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UltraData-Math-L1 · 86M rows

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train · 86M rows

content stringlengths 86 994k	meta stringlengths 288 619
Correction approaches for surrogate models • Alias: None • Arguments: None • Default: no surrogate correction Child Keywords: Required/Optional Description of Group Dakota Keyword Dakota Keyword Description zeroth_order Specify that truth values must be matched. Required (Choose One) Correction Order first_order Specify that truth values and gradients must be matched. second_order Specify that truth values, gradients and Hessians must be matched. additive Additive correction factor for local surrogate accuracy Required (Choose One) Correction Type multiplicative Multiplicative correction factor for local surrogate accuracy. combined Multipoint correction for a hierarchical surrogate Some of the surrogate model types support the use of correction factors that improve the local accuracy of the surrogate models. The correction specification specifies that the approximation will be corrected to match truth data, either matching truth values in the case of zeroth_order matching, matching truth values and gradients in the case of first_order matching, or matching truth values, gradients, and Hessians in the case of second_order matching. For additive and multiplicative corrections, the correction is local in that the truth data is matched at a single point, typically the center of the approximation region. The additive correction adds a scalar offset ( zeroth_order), a linear function ( ,first_order), or a quadratic function ( second_order) to the approximation to match the truth data at the point, and the multiplicative correction multiplies the approximation by a scalar ( zeroth_order), a linear function ( first_order), or a quadratic function ( second_order) to match the truth data at the point. The additive first_order case is due to [LN00] and the multiplicative first_order case is commonly known as beta correction [Haf91]. For the combined correction, the use of both additive and multiplicative corrections allows the satisfaction of an additional matching condition, typically the truth function values at the previous correction point (e.g., the center of the previous trust region). The combined correction is then a multipoint correction, as opposed to the local additive and multiplicative corrections. Each of these correction capabilities is described in detail in [EGC04]. The correction factors force the surrogate models to match the true function values and possibly true function derivatives at the center point of each trust region. Currently, Dakota supports either zeroth-, first-, or second-order accurate correction methods, each of which can be applied using either an additive, multiplicative, or combined correction function. For each of these correction approaches, the correction is applied to the surrogate model and the corrected model is then interfaced with whatever algorithm is being employed. The default behavior is that no correction factor is The simplest correction approaches are those that enforce consistency in function values between the surrogate and original models at a single point in parameter space through use of a simple scalar offset or scaling applied to the surrogate model. First-order corrections such as the first-order multiplicative correction (also known as beta correction [CHGK93]) and the first-order additive correction [LN00] also enforce consistency in the gradients and provide a much more substantial correction capability that is sufficient for ensuring provable convergence in SBO algorithms. SBO convergence rates can be further accelerated through the use of second-order corrections which also enforce consistency in the Hessians [EGC04], where the second-order information may involve analytic, finite-difference, or quasi-Newton Hessians. Correcting surrogate models with additive corrections involves f{equation} hat{f_{hi_{alpha}}}({bf x}) = f_{lo}({bf x}) + alpha({bf x}) f} where multifidelity notation has been adopted for clarity. For multiplicative approaches, corrections take the form f{equation} hat{f_{hi_{beta}}}({bf x}) = f_{lo}({bf x}) beta({bf x}) f} where, for local corrections, $\alpha({\bf x})$ and $\beta({\bf x})$ are first or second-order Taylor series approximations to the exact correction functions: f{eqnarray} alpha({bf x}) & = & A({bf x_c}) + nabla A({bf x_c})^T ({bf x} - {bf x_c}) + frac{1}{2} ({bf x} - {bf x_c})^T nabla^2 A({bf x_c}) ({bf x} - {bf x_c}) \ beta({bf x}) & = & B({bf x_c}) + nabla B({bf x_c})^T ({bf x} - {bf x_c}) + frac{1}{2} ({bf x} - {bf x_c})^T nabla^2 B({bf x_c}) ({bf x} - {bf x_c}) f} where the exact correction functions are f{eqnarray} A({bf x}) & = & f_{hi}({bf x}) - f_{lo}({bf x}) \ B({bf x}) & = & frac{f_{hi}({bf x})}{f_{lo}({bf x})} f} Refer to [EGC04] for additional details on the derivations. A combination of additive and multiplicative corrections can provide for additional flexibility in minimizing the impact of the correction away from the trust region center. In other words, both additive and multiplicative corrections can satisfy local consistency, but through the combination, global accuracy can be addressed as well. This involves a convex combination of the additive and multiplicative corrections: \[\hat{f_{hi_{\gamma}}}({\bf x}) = \gamma \hat{f_{hi_{\alpha}}}({\bf x}) + (1 - \gamma) \hat{f_{hi_{\beta}}}({\bf x})\] where $\gamma$ is calculated to satisfy an additional matching condition, such as matching values at the previous design iterate. It should be noted that in both first order correction methods, the function $\hat{f}(x)$ matches the function value and gradients of $f_{t}(x)$ at $x=x_{c}$ . This property is necessary in proving that the first order-corrected SBO algorithms are provably convergent to a local minimum of $f_{t}(x)$ . However, the first order correction methods are significantly more expensive than the zeroth order correction methods, since the first order methods require computing both $\nabla f_{t}(x_{c})$ and $\nabla f_{s}(x_{c})$ . When the SBO strategy is used with either of the zeroth order correction methods, or with no correction method, convergence is not guaranteed to a local minimum of $f_{t}(x)$ . That is, the SBO strategy becomes a heuristic optimization algorithm. From a mathematical point of view this is undesirable, but as a practical matter, the heuristic variants of SBO are often effective in finding local minima. Usage guidelines • Both the additive zeroth_order and multiplicative zeroth_order correction methods are “free” since they use values of $f_{t}(x_{c})$ that are normally computed by the SBO strategy. • The use of either the additive first_order method or the multiplicative first_order method does not necessarily improve the rate of convergence of the SBO algorithm. • When using the first order correction methods, the gradient-related response keywords must be modified to allow either analytic or numerical gradients to be computed. This provides the gradient data needed to compute the correction function. • For many computationally expensive engineering optimization problems, gradients often are too expensive to obtain or are discontinuous (or may not exist at all). In such cases the heuristic SBO algorithm has been an effective approach at identifying optimal designs [Giu02].	{"url":"https://snl-dakota.github.io/docs/6.19.0/users/usingdakota/reference/model-surrogate-ensemble-ordered_model_fidelities-correction.html","timestamp":"2024-11-09T07:28:50Z","content_type":"text/html","content_length":"29319","record_id":"<urn:uuid:93213758-2637-49e4-9acc-0f6deb584b9a>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00350.warc.gz"}
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Being Random and Trivial in Dagstuhl This week I'm at the Computability, Complexity and Randomness workshop at Dagstuhl in Germany. This meeting brings together two groups, complexity theorists and computability theorists, who share a common love of Kolmogorov complexity. From the logicians I learned about K-trivial sets. Let K(x) be the prefix-free Kolmogorov complexity of x, i.e., the size of the smallest program that generates x. There are several equivalent definitions of K-trivial sets, here is two of them. A are K-trivial if 1. For some constant c, for all x, K(x) ≤ K^A(x)+c, where K^A(x) is the smallest program generating x with access to oracle A. 2. For some constant c, for all n, K(A[1:n]) ≤ K(n)+c, where A[1:n] are the first n bits of the characteristic sequence of A. Lots of interesting properties about K-trivial sets. • All computable sets are K-trivial and there are K-trivial sets that are not computable. • There are only a countable number of K-trivial sets. In fact there are only a finite number of K-trivial sets for each fixed constant c above. • Every K-trivial set is super-low, that is the halting problem relative to a K-trivial set is non-adaptively reducible to the halting problem. • Every K-trivial non-adaptively reduces to a computably-enumerable set. • Every set reducible to a K-trivial is K-trivial. • The disjoint union of two K-trivial sets is K-trivial. • Random sets are still random relative to A. More about K-trivial sets and everything else computably random in a great book by Downey and Hirschfeldt. Us complexity theorists started thinking about polynomial-time versions of K-trivial sets but probably won't have as many nice properties. 7 comments: 1. Is there a result in Complexity Theory such that its statement has nothing to do with Kolmogorov Complexity, yet its proof uses Kolmogorov Complexity in a critical way? 2. Ignorant question (and I'm legitimately curious, not trying to be smarmy), what do computability researchers research these days? Are there new results? I always thought (or at least have read, not being in academia I don't have my own view on this) that computability theory has been stagnant for quite a while and most "related" results are in complexity theory now. Obviously I see this view is wrong, but what's cooking on on that end of things? 3. Doesn't Robin Moser's proof of the LLL use Kolmogorov complexity? 4. Two big non-stagnant topics in computability theory: 1) Reverse Mathematics- using computability theory to pin down how constructive certain theorems in math are. Here Computability theory is a tool for something else. (An earlier program which is more obviously using computability theory is Nerode's Recursive Mathematics program). For more see Stephen Simpson's home page. 2) Random sets, as Lance is talking about. That Superlow=K-trivial is a very nice result that connects the two fields. For more see Downey-Hirschfeld. 5. CORRECTION- my comment said Superlow=K-trivial. Actually its K-Trivail is proper Subset of Superlow. Rod Downey emailed me the corrections for which I thank him. 6. Attending this meeting it seemed to me that the computability school of K-complexity is slowly taking over the complexity school, with the majority of young people present studying computability aspects of K-complexity. I don't know if this an accurate description of the actual state of K-complexity, e.g. it may be due to the mere fact that the majority of the organizers have a computability background. I'm sure the organizers had good reasons, but I found the number of talks was on the low side, and I would have liked to hear more talks given the many "famous" people present. 7. Is there a result in Complexity Theory such that its statement has nothing to do with Kolmogorov Complexity, yet its proof uses Kolmogorov Complexity in a critical way? If you consider lower bounds as a hybrid between algorithms and complexity theory then there are tons of them.	{"url":"https://blog.computationalcomplexity.org/2012/01/being-random-and-trivial-in-dagstuhl.html?m=0","timestamp":"2024-11-09T14:23:35Z","content_type":"application/xhtml+xml","content_length":"187051","record_id":"<urn:uuid:c7e12be7-5878-4c77-a453-2ff4ecfb3f53>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00687.warc.gz"}
Camp Holtz There is something about Michigan summers – the best summers ever – especially up north on a lake. Ever since I can remember having a lake cottage has been a dream of mine. Luckily, I married Matt who was also born and raised in Michigan and so he understands this dream, and it was one of his too. Well, guess what? Matt and I have finally fulfilled this dream and bought a lake cottage, up north! We started at the beginning of summer, looked at 12+ places, put in 3 offers, and finally closed on the last day of August 2020. Our needs list included being no more than 2 hours away, lake front, sandy bottom, having room for guests, and a washer and dryer. And we found it in Harrison, Michigan, a chain of lakes, 1.5 hours away. The cottage is move in ready (they left everything), but does need some work. We are really looking forward to making it our own throughout the years. We have a big to-do list – but also want to enjoy our time at Camp Holtz! 29 Comments	{"url":"http://campholtz.com/","timestamp":"2024-11-01T19:11:35Z","content_type":"text/html","content_length":"53067","record_id":"<urn:uuid:c9853779-3595-4374-806d-c256d42b4b1b>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00203.warc.gz"}
Kindergarten Math - instruction and mathematics practice for Kindergarten Curriculum for Kindergarten From number recognition to counting and operations, students build the number sense and conceptual understanding they'll need for grade 1. MODULE 1 Numbers to 10 Topic A: Identifying Same, Different, and Alike Students consider the size, shape, and color of objects to determine whether they are the same, different, or alike. Additionally, they are introduced to dissimilar objects that "match" because of a related function, such as a sock and a shoe. Topic B: Sorting and Counting Similar Objects Students sort similar objects based on narrow categories (such as "bears"), broad categories (such as "clothes"), and self-determined categories. They identify objects that do not belong in a group without being given the name of the category. Finally, students count to determine the total number of objects in a group. Topic C: Numbers to 5 in Different Configurations, Math Drawings, and Expressions Students count in sequence to determine position or total up to 5. They relate objects to digits and see digits in a variety of fonts. They work with aligned objects, scattered objects, fingers, and the number line. Students learn strategies for counting with 1:1 match and learn that sets of objects with the same total can be aligned in different ways. Match a numbered set of 1, 2, or 3 cubes to an identical numbered set of cubes Match sets of cubes. A person with 1-3 cubes is shown. Then, there are three other people with sets of cubes. Select the set that matches the first set. The numbers are written in different fonts to practice number recognition Match numbered and non-numbered sets of cubes to a number 1-3 Select the pictures that show the stated number (1, 2, or 3) of cubes. Four pictures are given in each problem. The numbers are written in different fonts to practice number recognition. Some numbers are obscured to encourage counting the cubes Match sets of cubes to numbers 1-3 Given images with the numbers 1-3 and cards with sets of cubes, drag the correct set of cubes to each number. The numerals and the sets are in random order. Three sets of cubes and numbers are shown at a time Match numbers 1-3 to their positions on a number line labeled with numbers and dot patterns Put numbers on a number line. Drag the numbers 1-3 to their appropriate location on a number line. The number line spans the numbers 0 to 10 and is labeled with the numbers and dot patterns of each number. The numbers are presented in various fonts Identify a numbered set of cubes that matches an identical set of 4 or 5 numbered cubes Match sets of cubes. One person with 4 or 5 cubes is shown. Then, there are three other people with sets of cubes ranging from 2 to 5. Select the set that matches the first set. The numbers are written in different fonts to practice number recognition Match numbered and non-numbered sets of cubes to number 4 or 5 Select the pictures that show the stated number (4 or 5) of cubes. Four pictures are given in each problem. The numbers are written in different fonts to practice number recognition. Some numbers are obscured to encourage counting the cubes Count 2-5 aligned objects to determine the total Given a row of 2-5 cubes, type or select the number of cubes shown Match numbers 1-5 to arrangements of fingers displayed on a hand Match a hand with the a number of fingers displayed to the correct numeral. Three numbers and hands are shown at a time Match sets of cubes to numbers 1-5 Match a set of objects to the number. Given images with the numbers 1-5 and cards with sets of cubes, drag the correct set of cubes to each number. The numerals and the sets are in random order. Three sets of cubes and numbers are shown at a time Identify numbered sets of cubes that match a given total up to 5 Select the pictures that show the stated number (1-5) of cubes. Four pictures with sets of cubes are given in each problem. More than one picture may contain the desired number of cubes, so select all that apply Match numbers 1-5 to their positions on a number line labeled with numbers and dot patterns Drag the numbers 1-5 to their appropriate location on a number line. The number line spans the numbers 0 to 10 and is labeled with the numbers and dot patterns of each number. The numbers to drag are presented in various fonts Align scattered objects to count and determine the total A randomly arranged set of objects (1-5) is presented. Drag the objects to a box to line them up to make it easier to count. Then, type or select the number shown Count to find the total of scattered objects by matching 1:1 numbers to objects Count the number of objects shown. Click on them to change their color to mark that they have been counted. Then, select the number Identify a set of scattered objects that matches a given total up to 5 Count the number of objects shown. Type or select the number. Then, given a statement of how many objects to find, select the container that has the correct number of items in it Topic D: The Concept of Zero and Working with Numbers 0 - 5 Students use digits 1-5 to sequence objects and to determine position or total. They work with aligned objects, scattered objects, and the number line. They explore the composition of the number 3 and begin using +, -, and = signs. Topic E: Working with Numbers 6 - 8 in Different Configurations Students count in sequence to determine position or total up to 9. They relate objects to digits and see digits in a variety of fonts. They work with aligned objects, scattered objects, fingers, and the number line. Students learn strategies for counting with 1:1 match and learn that sets of objects with the same total can be aligned in different ways. Recognize that one more than 5 is 6 Learn the number 6. Five items are shown. Select the correct numeral to represent that. Then, another item is added, showing that the digit 6 comes after 5 Recognize that one more than 6 is 7 Learn the number 7. Six items are shown. Select the correct numeral to represent that. Then, another item is added, showing that the digit 7 comes after 6 Identify the number of fingers up to 7 displayed on two hands Two hands are displayed showing between 0 and 7 fingers. Click or type the number of fingers shown Match numbers 0-7 to arrangements of fingers displayed on two hands Three numbers and three pictures of hands displaying fingers are shown. Match the hands to the numbers 0-7 Identify a set of aligned objects that matches a given total up to 7 Identify the image showing up to 7 cubes. Three choices of rows of cubes are given. Click on the one that matches the number Match numbered patterns of dots to an identical numbered pattern of dots up to 7 Match numbered dot patterns. Small octopuses are holding flags with numbers and dot patterns. Drag all that match the father octopus' number and pattern over to him. The numbers are written in various fonts Identify sets of numbered cubes that match a given total of 6 or 7 (Part 1) Identify the number of cubes shown, out of an option of two numbers. Then, a cube is added. Identify the number again for a total of 6 or 7. Select the pictures with the same total of cubes. The digits on the pictures are written in a variety of fonts Identify sets of numbered cubes that match a given total of 6 or 7 (Part 2) Choose sets that contain a stated total. Select all pictures that contain the given number of cubes (6 or 7). The digits in the pictures are written in a variety of fonts and the cubes are arranged in various configurations Place numbers below 10 on a number line marked at intervals of 5 Drag a given number to its place on a number line marked at intervals of 5. Given numbers are from 1-7 Count objects as they move away from a set and identify the total Birds fly away from a tree, one at a time. Count the birds that leave and type or select the number of how many flew away Count scattered objects two different ways to arrive at the same total Learn that the order in which a set is counted does not change the total. Follow the arrows to click on beads to change their color and count them. Then, enter the number. Count the items in a different order and enter the number again Recognize that one more than 7 is 8 Learn the number 8. Seven items are shown. Select the correct numeral to represent that. Then, another item is added, showing that the digit 8 comes after 7 Identify the number of fingers up to 8 displayed on two hands Two hands are displayed showing between 0 and 8 fingers. Click or type the number of fingers shown Match numbers 0-8 to arrangements of fingers displayed on two hands Three numbers and three pictures of hands displaying fingers are shown. Match the hands to the numbers 0-8 Identify a set of aligned objects that matches a given total up to 8 Identify the image showing up to 8 cubes. Three choices of rows of cubes are given. Click on the one that matches the number Identify sets of numbered cubes that match a given total up to 9 (Part 1) Identify the number of cubes shown, out of an option of two numbers. Then, one cube at a time is added. Identify the number again for a total of 8. Three or four pictures are then shown. Select the ones with the same total of cubes. The digits on the pict Identify sets of numbered cubes that match a given total up to 9 (Part 2) Select all of the pictures that contain the given number of cubes, up to 9. The digits in the pictures are written in a variety of fonts and the cubes are arranged in various configurations Count objects as they move away from a set and identify the total Birds fly away from a tree, one at a time. Count the birds that leave and type or select the total of how many flew away Count scattered objects two different ways to arrive at the same total Learn that the order in which a set is counted does not change the total. Follow the arrows to click on beads to change their color and count them. Then, enter the number. Count them again in a different order and enter the number again Identify the total number of scattered objects after a known total has been rearranged Learn that the number of objects does not change when they are moved around. Count the items in a container and enter the total. Then, the items move around. Count them again and enter the total Topic F: Working with Numbers 9 - 10 in Different Configurations Students work increasingly with 0 and with numbers 6-10 to determine totals and recognize digits. They work with scattered objects, non-identical objects, fingers, and the number line. Topic G: One More with Numbers 0 - 10 Students use familiar representations (cubes, base-10 blocks, a number path, and real-world objects) to explore the concept of one more. They begin to count on rather than count all. Topic H: One Less with Numbers 0 - 10 Students use familiar representations (cubes, base-10 blocks, a number path, and real-world objects) to explore the concept of one less. Then, they use their understanding of one more and one less to solve +/- 1 equations. MODULE 2 Two-Dimensional and Three-Dimensional Shapes Topic A: Two Dimensional Flat Shapes Students become familiar with the appearance and names of two-dimensional shapes. They work with squares, circles, triangles, rectangles, and hexagons. Students identify examples from among non-examples. Our sound feature allows non-readers access to shape names. Identify positions above and below an object Learn to identify the position as above or below an object. Select the stated location by clicking on the correct position Position objects above and below an object Position 1 or 2 objects above or below to a stationary object. Select both the correct location and the correct color as directed Position an object in front of or behind an object (Part 1) Learn to identify the position as in front of or behind an object. Select the stated location by clicking on the correct position Position an object in front of or behind an object (Part 2) Identify a location as "in front of" or "behind" an object. Drag a character to the stated position Identify the lines Learn to identify lines. Select lines in various orientations from among a group containing lines and curves Identify curved lines Learn to identify curves. Select curves in various orientations and forms from among a group containing lines and curves Identify rectangles, triangles, or circles from among a set of shapes Identify rectangles, triangles, and circles from sets of shapes. Click on the shapes that match the stated figure Identify properties of triangles Learn the characteristics of a triangle. Then, select those shapes that are triangles Identify properties of rectangles Learn the characteristics of a rectangle. Then, select those shapes that are rectangles Identify hexagons from among a set of shapes Determine which of two shapes is a hexagon. Then, select all of the hexagons out of many shapes Identify properties of hexagons Learn the characteristics of a hexagon. Then, select those shapes that are hexagons to complete a picture Identify real world objects composed of 2D shapes Select real-world objects that contain a given shape out of a set of 6 objects. The shapes include circles, rectangles, and triangles Match real world objects to their 2D shapes Select the correct 2-dimensional figure that models a real-world objects. The object and its matching figure may be oriented in different directions Match a 2D shape to its name Given the name of a shape, match it to the correct shape. Click the x or checkmark next to each of the three shapes to indicate whether it matches the word Identify shapes with a given number of sides, corners, or curved lines Select the shapes that have the stated characteristic. Out of a set of 5 shapes, choose those that have the