overview of this subject. What is Algebra? The very word, Algebra, invokes pictures of alphabets like a, b, x, y etc. in my mind. This is because Algebra is a branch of mathematics that uses symbols and letters to represent an unknown. History of Algebra It is believed, that the ancient people of Babylonia, used an advanced system of calculations in their daily life. Such calculations are used today to solve equations in Algebra. So we can say that the Babylonians are a source of the origin of Algebra. However, the roots of Algebra has also been traced back to the ancient Indian Mathematics that had a direct influence on Muhammad bin Mūsā al – Khwārizmi (c 780 – 850), a Persian mathematician, who wrote a book in Arabic, Al – kitab al – Jabr wa – l – Muqabala, in the 9th century A.D. It is from ai - jabr that Algebra gets its name. Symbols and letters Algebra is full of symbols and letters as that is the very substance of this branch of mathematics. They are useful, as they are easier to write and save space and time. A list of few symbol used: │Symbol │ Meaning │ │ C │Composite numbers │ │ R │ Real numbers │ │ Q │ Rational numbers │ │ Z │ Integers │ │ W │ Whole numbers │ │ + │ Addition │ │ - │ Subtraction │ │ = │ Equal to │ │ > │ Greater than │ These are the alphabets that are used, to denote the unknown numbers or quantities, in Algebra. Variables may not always be alphabets; they can be anything, e.g., symbols. For example, to solve: ------- + 2 = 20 Instead of saying or writing “dash” or “empty space” every time we refer to the problem, we can use , x + 2 = 20 Here x is that unknown number which when added to 2 gives 20. Although nothing about variables has been told by the Persian mathematician in his book, variables were first used by a French mathematician, François Vièta, at the end of 16th century A.D. A few things related to Algebra Algebraic terms : These are the basic units of an algebraic expression. They consist of a number and one or more variables. For example, the term 25ab. 25 is called the Numerical co-efficient of the term. Algebraic expressions : They are a collection of algebraic terms that are separated by the various arithmetical operations and they do not have an “=” sign. For example, 25ab + 30ab – 12ab Algebraic equations : They are statements stating that two expressions are equal. So, they contain an “=” sign. For example, 7x + 4 = 12y - 9 Classifications of Algebra Elementary Algebra – This contains the basic concepts of algebra Abstract or Modern algebra – This is a further study of Elementary Algebra. Linear Algebra – This branch of algebra deals with some specific properties of vector space (including matrices). Universal Algebra – As the name suggests, it deals with the properties that are common to all algebraic structures. Algebraic number theory – Here we study the various properties of numbers as we find in the algebraic system. Algebraic combinatory : Here we use abstract algebraic methods to study combinatorial questions. Finally, to wrap it up, I would like to tell you that Algebra is that branch of mathematics that is used in Arithmetic and also in Geometry.	{"url":"http://www.icoachmath.com/article/Algebra.html","timestamp":"2024-11-05T15:11:57Z","content_type":"text/html","content_length":"37756","record_id":"<urn:uuid:9e5b04b3-f821-4b34-a8ae-38ba98a60bf0>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00349.warc.gz"}
Area and perimeter of 2D shapes Chapter 16: Area and perimeter of 2D shapes 16.1 Introduction Pi ($\pi$) is an irrational number. Expressed as a decimal number, it has an infinite number of decimals and no repeating pattern. Can you guess how many digits have been calculated for the value of $\pi$? In $2021$, a supercomputer calculated $\pi$ to $\text{62,8}$ trillion decimal places!	{"url":"https://www.siyavula.com/read/za/mathematics/grade-8/area-and-perimeter-of-2d-shapes/16-area-and-perimeter-of-2d-shapes-01","timestamp":"2024-11-03T17:15:02Z","content_type":"text/html","content_length":"92107","record_id":"<urn:uuid:230a3bde-8ed5-421b-9f76-2b5f524d12bb>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00003.warc.gz"}
Optimizing Array Search in JavaScript: Using "Set" for Efficient Lookups \| @iamtouha Optimizing Array Search in JavaScript: Using Set for Efficient Lookups When working with large datasets in JavaScript, efficiently searching for elements across two arrays can make a significant difference in performance. A common use case is finding the first common element between two large arrays. Without optimization, this can become a computationally expensive operation. Fortunately, using the Set data structure, we can greatly enhance the speed of such lookups. Let’s explore how. Problem Setup Imagine you have two large arrays: 1. list1: A array of 1,000,000 elements. 2. list2: A smaller array of 50,000 elements. 3. Elements in the two arrays are arranged such that the only common element is the last element of both arrays. this is done to find the maximum time taken to find the common element. Our goal is to find the first common element between these two lists efficiently. const list1 = Array.from({ length: 1000000 }, (_, i) => i + 1); const list2 = Array.from({ length: 50000 }, (_, i) => i + 1000000).toReversed(); In this case, the first common element is 1,000,000, but how do we find it quickly? The Naive Approach Without optimization, one might use a nested loop or methods like indexOf() or includes() to search through both arrays. However, this results in O(mn) complexity, which is far too slow for large const list1 = Array.from({ length: 1000000 }, (_, i) => i + 1); const list2 = Array.from({ length: 50000 }, (_, i) => i + 1000000).toReversed(); const common = list1.find((item) => list2.includes(item)); This approach takes about 2230 milliseconds to find the common element. The Optimized Approach: Using Set By converting one of the lists into a Set, we can optimize the search process. A Set provides O(1) average time complexity for lookups, which is much faster than the O(n) time complexity for arrays. In total this approach has a time complexity of O(m+n). Here’s how we can leverage it: 1. Convert list1 (larger array) into a Set for faster lookups. 2. Use the find() method on list2 to iterate over the elements and check if they exist in the Set. Here’s the complete code: const list1 = Array.from({ length: 1000000 }, (_, i) => i + 1); const list2 = Array.from({ length: 50000 }, (_, i) => i + 1000000).toReversed(); const set = new Set(list1); const common = list2.find((item) => set.has(item)); This optimized approach takes only about 4 millisecond to find the common element. Key Benefits of the Optimized Approach 1. Scalability: As data sizes grow, the performance impact of using a Set becomes more pronounced. The time complexity of the lookup operation is reduced from O(mn) to O(M+n). 2. Efficiency: By leveraging the strengths of Set, the optimized solution provides a much more efficient search process, suitable for handling large datasets. When faced with the challenge of finding common elements between two arrays, using Set in JavaScript is a powerful and efficient solution. It reduces the computational complexity of the operation, ensuring that your code can handle large datasets quickly and effectively. Next time you need to compare large arrays in JavaScript, remember to leverage the power of Set for faster lookups and efficient code execution!	{"url":"https://www.touha.dev/posts/optimizing-array-search-in-javascript-using-set-for-efficient-lookups","timestamp":"2024-11-13T19:06:44Z","content_type":"text/html","content_length":"31767","record_id":"<urn:uuid:a37d1bba-f84b-48d9-b406-47acf7b072cc>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00483.warc.gz"}
IF / AND argument Can anyone tell me what is wrong with the following formula? =IF(AS26<0,"G",IF(AS26>0<>AS26<5,"Y",IF(AS26>6,"R"))). The target cell is AV14. I want the target cell to equal one of three values (G,Y,R) given that the value of AS26 meets the following requirements: If AS26 is <0 = "G", If AS26 is >0 but <5 = "Y" and If AS26 is >6 = "R"... HELP! Excel Facts Round to nearest half hour? Use =MROUND(A2,"0:30") to round to nearest half hour. Use =CEILING(A2,"0:30") to round to next half hour. Hope this helps, Nov 7, 2006 Office Version 1. 2019 If the value of AS26 is BETWEEN 5 and 6 the result will be blank, is that what you want? Try this instead Chris - The value of >6 does not result in an "R" with your formula Special K99 - I'm looking for a "Y" value in that case. This for a variance report. If the variance value is below 0 ("G") if it is between 1 and 5 ("Y") and over 6 ("R") Chris - The value of >6 does not result in an "R" with your formula It does for me. Hmm - It results in a "Y" for me.... Excel 2010 I see what I did wrong.... The format is %. Once I added % to your formula, it works. Thanks for your input!	{"url":"https://www.mrexcel.com/board/threads/if-and-argument.803538/","timestamp":"2024-11-11T10:27:51Z","content_type":"text/html","content_length":"134418","record_id":"<urn:uuid:35ac46a8-d37e-4a5a-87b7-3bac4180e46f>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00377.warc.gz"}