given number of sides or angles Identify and sort triangles and rectangles Students will identify rectangles and triangles from a group of shapes. Then, students will sort triangles and rectangles Identify 2D shapes and move them into positions above, below, in front of, and behind Practice 2-dimensional shapes and position. Drag a specified shape to the directed location with respect to a static picture Topic B: Three-Dimensional Solid Shapes Students become familiar with the appearance and names of three-dimensional shapes. They work with cubes, spheres, cones, and cylinders. Students also reinforce their understanding of positions (above, below, in front of, next to, and behind). Our sound feature allows non-readers access to shape names. Sort shapes into the categories flat and solid Learn to differentiate between 2-dimensional and 3-dimensional figures. Then, sort them into their categories Match shapes to real-world objects Match real-world objects to 3-dimensional solid figures Identify cylinders from among 3-D shapes Learn to identify cylinders from a set of 3-dimensional solids. Click on cylinders among a group of other solids Identify cones from among 3-D shapes Practice identifying cones from a set of 3-dimensional solids. Click on cones among a group of other solids Identify cones and cylinders Students will match names to the corresponding 3-D shapes. Tasks focus on cones and cylinders Identify cubes from among 3-D shapes Learn to identify cubes from a set of 3-dimensional solids. Click on cubes among a group of other solids Identify cones, cubes, and cylinders Students will match names to the corresponding 3-D shapes. Tasks focus on cones, cubes, and cylinders Identify spheres from among 3-D shapes Practice identifying spheres from a set of 3-dimensional solids. Click on spheres among a group of other solids Identify cones, cubes, cylinders, and spheres Students will match names to the corresponding 3-D shapes. Tasks focus on cones, cubes, cylinders, and spheres Identify 3-D shapes by name Choose the correct name for a given solid. Two choices will be given Identify 3-D shapes among real-world objects Find the basic 3-dimensional solid shapes used in pictures of real-life objects. When presented with a set of less than 10 pictures, click the ones that contain the specified solid Identify and position 3D shapes above, below, in front of, or behind an object Students identify a specified 3D shape and practice positioning it correctly above, below, in front of, or behind a dinosaur friend Position 3-D shapes above, below, in front of, and behind an object Practice 3-dimensional solids and position. Drag a specified solid to the directed location with respect to a static picture Practice position terms Students will practice the terms "next to" and "far from" with regard to an object's position. They will select the objects "next to" or "far from" a reference object Deepen understanding of the terms "next to" and "far from" Students will learn that there is more than one position that can be "next to" an object. They will continue practicing identification of places that are "next to" and "far from" a reference object Practice using position terms in context Students will place objects "next to" and "far from" a reference object. Students will recognize that there are multiple locations that can be "next to" or "far from" a given place Identify objects in positions above, below, next to, and in front of Practice the position words "above", "below", "next to", and "in front of". Click in one of two characters that are in the correct position relative to the stated character Topic C: Two-Dimensional and Three-Dimensional Shapes Students apply their previous understanding of flat and solid shapes to distinguish between the two. MODULE 3 Comparison of Length, Weight, Capacity, and Numbers to 10 Topic A: Comparison of Length and Height Using familiar, real objects, students use the language of comparison (tallest, taller, longest, longer, shortest, shorter) as they explore length. Topic B: Comparison of Length and Height of Linking Cube Sticks Within 10 Students work with familiar linking cubes to count cubes to determine length of sticks. They compare lengths using longer, shorter, and the same as. Topic C: Comparison of Weight Using familiar, real objects, students use the language of comparison (heavier, lighter, about the same) as they explore weight. To do so, they employ a virtual balance scale to compare. Topic D: Comparison of Volume Students learn the word capacity and use it to measure the amount of liquid a container can hold. Using the unit "glasses," they measure by pouring glasses into a larger container and by pouring from a larger container into glasses. They also compare measured capacities of two containers. Topic E: Are There Enough? To prepare for comparing numbers, students work with real world objects to determine if there are "enough" to make pairs. They then determine whether a set of real objects has more than, fewer than, or the same as another set of objects. Topic F: Comparison of Sets Within 10 Students compare two sets of objects using the words "more," "fewer," "greater," and "less." They work with aligned objects and numbers. Topic G: Comparison of Numerals Students compare two sets of objects using the words more, fewer, and same. They work with similar and dissimilar objects, both aligned and scattered, in vertical and horizontal alignments. Finally, they begin to compare numbers. MODULE 4 Number Pairs, Addition and Subtraction to 10 Topic A: Addition with totals of 2, 3, 4, and 5 Students learn the basics of addition with totals to 5. They work with real-world objects and math cubes in two colors to represent addends. They learn the meaning and use of the + and = symbols and solve equations based on real-world and cube models. Students also compose number bonds and addition statements based on the models. Topic B: Addition to 6, 7, and 8 Students extend their addition skills to totals of 8 using familiar tools and methods. They use and build models using real-world objects and math cubes. They compose number bonds, addition statements, and equations. Topic C: Addition concepts to 8 Students deepen their understanding of addition by strengthening the connection between concrete objects, base-10 blocks, and equations. They determine both sums and missing addends. They record equations, including the + symbol. Topic D: Subtraction from Numbers to 8 This topic introduces formal subtraction concepts including writing and solving expressions and equations. Students acting out take away stories and working at the pictorial level. Then the concrete objects and pictorial representations are tied to or matched to the representative subtraction expression or equation using the minus sign with no unknown. Solve subtraction problems (5 or less objects) Students will use objects to solve subtraction problems involving numbers that are 5 or less Solve more subtraction problems (5 or less objects). Rephrase subtraction as "taking away." Students will solve more subtraction problems using 5 or less objects. Students will begin to understand subtraction as a process of "taking away." Solve subtraction problems (5 or less objects) phrased as "taking away." Students will solve more subtraction problems using 5 or less objects, phrased as taking away a smaller number of objects from a larger group Learn about the meanings of the minus and equals signs Students will learn that - represents taking away and = represents something having the same value. Students will select - and = signs hidden in a picture. They will click and drag numbers and these signs to complete subtraction equations Identify minus and plus signs Students will click and drag minus and plus signs to sort them. Then, students will select the equation with a minus sign from a given set of equations Rewrite word problems as subtraction equations Students will represent word problems in equation form using what they know about minus and equals signs. Then, students will solve the subtraction equations Solve word problems and number sentences involving subtraction within 5 Students will solve word problems involving subtraction within 5 using picture models and equations. Students then use picture models to fill in and solve number sentences also involving subtraction within 5 Solve subtraction problems within 5 using picture models for support Given a word problem, model, and incomplete number sentence, students will solve subtraction problems within 5 Match number sentences to associated word problems Given a word problem and picture model, students will select the correct number sentence that represents the problem Match linking cubes to a set of objects Given a set of objects, students will take away a certain amount of cubes to match the initial set. Then, children will be given a set of objects and select a corresponding set of cubes that represents the same amount Represent subtraction problems with linking cubes. Write number sentences to match models Students will use cubes to model taking away a certain number from a given set. Then, they will represent this model by writing a number sentence and solving the subtraction equation. Problems have 5 or less cubes Use models to complete subtraction number sentences within 5 Students will see a linking cube model (of 5 or less cubes) where several cubes are taken way. They will complete a given number sentence based on the model. There is a gradual release of support as the task progresses Solve subtraction equations with and without using linking cube models Students will view a linking cube model (of 5 or less cubes) and use it to solve a subtraction equation. Then, students will solve equations without models Solve expressions involving subtraction of numbers less than 5 Given an expression subtracting numbers less than 5, students will select the solution More practice solving subtraction equations within 5 Students will practice solving subtraction equations within 5. Students will solve equations without models or other supports Solving subtraction problems using greater numbers (6 or less) Students are given a picture model to represent a subtraction problem. Students complete a number sentence to show what number they began with, what was taken away, and what remains Solve subtraction problems with greater numbers (6 or less) using linking cubes for support Students are given linking cubes to represent a subtraction problem. Students will help create a number sentence to represent this problem and solve for the answer. Students review appropriate use of the minus sign in this task Create number sentences to represent word problems involving subtraction (7 or less) Given a word problem involving subtraction of numbers 7 or less, students will make a number sentence to represent and solve the problem More practice making number sentences representing subtraction of 7 or less objects Students will select a certain amount of objects to take away from a given set. Students will then make a number sentence representing the total amount, amount taken away, and what remains, and using the correct sign More practice using cubes to model subtraction problems and create number sentences Students will use cubes to model a subtraction problem. Students will create a number sentence to represent and solve the subtraction equation. Numbers in the task are 7 or less Select equations that represent a model of subtraction Given a picture model of a subtraction problem, students will select the correct equation that corresponds to the problem. Numbers in the task are 7 or less Write number sentences to represent subtraction problems with support Children will select matching pairs from a group of objects. Then, given a cube model of a subtraction problem, students will write a number sentence to represent the equation with support. Numbers in the task are 7 or less Topic E: Addition to and decomposition of 9 and 10 Students use the connections between concrete objects, base-10 blocks and equations to build upon their understanding of addition. They find missing addends and sums and record equations, including the + symbol. Topic F: Subtraction from 9 and 10 Students use the connections between concrete objects, number sentences and word problems to explore subtraction from 9 and 10. They learn to see subtraction as "taking away" and associate the minus sign with subtraction. Topic G: Patterns with Adding 0 and 1 and Making 10 Students use concrete objects, a number path, and equations to practice adding and subtracting 0 or 1 from a number. Students find number pairs that make 10 and use ten-frames and cube models to solve missing addend equations. MODULE 5 Numbers 10 - 20 and Counting to 100 Topic A: Count 10 Ones and Some Ones Students differentiate a "ten" from ones using a ten-frame or rod. They begin to explore the composition of 2-digit numbers and number names. These exercises form a foundation for later learning about place value and addition strategies. Students use a number line for the first time. Count objects and learn the concept of a ten-frame Students practice counting a variety of objects in ten-frames. Then, they practice identifying a ten-frame Learn the numbers 10-12 using a ten-frame and additional objects Students complete and count ten-frames before counting any remaining objects. Students count and learn the numbers from 10-12 as they go Use cubes and ten-rods to count to 11 and 12 Students count out 10 cubes and any remaining cubes. Then, students identify the numbers 11 and 12 Use a number line and practice the numbers 11 and 12 Students observe a number line and reinforce the numbers 11 and 12. Then, students fill in missing numbers on the number line. Next, students reinforce the names of numbers by hearing the number and then typing in the correct number Learn the numbers 13, 14, and 15 using ten-frames and remaining objects Students count out ten-frames and remaining objects to learn the numbers 13, 14, and 15. Then, students practice counting ten-frames and remaining objects altogether to count to 13, 14, and 15 Count and identify the numbers 13 through 15 using cubes and ten rods Students practice counting to 15 using ten rods and cubes. Then, students practice writing the numerals 13, 14, and 15. Additionally, students name the numbers by hearing or reading the correct word for 13, 14, and 15 Identify numbers on a number line and reinforce knowledge of numerals 1-15 Students practice matching numerals 13, 14 and 15 to the number name with a number line visualization. Then, students reinforce number knowledge by hearing a number between 1 and 15, and then typing the correct numeral Learn the numbers 16, 17, 18, and 19 using ten-frames and remaining objects Students count out ten-frames and remaining objects to learn the numbers 16, 17, 18, and 19. Students also practice typing the correct numeral for each of these new numbers Learn the number 20 Students use ten-frames to count to 19. Then, students learn that two ten frames are 20. In addition, students practice the numeral 20 Use a number line to fill in missing numbers, especially focusing on numbers 10-20 Using a number line visualization, students type in missing numbers 17, 18, 19, and 20, and name them. Then, students use a number line to review the order of numbers by filling in missing numbers from 1 to 20 Form 2-digit numbers within 20 using ten rods and additional cubes Students practice forming 2-digit numbers within 20 using ten rods and additional cubes Count to 2-digit numbers within 20 using partially filled ten frames and additional dots Students practice counting to 2-digit numbers within 20 using partially filled ten frames and additional dots Topic B: Decompose Numbers 11 - 20, and Count to Answer "How Many?" Students further their understanding of teen numbers by applying them to positions on a number line and simple equations. Topic C: Extend Count Sequence to 100 Students will explore how multiples of 10 are represented by 10-frames or number names. Students begin to order multiples of 10 on a number path to practice skip-counting by 10s. They begin to see 2-digit numbers as groups of 10 and some 1s. Use 10-frames to explore multiples of 10 (20, 30) Students will use full 10-frames to count groups of tens. They will learn about the numbers twenty and thirty and use 10-frames to understand that they represent groups of ten Use 10-frames to explore multiples of 10 (40, 50) Students will use full 10-frames to count groups of tens. They will learn about the numbers forty and fifty and use 10-frames to understand that they represent groups of ten Match number names to their numeric forms (0-50) Students will match number names to their numeric form, and vice versa. All numbers are 50 or less Skip count by 10s to 50, using 10-frames. Identify missing multiples of 10 in a set Students will identify the number shown by groups of 10-frames, then count by 10s in ascending order to 50. They will begin to notice how the value increases along with the numbers. Students then identify the missing multiple of 10 in a given set Use 10-frames to explore multiples of 10 (60, 70) Students will use full 10-frames to count groups of ten. They will learn about the numbers sixty and seventy and use 10-frames to understand that they represent groups of ten Use 10-frames to explore multiples of 10 (80, 90) Students will use full 10-frames to count groups of tens. They will learn about the numbers eighty and ninety and use 10-frames to understand that they represent groups of ten Given the number name, identify the numeric form of multiple of 10 Given the number name of a multiple of 10, students will identify and type the numeric form Use 10-frames to explore the concept of 100 Students will use full 10-frames to count groups of tens. They will learn that the number one hundred is ten tens, or represents 100 ones Match number names to their numeric forms (60-100) Students will match the names of multiples of 10 to their numeric form. Students will fill in the missing multiples of 10 on a number path from 0-100 Count by 1s and 10s Students will review counting by 1s to 10. Then, students will practice skip-counting by 10s to 100 Order multiples of 10 on a number path Students will place multiples of 10 in their correct space on a number path Begin to understand that 2-digit numbers are composed of multiples of 10 and some ones Students are given a set of full 10-frames and a few loose dots. Students will add the components together to find a 2-digit number. Then, students will begin to add a 1-digit number to a multiple of 10 to find a 2-digit number Given a number name, identify the numeric form of a 2-digit number Given the number name of a 2-digit number, students will identify and type the numeric form Order 2-digit numbers on a number path Students will complete number paths with missing 2-digit numbers. 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Getting Help with Proofs at the QSR Center and the Writing Center For help with the math: The tutors at the QSR are great, and they will happily help you think through the math that goes into a proof. In order to make sure that you are getting the best possible math help, here are some expectations about the kind of help you'll get at the QSR when it comes to your proofs assignments. We (the math faculty) ask the tutors to help you with the math, but not the writing, part of the Here are some guidelines for working on proofs at the QSR: • No working out the math in Overleaf at the QSR! (It's your job to typeset the math nicely after you have figured it out.) • As a corollary: you may only work things out using on paper (or whiteboard, or otherwise "by hand") with a tutor at the QSR. Why? Usually, thinking through a problem and understanding what's happening is necessary before you can write it up, and when you write it up, it may look a lot different than what you worked out on paper first. • Figuring out what properties you are using in a proof is a math problem; figuring out what order to write equations or how to break up the math and exposition is a writing challenge. Make sure you are asking about math, not writing. As long as you don't abuse the goodwill of the QSR tutors and director, you can also ask simple LaTeX questions if you can't find an answer on the internet or elsewhere. (Many of the tutors are also LaTeX gurus, and they can help troubleshoot your code, but don't forget that Google is a great debugger, too!) Conversely... for help with the writing: If you've figured out the math but you're having trouble writing it up, make an appointment to talk to a Writing Center peer counselor! There are usually a couple tutors at the Writing Center that have been recommended by math faculty, and we ask that they help you with the writing, but not the math, part of the proofs. The Upshot: You can use both centers for an assignment to get help with the math and the writing! But, you can't use one center for both. No comments:	{"url":"https://crgibbon.blogspot.com/2021/11/getting-help-with-proofs-at-qsr-center.html","timestamp":"2024-11-14T18:12:48Z","content_type":"application/xhtml+xml","content_length":"57086","record_id":"<urn:uuid:ab741b10-15bc-410b-abfd-4cf2590c4ea0>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00636.warc.gz"}
Fourth-order Runge-Kutta method - (Symplectic Geometry) - Vocab, Definition, Explanations \| Fiveable Fourth-order Runge-Kutta method from class: Symplectic Geometry The fourth-order Runge-Kutta method is a numerical technique used to approximate solutions of ordinary differential equations. It improves accuracy by calculating multiple slopes (or derivatives) at each step, allowing for a more precise estimate of the function's value at the next point. This method is especially useful in fields like physics and engineering, where complex systems need to be modeled accurately over time. congrats on reading the definition of fourth-order Runge-Kutta method. now let's actually learn it. 5 Must Know Facts For Your Next Test 1. The fourth-order Runge-Kutta method uses four evaluations of the derivative at each time step to produce a more accurate approximation than simpler methods. 2. This method's error decreases significantly with smaller step sizes, making it preferable for problems requiring high precision. 3. It is particularly well-suited for solving initial value problems where precise calculations are essential. 4. The algorithm is computationally more intensive than first or second-order methods due to the multiple derivative evaluations needed. 5. Applications of this method extend beyond mathematics, playing a key role in simulating physical systems in ray tracing and optical systems. Review Questions • How does the fourth-order Runge-Kutta method enhance the accuracy of approximating solutions for differential equations? □ The fourth-order Runge-Kutta method enhances accuracy by calculating the slope at multiple points within each time step. Specifically, it evaluates the derivative at the beginning, middle, and end of the interval, which allows it to better capture the behavior of the function. This approach helps in providing a more reliable estimate for the next value, making it superior to simpler methods that rely on fewer evaluations. • Discuss how the fourth-order Runge-Kutta method can be applied in ray tracing and optical systems to improve simulations. □ In ray tracing and optical systems, the fourth-order Runge-Kutta method can be used to accurately model light paths as they interact with various media. By approximating the differential equations governing light propagation with high precision, this method allows for more realistic simulations of optical phenomena such as reflection, refraction, and diffraction. The increased accuracy leads to better visual representations and enhances the effectiveness of simulations in design and analysis. • Evaluate the trade-offs involved when choosing to use the fourth-order Runge-Kutta method compared to lower-order methods in complex simulations. □ When choosing between the fourth-order Runge-Kutta method and lower-order methods, one must consider both accuracy and computational efficiency. While the fourth-order method provides a significant improvement in precision due to its multiple derivative evaluations, it also requires more computational resources and time. This trade-off is critical in complex simulations where higher accuracy might be necessary but could lead to longer processing times. Ultimately, selecting the appropriate method depends on balancing these factors against the specific requirements of the simulation being performed. "Fourth-order Runge-Kutta method" also found in: Â© 2024 Fiveable Inc. All rights reserved. APÂ® and SATÂ® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.	{"url":"https://library.fiveable.me/key-terms/symplectic-geometry/fourth-order-runge-kutta-method","timestamp":"2024-11-15T00:46:40Z","content_type":"text/html","content_length":"144623","record_id":"<urn:uuid:9ca644e0-3952-4206-9d3f-8a1213fb3854>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00334.warc.gz"}