Top 10 data mining algorithms in plain English - Hacker Bits Top 10 data mining algorithms in plain English Today, I'm going to explain in plain English the top 10 most influential data mining algorithms as voted on by 3 separate panels in this survey paper. Once you know what they are, how they work, what they do and where you can find them, my hope is you'll have this blog post as a springboard to learn even more about data mining. What are we waiting for? Let's get started! C4.5 constructs a classifier in the form of a decision tree. In order to do this, C4.5 is given a set of data representing things that are already classified. k-means creates $k$ groups from a set of objects so that the members of a group are more similar. It’s a popular cluster analysis technique for exploring a dataset. Support vector machine (SVM) learns a hyperplane to classify data into 2 classes. At a high-level, SVM performs a similar task like C4.5 except SVM doesn’t use decision trees at all. The Apriori algorithm learns association rules and is applied to a database containing a large number of transactions. In data mining, expectation-maximization (EM) is generally used as a clustering algorithm (like k-means) for knowledge discovery. PageRank is a link analysis algorithm designed to determine the relative importance of some object linked within a network of objects. AdaBoost is a boosting algorithm which constructs a classifier. As you probably remember, a classifier takes a bunch of data and attempts to predict or classify which class a new data element belongs to. kNN, or k-Nearest Neighbors, is a classification algorithm. However, it differs from the classifiers previously described because it’s a lazy learner. Naive Bayes is not a single algorithm, but a family of classification algorithms that share one common assumption: Every feature of the data being classified is independent of all other features given the class. CART stands for classification and regression trees. It is a decision tree learning technique that outputs either classification or regression trees. Like C4.5, CART is a classifier. Now it's your turn... Now that I've shared my thoughts and research around these data mining algorithms, I want to turn it over to you. • Are you going to give data mining a try? • Which data mining algorithms have you heard of but weren't on the list? • Or maybe you have a question about an algorithm? Let me know what you think by leaving a comment below right now. Thanks to Yuval Merhav and Oliver Keyes for their suggestions which I've incorporated into the post. Thanks to Dan Steinberg (yes, the CART expert!) for the suggested updates to the CART section which have now been added. Comments 150 Joe Guy Your explanation of SVM is the best I have ever seen. Thanks! Raymond Li Thanks, Joe. Definitely appreciate it! 🙂 I owe a lot of it to a few threads from Reddit and Yuval (both are linked in the post above). Heather Stark Roger Huang Really snappy and informative view into data mining algorithms. I clicked on a whole ton of links: always a mark of a resource done right! Kudos. Raymond Li Thanks, Roger. I’m happy you found it snappy and click-worthy. 🙂 Sometimes data mining resources can be a bit on the dry side. Thanks for the excellent compile This is what I was looking for as a starter. Ray Li Thanks, Lakshminarayanan! 4. Pingback: Els 10 primers algoritmes del Big data explicats en paraules \| Blog d'estadÃstica oficial 5. Pingback: LessThunk.com « Top 10 data mining algorithms in plain English — recommended for big data users Out of all the numerous websites about data mining algorithms I have gone through, this one is by far the best! Explaining everything in such casual terms really helps beginners like me. The examples were definitely apt and helpful. Thank you so much! You made my work a lot easier. 🙂 Ray Li I’m excited to hear this helped with your work, Meghana! And really appreciate the kind words. 🙂 Vrashabh Irde This is an awesome list! Thanks. Trying to dabble into ML myself and having a simple know how of everything is very useful Ray Li Very glad to hear you find it useful, Vrashabh. Thank you! 8. Pingback: Distilled News \| Data Analytics & R Excellent man, this is so well explained. Thanks!! Ray Li My pleasure, Kyle. 🙂 Thanks for the excellent share Ray Li My pleasure, Suanfazu! Thanks for exploring the blog and leaving your kind words. 🙂 Hey, great introduction! I would love to see more posts like this in our community; great way to grasp the concept of algorithms before diving into the hard math. Just one thing, though: On Step 2 in Naive Bayes you repeated P(Long \| Banana) twice. The third one should be P(Yellow \| Banana). Thanks again! Ray Li Hi Anonymous, Nice catch! I fixed it now, but have no one to attribute the fix to. 🙁 I totally agree about understanding the concepts of the algorithm before the hard math. I’ve always felt using concepts and examples as a platform for understanding makes the math part way Thanks again, Robert Klein This is a great resource. I’ve bookmarked it. Thanks for your work. I love using height-zip code to illustrate independence. That will be a go-to for me now. The only thing I can offer in return is a heads-up about the API we just released for ML preprocessing. It’s all about correlating themes in unstructured information streams. Hope it’s useful. Let us know what you think. Thanks Ray Li Thanks for bookmarking and the heads-up, Robert! 🙂 Hello Ray, Thanks for a great article. It looks like there is a typo in step 2 of Naive Bayes. One of the probabilities should be P(Yellow\|Banana). Thanks again! Ray Li My pleasure, Raghav. Thanks also for letting me know about the typo. It should be corrected now. Hello Raymond, first of all kudos for your sum up of data mining algos! I’ve been exploring this for a few weeks now (mainly using scikit learn and nltk in python). In the past few days I came up with the idea to create a classifier that is able to group products by their title to a corresponding product taxonomy. For that I crawled a German product marketplace for their category landingpages and created a corpus consisting of a taxonomy tree node in column “a” and a set of snowball stemmed relevant uni and bigram keywords ( appx. 50 per node) that have been extracted from all products on each category page (this is comma separated in column “b”). Now I would like to build a classifier from that with the idea in mind, that I could throw stemmed product titles at the classifier and let it return the most probable taxonomy node. Could you advise which would be the most appropriate one for the given task. I can email you the corpus… Hope to get some direction… to omit any detours / too much trial and error. Looking forward to your reply. Thanks again for your great article. Cheers from Cologne Germany Ray Li Hi Jens, Thanks for the kudos and taking the time to leave a comment. Short answer to your question… I don’t know. 🙂 It sounds like there’s a bunch I could learn from you! For example: You just taught me about stemming and the Snowball framework. Honestly, I’m amazed there are tools like Snowball that can create stemming algorithms. Very cool! Longer answer… I found the StackOverflow.com, stats.stackexchange.com and reddit.com forums invaluable when I was learning, researching and simplifying the algorithms to make them easier to describe. Sorry I couldn’t be more help, but I’m working to catch up… 🙂 Hi Ray, thanks for your feedback 🙂 I found a good solution in the meantime using a naive bayes approach. By the way your regular contact form does not work. There is an htaccess authentication popping up upon form submit. Ray Li Also, thanks for the heads up about the contact form. It should be fixed now. There’s a small issue with the confirmation message (some fields are not displayed), but no more auth pop-up and the message successfully sends. This goes in my bookmarks. Excellent simple explanation. Loved you have taken SVM. It would be great if you can put Neural network with various kernels. Ray Li Definitely appreciate the bookmark, Malhar! Thanks for your suggestion about the neural nets. I’ll definitely be diving into that one very soon. Exactly the same concern, Malhar. I was looking for information on Neural Networks as well. Man, I really wish I had this guide a few years ago! I was trying my hand at unsupervised categorization of email messages. I didn’t know what terms to google, so the only thing I used was LSM (latent semantic mapping). The problem is, when you have thousands of words and tens of thousands of emails, the N^2 matrix gets a little hard to handle, computationally. I ended up giving up on What I had never considered was using a different algorithm to pre-create groups, which would have helped a lot. This was a useful read. Ray Li Thanks for reading and your kind words, Serge! 17. Pingback: The Data Scientist - Professional Data Science in Singapore » 10 Data Science Algorithms Explained – In English Great article! Now, as a public service, how about a decision tree or categorization matrix for selecting the right algorithm? Ray Li Thanks, David. It’s a good call about selecting the right algorithm. From all the readings so far, I feel picking the right one is the hardest part. It’s one of the main reasons I was attracted to the original survey paper despite it being a bit outdated. Might as well dive into the ones the panelists thought were important, and then figure out why they use them. I certainly have a lot more to learn, and I’m already having some ideas on future posts. D Lego Good post. It is curious, I’m write one version in spanish about this same theme. Ray Li Thank you, D Lego. I’m curious — can you email me the link? michael davies Great work Raymond Ray Li Appreciate it, Michael! Sthitaprajna Sahoo Couldn’t ask for more simpler explanation. A very good collection and hoping more posts from you . Ray Li My pleasure, Sthitaprajna. Stephen Oman This is a really excellent article with some nice explanations. Looking forward to your piece on Artificial Neural Networks too! Ray Li Thanks, Stephen! Richard Grigonis Including Decision Forests would have been nice. Ray Li Although I haven’t used that one myself, that’s a good one, Richard! Daniel Zilber Thanks for the write up! Ray Li Appreciate it, Daniel. 🙂 Sylvio Allore It is a good review of things undergraduates learn but what about starting with just a single example of application in predicting stock returns, for example. Do you have an example of applying, for example, naive Bayes to predicting stock returns? That would be more useful that listing a set of methods one can find in most ML books. Ray Li Thanks, Sylvio. I appreciate the constructive comments. Depth and real-life applications are certainly something to improve on in this article series (Yep… I think it deserves to be a series!). Stay tuned… 🙂 Ray Li Super excited about this… Due to all your comments and sharing, this article has been reposted to KDnuggets, a leading resource on data mining: http://bit.ly/1AoicbW! There’s no way this could’ve happened without you reading, commenting and sharing. My sincerest thank you! 🙂 Matt Cairnduff Echoing all the sentiments above Ray. This is a tremendously useful resource that’s gone straight into my bookmarks. Really appreciate the informal writing style as well, which makes it nice and accessible, and easy to share with colleagues! Ray Li Thank you, Matt. I’m glad you found the writing style accessible and shareable. Please do share… 🙂 Adriana Wilde Excellent blogpost! Very accessible and rather complete (apart from multilayer perceptrons, which I hope you’ll touch in a follow up post). I found useful that you refer to the NFL theorem and list characteristics of each algorithm which make them more suited to one type of problem than another (e.g. lazy learners are faster in training but slower classifiers, and why). I also liked you explained which algorithms are for supervised and unsupervised learning. These are all things to take into account when choosing a classifier. Wish I read this 5 years ago! Ray Li Hi Adriana, Thank you for your kind words. I think I came across the standard perceptron while researching SVM. Definitely thinking about tackling MLPs and more recently all the buzz about deep learning at some point. Thanks for your insightful comment. brian piercy What an awesome article! I learned more from this than 20 hours of plowing through SciKit. Well done! Ray Li Appreciate it, Brian! 🙂 david berneda Thanks a lot Ray for your article ! I did a clustering library sometime ago, your article encourages me to try expanding it with more algorithms. Ray Li My pleasure, David. Martin Campbell This is a fantastic article and just what I needed as I start attempting to learn all this stuff. I’ll be shooting up the Kaggle rankings in now time (well, from 100,000 to 90,000 perhaps!). Ray Li Appreciate it, Martin. I’m really happy to hear that it helps to get the ball rolling for you. Your increased Kaggle ranking would be nice icing on the cake! 🙂 Yolande Tra Excellent overview. You have a gift in teaching complex topics into down-to earth terms. Here is my comment: when using data mining algorithm, in this list (classifiers) I am more concerned about accuracy. We can try and use each one of these but in the end we are interested in validation after training. Accuracy was only addressed with SVM and Adaboost. Ray Li Thank you for your kind words, Yolande. It’s a good point about the accuracy. I’ll definitely keep this in mind to explore accuracy in an upcoming post. Maksim Gayduk I didn’t quite understand the part about C4.5 pruning. In the link provided, it says that in order to decide whether to prune a tree or not, it calculates error rate of both pruned and unpruned tree and decides which one leads to the lower limit of confidence interval. It should work okey for already pruned trees, but how does it start? Usually decision tree algorhythms build the tree until it reaches entrophy = 0, which means zero error rate, and zero upper limit for confidence interval. In this case, such tree can never be pruned, using that logic … Ray Li This is a great question, Maksim. It got me thinking a bunch, but unfortunately I don’t have an answer that I’m satisfied with. My investigation so far indicates that the error rate for the training data is distinct from the estimated error rate for the unseen data. As you pointed out, this is what the confidence interval is meant to bound. Based on the formula in the link, given f=0, I’m also at a loss on how a pruned tree could beat the unpruned tree. If you’re up for it, CrossValidated or StackOverflow might be an awesome place to get your question answered. You or I could even post a link here for reference. Ilan Sharfer Ray, thanks a lot for this really useful review. Some of the algorithms are already familiar to me, others are new. So it surely helps to have them all in one place. As a practical application I’m interested in a data mining algorithm that can be used in investment portfolio selection based on historical data, that is, decide which stocks to invest in and make timely buy/sell orders. Can you recommend a suitable algorithm? Ray Li My pleasure, Ilan. Same here, I’ve come across a few of these algorithms before writing this article, and I had to teach myself the unfamiliar ones. I’m planning to go into more practical applications in an upcoming post. Stay tuned for that one… 🙂 On a side note, you might already be aware of them, and the “random walk hypothesis” and “efficient-market hypothesis” might be of interest to you. It doesn’t answer your question, but it is an alternate perspective on predicting future returns based on historical data. Awesome explanation! Ray Li Much appreciated, Zeeshan. Lalit A Patel This is an excellent blog. It is helping me digest what I have studied elsewhere. Thanks a lot. Ray Li Thank you, Lalit. I’m happy to hear the blog is helping you with your studies. Fantastic post ray. Nicely explained. Helped me enhancing my understanding. Please keep sharing the knowledge 🙂 It helps. Ray Li Thanks, Phaneendra. More is definitely on the way… 🙂 Adrian Cuyugan These are very good and simple explanation. Thank you for sharing! Ray Li Appreciate it, Adrian. 39. Pingback: BirdView (2) – Ranking Everything: an Overview of Link Analysis Using PageRank Algorithm \| datawarrior Peter Nour Thanks Ray! This is a fantastic post with great details and yet so simple to understand. Ray Li Much appreciated, Peter. Glad you liked the post. Awesome explanation of some of the oft-used data-mining algorithms. Are you thinking of doing something similar for some of the other algorithms (Discriminant Analysis, Neural Networks, etc.) as well? Would love to read your posts on them. Ray Li Thanks, Sanjoy. Those are good ones. NNs are definitely at the top of the list. Thanks Ray!! Awesome compilation and explanation. This truly helps me get started with learning and applying data science. Ray Li My pleasure, Suresh. I’m really happy to hear the post helped you start learning and applying. I’m afraid to be rather boring by having nothing to contribute than more of the well deserved praise to the quality of your article: thanks, really a great wrap-up and very good primer for the I shared the link to your post on the intranet of my company and rarely an article has received so many “likes” in no time. The only thing I was missing was a bit more visual support. You have an excellent video embedded for SVM. But for many of the other concepts, there are also rather straight forward visual representations possible (e.g. clustering, k-nearest-neighbour). I found the book “Data Science for Business” (http://www.data-science-for-biz.com/) a VERY good start into the subject (….though I would have prefered to have read your article beore, as it really wraps it up so well….). This book offers real real inspiration as to how the underlying concepts of the algorithms you explain can be visualized and thus be made more intuitively Enhancing your article with a bit more visual support would be the cherry on the icing on the cake 😉 Ray Li Hi Ulf, Really appreciate your kind words and you sharing it with your colleagues. 