Y-plus in Wall-Bounded Flows \| Importance, Calculation & Impact Y-plus in wall-bounded flows Explore the significance of Y-plus in CFD, its calculation, impact on turbulence modeling, and optimization strategies for accurate simulations. Understanding Y-plus in Wall-Bounded Flows Y-plus (Y^+) is a critical dimensionless parameter in Computational Fluid Dynamics (CFD), particularly in the study of wall-bounded flows. It plays a pivotal role in determining the accuracy and reliability of turbulence models used in CFD simulations. Understanding Y^+ is essential for engineers and researchers working in fields like aerodynamics, hydrodynamics, and process engineering. Importance of Y-plus in CFD Simulations Y^+ is used to quantify the relationship between the physical scale of a turbulent flow near a wall and the grid resolution of a CFD model. It is particularly relevant in the context of the boundary layer, a thin layer of fluid lying close to the surface of an object, where viscous effects are significant. The accuracy of turbulence modeling in this region is highly dependent on the Y^+ value. In simulations, a low Y^+ value (typically less than 5) indicates that the grid resolution is fine enough to resolve the viscous sublayer of the boundary layer. This allows for direct modeling of near-wall phenomena using low-Reynolds-number turbulence models. Conversely, a high Y^+ value suggests that the grid is too coarse, potentially leading to inaccurate predictions of wall shear stress and heat transfer. Calculation of Y-plus To calculate Y^+, the following formula is used: Y^+ = $ \frac{u_{\tau} \cdot y}{\nu} $ where $ u_{\tau} $ is the friction velocity, $ y $ is the distance from the wall, and $ \nu $ is the kinematic viscosity of the fluid. The friction velocity is a measure of shear stress at the wall and is calculated based on the flow conditions and surface roughness. Impact of Y-plus on Turbulence Modeling The choice of turbulence model in a CFD simulation is heavily influenced by the Y^+ value. For low Y^+ values, models like the Spalart-Allmaras or the k-omega SST (Shear Stress Transport) are often preferred due to their ability to accurately resolve near-wall effects. For higher Y^+ values, where resolving the entire boundary layer might not be computationally feasible, wall functions are employed to approximate the effects of the boundary layer. These wall functions bridge the gap between the log-law region of the boundary layer and the outer flow, allowing for simulations that can capture the essential physics without the computational expense of fully resolving the viscous sublayer. This approach is particularly useful in industrial applications where computational resources are limited. Thus, understanding and correctly interpreting Y^+ values is crucial for ensuring the fidelity of CFD simulations in wall-bounded flows. Optimizing Y-plus for Accurate CFD Simulations Optimizing the Y^+ value is a crucial step in setting up CFD simulations. Engineers must balance the need for accuracy in the boundary layer with computational limitations. A very low Y^+ (less than 1) might lead to extremely fine mesh requirements, increasing computational costs significantly. On the other hand, a Y^+ value too high can lead to loss of important flow details near the wall, affecting the accuracy of the simulation. Generally, a Y^+ value in the range of 30-300 is considered acceptable for most industrial applications using wall functions. Challenges and Solutions in Y-plus Calculation One of the primary challenges in using Y^+ is the estimation of the appropriate mesh density before the simulation. Iterative methods are often used, where initial simulations help in refining the mesh based on the calculated Y^+ values. Advanced meshing techniques and adaptive mesh refinement (AMR) can also be employed to optimize the mesh dynamically based on Y^+ values during the simulation Furthermore, the non-linear nature of fluid flow near walls can introduce complexities in accurately predicting Y^+. This necessitates a deep understanding of fluid dynamics and turbulence modeling to make informed decisions during the simulation setup. In conclusion, Y-plus (Y^+) is a fundamental parameter in CFD simulations of wall-bounded flows, crucial for the accurate prediction of flow characteristics near walls. It serves as a bridge between the physical flow phenomena and the numerical world of CFD, guiding the choice of turbulence models and mesh resolution. The careful calculation and optimization of Y^+ are essential for achieving a balance between computational efficiency and simulation accuracy. As CFD technology advances, the importance of understanding and effectively using Y^+ in simulations continues to grow, particularly in fields where precise flow modeling is critical.	{"url":"https://modern-physics.org/y-plus-in-wall-bounded-flows/","timestamp":"2024-11-05T15:20:35Z","content_type":"text/html","content_length":"160690","record_id":"<urn:uuid:43e07b33-ced1-46dc-9d29-2cdb456c2b66>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00868.warc.gz"}
Proof Market Definition \| CoinMarketCap Proof Market A proof market is a decentralized marketplace where users can buy and sell cryptographic proofs to verify ownership. What Is a Proof Market? A proof market is a marketplace where users can buy and sell proofs to verify the ownership of digital assets , the validity of a particular transaction and the authenticity of a piece of information or computation results. The buyer can then use the proofs to verify a statement or fact without having to independently verify it themselves. Why Is a Proof Market Needed? To understand why proof markets are needed, the concept of Zero Knowledge Proofs (ZK Proofs) needs to be understood. Zk proofs are paving the way forward for a more secure and privacy-focused future. However, one of the biggest challenges to widespread adoption is the time-consuming computation required to generate these proofs due to the complexity and heavy computations. These generators require significant investment to maintain and build up their infrastructure to keep up with generating complex proofs. Due to the costly nature of zk proofs, proof generation is generally only accessible to larger projects which may be better funded compared to smaller projects that are just starting up. This could lead to a high reliance on multiple large players and limit the widespread adoption of the technology by the developers' community. A proof market enables anyone to post a request with their stated price for a proof, and any party can offer to fill this request. How Does a Proof Market Work? A proof market hosts a marketplace for the proofs generated from zero-knowledge technology which enables a statement to be proven true or false without revealing any additional information beyond the fact. For example, a user might be interested to order a proof that can verify that they have a certain amount of funds in their DeFi wallet or that a particular computation with data has been executed correctly. These proofs are supplied by proof generators as a commodity on a proof market that is subjected to the conventional mechanism of supply and demand. A generator will have to take into consideration the cost and time required to produce a proof efficiently that is competitive with the market. This induces competition between proof generators to produce proofs with the smallest and /or cheapest generation cost. To generate a zero-knowledge proof, the proof requestor will create an order for zk-proof over a predefined circuit. The proof market will match the order to proof generators and send the order to the selected generator which will execute computations. The proof generator will send the generated proof into the proof market for validation. Once the proof has been verified, the proof market will send the validated proof to the proof requestor. The Benefits of a Proof Market 1. Cost savings: An easily accessible platform to buy pre-existing, high-quality proofs and conserve resources as opposed to investing in costly infrastructure for proof generators. 2. Wide variety of proofs: Proofs available on a proof market offer a range of different proofs to fit the needs of different proof systems with the assurance that the proofs being traded are of high quality. 3. Democratizing the production of ZK proofs: Projects, no matter the size, can now obtain high-quality proofs to build their protocols in a decentralized manner. 4. Low technical knowledge needed: The proof market platform simplifies the process of generating and verifying cryptographic proofs for any developer to access them without the need for extensive technical expertise. 5. Standardization of Proofs: Standardising zk proofs generation and verification across different applications and platforms not only helps facilitate but also the usability of zero-knowledge proofs in the blockchain industry. 6. Open market data: Users are able to analyze and understand the profitability of each order and make assumptions based on earnings due to the real-time nature of data across multiple dashboards. The Limitations of a Proof Market While a proof market has immense potential to enhance zero-knowledge technology, it is a new concept and may have some limitations. 1. Limited utility: The utility of a proof market is dependent on the specific use cases of cryptographic proofs, particularly on enhancing scalability, privacy and security in various blockchain applications. As these use cases extend, so will the availability of proofs on specific use cases grow. 2. Circuit development: To prove any computations with zk proofs, developers have to first come up with a proof definition for their code, otherwise known as circuits. In order to write a circuit, the user requires a deep knowledge of how a particular proof system works. 3. Proof generation economics: The infrastructure to generate high-quality proofs efficiently is expensive. In order for generators to see a return, there needs to be different avenues that require proofs for there to be enough demand to make proof generation economically viable. Other minor limitations may include potential scalability issues and regulatory compliance How Can a Proof Market Support the Adoption of Zero-Knowledge Technology? Zk proofs are a vital component for projects to validate and authenticate their protocols and so, enabling greater access to high-quality proofs for projects of all sizes is necessary. The decentralized nature of a proof market ensures that transactions are secure, transparent and free from central control or manipulation. By providing a user-friendly and accessible platform for buying and selling cryptographic proofs, a proof market can help to democratize access to this technology and drive wider adoption in the blockchain industry. Contributor: Mikhail Komarov, founder of =nil; Foundation. Mikhail is a leading developer of infrastructure for effective zero-knowledge proof (ZKP) generation. Mikhail is a researcher and developer in the fields of cryptography and database management systems (DBMS). His journey in technology began in 2013 when he began contributing to BitMessage, a peer-to-peer encrypted communications protocol. He then worked with the blockchain network BitShares, and Steemit, the first application built on the Steem blockchain. Mikhail worked on a fork of Steem from 2017 to 2018, before founding =nil; Foundation in April 2018	{"url":"https://coinmarketcap.com/academy/glossary/proof-market","timestamp":"2024-11-03T19:37:00Z","content_type":"text/html","content_length":"159063","record_id":"<urn:uuid:fcfc7fb4-1481-43dc-8d79-6b0f7634dcc7>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00128.warc.gz"}
Re: Change the default -1 1 coding of categorical variable to 0 1 codingRe: Change the default -1 1 coding of categorical variable to 0 1 coding Dear JMP community, I found that when fitting a multiple linear regression (y ~ X1 + X2, X1 is a continuous and X2 is a categorical variable), by default JMP defines the dummy variable X2 as -1 1. Is there a way to change it to 0 1, which will be the same as the default dummy variable coding as R? Thanks!	{"url":"https://community.jmp.com/t5/Discussions/Change-the-default-1-1-coding-of-categorical-variable-to-0-1/m-p/799499","timestamp":"2024-11-04T20:58:54Z","content_type":"text/html","content_length":"746892","record_id":"<urn:uuid:ea202797-917b-49b3-96e9-bcdf2416ae2c>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00075.warc.gz"}
Motion in a Straight LineMotion in a Straight Line Motion in a Straight Line These Motion in a Straight Line Numericals are very useful for the students preparing for school as well as competitive examinations.Video solution with answer for each question for chapter Motion in a Straight Line Numerical of 11th physics is given. Q1) A Body starts from rest and travels 120cm in the 8th second. then acceleration of the body is body is..[AFMC 1997] ANS. 0.16m/se Q2) A car travels first half distance between to places with a speed of 30km/hr and the remaining half with a speed of 50km/hr. the average speed of the car Ans. 37.5km/hr Q3) The displacement x of a particle moving along a straight line at time t is given by The acceleration of the particle is Ans. 2a2 Q4) Find the total displacement of a body in 8 seconds starting from rest with an acceleration of 20cm/sec2[AFMC 2000] Q5) A particle moves along a straight line OX. At a time (in second) the distance x (in meters) of the particle from O is given by How long would the particle travel before coming to rest (AFMC2000) Ans. 40m Q6) A bus start from rest with an acceleration of 1 meter/sec2 . A man who is 48 meter behind the bus with a uniform velocity of 10m/sec then the minimum time after which the man will catch the bus is (AFMC 2001) Ans. 8sec Q7) The displacement of a particle moving in straight line depends on time as 2+yt+ᶑ the ratio of initial acceleration to its initial velocity depends [AFMC 2002] Ans. Only on β and y Q8) A particle covers 150 meter in 8th second starting from rest , its acceleration is [AFMC 2003] Ans. 20 meter/sec2 Q9) A car is moving with uniform acceleration it covers 200 meter in 2 second and 220 meter in next 4 second . The velocity of car at the end of end of 7th second from start is Ans. 10 m/s Q10) A train of 150 meter long is going towards north direction at speed of 10 meter/second . A parrot flies at the speed of 5meter/second towards south direction parallel to the railway track . The time taken by the parrot to cross the train is [CBSE-PMT 1992] Ans. 10 second Q11) A bus travelling the first one third distance at the speed of 10km/s , the next ne third at 20km/s and the last one third at 60km/s , the average speed of the bus is …… [CBSE-PMT 1991] Ans. 18km/s Q12) A car accelerates from rest at a constant rate α for some time, after which it decelerated at constant rate β and comes to rest. If the total time elapsed is t, then the maximum velocity acquired by the car is.. [CBSE PMT 1994 ] Ans. αβt/α+β Q13) A particle moves along a straight line such that its displac time ‘t’ is given by s t3-6t2+3t+4) meters The velocity when the acceleration es zero is… [CBSE PMT 1994] Q14) A particle covers half of its total distance with speed v1 and the rest half distance with speed v2. Its average speed during the complete journey is..[CBSE-PMT 2O11] Ans. 2v1v2/v1+v2 Q15) A ball is dropped form a high rise platform at t=O starting from rest. After 6 seconds another ball is thrown downwards from the same platform with a speed y. The two balls meet at t=1 8s. What is the value of y? [g=1Om/s2) [CBSE-PMT 2011] Ans. 75m/sec Q16) A particle moving in a straight line with a constant acceleration changes its velocity from 10 m/s to 20m/s, while passing through 135 m in t second. The value of t is ... [CBSE-PMT 2008] Ans. 9 Q17) A particle starts its motion from rest under the action of force. if the distance covered in first 10 second is S1 and first 20 seconds is S2 then .. [CBSE-PMT 20091] Ans. S2 = 4S1 Q18) If a ball is thrown vertically upwards with speed of ‘u’ the distance covered during the last ‘t’ second of its ascent is.. [CBSE-PMT 2003] Ans. ½ gt2 Q19) A bus is moving with a speed of 10meter/second on a straight road . A scooterist wishes to over take the bus in 100second. If the bus is at a distance of 1km from the scooterist , with what speed should the scooterist chase the bus ? [CBSE-PMT 2009] Ans. 20 m/s No comments	{"url":"http://numerical.nawenduclasses.com/2019/08/motion-in-straight-line.html","timestamp":"2024-11-10T12:47:11Z","content_type":"application/xhtml+xml","content_length":"137241","record_id":"<urn:uuid:20c105a8-0774-4f68-a4b3-1f333f283296>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00286.warc.gz"}
Detektering av radarsignaler med matchande filter - Åbo Coding and Modulation for Communication Systems The World's most comprehensive professionally edited abbreviations and acronyms database All trademarks/service marks referenced on this site are properties of their respective owners. BF-AWGN: Acronym Finder [home, info] Words similar to bf awgn Usage examples for bf awgn Words that often appear near bf awgn Rhymes of bf awgn Invented words related to bf awgn: Search for bf awgn on Google or Wikipedia. Search completed in 0.066 seconds. Additive white Gaussian noise (AWGN) is a basic noise model used in information theory to mimic the effect of many random processes that occur in nature. The modifiers denote specific characteristics: Additive White Gaussian Noise (AWGN) The performance of a digital communication system is quantified by the probability of bit detection errors in the presence of thermal noise . In the context of wireless communications, the main source of thermal noise is addition of random signals arising from the vibration of atoms in the receiver This channel model is used to evaluate the difference between a LOS model and a channel model including fast-fading effects. Hence, the AWGN results can be seen as reference measurements. out = awgn (in,snr) adds white Gaussian noise to the vector signal in. This syntax assumes that the power of in is 0 dBW. example. TURBO TRELLIS CODED MODULATION - Dissertations.se Capacity of AWGN channels In this chapter we prove that the capacity of an AWGN channel with bandwidth W and signal-to-noise ratio SNR is W log2(1+SNR) bits per second (b/s). The proof that reliable transmission is possible at any rate less than capacity is based on Shannon’s random code ensemble, typical-set Quick Reference - Communication Home : www.sharetechnote.com. AWGN, AWGN vs SNR . AWGN stands for Additive WhiteGaussianNoise. Awgn & Rayleigh Fading Channel - MD Taslim Arefin - häftad Signaaliavaruusesitys ja optimaalisen vastaanottimen suunnittelu AWGN-kanavan tapauksessa. Multi-rate control over awgn channels via analog joint source-channel coding. OFDM is an or-thogonal frequency division multiplexing to reduce inter symbol interference problem. Simulation of QPSK signals is carried with both AWGN … Hi, SNR implies that you have a signal, in your call to awgn() you have no signal, just noise. If you use awgn() you should input your signal and awgn() will add noise at the specified SNR. 3 Comments. Show Hide 2 older comments. Performance Analysis for AWGN and Rayleigh Fading Channel under Different AND & OR Fusion Rules - written by Abdullah Al Zubaer , Sabrina Ferdous , Rohani Amrin published on 2020/09/09 download full article with reference data and citations AWGN Channel Section Overview. An AWGN channel adds white Gaussian noise to the signal that passes through it. 1 dollar till sek W-CDMA system in AWGN and Multipath Rayleigh Fading. 3. Multi-user W-CDMA system in AWGN and Multipath Rayleigh Fading. There are some parameters for multiple rays using QPSK and QAM in W-CDMA system models that will be obtained using MatLab. They are 1. An AWGN channel is the most basic model of a communication system. Some examples of systems operating largely in AWGN conditions are space Achieving 1/2 log (1+SNR) on the AWGN channel with lattice encoding and can achieve the additive white Gaussian noise (AWGN) channel capacity. Search completed in 0.066 seconds. Additive white Gaussian noise (AWGN) is a basic noise model used in information theory to mimic the effect of many random processes that occur in nature. The modifiers denote specific characteristics: Additive White Gaussian Noise (AWGN) The performance of a digital communication system is quantified by the probability of bit detection errors in the presence of thermal noise . In the context of wireless communications, the main source of thermal noise is addition of random signals arising from the vibration of atoms in the receiver electronics. A basic and generally accepted noise model is known as Additive White Gaussian Noise (AWGN), which imitates various random processes seen in nature. Antal barn i sverige Abbreviations used in this thesis are listed below: • AWGN Additive White Gaussian Noise. • BER Bit Error Rate. • BPSK Binary Phase Shift Keying. Noisecom PNG7000 Analog AWGN Noise Generator. The PNG7000A Series instruments generate white Gaussian noise (AWGN) and provide a summing input Acronym, Definition. AWGN, Additive White Gaussian Noise. AWGN, Adaptive White Gaussian Noise. A basic and generally accepted noise model is known as Additive White Gaussian Noise (AWGN), which imitates various random processes seen in nature. Let’s break each of those words down for further clarity: The AWGN channel is a well-known model to indicate line of sight (LOS) conditions. This channel model is used to evaluate the difference between a LOS model and a channel model including fast-fading effects. Hence, the AWGN results can be seen as reference measurements. out = awgn (in,snr) adds white Gaussian noise to the vector signal in. This syntax assumes that the power of in is 0 dBW. example. Concerta urinprov ringarnas herre ringens brödraskapnorian drillable injecttriss skatt på vinstfel i tal9000 dollar i kromorganisering statens vegvesenfakturera utan eget bolag Kvalitet hos Multimediatjänster - Sida 301 - Google böcker, resultat Makten över X antas vara 0 in software simulation, including band-limiting using pulse-shaping filters and insertion of both Additive White Gaussian Noise (AWGN) and coherent noise. Data Transmission over Radio Channels. 4.1. 187. Axfood rapportelektroteknik lth antagningspoäng Podcast - Meditera mera - Sveriges första interaktiva podcast Printer friendly. Menu Search. New search features Acronym Blog Free tools "AcronymFinder.com. Abbreviation to The AWGN assumption simplified the channel model and allowed us to learn the basic skeletal operation of NOMA. In this post, we will use MATLAB to simulate the capacity, outage and BER performance of a two user NOMA network by following a more realistic model namely, Rayleigh fading model. AWGN Channel Section Overview. An AWGN channel adds white Gaussian noise to the signal that passes through it. Ber - Yolk Music AWGN står för Tillsatsen vit Gaussisk buller. Control and Communication with Signal-to-Noise Pris: 588 kr. häftad, 2011. Skickas inom 5-7 vardagar. Köp boken Awgn & Rayleigh Fading Channel av MD Taslim Arefin (ISBN 9783639375985) hos Adlibris.	{"url":"https://valutawcwwhcw.netlify.app/76676/9047","timestamp":"2024-11-08T04:58:55Z","content_type":"text/html","content_length":"17480","record_id":"<urn:uuid:42350dad-97dc-4b9a-8ed4-d5fb4df51981>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00185.warc.gz"}
moment of inertia? ・In Japanese ＜Premise knowledge＞・Angular acceleration ■What is the moment of inertia? The moment of inertia expresses the ease of rotation of a rotating body, and is expressed by the following formula. For example, if the inertia is small, the rotating body is easy to rotate and stop, but if the inertia is large, the rotating body is difficult to rotate and stop. ■Relationship between torque, angular velocity, and inertia Torque is proportional to inertia and angular acceleration and is expressed below. This has the same relational expression to Newton's law F = ma, only the variables have changed. ■Confirm the influence of inertia by simulation See how inertia affects the movement of the rotating body. The above equation (1) is designed with scilab as follows. The simulation results are as follows. It simulates the movement of angular velocity when the same torque is applied to rotating bodies with different inertia. Even if the same torque is applied, the larger the inertia, the slower the speed rise.	{"url":"http://taketake2.com/P54_en.html","timestamp":"2024-11-10T17:35:50Z","content_type":"text/html","content_length":"10259","record_id":"<urn:uuid:6961c6fe-5c63-4a1f-af2b-a6eed149e6a8>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00115.warc.gz"}
Exponential Functions - Formula, Properties, Graph, Rules What’s an Exponential Function? An exponential function measures an exponential decrease or increase in a particular base. For instance, let us assume a country's population doubles annually. This population growth can be depicted as an exponential function. Exponential functions have many real-life uses. Mathematically speaking, an exponential function is displayed as f(x) = b^x. Here we discuss the fundamentals of an exponential function along with important examples. What’s the equation for an Exponential Function? The general formula for an exponential function is f(x) = b^x, where: 1. b is the base, and x is the exponent or power. 2. b is a constant, and x varies For example, if b = 2, we then get the square function f(x) = 2^x. And if b = 1/2, then we get the square function f(x) = (1/2)^x. In a situation where b is greater than 0 and unequal to 1, x will be a real number. How do you plot Exponential Functions? To plot an exponential function, we have to discover the dots where the function intersects the axes. This is known as the x and y-intercepts. Since the exponential function has a constant, we need to set the value for it. Let's focus on the value of b = 2. To discover the y-coordinates, one must to set the value for x. For example, for x = 2, y will be 4, for x = 1, y will be 2 In following this method, we achieve the domain and the range values for the function. After having the values, we need to graph them on the x-axis and the y-axis. What are the properties of Exponential Functions? All exponential functions share identical qualities. When the base of an exponential function is larger than 1, the graph would have the following characteristics: • The line intersects the point (0,1) • The domain is all positive real numbers • The range is greater than 0 • The graph is a curved line • The graph is on an incline • The graph is smooth and continuous • As x nears negative infinity, the graph is asymptomatic regarding the x-axis • As x advances toward positive infinity, the graph increases without bound. In events where the bases are fractions or decimals between 0 and 1, an exponential function exhibits the following properties: • The graph intersects the point (0,1) • The range is greater than 0 • The domain is entirely real numbers • The graph is declining • The graph is a curved line • As x approaches positive infinity, the line in the graph is asymptotic to the x-axis. • As x advances toward negative infinity, the line approaches without bound • The graph is level • The graph is unending There are some basic rules to remember when dealing with exponential functions. Rule 1: Multiply exponential functions with the same base, add the exponents. For instance, if we have to multiply two exponential functions that have a base of 2, then we can compose it as 2^x * 2^y = 2^(x+y). Rule 2: To divide exponential functions with an equivalent base, deduct the exponents. For example, if we have to divide two exponential functions that posses a base of 3, we can write it as 3^x / 3^y = 3^(x-y). Rule 3: To increase an exponential function to a power, multiply the exponents. For instance, if we have to raise an exponential function with a base of 4 to the third power, then we can write it as (4^x)^3 = 4^(3x). Rule 4: An exponential function that has a base of 1 is consistently equal to 1. For example, 1^x = 1 no matter what the worth of x is. Rule 5: An exponential function with a base of 0 is always equivalent to 0. For example, 0^x = 0 no matter what the value of x is. Exponential functions are usually used to signify exponential growth. As the variable increases, the value of the function rises quicker and quicker. Example 1 Let’s examine the example of the growing of bacteria. Let us suppose that we have a culture of bacteria that multiples by two every hour, then at the end of the first hour, we will have 2 times as many bacteria. At the end of hour two, we will have 4 times as many bacteria (2 x 2). At the end of the third hour, we will have 8 times as many bacteria (2 x 2 x 2). This rate of growth can be portrayed an exponential function as follows: f(t) = 2^t where f(t) is the number of bacteria at time t and t is measured in hours. Example 2 Also, exponential functions can portray exponential decay. Let’s say we had a dangerous material that decays at a rate of half its amount every hour, then at the end of hour one, we will have half as much material. After hour two, we will have 1/4 as much substance (1/2 x 1/2). At the end of hour three, we will have one-eighth as much material (1/2 x 1/2 x 1/2). This can be displayed using an exponential equation as below: f(t) = 1/2^t where f(t) is the amount of material at time t and t is calculated in hours. As demonstrated, both of these samples follow a similar pattern, which is why they are able to be depicted using exponential functions. In fact, any rate of change can be indicated using exponential functions. Recall that in exponential functions, the positive or the negative exponent is represented by the variable while the base continues to be fixed. This means that any exponential growth or decay where the base changes is not an exponential function. For instance, in the case of compound interest, the interest rate remains the same whilst the base varies in regular amounts of time. An exponential function is able to be graphed using a table of values. To get the graph of an exponential function, we have to plug in different values for x and then asses the equivalent values for Let's look at the example below. Example 1 Graph the this exponential function formula: y = 3^x To begin, let's make a table of values. As you can see, the values of y grow very fast as x rises. Imagine we were to plot this exponential function graph on a coordinate plane, it would look like the following: As shown, the graph is a curved line that rises from left to right ,getting steeper as it persists. Example 2 Plot the following exponential function: y = 1/2^x To begin, let's create a table of values. As you can see, the values of y decrease very quickly as x increases. The reason is because 1/2 is less than 1. If we were to graph the x-values and y-values on a coordinate plane, it is going to look like what you see below: This is a decay function. As shown, the graph is a curved line that decreases from right to left and gets smoother as it proceeds. The Derivative of Exponential Functions The derivative of an exponential function f(x) = a^x can be shown as f(ax)/dx = ax. All derivatives of exponential functions display particular characteristics whereby the derivative of the function is the function itself. The above can be written as following: f'x = a^x = f(x). Exponential Series The exponential series is a power series whose terms are the powers of an independent variable number. The general form of an exponential series is: Grade Potential is Able to Help You Learn Exponential Functions If you're fighting to grasp exponential functions, or just require a bit of extra help with math in general, consider working with a tutor. At Grade Potential, our Riverside math tutors are experts in their field and can supply you with the one-on-one support you need to triumph. Call us at (951) 337-4564 or contact us now to discover more about the ways in which we can assist you in reaching your academic potential.	{"url":"https://www.riversideinhometutors.com/blog/exponential-functions-formula-properties-graph-rules","timestamp":"2024-11-10T17:44:33Z","content_type":"text/html","content_length":"83839","record_id":"<urn:uuid:a1fc88e5-f583-4ee6-9fb5-7411550a009d>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00565.warc.gz"}