🙂 That’s a good point about visualizations… especially for visual learners. Like in the case of the SVM video, I found seeing it in action made it so much clearer. I definitely appreciate the book recommendation. From the sound of it, that book might be a fantastic reference not just for this article but for future articles covering this area. Thanks again, Praveen G S Thanks for your wonderful post. I like the way you describe the SVM, kNN, Bayes. Since you language is so user friendly and easy to understand. Can you also write a blog on the some of the ensembles like random forest which is one of the most popular machine learning algorithm and has a good predictive power compared to other algorithms Ray Li Thanks, Praveen. Those are good ones, and I’ll add them to my growing list of potential algorithms to dive into. Tom F Fantastic article. Thanks. One point: >> What do the balls, table and stick represent? The balls represent data points, and the red and blue color represent 2 classes. The stick represents the simplest hyperplane which is a line. The simplest (i.e. 1 dimensional) hyperplane is a point, not a line. Ray Li Thanks, Tom. Good “point” about the simplest hyperplane. I’ve modified the sentence to read “The stick represents the hyperplane which in this case is a line.” Hi Ray, All Algorithms are explained in a simple and neat manner. It will be extremely useful for beginners as well as pros if u could come up with a “cheat sheet”, explaining best and worst scenario, for each algorithms. ( I mean how to choose the best algorithm for a given data). Thank you Ray Li Appreciate your kind words, vdep! Thanks also for your suggestion about the “cheat sheet.” 🙂 Hi Ray, Thank you for your effort to explain such algorithms with such simplicity. Good to start on data science ! Ray Li My pleasure, Houssem! 48. Pingback: ‘PoesÃa eres tÃº’ se suma a la IA: ahora compone y recita poemas \| RubÃ©n Hinojosa Chapel - Blog personal 49. Pingback: Linkblog #6 \| Ivan Yurchenko 50. Pingback: DB Weekly No.59 \| ENUE Blog Excellent simplified approach! Ray Li Thanks, Paris! Much appreciated… 🙂 52. Pingback: Very interesting explainer: Top 10 data mining algorithms in plain English rayli.net/blog/data/top-10-dat… (via @TheBrowser) \| Stromabnehmer 53. Pingback: æœºå™¨å¦ä¹ (Machine Learning)&æ·±åº¦å¦ä¹ (Deep Learning)èµ„æ–™(Chapter 1) \| ~ Code flavor ~ 54. Pingback: Data Lab Link Roundup: python pivot tables, Hypothesis for testing, data mining algorithms in plain english and more… \| Open Data Aha! The latest downloadable Orange data mining suite and its Associate add-on doesn’t seem to be using Apriori for enumerating frequent itemsets but FP-growth algorithm instead. I must say it’s MUCH faster now. 😀 Ray Li Thanks, Kurac. 56. Pingback: Simulando, visualizando ML, algoritmos, cheatsheet y conjuntos de datos: Lecturas para el fin de semana \| To the mean! is there any searching technique algorithm in data mining ..please help me.. Ray Li Yes, even within the context of the 10 data mining algorithms, we are searching. The first 3 that come to mind are K-means, Apriori and PageRank. K-means groups similar data together. It’s essentially a way to search through the data and group together data that have similar attributes. Apriori attempts to search for relationships and patterns among a set of transactions. Finally, PageRank searches through a network in order to unearth the relative importance of an object in the network. Hope this helps! Ray Li However, if you’re looking for a search algorithm that finds specific item(s) that match certain attributes, these 10 data mining algorithms may not be a good fit. This article is so helpful! I’ve always have trouble understanding the Naive Bayes and SVM algorithms. Your article has done a really great job in explaining these two algorithms that now I have a much better understanding on these algorithms. Thanks alot! 🙂 Ray Li Glad you found the article helpful, Jenny. Thanks for the kind words! Thank you! David Millie very nice summary article … question – is the current implementation of Orange (still) using C4.5 as the classification tree algorithm … I cannot find any reference to it in the current Ray Li Thanks, David. This might help: http://orange.biolab.si/docs/latest/reference/rst/Orange.classification.tree.html. Orange includes multiple implementations of classification tree learners: a very flexible TreeLearner, a fast SimpleTreeLearner, and a C45Learner, which uses the C4.5 tree induction Hope this helps! Good job! 🙂 This is a great resource for a beginner like me. Ray Li Thank you, Mak! Jermaine Allgood THANK YOU!!!!!!! As a budding data scientist, this is really helpful. I appreciate it immensely!!!!! Ray Li Thanks, Jermaine! Good luck in your data scientist journey. 🙂 Bruno Ferreira Thank very much for this article. This is from a far the best page about the most used data-mining algorithms. As a data-mining student, this was very helpful. Ray Li My pleasure, Bruno. Thanks for the kind words! Great article, Ray, top level, thank you so much! This question could be a bit OT: which technique do you feel to suggest for the analysis of biological networks? Classical graph theory measures, functional cartography (by Guimera & Amaral), entropy and clustering are already used with good results. PageRank on undirected networks provides similar results to betweenness centrality, I am looking for innovative approaches to be compared with the mentioned ones. Thanks again! Ray Li Thank you, Paolo. Really appreciate it! From the techniques you’ve already mentioned, it sounds like you’re already deep into the area of biological network analysis. Although I don’t have any new approaches to add (and probably not as familiar with this area as you are), perhaps someone reading this thread could point us in the right direction. Wonderful list and even more wonderful explanations. Question though, you don’t think Random Forests merit a place on that list? Ray Li Thanks, Abdul! Random forests is a great one. However, the authors of the original 2007 paper describe how their analysis arrived at these top 10. If a similar analysis were done today, I’m sure random forest would be a strong contender. Ok. Fair enough Again, nice work I did not read the whole article, but the description of the Apriori algorithm is incorrect. It is said that there are three steps and that the second step is “Those itemsets that satisfy the support and confidence move onto the next round for 2-itemsets.” This is incorrect and it is not how the Apriori algorithm works.. The Apriori algorithms does NOT consider the confidence when generating itemsets. It only considers the confidence after finding the itemsets, when it is generating the rules. In other words, the Apriori algorithms first find the frequent itemsets by applying the three steps. Then it applies another algorithm for generating the rules from these itemsets. The confidence is only considered by the second algorithm. It is not considered during itemset generation. 67. Pingback: d204: Top 10 data mining algorithms explained in plain English [nd009 study materials] – AI 68. Pingback: Top 10 data mining algorithms in plain English \| rayli.net – Unstable Contextuality Research Aftab khan This information is very helpful for the students like me. I was searching for an algorithm for my final year project in data mining. Now i can easily select an algorithm to start my work on my final year project. Thanks Kirk Paul Lafler Fantastic explanation of the top data mining algorithms. Thank you for sharing! Sokolyk Petro Thank you, Mr. Ray Li. Your explanation is much easier to understand for beginners. This site uses Akismet to reduce spam. 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On the Stability of Shear Flows in Bounded Channels, I: Monotonic Shear Flows We discuss some of our recent work on the linear and nonlinear stability of shear flows as solutions of the 2D Euler equations in the bounded channel T×[0,1]. More precisely, we consider shear flows u=(b(y),0) given by smooth functions b:[0,1]→R. We prove linear inviscid damping and linear stability provided that b is strictly increasing and a suitable spectral condition involving the function b is satisfied. Then we show that this can be extended to full nonlinear inviscid damping and asymptotic nonlinear stability, provided that b is linear outside a compact subset of the interval (0, 1) (to avoid boundary contributions which are not compatible with inviscid damping) and the vorticity is smooth in a Gevrey space. In the second article in this series we will discuss the case of non-monotonic shear flows b with non-degenerate critical points (like the classical Poiseuille flow b:[-1,1]→R, b(y)=y^2). The situation here is different, as nonlinear stability is a major open problem. We will prove a new result in the linear case, involving polynomial decay of the associated stream function. All Science Journal Classification (ASJC) codes • 35B40 • 35P25 • 35Q31 • Euler equations • Linear inviscid damping • Monotonic shear flows Dive into the research topics of 'On the Stability of Shear Flows in Bounded Channels, I: Monotonic Shear Flows'. Together they form a unique fingerprint.	{"url":"https://collaborate.princeton.edu/en/publications/on-the-stability-of-shear-flows-in-bounded-channels-i-monotonic-s","timestamp":"2024-11-10T02:17:51Z","content_type":"text/html","content_length":"51263","record_id":"<urn:uuid:3d729cf4-a672-4ed4-b7f1-1002c7b77dc8>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00484.warc.gz"}