2.2 Variables and Data Types In this section, we will discuss variables and data types in Python. Variables are essential in programming as they allow you to store and manipulate data, while data types define the kind of data that can be stored in a variable. 2.2.1 Variables Variables in Python are used to store values for later use in your program. A variable is assigned a value using the assignment operator (=). The variable name must follow certain naming conventions: • Start with a letter or an underscore (_) • Consist of letters, numbers, or underscores • Be case-sensitive Examples of variable assignments: name = "John" age = 25 _height = 175.5 You can also perform multiple assignments in a single line: x, y, z = 1, 2, 3 2.2.2 Data Types Python has several built-in data types that allow you to work with various forms of data. Some of the most common data types are: • Integers (int): Whole numbers, such as 5, -3, or 42. • Floating-point numbers (float): Decimal numbers, such as 3.14, -0.5, or 1.0. • Strings (str): Sequences of characters, such as "hello", 'world', or "Python is fun!". • Booleans (bool): Logical values, either True or False. You can determine the type of a value or variable using the type() function: x = 42 print(type(x)) # Output: <class 'int'> y = 3.14 print(type(y)) # Output: <class 'float'> z = "Python" print(type(z)) # Output: <class 'str'> a = True print(type(a)) # Output: <class 'bool'> 2.2.3 Type Conversion You can convert between different data types using built-in Python functions, such as int(), float(), str(), and bool(). Examples of type conversion: x = 5.5 y = int(x) # Convert float to int, y becomes 5 a = 3 b = float(a) # Convert int to float, b becomes 3.0 c = 42 d = str(c) # Convert int to str, d becomes "42" e = "123" f = int(e) # Convert str to int, f becomes 123 It's important to note that not all conversions are possible. For example, converting a non-numeric string to an integer or float will result in a ValueError. Understanding variables and data types is crucial in Python programming, as it forms the basis for manipulating data and performing various operations. As you progress through this book, you will encounter more advanced data types, such as lists, dictionaries, and tuples, which will allow you to work with more complex data structures. Exercise 2.2.1: Celsius to Fahrenheit Converter In this exercise, you will write a Python program that converts a temperature in degrees Celsius to degrees Fahrenheit. You will practice using variables, data types, expressions, and the print() 1. Create a new Python file or open a Python interpreter. 2. Declare a variable named celsius and assign it the value of the temperature in degrees Celsius (e.g., 30). 3. Calculate the temperature in degrees Fahrenheit using the formula fahrenheit = (celsius * 9/5) + 32, and assign the result to a new variable named fahrenheit. 4. Use the print() function to display the temperature in degrees Fahrenheit, formatted as a string with a degree symbol (°F). Your final code should look something like this: celsius = 30 fahrenheit = (celsius * 9/5) + 32 print(f"{celsius}°C is equal to {fahrenheit}°F") When you run your program, you should see output similar to the following: 30°C is equal to 86.0°F Feel free to modify the celsius value to test your program with different temperatures. This exercise helps you become familiar with variables, data types, arithmetic expressions, and the print() function with f-strings for formatted output. Exercise 2.2.2: Calculate the Area and Perimeter of a Rectangle In this exercise, you will write a Python program that calculates the area and perimeter of a rectangle. You will practice using variables and data types. 1. Create a new Python file or open a Python interpreter. 2. Declare two variables: length and width. Assign them appropriate values (e.g., 5 and 3, respectively). 3. Calculate the area of the rectangle using the formula area = length * width, and assign the result to a variable named area. 4. Calculate the perimeter of the rectangle using the formula perimeter = 2 * (length + width), and assign the result to a variable named perimeter. 5. Use the print() function to display the calculated area and perimeter of the rectangle. Your final code should look something like this: length = 5 width = 3 area = length * width perimeter = 2 * (length + width) print(f"The area of a rectangle with length {length} and width {width} is {area}") print(f"The perimeter of a rectangle with length {length} and width {width} is {perimeter}") When you run your program, you should see output similar to the following: The area of a rectangle with length 5 and width 3 is 15 The perimeter of a rectangle with length 5 and width 3 is 16 Feel free to modify the length and width variables to practice with different rectangle dimensions. This exercise helps you become familiar with variables and data types in Python. Exercise 2.2.3: Simple Interest Calculator In this exercise, you will write a Python program that calculates the simple interest earned on an investment. You will practice using variables and data types. 1. Create a new Python file or open a Python interpreter. 2. Declare three variables: principal, rate, and time. Assign them appropriate values (e.g., 1000, 0.05, and 3, respectively). 3. Calculate the simple interest using the formula simple_interest = principal * rate * time, and assign the result to a variable named simple_interest. 4. Use the print() function to display the calculated simple interest. Your final code should look something like this: principal = 1000 rate = 0.05 time = 3 simple_interest = principal * rate * time print(f"The simple interest earned on an investment of ${principal} at a rate of {rate * 100}% over {time} years is ${simple_interest}") When you run your program, you should see output similar to the following: The simple interest earned on an investment of $1000 at a rate of 5.0% over 3 years is $150.0 Feel free to modify the variables to practice with different investment scenarios. These exercises help you become familiar with variables and data types in Python. 2.2 Variables and Data Types In this section, we will discuss variables and data types in Python. Variables are essential in programming as they allow you to store and manipulate data, while data types define the kind of data that can be stored in a variable. 2.2.1 Variables Variables in Python are used to store values for later use in your program. A variable is assigned a value using the assignment operator (=). The variable name must follow certain naming conventions: • Start with a letter or an underscore (_) • Consist of letters, numbers, or underscores • Be case-sensitive Examples of variable assignments: name = "John" age = 25 _height = 175.5 You can also perform multiple assignments in a single line: x, y, z = 1, 2, 3 2.2.2 Data Types Python has several built-in data types that allow you to work with various forms of data. Some of the most common data types are: • Integers (int): Whole numbers, such as 5, -3, or 42. • Floating-point numbers (float): Decimal numbers, such as 3.14, -0.5, or 1.0. • Strings (str): Sequences of characters, such as "hello", 'world', or "Python is fun!". • Booleans (bool): Logical values, either True or False. You can determine the type of a value or variable using the type() function: x = 42 print(type(x)) # Output: <class 'int'> y = 3.14 print(type(y)) # Output: <class 'float'> z = "Python" print(type(z)) # Output: <class 'str'> a = True print(type(a)) # Output: <class 'bool'> 2.2.3 Type Conversion You can convert between different data types using built-in Python functions, such as int(), float(), str(), and bool(). Examples of type conversion: x = 5.5 y = int(x) # Convert float to int, y becomes 5 a = 3 b = float(a) # Convert int to float, b becomes 3.0 c = 42 d = str(c) # Convert int to str, d becomes "42" e = "123" f = int(e) # Convert str to int, f becomes 123 It's important to note that not all conversions are possible. For example, converting a non-numeric string to an integer or float will result in a ValueError. Understanding variables and data types is crucial in Python programming, as it forms the basis for manipulating data and performing various operations. As you progress through this book, you will encounter more advanced data types, such as lists, dictionaries, and tuples, which will allow you to work with more complex data structures. Exercise 2.2.1: Celsius to Fahrenheit Converter In this exercise, you will write a Python program that converts a temperature in degrees Celsius to degrees Fahrenheit. You will practice using variables, data types, expressions, and the print() 1. Create a new Python file or open a Python interpreter. 2. Declare a variable named celsius and assign it the value of the temperature in degrees Celsius (e.g., 30). 3. Calculate the temperature in degrees Fahrenheit using the formula fahrenheit = (celsius * 9/5) + 32, and assign the result to a new variable named fahrenheit. 4. Use the print() function to display the temperature in degrees Fahrenheit, formatted as a string with a degree symbol (°F). Your final code should look something like this: celsius = 30 fahrenheit = (celsius * 9/5) + 32 print(f"{celsius}°C is equal to {fahrenheit}°F") When you run your program, you should see output similar to the following: 30°C is equal to 86.0°F Feel free to modify the celsius value to test your program with different temperatures. This exercise helps you become familiar with variables, data types, arithmetic expressions, and the print() function with f-strings for formatted output. Exercise 2.2.2: Calculate the Area and Perimeter of a Rectangle In this exercise, you will write a Python program that calculates the area and perimeter of a rectangle. You will practice using variables and data types. 1. Create a new Python file or open a Python interpreter. 2. Declare two variables: length and width. Assign them appropriate values (e.g., 5 and 3, respectively). 3. Calculate the area of the rectangle using the formula area = length * width, and assign the result to a variable named area. 4. Calculate the perimeter of the rectangle using the formula perimeter = 2 * (length + width), and assign the result to a variable named perimeter. 5. Use the print() function to display the calculated area and perimeter of the rectangle. Your final code should look something like this: length = 5 width = 3 area = length * width perimeter = 2 * (length + width) print(f"The area of a rectangle with length {length} and width {width} is {area}") print(f"The perimeter of a rectangle with length {length} and width {width} is {perimeter}") When you run your program, you should see output similar to the following: The area of a rectangle with length 5 and width 3 is 15 The perimeter of a rectangle with length 5 and width 3 is 16 Feel free to modify the length and width variables to practice with different rectangle dimensions. This exercise helps you become familiar with variables and data types in Python. Exercise 2.2.3: Simple Interest Calculator In this exercise, you will write a Python program that calculates the simple interest earned on an investment. You will practice using variables and data types. 1. Create a new Python file or open a Python interpreter. 2. Declare three variables: principal, rate, and time. Assign them appropriate values (e.g., 1000, 0.05, and 3, respectively). 3. Calculate the simple interest using the formula simple_interest = principal * rate * time, and assign the result to a variable named simple_interest. 4. Use the print() function to display the calculated simple interest. Your final code should look something like this: principal = 1000 rate = 0.05 time = 3 simple_interest = principal * rate * time print(f"The simple interest earned on an investment of ${principal} at a rate of {rate * 100}% over {time} years is ${simple_interest}") When you run your program, you should see output similar to the following: The simple interest earned on an investment of $1000 at a rate of 5.0% over 3 years is $150.0 Feel free to modify the variables to practice with different investment scenarios. These exercises help you become familiar with variables and data types in Python. 2.2 Variables and Data Types In this section, we will discuss variables and data types in Python. Variables are essential in programming as they allow you to store and manipulate data, while data types define the kind of data that can be stored in a variable. 2.2.1 Variables Variables in Python are used to store values for later use in your program. A variable is assigned a value using the assignment operator (=). The variable name must follow certain naming conventions: • Start with a letter or an underscore (_) • Consist of letters, numbers, or underscores • Be case-sensitive Examples of variable assignments: name = "John" age = 25 _height = 175.5 You can also perform multiple assignments in a single line: x, y, z = 1, 2, 3 2.2.2 Data Types Python has several built-in data types that allow you to work with various forms of data. Some of the most common data types are: • Integers (int): Whole numbers, such as 5, -3, or 42. • Floating-point numbers (float): Decimal numbers, such as 3.14, -0.5, or 1.0. • Strings (str): Sequences of characters, such as "hello", 'world', or "Python is fun!". • Booleans (bool): Logical values, either True or False. You can determine the type of a value or variable using the type() function: x = 42 print(type(x)) # Output: <class 'int'> y = 3.14 print(type(y)) # Output: <class 'float'> z = "Python" print(type(z)) # Output: <class 'str'> a = True print(type(a)) # Output: <class 'bool'> 2.2.3 Type Conversion You can convert between different data types using built-in Python functions, such as int(), float(), str(), and bool(). Examples of type conversion: x = 5.5 y = int(x) # Convert float to int, y becomes 5 a = 3 b = float(a) # Convert int to float, b becomes 3.0 c = 42 d = str(c) # Convert int to str, d becomes "42" e = "123" f = int(e) # Convert str to int, f becomes 123 It's important to note that not all conversions are possible. For example, converting a non-numeric string to an integer or float will result in a ValueError. Understanding variables and data types is crucial in Python programming, as it forms the basis for manipulating data and performing various operations. As you progress through this book, you will encounter more advanced data types, such as lists, dictionaries, and tuples, which will allow you to work with more complex data structures. Exercise 2.2.1: Celsius to Fahrenheit Converter In this exercise, you will write a Python program that converts a temperature in degrees Celsius to degrees Fahrenheit. You will practice using variables, data types, expressions, and the print() 1. Create a new Python file or open a Python interpreter. 2. Declare a variable named celsius and assign it the value of the temperature in degrees Celsius (e.g., 30). 3. Calculate the temperature in degrees Fahrenheit using the formula fahrenheit = (celsius * 9/5) + 32, and assign the result to a new variable named fahrenheit. 4. Use the print() function to display the temperature in degrees Fahrenheit, formatted as a string with a degree symbol (°F). Your final code should look something like this: celsius = 30 fahrenheit = (celsius * 9/5) + 32 print(f"{celsius}°C is equal to {fahrenheit}°F") When you run your program, you should see output similar to the following: 30°C is equal to 86.0°F Feel free to modify the celsius value to test your program with different temperatures. This exercise helps you become familiar with variables, data types, arithmetic expressions, and the print() function with f-strings for formatted output. Exercise 2.2.2: Calculate the Area and Perimeter of a Rectangle In this exercise, you will write a Python program that calculates the area and perimeter of a rectangle. You will practice using variables and data types. 1. Create a new Python file or open a Python interpreter. 2. Declare two variables: length and width. Assign them appropriate values (e.g., 5 and 3, respectively). 3. Calculate the area of the rectangle using the formula area = length * width, and assign the result to a variable named area. 4. Calculate the perimeter of the rectangle using the formula perimeter = 2 * (length + width), and assign the result to a variable named perimeter. 5. Use the print() function to display the calculated area and perimeter of the rectangle. Your final code should look something like this: length = 5 width = 3 area = length * width perimeter = 2 * (length + width) print(f"The area of a rectangle with length {length} and width {width} is {area}") print(f"The perimeter of a rectangle with length {length} and width {width} is {perimeter}") When you run your program, you should see output similar to the following: The area of a rectangle with length 5 and width 3 is 15 The perimeter of a rectangle with length 5 and width 3 is 16 Feel free to modify the length and width variables to practice with different rectangle dimensions. This exercise helps you become familiar with variables and data types in Python. Exercise 2.2.3: Simple Interest Calculator In this exercise, you will write a Python program that calculates the simple interest earned on an investment. You will practice using variables and data types. 1. Create a new Python file or open a Python interpreter. 2. Declare three variables: principal, rate, and time. Assign them appropriate values (e.g., 1000, 0.05, and 3, respectively). 3. Calculate the simple interest using the formula simple_interest = principal * rate * time, and assign the result to a variable named simple_interest. 4. Use the print() function to display the calculated simple interest. Your final code should look something like this: principal = 1000 rate = 0.05 time = 3 simple_interest = principal * rate * time print(f"The simple interest earned on an investment of ${principal} at a rate of {rate * 100}% over {time} years is ${simple_interest}") When you run your program, you should see output similar to the following: The simple interest earned on an investment of $1000 at a rate of 5.0% over 3 years is $150.0 Feel free to modify the variables to practice with different investment scenarios. These exercises help you become familiar with variables and data types in Python. 2.2 Variables and Data Types In this section, we will discuss variables and data types in Python. Variables are essential in programming as they allow you to store and manipulate data, while data types define the kind of data that can be stored in a variable. 2.2.1 Variables Variables in Python are used to store values for later use in your program. A variable is assigned a value using the assignment operator (=). The variable name must follow certain naming conventions: • Start with a letter or an underscore (_) • Consist of letters, numbers, or underscores • Be case-sensitive Examples of variable assignments: name = "John" age = 25 _height = 175.5 You can also perform multiple assignments in a single line: x, y, z = 1, 2, 3 2.2.2 Data Types Python has several built-in data types that allow you to work with various forms of data. Some of the most common data types are: • Integers (int): Whole numbers, such as 5, -3, or 42. • Floating-point numbers (float): Decimal numbers, such as 3.14, -0.5, or 1.0. • Strings (str): Sequences of characters, such as "hello", 'world', or "Python is fun!". • Booleans (bool): Logical values, either True or False. You can determine the type of a value or variable using the type() function: x = 42 print(type(x)) # Output: <class 'int'> y = 3.14 print(type(y)) # Output: <class 'float'> z = "Python" print(type(z)) # Output: <class 'str'> a = True print(type(a)) # Output: <class 'bool'> 2.2.3 Type Conversion You can convert between different data types using built-in Python functions, such as int(), float(), str(), and bool(). Examples of type conversion: x = 5.5 y = int(x) # Convert float to int, y becomes 5 a = 3 b = float(a) # Convert int to float, b becomes 3.0 c = 42 d = str(c) # Convert int to str, d becomes "42" e = "123" f = int(e) # Convert str to int, f becomes 123 It's important to note that not all conversions are possible. For example, converting a non-numeric string to an integer or float will result in a ValueError. Understanding variables and data types is crucial in Python programming, as it forms the basis for manipulating data and performing various operations. As you progress through this book, you will encounter more advanced data types, such as lists, dictionaries, and tuples, which will allow you to work with more complex data structures. Exercise 2.2.1: Celsius to Fahrenheit Converter In this exercise, you will write a Python program that converts a temperature in degrees Celsius to degrees Fahrenheit. You will practice using variables, data types, expressions, and the print() 1. Create a new Python file or open a Python interpreter. 2. Declare a variable named celsius and assign it the value of the temperature in degrees Celsius (e.g., 30). 3. Calculate the temperature in degrees Fahrenheit using the formula fahrenheit = (celsius * 9/5) + 32, and assign the result to a new variable named fahrenheit. 4. Use the print() function to display the temperature in degrees Fahrenheit, formatted as a string with a degree symbol (°F). Your final code should look something like this: celsius = 30 fahrenheit = (celsius * 9/5) + 32 print(f"{celsius}°C is equal to {fahrenheit}°F") When you run your program, you should see output similar to the following: 30°C is equal to 86.0°F Feel free to modify the celsius value to test your program with different temperatures. This exercise helps you become familiar with variables, data types, arithmetic expressions, and the print() function with f-strings for formatted output. Exercise 2.2.2: Calculate the Area and Perimeter of a Rectangle In this exercise, you will write a Python program that calculates the area and perimeter of a rectangle. You will practice using variables and data types. 1. Create a new Python file or open a Python interpreter. 2. Declare two variables: length and width. Assign them appropriate values (e.g., 5 and 3, respectively). 3. Calculate the area of the rectangle using the formula area = length * width, and assign the result to a variable named area. 4. Calculate the perimeter of the rectangle using the formula perimeter = 2 * (length + width), and assign the result to a variable named perimeter. 5. Use the print() function to display the calculated area and perimeter of the rectangle. Your final code should look something like this: length = 5 width = 3 area = length * width perimeter = 2 * (length + width) print(f"The area of a rectangle with length {length} and width {width} is {area}") print(f"The perimeter of a rectangle with length {length} and width {width} is {perimeter}") When you run your program, you should see output similar to the following: The area of a rectangle with length 5 and width 3 is 15 The perimeter of a rectangle with length 5 and width 3 is 16 Feel free to modify the length and width variables to practice with different rectangle dimensions. This exercise helps you become familiar with variables and data types in Python. Exercise 2.2.3: Simple Interest Calculator In this exercise, you will write a Python program that calculates the simple interest earned on an investment. You will practice using variables and data types. 1. Create a new Python file or open a Python interpreter. 2. Declare three variables: principal, rate, and time. Assign them appropriate values (e.g., 1000, 0.05, and 3, respectively). 3. Calculate the simple interest using the formula simple_interest = principal * rate * time, and assign the result to a variable named simple_interest. 4. Use the print() function to display the calculated simple interest. Your final code should look something like this: principal = 1000 rate = 0.05 time = 3 simple_interest = principal * rate * time print(f"The simple interest earned on an investment of ${principal} at a rate of {rate * 100}% over {time} years is ${simple_interest}") When you run your program, you should see output similar to the following: The simple interest earned on an investment of $1000 at a rate of 5.0% over 3 years is $150.0 Feel free to modify the variables to practice with different investment scenarios. These exercises help you become familiar with variables and data types in Python.	{"url":"https://www.cuantum.tech/app/section/22-variables-and-data-types-36a9e726a33d4bef85e6894ae559e15f","timestamp":"2024-11-06T09:15:48Z","content_type":"text/html","content_length":"95983","record_id":"<urn:uuid:c104395e-97f3-45f0-bd87-3f660730d67c>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00067.warc.gz"}