Powerful and Useful Online Calculators - Your Go-To Hub for All Essential Calculations - Page 2 Discover how a personal loan calculator can help you understand monthly payments, total loan cost, and APR. Empower your financial decisions with this comprehensive guide. Understanding Binary Numbers for Binary Calculations Posted on: Understanding Binary Numbers for Binary Calculations: Learn the basics of binary numbers, essential operations, conversions, and real-life applications in computing and digital devices. Future Value Calculator Posted on: Curious about your investment’s future worth? Our Future Value Calculator article guides you through optimizing financial decisions to maximize returns. Learn more now! Science & Technology Preview Area Calculator Widget Posted on: Quickly find the surface area of shapes like rectangles, triangles, circles, and more with the Area Calculator Widget. Easy to use with step-by-step solutions! APR Calculator for Accurate Loan Assessments Posted on: Understand loan costs with our detailed guide on using an APR calculator for accurate assessments. Learn to calculate real APR and make informed financial decisions. Percentage Difference Calculator for Accurate Math Solutions Posted on: Easily compare numbers with our Percentage Difference Calculator. Our guide helps you accurately measure differences in prices, stats, or any values. Read more! Return on Investment (ROI) Calculator Posted on: Calculate your investment profitability easily with our ROI Calculator. Get insights on returns, risks, and make informed financial decisions. Dive into your investment performance today! Using the Right Triangle Calculator Posted on: Learn how to find missing measurements of right triangles using an online right triangle calculator. Solve for side lengths, angles, area, perimeter, and more easily! Business Loan Calculator: A Financial Tool for Entrepreneurs Posted on: Discover how a business loan calculator helps entrepreneurs estimate payments, calculate interest, and understand the true cost of borrowing for informed financial decisions. Mixed Fraction Calculator Posted on: Mixed Fraction Calculator: Simplify conversions of mixed numbers to improper fractions. Learn the step-by-step guide and use the calculator for effortless results!	{"url":"https://calculatorbeast.com/page/2/","timestamp":"2024-11-13T11:01:46Z","content_type":"text/html","content_length":"134609","record_id":"<urn:uuid:2f5e3159-bdca-46ef-990f-810dc64b2543>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00262.warc.gz"}
Do you find some mathematical activities useless? 04-01-2018, 10:09 PM Post: #23 John Keith Posts: 1,064 Senior Member Joined: Dec 2013 RE: Do you find some mathematical activities useless? (04-01-2018 02:30 PM)rprosperi Wrote: I never purchase top-of-the-line graphics cards or CPUs, they are always priced stupidly high for 1%-ers that want that kind of edge for gaming (or more recently, mining), however as the top of the line moves up, all the mid-range move up as well, and typically at the same price points, so this year's mid-range is as much as 75-100% faster than last years, but at the same price. So I get the benefit of those silly top-end components, without buying one. Bottom line is I'm glad all those folks are buying top-end components for mining. That can be said for computer hardware in general. The "state of the art" almost always has a lousy price-performance ratio and is "obsolete" a week after you buy it. Back on topic, I use only simple math in my day job but I enjoy programming and solving math problems in my spare time. It helps to prevent my aging brain from fossilizing. User(s) browsing this thread: 1 Guest(s)	{"url":"https://hpmuseum.org/forum/showthread.php?tid=10412&pid=94179&mode=threaded","timestamp":"2024-11-03T10:48:41Z","content_type":"application/xhtml+xml","content_length":"35709","record_id":"<urn:uuid:c3585633-cbfb-4d94-90f0-c92a15768670>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00124.warc.gz"}
Lab 1G: What’s the FREQ? Lab 1G - What's the FREQ? Directions: Follow along with the slides, completing the questions in blue on your computer, and answering the questions in red in your journal. Clean it up! • In Lab 1F, we saw how we could clean data to make it easier to use and analyze. – You cleaned a small set of variables from the American Time Use (ATU) survey. – The process of cleaning and then analyzing data is very common in Data Science. • In this lab, we'll learn how we can create frequency tables to detect relationships between categorical variables. – For the sake of consistency, rather than using the data that you cleaned, you will use the pre-loaded ATU data. – Use the data() function to load the atu_clean data file to use in this lab. How do we summarize categorical variables? • When we're dealing with categorical variables, we can't just calculate an average to describe a typical value. – (Honestly, what's the average of categories orange, apple and banana, for instance?) • When trying to describe categorical variables with numbers, we calculate frequency tables Frequency tables? • When it comes to categories, about all you can do is count or tally how often each category comes up in the data. • Fill in the blanks below to answer the following: How many more females than males are there in our ATU data? tally(~ ____, data = ____) 2-way Frequency Tables • Counting the categories of a single variable is nice, but often times we want to make comparisons. • For example, what if we wanted to answer the question: – Does one gender seem to have a higher occurrence of physical challenges than the other? • We could use the following plot to try and answer this question: bargraph(~phys_challenge \| gender, data = atu_clean) • The split bargraph helps us get an idea of the answer to the question, but we need to provide precise values. • Use a line of code, that’s similar to how we facet plots, to obtain a tally of the number of people with physical challenges and their genders. □ Write down the resulting table. Interpreting 2-way frequency tables • Recall that there were 1153 more women than men in our data set. – If there are more women, then we might expect women to have more physical challenges (compared to men). • Instead of using counts we use percentages. • Include: format = "percent" as an option to the code you used to make your 2-way frequency table. – Does one gender seem to have a higher occurrence of physical challenges than the other? If so, which one and explain your reasoning? • It’s often helpful to display totals in our 2-way frequency tables. – To include them, include margins = TRUE as an option in the tally function. Conditional Relative Frequencies • There is as difference between phys_challenge \| gender and gender \| phys_challenge! tally(~phys_challenge \| gender, data = atu_clean, margin = TRUE) ## gender ## phys_challenge Male Female ## No difficulty 4140 5048 ## Has difficulty 530 775 ## Total 4670 5823 tally(~gender \| phys_challenge, data = atu_clean, margin = TRUE) ## phys_challenge ## gender No difficulty Has difficulty ## Male 4140 530 ## Female 5048 775 ## Total 9188 1305 Conditional Relative Frequencies, continued • At first glance, the two-way frequency tables might look similar (especially when the margin option is excluded). Notice, however, that the totals are different. • The totals are telling us that R calculates conditional frequencies by column! • What does this mean? – In the first two-way frequency table the groups being compared are Male and Female on the distribution of physical challenges. – In the second two-way frequency table the groups being compared are the people with No difficulty and those that Has difficulty on the distribution of gender. • Add the option format = "percent" to the first tally function. How were the percents calculated? Interpret what they mean. On your own • Describe what happens if you create a 2-way frequency table with a numerical variable and a categorical variable. • How are the types of statistical investigative questions that 2-way frequency tables can answer different than 1-way frequency tables? • Which gender has a higher rate of part time employment?	{"url":"https://curriculum.idsucla.org/unit1/lab1g/","timestamp":"2024-11-11T08:21:48Z","content_type":"text/html","content_length":"75847","record_id":"<urn:uuid:08891c7c-4cc8-4b1f-b701-ca052bb0d407>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00435.warc.gz"}
simplify sqrt(x/y^2)y simplify sqrt(x/y^2)y Hello my problem is with a bigger equation but the problem is bassicly this : sage: var('x','y', domain=real) sage: assume(x>0) sage: assume(y>0) sage: e = sqrt(1/y^2)y sage: e.simplify_full() sage: e = sqrt(x/y^2)y sage: e.simplify_full() Anyone explain me how to get just sqrt(x) and why it doesn't do directly? Thanks 2 Answers Sort by ยป oldest newest most voted I hope there is no way to coerce Sage into returning just x, because your expression is only equal to that for a very limited number of choices for x. Sage (Maxima, really) can be convinced to make branch choices to simplify the expression to sqrt(x), but not via assumptions, apparently. This does work: sage: var('x,y') (x, y) sage: E=sqrt(x/y^2)*y sage: E.canonicalize_radical() See the documentation of canonicalize_radical for details. edit flag offensive delete link more thank you, I edited miguelython ( 2015-09-27 11:43:56 +0100 )edit Yes, I mean just "x" part so sqrt(x)... sorry for not to be clear. Thanks. It works, but I wonder why canonicalize_radical isn't part of simplify_full? edit flag offensive delete link more This is because sage: x = var('x', domain='real') sage: assume(x<0) sage: sqrt(x^2).canonicalize_radical() and I guess you don't want this (note the x<0 assumption). eric_g ( 2015-09-20 23:28:18 +0100 )edit Miguel, note that sqrt(x^2) != x for x<0. rws ( 2015-09-21 09:50:22 +0100 )edit I don't want this beacause is false. I want sqrt(x^2)=abs(x) or in case ofassume(x>0) or assume(x<0) : sqrt(x^2)=x or sqrt(x^2)=-x resp miguelython ( 2015-09-27 12:17:05 +0100 )edit I mean if I had the time and knowleage, I change it in the source code. Nevertheless, I'm thankful to all community of sagemath miguelython ( 2015-09-27 12:24:09 +0100 )edit A fix should be possible with pynac-0.4.x (pynac is part of Sage); pynac git master already does sqrt(x^2) --> x for x>0 as side effect of other changes. rws ( 2015-09-27 16:39:58 +0100 )edit	{"url":"https://ask.sagemath.org/question/29514/simplify-sqrtxy2y/","timestamp":"2024-11-12T13:42:37Z","content_type":"application/xhtml+xml","content_length":"68349","record_id":"<urn:uuid:5876d2b2-2b73-48ef-adb9-8645f3e5b63c>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00290.warc.gz"}
1D maps The period-doubling accumulation point in unimodal map of 8-th power. For 1D maps with three extrema - the limit of period-doubling on a curve in the 3D parameter space defined by the condition "extremum to extremum to extremum". More general systems E-point may appear generically in codimension 7. In some cases the pseudo-E type is expected to occur in codimension 3 (as an intermediate asymptotics). RG equation The fixed point The orbital scaling factor Critical multiplier Relevant eigenvalues: CoDim=7 (restr. 3)	{"url":"https://sgtnd.narod.ru/science/alphabet/eng/doubling/e.htm","timestamp":"2024-11-06T14:49:11Z","content_type":"text/html","content_length":"7115","record_id":"<urn:uuid:e1f6c995-e214-4614-878e-41f062de68f8>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00099.warc.gz"}
the slope of demand curve is not used to measure the price elasticity of demand the slope of line cannot have negative \| Question AI The slope of a demand curve is not used to measure the price elasticity of demand because the slope of a line cannot have a negative value. the measurement of slope is sensitive to the units chosen for price and quantity. the slope of a linear demand curve is not constant. the slope of the demand curve does not tell us how much quantity changes as price changes. Not your question?Search it Lucretia WuProfessional · Tutor for 6 years # Explanation:<br />## Step 1: Understanding the concept of slope of a demand curve<br />The slope of a demand curve is defined as the rate of change of quantity demanded with respect to the price. Mathematically, slope of a line is calculated as the change in y-values (quantity in this case) for every unit change in x-values (price in this case).<br /> <br />## Step 2: Understanding the concept of price elasticity of demand<br />Price elasticity of demand is a measure of the responsiveness or sensitivity of quantity demanded to a change in price. It's calculated as the percentage change in quantity demanded divided by the percentage change in price.<br /><br />## Step 3: Analyzing the difference between the concepts<br />While both the slope of a demand curve and price elasticity of demand measure the responsiveness of quantity demanded to a change in price, the latter accounts for the relative changes (percentage changes) rather than absolute changes. Therefore, the price elasticity measure isn't influenced by the unit of measurement for price and quantity. <br /><br />## Step 4: Conclusion <br />The measurement of slope is sensitive to the units chosen for price and quantity which leads to differences in interpretation if the units vary. In contrast, the measure of price elasticity of demand gives a unit-free measure ensuring consistency in interpretation regardless of the units used. This explains why the slope of a demand curve is not used to measure the price elasticity of demand. <br /><br /># Answer: <br />The measurement of slope is sensitive to the units chosen for price and quantity. Step-by-step video The slope of a demand curve is not used to measure the price elasticity of demand because the slope of a line cannot have a negative value. the measurement of slope is sensitive to the units chosen for price and quantity. the slope of a linear demand curve is not constant. the slope of the demand curve does not tell us how much quantity changes as price changes. All Subjects Homework Helper	{"url":"https://www.questionai.com/questions-tpSfwqBtYY/slope-demand-curve-used-measure-price-elasticity-demand","timestamp":"2024-11-10T22:18:49Z","content_type":"text/html","content_length":"103031","record_id":"<urn:uuid:f381c8e0-753d-4dab-9119-9aec73216ee5>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00205.warc.gz"}
Reflection Loss The field strength $\underset{_}{{E}_{r}}$ of the reflected wave is computed using the reflection factor $\underset{_}{{R}_{12}}$ and the field strength $\underset{_}{{E}_{i}}$ of the incident wave^1 The reflection factor $\underset{_}{\|{\underset{_}{{R}_{12}}}^{"}\|}$ is computed from the transmission factor $\underset{_}{\|{T}_{12}\|}$: Additionally, the roughness is considered by: $\|\underset{_}{{R}_{12}}\|=\|{\underset{_}{{R}_{12}}}^{"}\|\cdot {\rho }_{0}$ A smooth surface has a value of ${\rho }_{0}=1.0$ The value of $\|{R}_{12}\|$ is shown in column L in the table. The reflection loss is then computed as follows: ${L}_{R}=20\cdot \mathrm{log}\|\underset{_}{{R}_{12}}\|$ G. Wölfle: Adaptive Modelle für die Funknetzplanung und zur Berechnung der Empfangsqualität in Gebäuden. (Adaptive Propagation Models for the Planning of Wireless Communication Networks and for the Computation of the Reception Quality inside Buildings). PhD Thesis, University of Stuttgart, published by Shaker Verlag, 2000.	{"url":"https://help.altair.com/winprop/topics/winprop/user_guide/wallman_tuman/materials/reflection_loss_winprop.htm","timestamp":"2024-11-08T11:25:31Z","content_type":"application/xhtml+xml","content_length":"44380","record_id":"<urn:uuid:22bbd202-50a5-4e58-8f84-3fb446af9795>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00292.warc.gz"}
Class 7 Chapter 13: Visualizing Solid Shapes Important Concepts and Formulas 2. The flat surfaces of a solid shape are called the faces and the corners are called the vertices. 5. There are two different types of sketches that are possible for a solid shape. They are oblique sketch and isometric sketch. 6. An oblique sketch conveys all important aspects of the appearance of a 3D shape, though it does not have proportional lengths of its edges. 7. In an isometric sketch of a 3D shape, the measurements are kept proportional. 8. We can view a solid shape in different ways. A solid shape looks different if we view it from different angles. Please do not enter any spam link in the comment box. Post a Comment (0)	{"url":"https://www.maths-formula.com/2020/03/class-7-chapter-13-visualizing-solid.html","timestamp":"2024-11-02T19:05:50Z","content_type":"application/xhtml+xml","content_length":"237974","record_id":"<urn:uuid:2711473f-9710-416d-a28f-7894d2fa8cc5>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00110.warc.gz"}