How to find the average densityHow to find the average density 🚩 how to calculate average density 🚩 Science. You will need • - scales; • - measuring cylinder; • - table of densities of various substances. If the body does not consist of the same stuff, find it with weights its weight, and then measure the volume. If it is fluid, perform measurement using the measuring cylinder. If it is a solid (cube, prism, polyhedron, sphere, cylinder, etc.), find its volume by geometric methods. If the body is irregular in shape, submerge it in the water, which poured into the measuring cylinder, and its ascent will determine body volume. Divide the measured mass of a body on its volume, the result is the average density of the body ρ=m/V. If the mass is measured in kilograms, volume in m3 Express, if in grams in cm3. Accordingly, the density is obtained in kg/m3 or g/ cm3. If you weigh the body is not possible, find out the density of the materials of which it is composed, then measure the volume of each part of the body. Then find the mass of materials which compose the body, multiplying their density on the volume and the total volume of the body, adding the volumes of its constituent parts, including voids. Divide the total mass of a body on its volume and get the average density of the body ρ= (ρ1•V1+ ρ2•V2+...)/(V1+V2+...). If the body can be immersed in water, find its weight in water with the help of dynamometer. Determine the amount of ejected water, which is equal to the volume submerged in her body. In the calculations, keep in mind that the density of water is 1000 kg/m3. To find the average density of the body immersed in water, to its weight in Newtons, add the product of the number 1000 (density of water) on the acceleration of gravity 9.81 m/S2 and the body volume in m3. The number you divide by the product of the volume of the body and of 9.81 ρ=(R+ RV•V•9,81)/(9,81• V). When a body floats in water, find the volume of the ejected fluid, the volume of the body. Then, the average density of a body is equal to the ratio of the product of density of water on her body ejected the volume and of the volume of the body ρ= RV•V/Vт.	{"url":"https://eng.kakprosto.ru/how-78793-how-to-find-the-average-density","timestamp":"2024-11-08T01:17:26Z","content_type":"text/html","content_length":"31499","record_id":"<urn:uuid:a7379f94-9a34-4104-a65b-d797438b50d6>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00719.warc.gz"}
How does thermodynamics explain the behavior of semiconductors? \| Do My Chemistry Online Exam How does thermodynamics explain the behavior of semiconductors? As a mathematician, I learn about thermodynamics to find the smallest possible values of the exponentic characteristic function. Also, I was curious what it means to enter some examples of thermodynamic states and how simple this fits the theory of heat transfer. These questions help me understand related ideas and answer these questions. Definition of the concept of thermodynamics consists in defining variables describing how the thermodynamic features of the system are described, and their properties. I can not work only to define most popular names for thermodynamic states or concepts given in textbooks and related books, but also to discuss the thermodynamics of many macroscopic systems. These topics are related to the phenomena of matter and materials in nature, and to physical theories of matter and materials in nature as well. Consider a set of thermodynamic states of a simple insulator as being a natural candidate for thermodynamic states. Consider a material that is pure and free or is free from stress. A particular stress/stress associated to a macroscopic system described in the thermodynamic state is called a stress in a particular thermodynamic state. We say the Stress state is a natural choice for a thermodynamic state if there are two different stresses associated to the same macroscopic state. Thus, a stress is a homogeneous point of stress: the point when all the stresses are equal to zero, which indicates the symmetric states (these states are noninteracting to be critical in the thermodynamic limit). We say that a particular stress has two dissimilar stresses associated to two different states: the set of all possible all the stress dissimilarities is called a dissimilar stress state and the set of all possible all the dissimilarities is called a dissimilar stress state. A dissimilar stress state is a superset of all dissimilar stress, and one of these wikipedia reference site web two different stresses. Typical examples that lead to $P$ and $Q$ isHow does thermodynamics explain the behavior of semiconductors? The basic principles of thermodynamics and click for info chemistry are not identical – for example, we have those that obey thermodynamics, quantum coherence and thermodynamic properties. However, even fundamental information, as in the quantum theory of matter, shows a change in the thermodynamic behavior. How can thermodynamics describe the behavior of materials? Well, one can think of all that comes down to information. Thus, everything, from electrons and ions to molecules to supercapacitors and the like, are linked to information. It must be possible to make such a link. When the thermodynamics of materials works, the information of materials, such as the charge of atoms and electrons, must be propagated from the charge and the time necessary for translation into the time needed for the experiment to work. This is because information cannot be transmitted from one material to another, and when a material goes and enters a quencher’s pocket, the charge stays at the same place. Complete My Homework In other words, no system can withstand the energy required by light, an analog of which in time is the electron in a hydrogen atom or its constituent atoms. The essence of the energy-momentum relation in large scale physics is that the kinetic energy of a material can never get away from the energy of light, because in a proton the momentum can only be conserved if the energy is sufficiently high. But if the momentum of a mass is relatively large compared to its energy and therefore a material can pass into an atomic nucleus – and this means that its kinetic energy is very small – then the process of creation of a particle is self-consistent. The first time-moment can only be created if other information are available that means that these will not contribute to the process of creation. When a superconductor and a Bose-Einstein condensate are in conical overlap, all these radiation can never contribute to creation of a particle, and a perfect thermalHow does thermodynamics explain the behavior of semiconductors? \| What makes them soluble? \| How do they relate to the thermodynamic character of semiconductors? \| The view that thermodynamics can explain matters like concentration or crystal structures \| The view that thermodynamics can explain the appearance of certain types of crystalline structures \| What is a model? (In this book, we call it a model) \| What is a transition? Post navigation 55 thoughts on “Interpret that as a theory: An idea really?” It seems to me that one should try and think about this idea and only move forward with understanding how the mathematical formalism and the physical theory make sense. If it turns out that one is going to adopt a highly mathematical formalism while thinking about possible computational models, as opposed to what makes sense in physics, then it is very important to understand the underlying physical relationships in the language of mathematics. The only way to approach the problem of understanding the ultimate fact is to change perspective. _________________ One may be a beginner, but I learned a lot over my years of studying semiconductors. I definitely understand the idea of semiconductance and we can only discuss it one bit. Hope there is something in there!	{"url":"https://chemistryexamhero.com/how-does-thermodynamics-explain-the-behavior-of-semiconductors","timestamp":"2024-11-01T23:31:44Z","content_type":"text/html","content_length":"129238","record_id":"<urn:uuid:30368b00-063e-45a7-bb54-ae52596142db>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00325.warc.gz"}
How to Create Tax Formula in Oracle Fusion Tax - My Techno Journal How to Create Tax Formula in Oracle Fusion Tax Tax Formula Overview Tax formulas are used in the tax calculation process. They help to determine the taxable basis of a transaction line and the calculation methodology that must be applied to obtain the tax amount. When the parameters available on a transaction do not satisfy the rule conditions, the default tax formulas defined for the tax are applicable. There are two types of tax formulas: • Taxable basis tax formula • Tax calculation tax formula Taxable Basis Tax Formula The taxable basis tax formula is used in the tax calculation process. They help to determine the amount or quantity that should be considered as the taxable basis of a transaction line. The tax rate is applied on the taxable basis amount to derive the basic tax amount on a transaction line. The taxable basis type, defined in the taxable basis formula, decides the characteristics of the taxable basis amount. The various taxable basis types are: • Line amount • Assessable value • Prior tax • Quantity The following standard predefined taxable basis tax formulas are available: • STANDARD_QUANTITY • STANDARD_TB • STANDARD_TB_DISCOUNT Line amount Use line amount when the transaction line amount is to be treated as the taxable basis to calculate tax. By Default, Taxable Basis is Line Amount (STANDARD_TB) in tax setup Step 1: Search “Manage Tax Formulas” task Step 2: Select the Tax Formula Type Step-3 Search the Tax Formula Code Scenario: STANDARD_TB Taxable Basis is Line Amount (STANDARD_TB) Line Amount = 1000 Tax Calculation = Taxable Basis Rate% Tax Calculation = Line Amount 10% Tax = 1000 10% Tax =100 Base Rate Modifier The transaction line amount is increased or decreased based on the percentage value given. Scenario 1: Taxable Basis = Line Amount + Base Rate Modifier Step 1: Creating Taxable Basis Formula: Navigation: Manage Tax Formulas →Create Taxable Basis Formula Step 2-Assign the user defined taxable basis formula in the tax setup. Line Amount = 1000 Base Rate Modifier = 50% of line amount = 500 Taxable Amount = Line Amount + Base Rate Modifier = 1000 + 500 = 1500 Tax Calculation = Taxable Amount Rate% = 1500*10% Tax = 150 Step 3: Create transaction and check the taxable amount Scenario 2: Taxable Basis = Line Amount – Base Rate Modifier Another example of Base Rate Modifier: Calculate two taxes on a transaction. Tax1 should be 3% of line amount and Tax2 should be (Line Amount – Tax1) X 0.1% Line amount = 20000 Tax1 = 200003% Tax1 = 600 Taxable amount for Tax2 is (Line Amount –Tax1) i.e. 20000-600= 19400 Tax2 = 194000.1% = 19.4 Step 1: Create Tax1 and select STANDARD_TB as taxable basis formula Step 2: Create Taxable Basis Formula for Tax2. Step 3: Create Tax2 and select above custom taxable basis formula. Step 4: Create a transaction and check the taxable amount for Tax2 Line amount = 20000 Tax1 = 200003% Tax1 = 600 Taxable amount for Tax2 is (Line Amount –Tax1) i.e. 20000-600= 19400 Tax2 = 194000.1% = 19.4 In Next session we will discuss Tax formula compounding	{"url":"https://mytechnojournal.com/how-to-create-tax-formula-in-oracle-fusion-tax/","timestamp":"2024-11-07T07:33:00Z","content_type":"text/html","content_length":"84969","record_id":"<urn:uuid:4447bb9c-3db9-4151-a3b7-4ce389e8d2b2>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00293.warc.gz"}
Approximating Normality Unicorn space race with always sometimes nevers in partners. #teach180 https://t.co/EmoblCM9wC via http://twitter.com/DavidGriswoldHH at http://twitter.com/DavidGriswoldHH/status/672874845693157376 at December 04, 2015 at 02:27PM I’m struggling teaching probability in AP Statistics for a lot of reasons, that I may enumerate and reflect on in a later post, but for now, here’s a problem from my text’s tests (numbers and exact context changed) that I found particularly troublesome for students. “A grocery store examines its shoppers product selection and calculates the following: TheRead More Today in class I had one of those fun (and sadly rare) Great Activities I Didn’t Really Plan moments. We had a list of Always-Sometimes-Nevers about parallelograms that I took from the CME Geometry book (thanks y’all!). I decided I wanted to spend about 25 minutes looking and thinking about those, but didn’t really planRead More If there’s one thing that Intro to CS is teaching me this year, it’s that there is so much more math in programming and computers than I even notice. I’ve never taught a full-semester CS course before, and i’m taking it slow and easy. We have definitely not done as much as I could beRead More I do not have enough white board space in my geometry classroom. I am working on getting an entire wall covered, but I can’t just put up my own stuff, and it is a slow process. Until it happens, I can’t really do full-fledged Vertical Non-Permanent Surfaces work the way I would sometimes like. ThoughRead More I’m trying a couple of new things in the triangle congruence / quadrilateral properties quarter of my geometry class this year. First, I’m implementing CME Geometry’s methods of analyzing proofs, which uses flowcharts as one way to figure out a proof. I’ve always used flowcharts more as a way to actually present a proof, as done in DiscoveringRead More This was my first year doing meet the teacher night in my current school and division; I taught middle school science here for three years, but moved up to upper school math last year. However, I did not attend meet the teacher night last year as I was on paternity leave. So I needed somethingRead More I know that there are plenty of teachers using standards based grading in AP classes, but despite that I didn’t really know how I was going to implement it in my classroom. But I decided to give it a whirl, inspired by the technology at my disposal: the Ultimate Gamified Learning Goal Assessment Spreadsheet thatRead More I will soon be implementing Christopher Danielson’s Hierarchy of Hexagons in my high school geometry class, and would love suggestions. Some necessary background: I teach in an all-girls private school with a pretty high academic reputation. The girls at my school are generally very academically motivated. We do have tracking, and my class is the “standard”Read More I have started working on a Star-making activity that has, I think, great potential in a computer programming OR a geometry class. I am currently doing it with my very small computer programming class (though I haven’t figured everything out yet – first time teaching a class and a small willing audience means we canRead More Interesting / painful road block in Statistics question Google Slides with Always/Sometimes/Nevers -fascinating student interactions Graph Theory and other math topics in Intro to CS Using Google Slides for examining student work in class Virtual Proof Blocks and a Proof Portfolio Meet the teacher night 2015 SBG in AP Stat – The melding of two assessment styles Doing Hierarchy of Hexagons soon – suggestions? Starmaking – A Programming and/or Geometry project	{"url":"https://davidgriswoldhh.mtbos.org/page/5/","timestamp":"2024-11-11T05:16:25Z","content_type":"text/html","content_length":"48116","record_id":"<urn:uuid:8adee565-a767-4d52-b60c-bc460c1b6219>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00095.warc.gz"}
The goal of airt is to evaluate a portfolio of algorithms using Item Response Theory (IRT). To fit the IRT models, I have used the R packages mirt and EstCRM. The fitting function is EstCRM is slightly changed to accommodate for the algorithm evaluation setting. You can install the released version of airt from CRAN with: You can install the development version from GitHub with: Example - Continuous Performance Data Let us consider some synthetic performance data. For this example, I will generate performance values for 4 algorithms that are correlated with each other. algo1 <- runif(200) algo2 <- 2algo1 + rnorm(200, mean=0, sd=0.1) algo2 <- (algo2 - min(algo2))/(max(algo2) - min(algo2)) algo3 <- 1 - algo1 + rnorm(200, mean=0, sd=0.1) algo3 <- (algo3 - min(algo3))/(max(algo3) - min(algo3)) algo4 <- 2 + 0.2algo1 + rnorm(200, mean=0, sd=0.1) algo4 <- algo4/max(algo4) df <- cbind.data.frame(algo1, algo2, algo3, algo4) colnames(df) <- c("A", "B", "C", "D") #> A B C D #> 1 0.2655087 0.2726160 0.7842857 0.8640853 #> 2 0.3721239 0.3988125 0.5474863 0.8573372 #> 3 0.5728534 0.5370545 0.6271255 0.8370052 #> 4 0.9082078 0.8881757 0.1776022 0.9152130 #> 5 0.2016819 0.2134300 0.8944309 0.8970685 #> 6 0.8983897 0.9519739 0.3346240 0.9807495 This dataframe gives the performances of algorithms A, B, C and D. Each row is an dataset/instance and each column denotes an algorithm. Let us fit a continuous IRT model to this data. The input to the airt model is the dataframe df. Fitting the IRT model modout <- cirtmodel(df) paras <- modout$model$param #> a b alpha #> A 2.8527066 -0.05943140 1.6853860 #> B 3.8607317 -0.12775543 1.2914311 #> C -1.7297386 0.09323845 -1.0843468 #> D 0.5348669 -0.46017926 0.4229044 Now we have our AIRT model. The paras contain the standard IRT parameters from the EstCRM R package. Here a denotes discrimination, b difficulty and alpha the scaling parameter. From these parameters we construct algorithm attributes/features. They are anomalousness, consistency and difficulty limit. These are AIRT algorithm attributes and are available in the output modout. cbind.data.frame(anomalousness = modout$anomalous, consistency = modout$consistency, difficulty_limit = modout$difficulty_limit) #> anomalousness consistency difficulty_limit #> A 0 0.3505443 0.05943140 #> B 0 0.2590183 0.12775543 #> C 1 0.5781220 -0.09323845 #> D 0 1.8696240 0.46017926 The anomalous feature is either 1 or 0. If it is 1, the algorithm is anomalous and if it is 0, it is not. We can see that algorithm C is anomalous. That is, it performs well for difficult datasets/ problems and poorly for easy datasets/problems. The other algorithm perform in a normal way, they gives good performances for easy problems and poor performances for difficult problems. The difficultly limit tells us the highest difficulty level the algorithm can handle. If an algorithm has a high difficulty limit, it is good because, then it can handle very hard problems. In this case, Algorithm D has the highest difficulty limit. Thus, it can handle very difficult problems. Also observe that C has a negative difficulty limit. This is because it is anomalous. The consistency feature tells us how consistent an algorithm is. Some algorithms are consistently good for many problems and some are consistently poor. Others can vary a lot depending on the problem. Algorithm D has the highest consistency in the portfolio. That means, it is consistently good or consistently bad. Because its difficulty limit is very high, we know it is consistently good. But we can’t say it by just looking at the consistency value. In this portfolio, the least consistent algorithm is A. That is, it fluctuates a lot depending on the problem. By plotting heatmaps we can visually see these characteristics. The figure above shows the probability density heatmaps for algorithms. The y axis denotes the normalized performance values. The x axis gives the problem easiness. Lower x/Theta values denote more difficult problems and higher Theta values denote easier problems. For algorithm A, B and C we see lines across the panes. These lines denote high density regions. Let’s say we look at Theta = 0 for algorithm A. The place where the yellow line intersects Theta = 0 (vertical line, not drawn) is the highest probable region. That denotes the the most probably performance for that type of problem. Generally, sharper lines indicate more discriminating algorithms and blurry lines or no lines (algorithm D) indicate more stable algorithms. If the line has a positive slope, then it is not anomalous. If the line has a negative slope (algorithm C), then it is anomalous. We see that algorithms A and B obtain high performance values for high Theta values. That is because these algorithms are normal, i.e., not anomalous. However, algorithm C obtains high performance values for low Theta values. These are difficult datasets. Hence, algorithm C is anomalous. Algorithm D, does not discriminate among the datasets, as such it is a more consistent algorithm. These are the insights we can get from the heatmaps. The Problem Difficulty Space and Algorithm Performance Next, let’s do a latent trait analysis. Latent trait analysis looks at the algorithm performance with respect to the dataset difficulty. The datasets are ordered in the latent trait according to their difficulty. IRT does this for us. Let’s look at the latent trait. obj <- latent_trait_analysis(df, modout$model$param, epsilon = 0 ) #> Warning: The `x` argument of `as_tibble.matrix()` must have unique column names if #> `.name_repair` is omitted as of tibble 2.0.0. #> ℹ Using compatibility `.name_repair`. #> ℹ The deprecated feature was likely used in the airt package. #> Please report the issue to the authors. #> This warning is displayed once every 8 hours. #> Call `lifecycle::last_lifecycle_warnings()` to see where this warning was #> generated. #> Joining with `by = join_by(group)` #> Joining with `by = join_by(group)` autoplot(obj, plottype = 1) The figure above shows the performance of the 4 algorithms on different datasets ordered by dataset difficulty. Again, we see that the performance of algorithms A and B decrease with dataset difficulty while the performance of algorithm C increases with dataset difficulty. Even though the performance of Algorithm D somewhat decreases with dataset difficulty it is very consistent. We can plot it by algorithm (if the portfolio has lots of algorithms) by using plottype = 2. strengths and weaknesses, we first fit smoothing splines to the data above. We can do this by using plottype = 3. Strengths and Weaknesses of Algorithms We can also compute the proportion of the latent trait spectrum occupied by each algorithm. We call this the latent trait occupancy (LTO). #> # A tibble: 3 × 4 #> group Proportion algorithm colour #> <dbl> <dbl> <chr> <chr> #> 1 4 0.855 D #C77CFF #> 2 3 0.075 C #00BFC4 #> 3 1 0.07 A #F8766D In the above table LTO is given by the Proportion column. We see that algorithms D, A, C and D occupy 0.865, 0.070 and 0.0.065 of the latent trait respectively. Just like the strengths, we can look at the weaknesses. An algorithm is weak for a given problem difficulty, if the associated curve is at the bottom. From the graph above we see that Algorithm C is weak for easy problems, and Algorithm A is weak for difficulty problems. We can get latent trait occupancy (LTO) for weaknesses as well. #> # A tibble: 2 × 4 #> group Proportion algorithm colour #> <dbl> <dbl> <chr> <chr> #> 1 1 0.52 A #F8766D #> 2 3 0.48 C #00BFC4 We can put these in a strengths and weaknesses diagram by using plottype = 4. epsilon = 0, which only considers one algorithm for a given problem difficulty. If we want to consider the algorithms that give close enough performances for a given problem difficulty, we can set epsilon to a different value. Say we let epsilon = 0.05. This would consider algorithms having similar performances up to 0.05. We can do the same analysis as before. The strengths would be slightly different in this case obj2 <- latent_trait_analysis(df, modout$model$param, epsilon = 0.05 ) #> Joining with `by = join_by(group)` #> Joining with `by = join_by(group)` #> # A tibble: 4 × 4 #> group Proportion algorithm colour #> <dbl> <dbl> <chr> <chr> #> 1 4 0.955 D #C77CFF #> 2 1 0.135 A #F8766D #> 3 2 0.125 B #7CAE00 #> 4 3 0.125 C #00BFC4 Now, all 4 algorithms have strengths. We see that the latent trait occupancy (proportion in the table above) of D has increased to 0.94, that is a large proportion of the problem space. And if we add up the proportions, the sum is greater than 1. This is because there is an overlap of strong algorithms in the latent trait. Let us look at the strengths and weaknesses plot. Next we look at discrete example. Discrete models are called polytomous in IRT literature. Example - Polytomous (discrete) Performance Data Polytomous data is ordered. A bit like grades A, B, C, D where A > B > C > D. We can think if it as discrete. # Generating data algo1 <- sample(1:5, 100, replace = TRUE) inds1 <- which(algo1 %in% c(4,5)) algo2 <- rep(0, 100) algo2[inds1] <- sample(4:5, length(inds1), replace = TRUE) algo2[-inds1] <- sample(1:3, (100-length(inds1)), replace = TRUE) algo3 <- rep(0, 100) algo3[inds1] <- sample(1:2, length(inds1), replace = TRUE) algo3[-inds1] <- sample(3:5, (100-length(inds1)), replace = TRUE) algorithms <- cbind.data.frame(algo1, algo2, algo3) # Fitting the polytomous model mod <- pirtmodel(algorithms, vpara = FALSE) # Tracelines for each algorithm gdf <- tracelines_poly(mod) # Plotting # AIRT metrics cbind.data.frame(anomalousness = mod$anomalous, consistency = mod$consistency, difficulty_limit = mod$difficulty_limit[ ,1]) #> anomalousness consistency difficulty_limit #> algo1 0 0.6132055 0.8049593 #> algo2 0 0.4744101 0.7410049 #> algo3 1 0.2955284 -1.0099760 We see that algo3 is anomalous. That is, it performs well on test instances that others perform poorly. For algo1 and algo_2, the highest level of performance P5 is achieved for high values of $\dpi {110}&space;\bg_white&space;\theta$. But for algo3 the P5 is achieved for low values of $\dpi{110}&space;\bg_white&space;\theta$. More on airt The pkgdown site describes the functionality of airt : https://sevvandi.github.io/airt/. More details are available at (Kandanaarachchi and Smith-Miles 2023). Firstly, thanks to Rob Hyndman for coming up with the name airt, which is an old Scottish word meaing to guide. Also, thanks to Phil Chalmers for being very quick in responding to emails about his R package mirt. Many people helped me with the hex sticker. A big thank you to Patricia Menendez Galvan, Di Cook, Emi Tanaka, Nishka and Sashenka Fernando for giving me great feedback. Zopluoglu C (2022). EstCRM: Calibrating Parameters for the Samejima’s Continuous IRT Model. R package version 1.6, https://CRAN.R-project.org/package=EstCRM. R. Philip Chalmers (2012). mirt: A Multidimensional Item Response Theory Package for the R Environment. Journal of Statistical Software, 48(6), 1-29. Kandanaarachchi, Sevvandi, and Kate Smith-Miles. 2023. “Comprehensive Algorithm Portfolio Evaluation Using Item Response Theory.” Journal of Machine Learning Research, Accepted.	{"url":"https://cran.opencpu.org/web/packages/airt/readme/README.html","timestamp":"2024-11-04T21:55:35Z","content_type":"application/xhtml+xml","content_length":"37977","record_id":"<urn:uuid:6667df16-3658-4087-b35f-268614aadde6>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00241.warc.gz"}