Assignment: Growth Models Writing Task Search the Internet to find a graph or table of values that shows how something is changing over time. The data should be real measured data (not some made-up values for a math problem example or something), and should show something that is changing linear or exponentially. • The graph or table should show data for at least 4 time periods (years, months, etc.) • Find data that appears to have a linear or exponential trend. • Find data that someone hasn’t already used. What You Will Turn In • Construct a document that includes a link to the data or graph you used (+2 pt for including, -10 for not including) • Recreate the table (if it’s huge, just include at least 4 time periods), or create a table from the graph you chose and include it in your document (6 pts.) • State whether the data appears to be changing linearly or exponentially, and give a reason (4 pts) • Find an explicit equation to model the data, clearly defining your variables and showing your work (10 pts) • Use your model to make a prediction about the future (4 pts) Table 2 on page 2 of this document from Snohomish County, WA shows the incidents of sexual orientation motivated hate crimes in Washington. While not perfectly linear, the trend appears roughly linear [don’t just assume linearity! – make sure you can give a reason. If I graph your data and it looks exponential, you WILL lose points]. Using the 1996 as n=0, and using data from 1996 and 2001, (1996) P[0] = 1016 (2001) P[5] = 1393 [FROM HERE, you would then follow the examples in the book to calculate the equation. IF your data is changing linearly, the procedure will be similar to the “population of elk” example. IF your data is changing exponentially, the procedure will be similar to the “carbon dioxide emissions” example. Suppose I followed the procedure, showing my steps, and came up with this equation:] P[n] = 1016 + 75n where n is years after 1996 Predicting in 2010 (n=14): P[14] = 1016 + 75(14) = 2066 So if this trend were to continue at this rate, this model predicts that in 2010 there would be 2066 incidents of sexual orientation motivated hate crimes in Washington. Again, this example was specific to linear growth. For an example using exponential growth, see the carbon dioxide emissions example in the book. A note on determining if the trend is linear or exponential Linear trends increase by approximately the same amount each year (or month or whatever the time unit is), so they have the shape of straight lines. It is important to remember, though, that the world is not perfect, so data is rarely a perfect line. The gas consumption example in the book is an example of data that is not perfectly in a linear, but appears to have an approximately linear trend; in other words, a line fits it pretty well. Exponential trends are ones that increase by the same percent each year (or whatever the time unit is). If the data is increasing, they have a shape that curves upwards. Sometimes that curve upwards is subtle (like in the first graph of the fish population in the book), and sometimes it’s very pronounced (like in the second graph of the fish population). If the data is a decreasing trend, an exponential trend would look something like this: In some cases, it’s hard to tell if the trend is linear or exponential. In that case, think about how the quantity is changing. Is it likely to be increasing by the same number each year? Or is likely to be increasing by the same percent each year? Keep in mind that there’s always a third possibility: that the data is not changing linearly or exponentially. Here’s a great example: In this graph, the blue graph looks approximately linear, but the red graph is neither linear or exponential (and you shouldn’t use data like that for this assignment!) Download the assignment from one of the links below (.docx or .rtf): Growth Models Writing Task: Word Document	{"url":"https://courses.lumenlearning.com/wmopen-mathforliberalarts/chapter/assignment-growth-models-writing-task/","timestamp":"2024-11-11T14:25:30Z","content_type":"text/html","content_length":"51316","record_id":"<urn:uuid:beaac3fc-c239-4dcf-86c1-a59e541a316a>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00819.warc.gz"}
Elementary Statistics 3rd Edition Navidi Solutions Manual [DEL:$50.00:DEL] (-46%) Elementary Statistics 3rd Edition Navidi Solutions Manual. Elementary Statistics 3rd Edition Navidi Solutions Manual Product details: • ISBN-10 : 1259969452 • ISBN-13 : 978-1259969454 • Author: William Navidi Elementary Statistics, Third Edition is a conceptual and procedural course in introductory statistics. It has been developed around three central themes: clarity, quality, and accuracy, based on extensive market research and feedback from statistics instructors across the country. High-quality exercises, clear examples, author-created supplements, and fully integrated technology make this one of the more masterful elementary statistics courses available. The text and supplements are flexible enough to work effectively with a wide variety of instructor styles; for example, the text covers both P-value and critical value approaches to hypothesis testing. Improvements to this third edition include a new objective on the weighted mean, recent real data in new exercises and case studies, and a new supplement focused on prerequisite skills.Because statistics instructors universally agree that using real data better engages students, most examples and exercises in this text use real-life data; in fact, more than 750 actual applications of statistics appear within the book, and topics and their pages can be found in an index. Each section contains step-by-step instructions explaining how to use multiple forms of technology to carry out the procedures explained in the text. Table contents: 1 Basic Ideas 2 Graphical Summaries of Data 3 Numerical Summaries of Data 4 Summarizing Bivariate Data 5 Probability 6 Discrete Probability Distributions 7 The Normal Distribution 8 Confidence Intervals 9 Hypothesis Testing 10 Two-Sample Confidence Intervals 11 Two-Sample Hypothesis Tests 12 Tests with Qualitative Data 13 Inference in Linear Models 14 Analysis of Variance 15 Nonparametric Statistics Appendix A: Tables Appendix B: TI-84 PLUS Stat Wizards People also search: elementary statistics navidi and monk 3rd edition elementary statistics 13th edition answers elementary statistics california 3rd ed elementary statistics third edition answers elementary statistics larson statistics q3 Instant download after Payment is complete	{"url":"https://testbankdeal.com/product/elementary-statistics-3rd-edition-navidi-solutions-manual/","timestamp":"2024-11-03T17:21:46Z","content_type":"text/html","content_length":"106422","record_id":"<urn:uuid:c877181a-f163-4ab7-8dfb-fab3154b7ce0>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00498.warc.gz"}
ESP Biography JESSICA OEHRLEIN, Fitchburg State University math/stats professor Major: Applied Mathematics College/Employer: Fitchburg State University Year of Graduation: Not available. Brief Biographical Sketch: I'm a statistics and applied math professor who studied atmospheric science in grad school and mechanical engineering as an undergrad. Romeo and Juliet is my favorite ballet (but one of my least favorite plays), and I've seen ten versions of it performed in four countries. My favorite contra dance figures are a hey for four and a ladies chain with a courtesy turn. 193 distinct roller coasters and counting! Past Classes (Clicking a class title will bring you to the course's section of the corresponding course catalog) A14741: From Swans to Spartacus: Ballet in the Soviet Union in Splash 2021 (Nov. 20 - 21, 2021) We often associate classical ballet with Imperial Russia -- long, royal-themed stories, extravagant costumes, technical choreography that is still revered today. However, ballet was also culturally important during the Soviet period, and balletic developments in the Soviet Union were really different from those elsewhere. We'll talk about the history of Soviet ballet, how it influenced and was influenced by ballet in the West, and why only a few Soviet ballets survived the fall of the Soviet Union. M14213: Mathematical Modeling in Splash 2020 (Nov. 14 - 15, 2020) Math modeling is how we use mathematics to study open-ended questions about real-world phenomena. What's the best location for a food truck? How does an invasive species affect an ecosystem? How do we clean up space debris? These are all questions that we can start to answer with math modeling. The goal of this class is to introduce you to the modeling process. By the end, you'll have developed models to answer questions about a couple of different scenarios, and you'll know about some of the tools you can use to tackle more significant modeling problems. Z14214: History of Ballet: 1900-Present in Splash 2020 (Nov. 14 - 15, 2020) This class will cover the past century of ballet, focusing on the westward spread of Russian-style classical ballet, the establishment of new major ballet traditions in England and the United States, and the rise of contemporary ballet. We'll look at photos and video and discuss as a class how different techniques and styles emerged in different parts of the world. We'll also talk about where ballet is headed now. M13459: How to (Mathematically) Guard an Art Gallery in Splash 2019 (Nov. 23 - 24, 2019) Suppose you have a polygonal art gallery with $$n$$ sides that you want to guard