Harmonic maps and heat flows on hyperbolic spaces Printable PDF Department of Mathematics, University of California San Diego Joint UCI-UCR-UCSD Southern California Differential Geometry Seminar Vlad Markovic Cal Tech Harmonic maps and heat flows on hyperbolic spaces We prove that any quasi-isometry between hyperbolic manifolds is homotopic to a harmonic quasi-isometry. November 3, 2016 4:00 PM UC Irvine, Rolland Hall, Room 306	{"url":"https://math.ucsd.edu/seminar/harmonic-maps-and-heat-flows-hyperbolic-spaces","timestamp":"2024-11-02T23:48:56Z","content_type":"text/html","content_length":"32759","record_id":"<urn:uuid:ca616810-e481-4f1b-aa41-0a557c42c216>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00460.warc.gz"}
What is NumPy? Explaining how it works in Python What is NumPy? Explaining how it works in Python NumPy is an open source mathematical and scientific computing library for Python programming tasks. The name NumPy is shorthand for Numerical Python. The NumPy library offers a collection of high-level mathematical functions including support for multi-dimensional arrays, masked arrays and matrices. NumPy also includes various logical and mathematical capabilities for those arrays such as shape manipulation, sorting, selection, linear algebra, statistical operations, random number generation and discrete Fourier transforms. As with any programming library, NumPy needs to be added only to an existing Python installation, and programmers can easily write Python code that makes calls and exchanges data with NumPy features and functions. The NumPy library was first released in 2006. Today, the scientific computing community supports the open source library, and NumPy is currently available through GitHub. Development of the NumPy library is active and ongoing. How are NumPy arrays different from Python lists? In simplest terms, a Python list is most suitable for data storage and not ideally intended for math tasks, while a NumPy list can easily support math-intensive tasks. Python provides a basic data structure called a list. A Python list supports a changeable -- or mutable -- ordered sequence of data elements or values called items. A single list can also contain many different data types. This makes lists handy for storing multiple data items as a single variable -- such as customer contact information and account numbers. However, lists are potentially inefficient, using significant amounts of memory and posing problems attempting to process mathematical operations on varied item types. By comparison, NumPy is built around the idea of a homogeneous data array. Although a NumPy array can specify and support various data types, any array created in NumPy should use only one desired data type -- a different array can be made for a different data type. This approach requires less memory and allows more efficient system performance when processing mathematical operations on array What are ndarrays and how to use them The primary data structure in NumPy is the N-dimensional array -- called an ndarray or simply an array. Every ndarray is a fixed-size array that is kept in memory and contains the same type of data such as integer or floating-point numbers. An ndarray can possess up to three dimensions including array length, width and height or layers. Ndarrays use the shape attribute to return a tuple (an ordered sequence of numbers) stipulating the dimensions of the array. The data type used in the array is specified through the dtype attribute assigned to the array. These can include integers, strings, floating-point numbers and so on. As a simple example, the following code snippet creates an array and uses shape and dtype attributes to render a two-dimensional array using two rows of three elements in the array, each intended to hold 4-byte (32-bit) integer numbers: g = np.array([[1, 2, 3], [4, 5, 6]], np.int32) <type 'numpy.ndarray'> (2, 3) Several NumPy functions can easily create arrays that are empty or prefilled with either zeros or ones. Different NumPy arrays can also be stacked (combined) either vertically or horizontally. An established ndarray can be indexed to access its contents. For example, to read the second row and third column of the array above, try the indexing function such as: The return value should be 6. Remember that the first row and column would be [0,0], so the second row and third column would be [1,2]. Once created, programmers can perform mathematical operations on the contents of an ndarray. As examples, simple math functions include the following: • Addition add the elements of an array: numpy.add(x,y) • Subtraction subtract the elements of an array: numpy.subtract(x,y) • Multiplication multiply the elements of an array: numpy.multiply(x,y) • Division divide the elements of an array: numpy.divide(x,y) • Power raise one array element to the power of another: numpy.power(x,y) • Matrix multiply apply matrix multiplication to the array: numpy.matmul(x,y) The following simple example creates two one-dimensional arrays and then adds the elements of one array to the elements of a second array: array1 = numpy.array([1, 2, 3]) array2 = numpy.array([4, 5, 6]) result = numpy.add(array1, array2) The addition's resulting printed (displayed) output would then be [5 7 9]. Over 60 basic math functions and many other complex functions support logic, algebra, trigonometry and calculus. NumPy documentation provides detailed lists of available functions and code examples for programmers to learn and implement NumPy capabilities. Common NumPy applications and uses The NumPy mathematical library can be used by any software developer (at any experience level) seeking to integrate complex numerical computing functions into their Python codebase. NumPy is also routinely used in many different data science, machine learning (ML) and scientific Python software packages including the following: • Matplotlib. • Pandas. • scikit-image. • scikit-learn. • SciPy. NumPy is regularly applied in a wide range of use cases including the following: • Data manipulation and analysis. NumPy can be used for data cleaning, transformation and aggregation. The data can then be readily processed through varied NumPy mathematical operations such as statistical analysis, Fourier analysis and linear algebra -- all vital for advanced data analytics and data science tasks. • Scientific computing. NumPy handles advanced mathematical operations such as matrix multiplication, eigenvalue calculation and differential equations. This makes NumPy particularly valuable across a vast range of simulation, modeling, visualization, computational processing and other scientific computing tasks. • Machine learning. ML tasks are math-intensive, and ML libraries -- such as TensorFlow and scikit-learn -- readily use NumPy for mathematical computations needed to support ML algorithms and model • Signal and image processing. Signals and images can readily be represented as data arrays, and NumPy can provide tools needed to perform important processing tasks on that data for purposes such as enhancement and filtering. Limitations of NumPy Although NumPy offers some noteworthy advantages for mathematical software designers, several limitations must be considered. Common disadvantages include the following: • Lower flexibility. The focus on numerical and homogeneous data types is key to NumPy performance and efficiency, but it can also limit the flexibility of NumPy compared to other storage array mechanisms where heterogeneous data types must be supported. Additionally, NumPy lacks support for missing values, requiring careful validation and vetting of the data set. • Non-numerical data types. NumPy can support many different data types, but its primary focus is on numerical data types, such as floating-point numbers, and non-numerical data types, such as text strings, which might see little benefit from NumPy array storage compared to other array storage mechanisms such as Python lists. • Demands of change. NumPy and Python lists are both mutable -- array contents can be appended, extended and combined. However, NumPy is inefficient in handling such tasks, and routines designed to change, add, combine or delete data within the array can suffer performance limitations because of how NumPy allocates and uses memory. Ultimately, NumPy provides a powerful platform for scientific computation, but it is not a replacement for all array programming tasks. Software designers must carefully evaluate the nature and requirements of the array before selecting NumPy versus Python lists or other array mechanisms. How to install and import NumPy The only prerequisite for NumPy is a Python distribution. The Anaconda distribution of Python already includes Python and NumPy and might be easier for users just getting started with NumPy and scientific computing projects. For users already employing a version of Python, NumPy can be installed using Conda (a package, dependency, and environment management tool) or Pip (a package manager for Python modules). Simple installation commands for each package manager can appear as: conda install numpy pip install numpy Once installed, the NumPy library can be added or connected to the Python codebase using the Python import command such as: import numpy The module name ("numpy" in this case) can also be renamed through the import command to make the module name more relevant, readable or easier to reference. As an example, to import NumPy as "np" use the as switch such as: import numpy as np Thus, any calls made to NumPy can be made using the abbreviation np instead. Detailed instructions on installing, importing and using NumPy can be found in NumPy documentation available from NumPy. This was last updated in July 2024 Continue Reading About What is NumPy? Explaining how it works in Python	{"url":"https://www.techtarget.com/whatis/definition/What-is-NumPy-Explaining-how-it-works-in-Python","timestamp":"2024-11-03T03:44:30Z","content_type":"text/html","content_length":"351573","record_id":"<urn:uuid:559a8ca3-1d0d-4711-8dd0-3b3cf26a81c2>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00321.warc.gz"}
Add Two Numbers View Add Two Numbers on LeetCode Time Spent Coding 5 minutes Time Complexity O(n+m+k) - We must iterate through each digit of the number in l1 (n) the number in l2 (m) and the sum of those numbers (k), resulting in the O(n+m+k) time complexity. Space Complexity O(k) - The length of the sum of the numbers stored in l1 and l2 is the number of ListNode’s needed, resulting in the O(k) space complexity. Runtime Beats 56.69% of other submissions Memory Beats 69.95% of other sumbissions Since each input is a linked list and not a number, the function buildNum is created to convert the input linked list to a number. The function buildNum takes in a linked list and builds the entire number it contains. It does this by converting each digit from the linked list into a string and adding it to the front of the res string since the number is represented backward in the linked lists (e.g. 1 -> 2 = 21). Once the number is built, it is converted into an integer and returned. Run this function for l1 and l2 and store their results in n1 and n2. Add them, store the result in tot, and convert it into a string. Initialize head as a ListNode and create a pointer to help incrementally add to the linked list. Begin iterating backward ([::-1]) through each digit in the tot string and place its integer representation into a new ListNode. Then make the node at the pointer point to the new node, and move the pointer to this new node. Once the tot string has been fully iterated, return head.next this is because the head node contains an empty ListNode, which would add a 0 to the front of the linked list. Data Structure Used Linked List - Similar to a list or an array, except the data allocated for the list is generated as each new node is added, meaning the memory for the entire list may not be in consecutive memory. A linked list can not be indexed and must be traversed using a pointer. 1 # Definition for singly-linked list. 2 # class ListNode: 3 # def __init__(self, val=0, next=None): 4 # self.val = val 5 # self.next = next 6 class Solution: 7 def addTwoNumbers(self, l1: Optional[ListNode], l2: Optional[ListNode]) -> Optional[ListNode]: 8 def buildNum(linked_list): 9 res = '' 10 while linked_list != None: 11 res = str(linked_list.val) + res 12 linked_list = linked_list.next 14 return int(res) 16 n1 = buildNum(l1) 17 n2 = buildNum(l2) 19 tot = str(n1 + n2) 21 head = ListNode() 22 pointer = head 24 for n in tot[::-1]: 25 new = ListNode() 26 new.val = int(n) 27 pointer.next = new 28 pointer = pointer.next 30 return head.next	{"url":"https://douglastitze.com/posts/add-two-numbers/","timestamp":"2024-11-13T14:58:47Z","content_type":"text/html","content_length":"28494","record_id":"<urn:uuid:99cd0360-0c24-412d-bd8f-b44676af5df8>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00581.warc.gz"}
Correlating Newtonian Model with Einstein's GR Well, this is your view, how do you look at it. No, that isn't my view, that's what the law states. These are standard physics terms already well defined. So new definition is necessary. Here A can be considered as area. "stress" is not a notation I'm familiar with, but I see that it is used on Wikipedia too: One problem is that you cannot have a strain on a particle; a particle's length is by definition so small its irrelevant. But the real issue is that this is just some generic definition of stress, not something specific to this situation. Please give the equation and its derivation of this "Lorentz-stress", where the velocity of the particle is used as an input. If it is not a straight line, as per Newton's Law of Inertia, it will be subject to a force. Irrelevant. Do you agree with me that a not-straight line isn't necessarily a loop, yes or no? "Equivalent of a straight line" and "a straight line" does not mean the samething. Technically, you are right, but that's really not the point here. A geodesic in GR is what a straight line is in flat space. In that regard, they are the same thing. It is the thumb rule, which bends magnetic field lines around a current carrying wire. The "thumb rule" is not a force, so that's wrong. Please give a force. "can be considered as" and "is" are not the same thing. As I already said, that's an imperfect analogue, so it will break down in places. You have described it literally as quantization: How will you define continuity of a function? Say consider f(x)=x. Is it continuous or quantized? This is a very basic calculus question. Off the top of my head: if a function has a well-defined derivative on its entire domain, it's continuous on that entire domain. f'(x)=1, so yes, that's A quantized (or as it's called in mathematics, "discreet") function will not be continuous, in general. Well, Hubble observed expansion from the Earth. No, Hubble observed galaxies moving away from us, not the expansion directly. You've missed the point: rulers stretch too, so against what ruler are you measuring this expansion? I'm not saying it's impossible, but one has to be very carefully to define these things properly. The rubber sheet analogue doesn't work properly in these cases. Why they will remain stationary with the rubber sheet? No force is applied to the balls. First of all, there's Newton's First Law. But in reality, it's more complicated than that. Pick a ball. The sheet is being stretch in all directions, out from underneath this ball. There is no force pushing the ball around; it remains stationary compared to the sheet directly underneath. The ball doesn't move. Pick another ball. The sheet is being stretch in all directions, out from underneath this ball. There is no force pushing the ball around; it remains stationary compared to the sheet directly underneath. The ball doesn't move. Both balls don't move with respect to the sheet underneath, because there's no force. Yet, the distance between the balls increases! "The mechanical energy can be expressed as follows, using the notation of Carroll and Ostlie: The above expression, I copy pasted from that link. OK, so the first link you gave was indeed wrong. Mechanical energy isn't the same as work, so the quoted part doesn't talk about work either.	{"url":"https://sciforums.com/threads/correlating-newtonian-model-with-einsteins-gr.159332/page-14","timestamp":"2024-11-07T07:23:07Z","content_type":"text/html","content_length":"55884","record_id":"<urn:uuid:586aad1b-b53b-463d-ad5e-1c2cd35a2259>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00240.warc.gz"}
Create a boolean mask between arrays of different size I would like to do the same thing as in python: import numpy as np >>> mask array([False, True, False, True, False, True, False]) In julia I can: mask = indexin(a,b) 7-element Vector{Union{Nothing, Int64}}: But I guess that there must be a way to get a simpler boolean array . One way is [x in b for x in a] or shorter with dot broadcasting in.(a, Ref(b)) 2 Likes Thanks !!! In your example, a and b are both sorted. That permits a fast implementation that scans over both. If that applies to your true problem, then you should consider doing that instead of the one-liner. I don’t use sorted array/matrix here. I tried both solutions, they are very slow compared to python. a have around 25 million elements. b have 200 000 elements. I think the best solutions depends on what are you eventually doing. Can you show some minimal example of what you’re benchmarking exactly in Numpy? Oh it’s quite simple, that’s two arrays of floats. In [1]: import numpy as np In [2]: a = np.random.rand(25106) In [3]: b = np.random.rand(210*5) In [4]: %timeit np.isin(a,b) 3.88 s ± 119 ms per loop (mean ± std. dev. of 7 runs, 1 loop each) like so? Also, I was asking what do you do after you get the mask – it may not be the best thing to do depends on what your ultimate goal is. or probably they sort b internally: In [1]: import numpy as np In [2]: a = np.random.rand(2510*6) In [3]: b = np.random.rand(210**5) In [4]: %timeit np.isin(a,b) 3.88 s ± 119 ms per loop (mean ± std. dev. of 7 runs, 1 loop each) edit: they sort it: in Julia you can also just use insorted(), notice the time includes the time to sort b julia> @be insorted.(a, Ref(sort(b))) evals=10 Benchmark: 1 sample with 10 evaluations 2.566 s (13 allocs: 6.040 MiB, 0.02% gc time) 2 Likes Ah thanks for the information, it’s indeed way faster. An alternative, which is much faster on my machine: in.(a, Ref(Set(b))) 3 Likes	{"url":"https://discourse.julialang.org/t/create-a-boolean-mask-between-arrays-of-different-size/121986","timestamp":"2024-11-06T18:26:14Z","content_type":"text/html","content_length":"37186","record_id":"<urn:uuid:abb43576-d071-4913-8528-4c722a37b53c>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00796.warc.gz"}
Lux to Watts - bimmodeller.com - BIM Modeling services Provider • Light or visible light is electromagnetic radiation that can be perceived by the human eye. • Visible light is usually defined as having wavelengths in the range of 400–700 nanometres, corresponding to frequencies of 750–420 terahertz, between the infrared and the ultraviolet. • Then, the symbol of lux is lx. • The watt is a unit of power or radiant flux.In the International System of Units and then it is defined as a derived unit of 1 kg·m²·s³ or, equivalently, 1 joule per second. • It is used to quantify the rate of energy transfer. • Then, the symbol fo watts is W. • Watt and lux units represent different physical quantities, so you can’t convert them directly. • Our application calculates and converts watts to lux to show how power affects the light efficiency on a given surface. • A higher number of lux makes the source more efficient in spreading (the surface is brighter), and a lower one means that your surface is darker, dimmer. • The formula used to calculate the following calculator lux to watts is given below. • The calculation formula, □ So, Feet = ( Illuminance in lux * Surface area / Luminous efficacy in lumens per watt ) * 0.09290304. □ Then, Meter = ( Illuminance in lux * Surface area / Luminous efficacy in lumens per watt ). • Besides the formula this calculator works. • First, Enter Illuminance in lux. • Second, Either select light source or enter Luminous efficacy in lumens per watt. • Third, Select Feet or Meter. • Fourth, Either enter spherical radius or enter surface area. • Click “Calculate” to get the result and then if you want to clear values Click “Reset” in addition to clear the values. • Finally, follow these steps to get the results. • Then you will get as a power result in watts.	{"url":"https://www.bimmodeller.com/lux-to-watts/","timestamp":"2024-11-13T05:48:40Z","content_type":"text/html","content_length":"280395","record_id":"<urn:uuid:f1024923-e00b-4aeb-a301-fb403f92ef8a>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00509.warc.gz"}
HCUP Nationwide Inpatient Sample TECHNICAL SUPPLEMENT 8: DESIGN OF THE HCUP NATIONWIDE INPATIENT SAMPLE, RELEASE 4 The Nationwide Inpatient Sample (NIS) of the Healthcare Cost and Utilization Project (HCUP) was established to provide analyses of hospital utilization across the United States. The NIS, Release 1 covers calendar years 1988-1992. The NIS, Release 2 covers calendar year 1993, the NIS, Release 3 covers calendar year 1994, and the NIS, Release 4 covers calendar year 1995. The target universe includes all acute-care discharges from all community hospitals in the United States; the NIS comprises all discharges from a sample of hospitals in this target universe. This fourth release of the NIS contains 6.7 million discharges from a sample of 938 hospitals in 19 states. The first release (1988 through 1992) contains 5.2 to 6.2 million discharges per year from a sample of 758 to 875 hospitals per year in 11 states (8 states for 1988). The second release of the NIS contains 6.5 million discharges from a sample of 913 hospitals in 17 states. The third release of the NIS contains 6.4 million discharges from a sample of 904 hospitals in 17 states. Thus, the NIS supports both cross-sectional and longitudinal analyses. Potential research issues focus on both discharge- and hospital-level outcomes. Discharge outcomes of interest include trends in inpatient treatments with respect to: • frequency, • costs, • lengths of stay, • effectiveness, • appropriateness, and • access to hospital care. Hospital outcomes of interest include: • mortality rates, • complication rates, • patterns of care, • diffusion of technology, and • trends toward specialization. These and other outcomes are of interest for the nation as a whole and for policy-relevant inpatient subgroups defined by geographic regions, patient demographics, hospital characteristics, physician characteristics, and pay sources. This report provides a detailed description of the NIS, Release 4 sample design, as well as a summary of the resultant hospital sample. Sample weights were developed to obtain national estimates of hospital and inpatient parameters. These weights and other special-use weights are described in detail. Tables include cumulative information for NIS, Release 1 (1988 through 1992); NIS, Release 2 (1993); NIS, Release 3 (1994); and NIS, Release 4 (1995) to provide a longitudinal view of the database. The hospital universe is defined by all hospitals that were open during any part of the calendar year and were designated as community hospitals in the American Hospital Association (AHA) Annual Survey of Hospitals. For purposes of the NIS, the definition of a community hospital is that used by the AHA: "all nonfederal short-term general and other specialty hospitals, excluding hospital units of institutions." Consequently, Veterans Hospitals and other federal hospitals are excluded. Table 29 shows the number of universe hospitals for each year based on the AHA Annual Survey. Table 29: Hospital Universe^1 │ Year │ Number of Hospitals │ │ 1988 │ 5,607 │ │ 1989 │ 5,548 │ │ 1990 │ 5,468 │ │ 1991 │ 5,412 │ │ 1992 │ 5,334 │ │ 1993 │ 5,313 │ │ 1994 │ 5,290 │ │ 1995 │ 5,260 │ Hospital Merges, Splits, and Closures All hospital entities that were designated community hospitals in the AHA hospital file were included in the hospital universe. Therefore, if two or more community hospitals merged to create a new community hospital, the original hospitals and the newly-formed hospital were all considered separate hospital entities in the universe for the year of the merge. Likewise, if a community hospital split, the original hospital and all newly created community hospitals were separate entities in the universe for the year of the split. Finally, community hospitals that closed during a year were included as long as they were in operation during some part of the calendar year. Stratification Variables To help ensure representativeness, sampling strata were defined based on five hospital characteristics contained in the AHA hospital files. The stratification variables were as follows: 1. Geographic Region — Northeast, Midwest, West, and South. This is an important stratifier because practice patterns have been shown to vary substantially by region. For example, lengths of stay tend to be longer in East Coast hospitals than in West Coast hospitals. 2. Control — government nonfederal, private not-for-profit, and private investor-owned. These types of hospitals tend to have different missions and different responses to government regulations and 3. Location — urban or rural. Government payment policies often differ according to this designation. Also, rural hospitals are generally smaller and offer fewer services than urban hospitals. 4. Teaching Status — teaching or nonteaching. The missions of teaching hospitals differ from nonteaching hospitals. In addition, financial considerations differ between these two hospital groups. Currently, the Medicare DRG payments are uniformly higher to teaching hospitals than to nonteaching hospitals. A hospital is considered to be a teaching hospital if it has an AMA-approved residency program or is a member of the Council of Teaching Hospitals (COTH). 5. Bedsize — small, medium, and large. Bedsize categories are based on hospital beds, and are specific to the hospital's location and teaching status, as shown in Table 30. Table 30. Bed Size Categories │ │ Hospital Bed Size │ │ Location and Teaching Status ├───────┬──────────┬───────┤ │ │ Small │ Medium │ Large │ │ Rural │ 1-49 │ 50-99 │ 100+ │ │ Urban, non-teaching │ 1-99 │ 100-199 │ 200+ │ │ Urban, teaching │ 1-299 │ 300-499 │ 500+ │ Rural hospitals were not split according to teaching status, because rural teaching hospitals were rare. For example, in 1988 there were only 20 rural teaching hospitals. The bedsize categories were defined within location and teaching status because they would otherwise have been redundant. Rural hospitals tend to be small; urban nonteaching hospitals tend to be medium-sized; and urban teaching hospitals tend to be large. Yet it was important to recognize gradations of size within these types of hospitals. For example, in serving rural discharges, the role of "large" rural hospitals (particularly rural referral centers) often differs from the role of "small" rural hospitals. The cut-off points for the bedsize categories are consistent with those used in Hospital Statistics, published annually by the AHA. To further ensure geographic representativeness, implicit stratification variables included state and three-digit zip code (the first three digits of the hospital's five-digit zip code). The hospitals were sorted according to these variables prior to systematic sampling. For each year, the universe of hospitals was established as all community hospitals located in the U.S. However, it was not feasible to obtain and process all-payer discharge data from a random sample of the entire universe of hospitals for at least two reasons. First, all-payer discharge data were not available from all hospitals for research purposes. Second, based on the experience of prior hospital discharge data collections, it would have been too costly to obtain data from individual hospitals, and it would have been too burdensome to process each hospital's unique data Therefore, the NIS sampling frame was constructed from the subset of universe hospitals that released their discharge data for research use. Two sources for all-payer discharge data were state agencies and private data organizations, primarily state hospital associations. At the time when the sample was drawn, the Agency for Health Care Policy and Research (AHCPR) had agreements with 22 data sources that maintain statewide, all-payer discharge data files to include their data in the HCUP database. However, only 8 states in 1988 and 11 states in 1989-1992 could be included in the first release of the NIS, an additional 6 states have been included in the second and the third release of the NIS, and an additional 2 states have been included in the fourth release of the NIS, as shown in Table 31. Table 31. States in the Frame for the NIS, Release 1; NIS, Release 2; and NIS, Release 3; and NIS, Release 4 │ Years │ States in the Frame │ │ NIS, Release 1 │ │ 1988 │ California, Colorado, Florida, Iowa, Illinois, Massachusetts, New Jersey, and Washington │ │ 1989-1992 │ Add Arizona, Pennsylvania, and Wisconsin │ │ NIS, Release 2 │ │ 1993 │ Add Connecticut, Kansas, Maryland, New York, Oregon, South Carolina │ │ NIS, Release 3 │ │ 1994 │ No new additions │ │ NIS, Release 4 │ │ 1995 │ Add Missouri, Tennessee │ The list of the entire frame of hospitals was composed of all AHA community hospitals in each of the frame states that could be matched to the discharge data provided to HCUP, with restrictions on the hospitals that could be included from Illinois, South Carolina, Missouri, and Tennessee. If an AHA community hospital could not be matched to the discharge data provided by the data source, it was eliminated from the sampling frame (but not from the universe). The Illinois Health Care Cost Containment Council stipulated that no more than 40 percent of the discharges provided by Illinois could be included in the database for any calendar quarter. Consequently, a systematic random sample of Illinois hospitals was drawn for the frame. This prevented the sample from including more than 40 percent of Illinois discharges. South Carolina and Tennessee stipulated that only hospitals that appear in sampling strata with two or more hospitals were to be included in the NIS. Four South Carolina hospitals were excluded from the frame since there were fewer than 2 South Carolina hospitals in 4 sampling frame strata. The remaining 59 South Carolina community hospitals are included in the frame. Six Tennessee hospitals were excluded from the frame since there were fewer than 2 Tennessee hospitals in 6 sampling frame strata. The remaining 66 Tennessee community hospitals are included in the frame. Missouri stipulated that only hospitals that had signed releases for public use should be included in the NIS, Release 4. Thirty-five Missouri hospitals signed releases for confidential use only. These hospitals were excluded from the sampling frame, leaving 80 hospitals in the frame. The number of frame hospitals for each year is shown in Table 32. Table 32. Hospital Frame │ Year │ Number of Hospitals │ │ 1988 │ 1,247 │ │ 1989 │ 1,658 │ │ 1990 │ 1,620 │ │ 1991 │ 1,604 │ │ 1992 │ 1,591 │ │ 1993 │ 2,168 │ │ 1994 │ 2,135 │ │ 1995 │ 2,284 │ Design Requirements The NIS is a stratified probability sample of hospitals in the frame, with sampling probabilities calculated to select 20 percent of the universe contained in each stratum. The overall objective was to select a sample of hospitals "generalizable" to the target universe, which includes hospitals outside the frame (zero probability of selection). Moreover, this sample was to be geographically dispersed, yet drawn from the subset of states with inpatient discharge data that agreed to provide such data to the project. It should be possible, for example, to estimate DRG-specific average lengths of stay over all U.S. hospitals using weighted average lengths of stay, based on averages or regression estimates from the NIS. Ideally, relationships among outcomes and their correlates estimated from the NIS should generally hold across all U.S. hospitals. However, since only 19 states contributed data to this fourth release, some estimates may differ from estimates from comparative data sources. When possible, estimates based on the NIS should be checked against national benchmarks, such as Medicare data or data from the National Hospital Discharge Survey to determine the appropriateness of the NIS for specific analyses. The target sample size was 20 percent of the total number of community hospitals in the U.S. for 1995. This sample size was determined by AHCPR based on their experience with similar research Alternative stratified sampling allocation schemes were considered. However, allocation proportional to the number of hospitals is preferred for several reasons: • Fewer than 10 percent of government-planned database applications will produce nationwide estimates. The major government applications will investigate relationships among variables. For example, government researchers will do a substantial amount of regression modeling with these data. • The HCUP-2 sample^2 used the same stratification and allocation scheme, and it has served AHCPR analysts well. Moreover, the large number of sample hospitals and discharges seemingly reduced the need for variance-reducing allocation schemes. • AHCPR researchers wanted a simple, easily understood sampling methodology. It was an appealing idea that the NIS sample could be a "miniaturization" of the universe of hospitals (with the obvious geographical limitations imposed by data availability). • AHCPR statisticians considered other optimal allocation schemes, including sampling hospitals with probabilities proportional to size (number of discharges), and they concluded that sampling with probability proportional to the number of hospitals was preferable. Even though it was recognized that the approach chosen would not be as efficient, the extremely large sample sizes yield good estimates. Furthermore, because the data are to be used for purposes other than producing national estimates, it is critical that all hospital types (including small hospitals) are adequately Hospital Sampling Procedure Once the universe of hospitals was stratified, up to 20 percent of the total number of U.S. hospitals was randomly selected within each stratum. If too few frame hospitals were in the stratum, then all frame hospitals were selected for the NIS, subject to sampling restrictions specified by states. To simplify variance calculations, at least two hospitals were drawn from each stratum. If fewer than two frame hospitals were contained in a stratum, then that stratum was merged with an "adjacent" stratum containing hospitals with similar characteristics. A systematic random sample was drawn from each stratum, after sorting hospitals by state within each stratum, then by the three-digit zip code (the first three digits of the hospital's five-digit zip code) within each state, and then by a random number within each three-digit zip code. These sorts ensured further geographic generalizability of hospitals within the frame states, and random ordering of hospitals within three-digit zip codes. Generally, three-digit zip codes that are near in value are geographically near within a state. Furthermore, the U.S. Postal Service locates regional mail distribution centers at the three-digit level. Thus, the boundaries tend to be a compromise between geographic size and population size. 1995 NIS Hospital Sampling Procedure The 1995 sample was drawn by a procedure that retained most of the 1994 hospitals, while allowing hospitals new to the frame an opportunity to enter the 1995 NIS. Even in frame states that were present in the 1994 sample, hospitals that opened in 1995 needed a chance to enter the sample. Also, hospitals that changed strata between 1994 and 1995 were considered new to the 1995 frame. Consequently, a recursive procedure was developed to update the sample from year to year in a way that properly accounted for changes in stratum size, composition, and sampling rate. The goal of this procedure was to maximize the year-to-year overlap among sample hospitals, yet keep the sampling rate constant for all hospitals within a stratum. The following procedure provides rules for creating a "year 2" sample, given that a "year 1" sample had already been drawn. In this example, year 1 would be 1994 and year 2 would be 1995. All notation is assumed to refer to sizes and probabilities within a particular stratum. Probabilities P[1] and P[2] were calculated for sampling hospitals from the frame within the stratum for year 1 and year 2, respectively, based on the frame and universe for year 1 and year 2, respectively. These probabilities were set by the same algorithm used to calculate P for the 1988 hospital sample (see Technical Supplement: Design of the HCUP Nationwide Inpatient Sample, Release 1, section "1988 NIS Hospital Sampling Procedure.") Now consider the three possibilities associated with changes between years 1 and 2 in the stratum-specific hospital sampling probabilities: 1. P[2] = P[1]: The target probability was unchanged. 2. P[2] ‹ P[1]: The target probability decreased. 3. P[2] > P[1]: The target probability increased. Below is the procedure used for each of these three cases with one exception: if the stratum-specific probability of selection P[2] was equal to 1, then all frame hospitals were selected for the year 2 sample, regardless of the value of P[1]. Stratum-Specific Sampling Rates the Same (P[2] = P[1]). If the probability P[2] was the same as P[1], all hospitals in the year 1 sample that remained in the year 2 frame were retained for the year 2 sample. Any new frame hospitals (those in the year 2 frame but not in the year 1 frame) were selected at the rate P[2], using the systematic sampling method described for the 1988 sample selection in Technical Supplement: Design of the HCUP Nationwide Inpatient Sample, Release 1. Stratum-Specific Sampling Rate Decreased (P[2] ‹ P[1]). Now consider the case where the probability of selection decreased between years 1 and 2. First, hospitals new to the frame were sampled with probability P[2]. Second, hospitals previously selected for the year 1 sample (that remained in the year 2 frame) were selected for the year 2 sample with probability P[2] ÷ P[1]. The justification for this second procedure was straightforward. For the year 1 sample hospitals that stayed in the frame, the year 1 sample was viewed as the first stage of a two-stage sampling process. The first stage was carried out at the sampling rate of P[1]. The second stage was carried out at the sampling rate of P[2] ÷ P[1]. Consequently, the "overall" probability of selection was P [1] x P[2] ÷ P[1] = P[2]. Stratum-Specific Sampling Rate Increased (P[2] > P[1]). The procedures associated with the case in which the probability of selection was increased between year 1 and year 2 were equally straightforward. First, hospitals new to the frame were sampled with probability P[2]. Second, hospitals that were selected in year 1 (that remained in the year 2 frame) were selected for the year 2 sample. Third, hospitals that were in the frame for both years 1 and 2, but not selected for the year 1 sample, were selected for the year 2 sample with probability (P[2]-P[1]) ÷ (1-P[1]). The justification for this sampling rate, (P[2]-P[1]) ÷ (1-P[1]), is somewhat complex. In year 1 certain frame hospitals were included in the sample at the rate P[1]. This can also be viewed as having excluded a set of hospitals at the rate (1-P[1]). Likewise, in year 2 it was imperative that each hospital excluded from the year 1 sample be excluded from the year 2 sample at an overall rate of (1-P[2]). Since P[2] > P[1], then (1-P[2]) ‹ (1-P[1]). Therefore, just as was done for the case of P[2] ‹ P[1], multistage selection was implemented. However, it was implemented for exclusion rather than Therefore, those hospitals excluded from the year 1 sample were also excluded from the year 2 sample at the rate S = (1-P[2]) ÷ (1-P[1]). This gave them the desired overall exclusion rate of (1-P[1]) x (1-P[2]) ÷ (1-P[1]) = (1-P[2]). Consequently, the inclusion rate for these hospitals was set at 1-S = (P[2]-P[1]) ÷ (1-P[1]). Zero-Weight Hospitals The 1995 sample contains no zero-weight hospitals. For a description of zero-weight hospitals in the 1988-1992 sample, see the Technical Supplement: Design of the HCUP Nationwide Inpatient Sample, Release 1. Ten Percent Subsamples Two nonoverlapping 10 percent subsamples of discharges were drawn from the NIS file for each year. The subsamples were selected by drawing every tenth discharge starting with two different starting points (randomly selected between 1 and 10). Having a different starting point for each of the two subsamples guaranteed that they would not overlap. Discharges were sampled so that 10 percent of each hospital's discharges in each quarter were selected for each of the subsamples. The two samples can be combined to form a single, generalizable 20 percent subsample of discharges. The annual numbers of hospitals and discharges in NIS, Release 1; NIS, Release 2; NIS, Release 3; and NIS, Release 4 are shown in Table 33, for both the regular NIS sample and the total sample (which includes zero-weight hospitals for 1988-1992). Table 33. NIS Hospital Sample │ │ Regular Sample │ Total Sample │ │ Year │ Number of Hospitals │ Number of Discharges │ Number of Hospitals │ Number of Discharges │ │ NIS, Release 1 │ │ 1988 │ 758 │ 5,242,904 │ 759 │ 5,265,756 │ │ 1989 │ 875 │ 6,067,667 │ 882 │ 6,110,064 │ │ 1990 │ 861 │ 6,156,638 │ 871 │ 6,268,515 │ │ 1991 │ 847 │ 5,984,270 │ 859 │ 6,156,188 │ │ 1992 │ 838 │ 6,008,001 │ 856 │ 6,195,744 │ │ NIS, Release 2 │ │ 1993 │ 913 │ 6,538,976 │ 913 │ 6,538,976 │ │ NIS, Release 3 │ │ 1994 │ 904 │ 6,385,011 │ 904 │ 6,385,011 │ │ NIS, Release 4 │ │ 1995 │ 938 │ 6,714,935 │ 938 │ 6,714,935 │ │ Total │ │ 49,098,402 │ │ 49,635,189 │ A more detailed breakdown of the 1995 NIS hospital sample by geographic region is shown in Table 34. For each geographic region, Table 34 shows the number of: • universe hospitals (Universe), • frame hospitals (Frame), • sampled hospitals (Sample), • target hospitals (Target = 20 percent of the universe), and • shortfall hospitals (Shortfall = Sample - Target). Table 34. Number of Hospitals in the Universe, Frame, Regular Sample, Target, and Shortfall by Region, 1995 │ Region │ Universe │ Frame │ Sample │ Target │ Shortfall │ │ NE │ 772 │ 648 │ 162 │ 154 │ 8 │ │ MW │ 1,507 │ 557 │ 317 │ 302 │ 15 │ │ S │ 2,004 │ 375 │ 278 │ 401 │ -123 │ │ W │ 977 │ 704 │ 181 │ 195 │ -14 │ │ Total │ 5,260 │ 2,284 │ 938 │ 1,052 │ -114 │ For example, in 1995 the Northeast region contained 772 hospitals in the universe. It also contained 648 hospitals in the frame, of which 162 hospitals were drawn for the sample. This was 8 hospitals more than the target sample size of 154. Table 35 shows the number of hospitals in the universe, frame, and regular sample for each state in the sampling frame for 1995. In all states except Illinois, South Carolina, Tennessee, and Missouri, the difference between the universe and the frame represents the difference in the number of community hospitals in the 1995 AHA Annual Survey of Hospitals and the number of community hospitals for which data were supplied to HCUP. As explained earlier, the number of hospitals in the Illinois frame is approximately 53 percent of the hospitals in the Illinois universe in order to comply with the agreement with the data source concerning the restriction on the number of Illinois discharges. The number of hospitals in the South Carolina frame is 7 fewer than the South Carolina universe. Four hospitals were excluded because of sampling restrictions stipulated by South Carolina, and 3 hospitals were not included in the data supplied to HCUP. The number of hospitals in the Tennessee frame is 63 fewer than the Tennessee universe. Six hospitals were excluded because of sampling restrictions stipulated by Tennessee, and 57 hospitals were not included in the data supplied to HCUP. The number of hospitals in the Missouri frame is 49 fewer than the Missouri universe. Thirty-five hospitals were excluded because they signed release for confidential use only, and 14 hospitals were not included in the data supplied to HCUP. The number of hospitals in the NIS hospital samples that continue across multiple sample years is shown in Table 36. This table will be of interest to those who may combine Releases 1, 2, 3, and 4 of the NIS. Table 36 shows that longitudinal cohorts that span several years and include 1988 and 1993 are the lowest in number of continuing sample hospitals. For example, if 1988 is taken as a starting year, only 38.1 percent of the 1988 hospital sample continued in the 1995 sample (289 of 758). Table 35 Number of Hospitals in the Universe, Frame, and Regular Sample for States in the Sampling Frame: 1995 │ State │ Universe │ Frame │ Sample │ │ AZ │ 62 │ 60 │ 15 │ │ CA │ 426 │ 425 │ 105 │ │ CO │ 70 │ 69 │ 22 │ │ CT │ 34 │ 32 │ 9 │ │ FL │ 215 │ 200 │ 141 │ │ IA │ 116 │ 116 │ 54 │ │ IL │ 207 │ 110 │ 73 │ │ KS │ 134 │ 124 │ 61 │ │ MA │ 96 │ 82 │ 25 │ │ MD │ 50 │ 50 │ 39 │ │ MO │ 129 │ 80 │ 49 │ │ NJ │ 92 │ 85 │ 18 │ │ NY │ 231 │ 230 │ 59 │ │ OR │ 64 │ 62 │ 17 │ │ PA │ 225 │ 219 │ 51 │ │ SC │ 66 │ 59 │ 46 │ │ TN │ 129 │ 66 │ 52 │ │ WA │ 89 │ 88 │ 22 │ │ WI │ 127 │ 127 │ 80 │ │ TOTAL │ 2,562 │ 2,284 │ 938 │ Table 36. Number of Hospitals and Discharges in Longitudinal Cohort │ Number of Years │ Calendar Years │ Longitudinal Regular Sample Hospitals │ % of Base Year Sample │ Longitudinal Regular Sample Discharges │ │ 2 │ 1988-1989 │ 610 │ 80.5 │ 8,492,039 │ │ ├────────────────┼───────────────────────────────────────┼───────────────────────┼─────────────────────────────────────────┤ │ │ 1989-1990 │ 815 │ 93.1 │ 11,525,749 │ │ ├────────────────┼───────────────────────────────────────┼───────────────────────┼─────────────────────────────────────────┤ │ │ 1990-1991 │ 802 │ 93.1 │ 11,297,175 │ │ ├────────────────┼───────────────────────────────────────┼───────────────────────┼─────────────────────────────────────────┤ │ │ 1991-1992 │ 781 │ 92.2 │ 11,272,981 │ │ ├────────────────┼───────────────────────────────────────┼───────────────────────┼─────────────────────────────────────────┤ │ │ 1992-1993 │ 609 │ 72.7 │ 8,804,638 │ │ ├────────────────┼───────────────────────────────────────┼───────────────────────┼─────────────────────────────────────────┤ │ │ 1993-1994 │ 693 │ 75.9 │ 10,271,404 │ │ ├────────────────┼───────────────────────────────────────┼───────────────────────┼─────────────────────────────────────────┤ │ │ 1994-1995 │ 762 │ 84.3 │ 10,747,682 │ │ 3 │ 1988-1990 │ 573 │ 75.6 │ 12,168,677 │ │ ├────────────────┼───────────────────────────────────────┼───────────────────────┼─────────────────────────────────────────┤ │ │ 1989-1991 │ 763 │ 87.2 │ 16,074,381 │ │ ├────────────────┼───────────────────────────────────────┼───────────────────────┼─────────────────────────────────────────┤ │ │ 1990-1992 │ 745 │ 86.5 │ 16,085,651 │ │ ├────────────────┼───────────────────────────────────────┼───────────────────────┼─────────────────────────────────────────┤ │ │ 1991-1993 │ 570 │ 67.3 │ 12,559,421 │ │ ├────────────────┼───────────────────────────────────────┼───────────────────────┼─────────────────────────────────────────┤ │ │ 1992-1994 │ 540 │ 64.4 │ 11,279,667 │ │ ├────────────────┼───────────────────────────────────────┼───────────────────────┼─────────────────────────────────────────┤ │ │ 1993-1995 │ 598 │ 65.5 │ 13,241,070 │ │ 4 │ 1988-1991 │ 542 │ 71.5 │ 15,096,807 │ │ ├────────────────┼───────────────────────────────────────┼───────────────────────┼─────────────────────────────────────────┤ │ │ 1989-1992 │ 709 │ 81.0 │ 20,340,970 │ │ ├────────────────┼───────────────────────────────────────┼───────────────────────┼─────────────────────────────────────────┤ │ │ 1990-1993 │ 548 │ 63.6 │ 16,023,500 │ │ ├────────────────┼───────────────────────────────────────┼───────────────────────┼─────────────────────────────────────────┤ │ │ 1991-1994 │ 508 │ 60.0 │ 14,481,319 │ │ ├────────────────┼───────────────────────────────────────┼───────────────────────┼─────────────────────────────────────────┤ │ │ 1992-1995 │ 464 │ 55.4 │ 12,712,613 │ │ 5 │ 1988-1992 │ 502 │ 66.2 │ 18,106,098 │ │ ├────────────────┼───────────────────────────────────────┼───────────────────────┼─────────────────────────────────────────┤ │ │ 1989-1993 │ 523 │ 59.8 │ 19,000,777 │ │ ├────────────────┼───────────────────────────────────────┼───────────────────────┼─────────────────────────────────────────┤ │ │ 1990-1994 │ 490 │ 56.9 │ 17,437,229 │ │ ├────────────────┼───────────────────────────────────────┼───────────────────────┼─────────────────────────────────────────┤ │ │ 1991-1995 │ 439 │ 51.8 │ 15,405,253 │ │ 6 │ 1988-1993 │ 378 │ 49.9 │ 16,906,818 │ │ ├────────────────┼───────────────────────────────────────┼───────────────────────┼─────────────────────────────────────────┤ │ │ 1989-1994 │ 471 │ 53.8 │ 19,987,910 │ │ ├────────────────┼───────────────────────────────────────┼───────────────────────┼─────────────────────────────────────────┤ │ │ 1990-1995 │ 422 │ 49.0 │ 14,817,797 │ │ 7 │ 1988-1994 │ 335 │ 44.2 │ 17,128,064 │ │ ├────────────────┼───────────────────────────────────────┼───────────────────────┼─────────────────────────────────────────┤ │ │ 1989-1995 │ 408 │ 46.6 │ 19,924,107 │ │ 8 │ 1988-1995 │ 289 │ 38.1 │ 16,658,485 │ Although the sampling design was simple and straightforward, it is necessary to incorporate sample weights to obtain state and national estimates. Therefore, sample weights were developed separately for hospital- and discharge-level analyses. Three hospital-level weights were developed to weight NIS sample hospitals to the state, frame, and universe. Similarly, three discharge-level weights were developed to weight NIS sample discharges to the state, frame, and universe. Hospital-Level Sampling Weights Universe Hospital Weights. Hospital weights to the universe were calculated by post-stratification. For each year, hospitals were stratified on the same variables that were used for sampling: geographic region, urban/rural location, teaching status, bedsize, and control. The strata that were collapsed for sampling were also collapsed for sample weight calculations. Within stratum s, each NIS sample hospital's universe weight was calculated as: W[s](universe) = N[s](universe) ÷ N[s](sample), where N[s](universe) and N[s](sample) were the number of community hospitals within stratum s in the universe and sample, respectively. Thus, each hospital's universe weight is equal to the number of universe hospitals it represented during that year. Frame Hospital Weights. Hospital-level sampling weights were also calculated to represent the entire collection of states in the frame using the same post-stratification scheme as described above for the weights to represent the universe. For each year, within stratum s, each NIS sample hospital's frame weight was calculated as: W[s](frame) = N[s](frame) ÷ N[s](sample). N[s](frame) was the total number of universe community hospitals within stratum s in the states that contributed data to the frame. N[s](sample) was the number of sample hospitals selected for the NIS in stratum s. Thus, each hospital's frame weight is equal to the number of universe hospitals it represented in the frame states during that year. State Hospital Weights. For each year, a hospital's weight to its state was calculated in a similar fashion. Within each state, strata often had to be collapsed after sample selection for development of weights to ensure a minimum of two sample hospitals within each stratum. For each state and each year, within stratum s, each NIS sample hospital's state weight was calculated as: W[s](state) = N[s](state) ÷ N[s](state sample). N[s](state) was the number of universe community hospitals in the state within stratum s. N[s](state sample) was the number of hospitals selected for the NIS from that state in stratum s. Thus, each hospital's state weight is equal to the number of hospitals that it represented in its state during that year. All of these hospital weights can be rescaled if necessary for selected analyses, to sum to the NIS hospital sample size each year. Discharge-Level Sampling Weights The calculations for discharge-level sampling weights were very similar to the calculations of hospital-level sampling weights. The discharge weights usually are constant for all discharges within a The only exceptions were for strata with sample hospitals that, according to the AHA files, were open for the entire year but contributed less than their full year of data to the NIS. For those hospitals, we adjusted the number