with 360-degree cameras at some of the polygon's vertices. What is the least number of cameras you could use? This is the classic art gallery problem, and it uses a lot of ideas from the mathematical field of graph theory. We'll cover some basic graph theory concepts and then tackle the art gallery problem! M13460: Bridges, Maps, and Networks: An Introduction to Graph Theory in Splash 2019 (Nov. 23 - 24, 2019) Graph theory is a relatively young area of mathematics, focused on studying structures that show the relationships among people, places, or objects. We'll talk about two of the first key questions in graph theory, the Königsberg bridge problem and the Four Color Theorem. We'll also explore some applications of graph theory, such as modeling social networks or the spread of information or disease. P12812: A Brief Tour of the Stratosphere in Spark 2019 (Mar. 16 - 17, 2019) What is the ozone hole, and when will it recover? What did a scientist actually observe when he noticed an "explosion-like warming" over the Arctic? Why did 1883 and 1908 data show tropical winds going in opposite directions? All of these questions are about phenomena that happen in the stratosphere, the layer of the atmosphere about 10-50 km above us. In groups, you'll explore questions like these, the related stratospheric phenomena, and their impacts on us. We'll put them all together to create a coherent picture of the stratosphere. M12863: Disease Modeling in Spark 2019 (Mar. 16 - 17, 2019) When studying infectious diseases like the flu, we can use math to describe how the illness spreads. Those descriptions or sets of equations are called a mathematical model of the disease. We'll talk about a common model for infectious disease, and then we'll simulate that model together and discuss the results. We'll also come up with some possible variations on that model. M12864: How to (Mathematically) Guard an Art Gallery in Spark 2019 (Mar. 16 - 17, 2019) Suppose you had a polygonal art gallery with $$n$$ sides that you wanted to guard with 360-degree cameras at some of the polygon's vertices. What is the least number of cameras you could use? This is the classic art gallery problem, and it uses a lot of basic concepts in the mathematical field of graph theory. We'll cover some basic graph theory concepts and then tackle the art gallery problem! Z12865: History of Ballet: 1900-Present in Spark 2019 (Mar. 16 - 17, 2019) This class will cover the past century of ballet, focusing on the westward spread of Russian-style classical ballet, the establishment of new major ballet traditions in England and the United States, and the rise of contemporary ballet. We'll look at photos and video and discuss as a class how different techniques and styles emerged in different parts of the world. We'll also talk about where ballet is headed now. M12358: Population Modeling in Splash 2018 (Nov. 17 - 18, 2018) Can trapping invasive crawfish save a newt population? How do regional or minority languages like Galician survive? When can two species with the same food sources coexist? In this class, we'll build mathematical models of populations to answer questions like these. In groups, you will mathematically describe different ways in which populations grow, decline, and interact. Each group's model will answer key questions about population behavior or control. We'll also discuss challenges and alternate methods for modeling. M12359: Bridges, Maps, and Networks: An Introduction to Graph Theory in Splash 2018 (Nov. 17 - 18, 2018) Graph theory is a relatively young area of mathematics, focused on studying structures that show the relationships among people, places, or objects. We'll talk about two of the first key questions in graph theory, the Königsberg bridge problem and the Four Color Theorem. We'll also explore some applications of graph theory, such as modeling social networks or the spread of information or disease. M12084: Mathematical Modeling in Spark 2018 (Mar. 17 - 18, 2018) Math modeling is how we use mathematics to study open-ended questions about real-world phenomena. What's the best location for a food truck? What would be the effects of sea level rise? How do we best distribute medicine to control a disease outbreak? These are all questions that we can start to answer with math modeling. The goal of this class is to introduce you to the modeling process. By the end, you'll have developed models to answer questions about a couple of different scenarios, and you'll know about some of the tools you can use to tackle more significant modeling problems. M12085: Bridges, Maps, and Networks: An Introduction to Graph Theory in Spark 2018 (Mar. 17 - 18, 2018) Graph theory is a relatively young area of mathematics, focused on studying structures that show the relationships among people, places, or objects. We'll talk about two of the first key questions in graph theory, the Königsberg bridge problem and the Four Color Theorem. We'll also explore some applications of graph theory, such as modeling social networks or the spread of information or disease. P12094: A Brief Tour of the Stratosphere in Spark 2018 (Mar. 17 - 18, 2018) We live in the layer of the atmosphere called the troposphere, where nearly all weather happens. But the stratosphere, just above the troposphere, is also important for climate! We'll talk about the ozone layer, why the direction of the winds in the tropics switches direction every 28 months, and what a scientist actually observed when he noticed an "explosion-like warming" over the Arctic. A11518: From Swans to Spartacus: Ballet in the Soviet Union in Splash 2017 (Nov. 18 - 19, 2017) We often associate classical ballet with Imperial Russia. However, ballet was also culturally important during the Soviet period, and balletic developments in the Soviet Union were really different from those elsewhere. We'll talk about the history of Soviet ballet, how it influenced and was influenced by ballet in the West, and why only a few Soviet ballets survived the fall of the Soviet M11651: Mathematical Modeling in Splash 2017 (Nov. 18 - 19, 2017) Math modeling is how we use mathematics to study open-ended questions about real-world phenomena. What's the best location for a food truck? How does an invasive species affect an ecosystem? How do we clean up space debris? These are all questions that we can start to answer with math modeling. The goal of this class is to introduce you to the modeling process. By the end, you'll have developed models to answer questions about a couple of different scenarios, and you'll know about some of the tools you can use to tackle more significant modeling problems. A11258: From Swans to Spartacus: Ballet in the Soviet Union in Spark 2017 (Mar. 11 - 12, 2017) We often associate classical ballet with Imperial Russia. However, ballet was also culturally important during the Soviet period, and balletic developments in the Soviet Union were really different from those elsewhere. We'll talk about the history of Soviet ballet, how it influenced and was influenced by ballet in the West, and why only a few Soviet ballets survived the fall of the Soviet P11259: Understanding Weather Data in Spark 2017 (Mar. 11 - 12, 2017) Atmospheric sounding charts are generated from weather balloon data, and they help us understand and predict weather conditions. Come learn what temperature through the atmosphere looks like when there's freezing rain and how to predict whether there will be a thunderstorm soon! B10561: Introduction to Hungarian Through Song in Splash 2016 (Nov. 19 - 20, 2016) We'll cover basic Hungarian* by singing (mostly children's) songs! You'll learn very important vocabulary words like yellow, raspberry, icicle, and animal. *I do not guarantee that you'll be able to hold any kind of reasonable conversation. M10632: Mathematical Modeling in Splash 2016 (Nov. 19 - 20, 2016) Math modeling is how we use mathematics to study open-ended questions about real-world phenomena. What's the best location for a food truck? How does an invasive species affect an ecosystem? How do we clean up space debris? These are all questions that we can start to answer with math modeling. The goal of this class is to introduce you to the modeling process. By the end, you'll have developed models to answer questions about a couple of different scenarios, and you'll know about some of the tools you can use to tackle more significant modeling problems. B7794: Introductory Azerbaijani in Splash! 2013 (Nov. 23 - 24, 2013) Come and learn Azerbaijani! Azerbaijan was along the Silk Roads, so the country and the language have lots of different influences. Azerbaijani is also the second most commonly spoken language in Iran! We'll also cover a little history along the way.	{"url":"https://esp.mit.edu/teach/teachers/jessoehrlein/bio.html","timestamp":"2024-11-13T15:20:35Z","content_type":"application/xhtml+xml","content_length":"28977","record_id":"<urn:uuid:67db519f-611d-4c8b-85f7-ecdaf7e3c5c1>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00594.warc.gz"}