of observed discharges by a factor 4 ÷ Q, where Q was the number of calendar quarters that the hospital contributed discharges to the NIS. For example, when a sample hospital contributed only two quarters of discharge data to the NIS, the adjusted number of discharges was double the observed number. With that minor adjustment, each discharge weight is essentially equal to the number of reference (universe, frame, or state) discharges that each sampled discharge represented in its stratum. This calculation was possible because the number of total discharges was available for every hospital in the universe from the AHA files. Each universe hospital's AHA discharge total was calculated as the sum of newborns and total facility discharges. Universe Discharge Weights. Discharge weights to the universe were calculated by post-stratification. Hospitals were stratified just as they were for universe hospital weight calculations. Within stratum s, for hospital i, each NIS sample discharge's universe weight was calculated as: DW[is](universe) = [DN[s](universe) ÷ ADN[s](sample)] * (4 ÷ Q[i]), where DN[s](universe) was the number of discharges from community hospitals in the universe within stratum s; ADN[s](sample) was the number of adjusted discharges from sample hospitals selected for the NIS; and Q[i] was the number of quarters of discharge data contributed by hospital i to the NIS (usually Q[i] = 4). Thus, each discharge's weight is equal to the number of universe discharges it represented in stratum s during that year. Frame Discharge Weights. Discharge-level sampling weights were also calculated to represent all discharges from the entire collection of states in the frame using the same post-stratification scheme described above for the discharge weights to represent the universe. For each year, within stratum s, for hospital i, each NIS sample discharge's frame weight was calculated as: W[is](frame) = [DN[s](frame) ÷ ADN[s](sample)] * (4 ÷ Q[i]), DN[s](frame) was the number of discharges from all community hospitals in the states that contributed to the frame within stratum s. ADN[s](sample) was the number of adjusted discharges from sample hospitals selected for the NIS in stratum s. Q^i was the number of quarters of discharge data contributed by hospital i to the NIS (usually Q[i] = 4). Thus, each discharge's frame weight is equal to the number of discharges it represented in the frame states during that year. State Discharge Weights. A discharge's weight to its state was similarly calculated. Strata were collapsed in the same way as they were for the state hospital weights to ensure a minimum of two sample hospitals within each stratum. Within stratum s, for hospital i, each NIS sample discharge's state weight was calculated as: W[is](state) = [DN[s](state) ÷ ADN[s](state sample)] * (4 ÷ Q[i]), DN[s](state) was the number of discharges from all community hospitals in the state within stratum s. ADN[s](state sample) was the adjusted number of discharges from hospitals selected for the NIS from that state in stratum s. Q[i] was the number of quarters of discharge data contributed by hospital i to the NIS (usually Q[i] = 4). Thus, each discharge's state weight is equal to the number of discharges that it represented in its state during that year. All of these discharge weights can be rescaled if necessary for selected analyses, to sum to the NIS discharge sample size each year. Discharge Weights for 10 Percent Subsamples In the 10 percent subsamples, each discharge had a 10 percent chance of being drawn. Therefore, the discharge weights contained in the Hospital Weights file can be multiplied by 10 for each of the subsamples, or multiplied by 5 for the two subsamples combined. Variance Calculations It may be important for researchers to calculate a measure of precision for some estimates based on the NIS sample data. Variance estimates must take into account both the sampling design and the form of the statistic. The sampling design was a stratified, single-stage cluster sample. A stratified random sample of hospitals (clusters) were drawn and then all discharges were included from each selected hospital. If hospitals inside the frame were similar to hospitals outside the frame, the sample hospitals can be treated as if they were randomly selected from the entire universe of hospitals within each stratum. Standard formulas for a stratified, single-stage cluster sampling without replacement could be used to calculate statistics and their variances in most applications. A multitude of statistics can be estimated from the NIS data. Several computer programs are listed below that calculate statistics and their variances from sample survey data. Some of these programs use general methods of variance calculations (e.g., the jackknife and balanced half-sample replications) that take into account the sampling design. However, it may be desirable to calculate variances using formulas specifically developed for some statistics. In most cases, computer programs are readily available to perform these calculations. For instance, OSIRIS IV, developed at the University of Michigan, and SUDAAN, developed at the Research Triangle Institute, do calculations for numerous statistics arising from the stratified, single-stage cluster sampling design. An example of using SUDAAN to calculate variances in the NIS is presented in Technical Supplement: Calculating Variances Using Data from the HCUP Nationwide Inpatient Sample.^3 These variance calculations are based on finite-sample theory, which is an appropriate method for obtaining cross-sectional, nationwide estimates of outcomes. According to finite-sample theory, the intent of the estimation process is to obtain estimates that are precise representations of the nationwide population at a specific point in time. In the context of the NIS, any estimates that attempt to accurately describe characteristics (such as expenditure and utilization patterns or hospital market factors) and interrelationships among characteristics of hospitals and discharges during a specific year from 1988 to 1995 should be governed by finite-sample theory. Alternatively, in the study of hypothetical population outcomes not limited to a specific point in time, analysts may be less interested in specific characteristics from the finite population (and time period) from which the sample was drawn, than they are in hypothetical characteristics of a conceptual "superpopulation" from which any particular finite population in a given year might have been drawn. According to this superpopulation model, the nationwide population in a given year is only a snapshot in time of the possible interrelationships among hospital, market, and discharge characteristics. In a given year, all possible interactions between such characteristics may not have been observed, but analysts may wish to predict or simulate interrelationships that may occur in the future. Under the finite-population model, the variances of estimates approach zero as the sampling fraction approaches one, since the population is defined at that point in time, and because the estimate is for a characteristic as it existed at the time of sampling. This is in contrast to the superpopulation model, which adopts a stochastic viewpoint rather than a deterministic viewpoint. That is, the nationwide population in a particular year is viewed as a random sample of some underlying superpopulation over time. Different methods are used for calculating variances under the two sample theories. Under the superpopulation (stochastic) model, procedures (such as those described by Potthoff, Woodbury, and Manton ^4) have been developed to draw inferences using weights from complex samples. In this context, the survey weights are not used to weight the sampled cases to the universe, because the universe is conceptually infinite in size. Instead, these weights are used to produce unbiased estimates of parameters that govern the superpopulation. In summary, the choice of an appropriate method for calculating variances for nationwide estimates depends on the type of measure and the intent of the estimation process. Computer Software for Variance Calculations The hospital weights will be useful for producing hospital-level statistics for analyses that use the hospital as the unit of analysis, and the discharge weights will be useful for producing discharge-level statistics for analyses that use the discharge as the unit of analysis. These would be used to weight the sample data in estimating population statistics. Several statistical programming packages allow weighted analyses.^5 For example, nearly all SAS (Statistical Analysis System) procedures incorporate weights. In addition, several publicly available subroutines have been developed specifically for calculating statistics and their standard errors from survey data: • OSIRIS IV was developed by L. Kish, N. Van Eck, and M. Frankel at the Survey Research Center, University of Michigan. It consists of two main programs for estimating variances from complex survey • SUDAAN, a set of SAS subroutines, was developed at the Research Triangle Institute by B. V. Shah. It is adequate for handling most survey designs with stratification. The procedures can handle estimation and variance estimation for means, proportions, ratios, and regression coefficients. • SUPER CARP (Cluster Analysis and Regression Program) was developed at Iowa State University by W. Fuller, M. Hidiroglou, and R. Hickman. This program computes estimates and variance estimates for multistage, stratified sampling designs with arbitrary probabilities of selection. It can handle estimated totals, means, ratios, and regression estimates. The NIS database includes a Hospital Weights file with variables required by these programs to calculate finite population statistics. In addition to the sample weights described earlier, hospital identifiers (PSUs), stratification variables, and stratum-specific totals for the numbers of discharges and hospitals are included so that finite-population corrections (FPCs) can be applied to variance estimates. In addition to these subroutines, standard errors can be estimated by validation and cross-validation techniques. Given that a very large number of observations will be available for most analyses, it may be feasible to set aside a part of the data for validation purposes. Standard errors and confidence intervals can then be calculated from the validation data. If the analytical file is too small to set aside a large validation sample, cross-validation techniques may be used. For example, tenfold cross-validation would split the data into ten equal-sized subsets. The estimation would take place in ten iterations. At each iteration, the outcome of interest is predicted for one-tenth of the observations by an estimate based on a model fit to the other nine-tenths of the observations. Unbiased estimates of error variance are then obtained by comparing the actual values to the predicted values obtained in this manner. Finally, it should be noted that a large array of hospital-level variables are available for the entire universe of hospitals, including those outside the sampling frame. For instance, the variables from the AHA surveys and from the Medicare Cost Reports are available for nearly all hospitals. To the extent that hospital-level outcomes correlate with these variables, they may be used to sharpen regional and nationwide estimates. As a simple example, each hospital's number of C-sections would be correlated with their total number of deliveries. The number of C-sections must be obtained from discharge data, but the number of deliveries is available from AHA data. Thus, if a regression can be fit predicting C-sections from deliveries based on the NIS data, that regression can then be used to obtain hospital-specific estimates of the number of C-sections for all hospitals in the universe. Longitudinal Analyses As previously shown in Table 36, hospitals that continue in the NIS for multiple consecutive years are a subset of the hospitals in the NIS for any one of those years. Consequently, longitudinal analyses of hospital-level outcomes may be biased if they are based on any subset of NIS hospitals limited to continuous NIS membership. In particular, such subsets would tend to contain fewer hospitals that opened, closed, split, merged, or changed strata. Further, the sample weights were developed as annual, cross-sectional weights rather than longitudinal weights. Therefore, different weights might be required, depending on the statistical methods employed by the analyst. One approach to consider in hospital-level longitudinal analyses is to use repeated-measure models that allow hospitals to have missing values for some years. However, the data are not actually missing for some hospitals, such as those that closed during the study period. In any case, the analyses may be more efficient (e.g., produce more precise estimates) if they account for the potential correlation between repeated measures on the same hospital over time, yet incorporate data from all hospitals in the sample during the study period. Discharge Subsamples The two nonoverlapping 10 percent subsamples of discharges were drawn from the NIS file for each year for several reasons pertaining to data analysis. One reason for creating the subsamples was to reduce processing costs for selected studies that will not require the entire NIS. Another reason is that the two subsamples may be used to validate models and obtain unbiased estimates of standard errors. That is, one subsample may be used to estimate statistical models, and the other subsample may be used to test the fit of those models on new data. This is a very important analytical step, particularly in exploratory studies, where one runs the risk of fitting noise. For example, it is well known that the percentage of variance explained by a regression, R2, is generally overestimated by the data used to fit a model. The regression model could be estimated from the first subsample and then applied to the second subsample. The squared correlation between the actual and predicted value in the second subsample is an unbiased estimate of the model's true explanatory power when applied to new data. 1. Most AHA surveys do not cover a January-to-December calendar year. The number of hospitals for 1988-1991 are based on the HCUP calendar-year version of the AHA Annual Survey files. To create a calendar-year reporting period, data from the AHA surveys must be apportioned in some manner across calendar years. Survey responses were converted to calendar-year periods for 1988-1991 by merging data from adjacent survey years. The number of hospitals for 1992-1994 are based on the AHA Annual Survey files. 2. Coffey, R. and D. Farley (1988, July). HCUP-2 Project Overview, (DHHS Publication No. (PHS) 88-3428. Hospital Studies Program Research Note 10, National Center for Health Services Research and Health Care Technology Assessment, Rockville, MD: Public Health Service 3. Duffy, S.Q. and J.P. Sommers (1996, March). Calculating Variances Using Data from the HCUP Nationwide Inpatient Sample. Rockville, MD: Agency for Health Care Policy and Research. 4. Potthoff, R.F., M.A. Woodbury, and K.G. Manton (1992). "Equivalent Sample Size" and "Equivalent Degrees of Freedom" Refinements for Inference Using Survey Weights Under Superpopulation Models. Journal of the American Statistical Association, Vol. 87, 383-396. 5. Carlson, B.L., A.E. Johnson, and S.B. Cohen (1993). An Evaluation of the Use of Personal Computers for Variance Estimation with Complex Survey Data. Journal of Official Statistics, Vol. 9, No. 4,	{"url":"https://hcup-us.ahrq.gov/db/nation/nis/reports/NIS_1995_Design_Report.jsp","timestamp":"2024-11-06T22:01:17Z","content_type":"text/html","content_length":"96390","record_id":"<urn:uuid:4595f2a4-f557-4073-8803-9e6bf999ca22>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00152.warc.gz"}
In the given figure sides AB and AC of ΔABC are extended to points P and Q respectively. Also, ∠PBC < ∠QCB. Show that AC > AB A day full of math games & activities. Find one near you. A day full of math games & activities. Find one near you. A day full of math games & activities. Find one near you. A day full of math games & activities. Find one near you. In Fig. 7.48, sides AB and AC of Δ ABC are extended to points P and Q respectively. Also, ∠PBC < ∠QCB. Show that AC > AB. Let's look into the solution below. In the given figure, ∠ABC + ∠PBC = 180° [Linear pair of angles] Also, ∠ABC = 180° - ∠PBC ... (1) ∠ACB + ∠QCB = 180° [Linear pair] ∠ACB = 180° - ∠QCB ... (2) As ∠PBC < ∠QCB (given), 180^o - ∠PBC > 180^o - ∠QCB ∠ABC > ∠ACB [From Equations (1) and (2)] Thus, AC > AB (Side opposite to the larger angle is larger). Hence proved, AC > AB. ☛ Check: NCERT Solutions for Class 9 Maths Chapter 7 Video Solution: In Fig. 7.48, sides AB and AC of Δ ABC are extended to points P and Q respectively. Also, ∠PBC < ∠QCB. Show that AC > AB. NCERT Maths Solutions Class 9 Chapter 7 Exercise 7.4 Question 2: If sides AB and AC of ΔABC are extended to points P and Q respectively, ∠PBC < ∠QCB, we have proved that AC > AB ☛ Related Questions: Math worksheets and visual curriculum	{"url":"https://www.cuemath.com/ncert-solutions/in-fig-748-sides-ab-and-ac-of-abc-are-extended-to-points-p-and-q-respectively-also-pbc-qcb-show-that-ac-ab/","timestamp":"2024-11-02T17:49:43Z","content_type":"text/html","content_length":"198807","record_id":"<urn:uuid:0b90319b-cf14-45c6-a47e-39ab54c880a7>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00772.warc.gz"}
2Ax’s mid-range driver I own a pair of 2Ax’s and was wondering about the quality of the mid-range driver. I have gone through two of these derivers recently and each have exhibited similar forms of distortion - what sounds like cone break-up. It is not easily noticed and only occurs under certain conditions. Would any one like to comment on the 2Ax’s mid-range design and their experiences with these drivers? The conditions for the “break-up” distortion occur at higher frequency and usually under complex musical passages – e.g. certain massed string symphonic passages. The crossover caps have been replaced as have the control potentiometers- with L-pads, as per the AR 3 restoration manual. I have used different amps and reversed the sides for each speaker. It is, as far as I can tell, the mid-range drivers. I have been careful not to over drive the speaker and used a 55-watt amp and an 80 watt one as well. But with the L-pad all the way up and the preamp volume set at 11:00 there is distortion in the two drivers that I have used in the same speaker. My other speaker is fine. Thank you. I own a pair of 2Ax’s and was wondering about the quality of the mid-range driver. I have gone through two of these derivers recently and each have exhibited similar forms of distortion - what sounds like cone break-up. It is not easily noticed and only occurs under certain conditions. Would any one like to comment on the 2Ax’s mid-range design and their experiences with these drivers? The conditions for the “break-up” distortion occur at higher frequency and usually under complex musical passages – e.g. certain massed string symphonic passages. The crossover caps have been replaced as have the control potentiometers- with L-pads, as per the AR 3 restoration manual. I have used different amps and reversed the sides for each speaker. It is, as far as I can tell, the mid-range drivers. I have been careful not to over drive the speaker and used a 55-watt amp and an 80 watt one as well. But with the L-pad all the way up and the preamp volume set at 11:00 there is distortion in the two drivers that I have used in the same speaker. My other speaker is fine. Thank you. I rescued and restored a pair of AR2axs just as they were headed for the curb. One of them had the same problem. I also rescued a pair of AR2as which have the old jensen 5" drivers I didn't like and the cones are badly warped. At first I thought it was the tweeter in the AR2ax was bad but it turned out to be the midrange. I bought three on e-bay. I've only fully restored the AR2axs so far and the replacement driver turned out to work just fine, surprisingly well in fact. The tweeters seem to function perfectly. I found that I needed to set the tweeters to their max output and the midranges to slightly less than half output to get the most accurate sounding results. I then equalized it by ear and the results come very close to what you would expect from combining the FRs of the drivers shown in the library, a boost in the deep bass and in the treble. There's also a slight cut required in the midrange. When the FR is flattened this way, I've been amazed at how well the speaker perform, much better than I'd ever heard them in the past when they were current. Very accurate, very clear, excellent high frequency dispersion with no perceived changed to spectral balance at 45 degrees off axis, the maximum I can go with the speakers in the corners of a room toed in 45 degrees. Bass is excellent too but I did damage one AR2a 10" driver so it's not nearly as robust as the 12" version. BTW, the edge to edge diameter across the inside of the outer suspension is 7" which is exactly the same as for KLH model 6. I haven't applied any sealant to the KLH 6 cloth suspensions and at the moment the AR2ax strikes me as having better bass between the I'll probably replace the Jensen drivers in the AR2as with the AR2ax midranges but I don't know what I will do about tweeters. The output from the 8 ohm version of the AR3 driver seems very weak. That's always the hard part in restoring these, the tweeters. I've got all kinds including some Tonegen ribbons. Try to find some of these AR2ax midranges on ebay, there seemed to be plenty of them when I was looking around a year ago. Othewise you can try a 3 1/2 " midrange driver from Parts Express, they aren't expensive but it won't be an exact match. Addendum to my last post: Please see the attached schematic for the AR 2ax and note the 2-ohm resistor in series with the mid- range driver. Also please see pages 10 and pages 14, and 15th of the AR 3a “Restoring the AR-3a” manual. The crossovers for the two speakers are of course different. The 2ax’s speakers that I bought had the AR 3a L-pad, parallel 25 ohm mid and high driver resistors installed but is missing the 2-ohm series resistor on the mid. Should I remove the 25-ohm resistors and/or reinstall the 2-ohm resistor back in series with the mid- range driver, leaving the 8-ohm L-Pad? Any thoughts? Thank much. I rescued and restored a pair of AR2axs just as they were headed for the curb. One of them had the same problem. I also rescued a pair of AR2as which have the old jensen 5" drivers I didn't like and the cones are badly warped. At first I thought it was the tweeter in the AR2ax was bad but it turned out to be the midrange. I bought three on e-bay. I've only fully restored the AR2axs so far and the replacement driver turned out to work just fine, surprisingly well in fact. The tweeters seem to function perfectly. I found that I needed to set the tweeters to their max output and the midranges to slightly less than half output to get the most accurate sounding results. I then equalized it by ear and the results come very close to what you would expect from combining the FRs of the drivers shown in the library, a boost in the deep bass and in the treble. There's also a slight cut required in the midrange. When the FR is flattened this way, I've been amazed at how well the speaker perform, much better than I'd ever heard them in the past when they were current. Very accurate, very clear, excellent high frequency dispersion with no perceived changed to spectral balance at 45 degrees off axis, the maximum I can go with the speakers in the corners of a room toed in 45 degrees. Bass is excellent too but I did damage one AR2a 10" driver so it's not nearly as robust as the 12" version. BTW, the edge to edge diameter across the inside of the outer suspension is 7" which is exactly the same as for KLH model 6. I haven't applied any sealant to the KLH 6 cloth suspensions and at the moment the AR2ax strikes me as having better bass between the two. I'll probably replace the Jensen drivers in the AR2as with the AR2ax midranges but I don't know what I will do about tweeters. The output from the 8 ohm version of the AR3 driver seems very weak. That's always the hard part in restoring these, the tweeters. I've got all kinds including some Tonegen ribbons. Try to find some of these AR2ax midranges on ebay, there seemed to be plenty of them when I was looking around a year ago. Othewise you can try a 3 1/2 " midrange driver from Parts Express, they aren't expensive but it won't be an exact match. Thank you for the information. I have gone through three drivers now. Looking into my situation a little deeper I found my AR 2ax's are missing the 2-ohm resistor in series with the mid- range driver. Also – they may have the wrong AR 3ax’s 25-ohm parallel resistors applied – as per the restoring manual. Cheers - Dan Addendum to my last post: Please see the attached schematic for the AR 2ax and note the 2-ohm resistor in series with the mid- range driver. Also please see pages 10 and pages 14, and 15th of the AR 3a “Restoring the AR-3a” manual. The crossovers for the two speakers are of course different. The 2ax’s speakers that I bought had the AR 3a L-pad, parallel 25 ohm mid and high driver resistors installed but is missing the 2-ohm series resistor on the mid. Should I remove the 25-ohm resistors and/or reinstall the 2-ohm resistor back in series with the mid- range driver, leaving the 8-ohm L-Pad? Any thoughts? Thank much. If your speakers are using l-pads (with or without 25 ohm resistors in parallel with mids and tweeters), be aware that when the l-pad is set to maximum, all (level control) resistance is removed from the circuit. This is different than when an original pot is set to max. The original pot set to maximum or fully "increased" retains 15 to 16 ohms in parallel with the driver, resulting in less driver output than a maxed l-pad provides. The bottom line is that the 2ax mid and tweeter can easily be pushed beyond their limits with l-pads set to maximum at high volume. They were never meant to be driven that hard. That 2 ohm resistor in the schematic you are referring to is not part of the original design...and will not address your problem. You would be better off turning your mid level control down a bit. In my experience most folks don't have the 2ax mid level controls (especially when they are l-pads) set to maximum. If your speakers are using l-pads (with or without 25 ohm resistors in parallel with mids and tweeters), be aware that when the l-pad is set to maximum, all (level control) resistance is removed from the circuit. This is different than when an original pot is set to max. The original pot set to maximum or fully "increased" retains 15 to 16 ohms in parallel with the driver, resulting in less driver output than a maxed l-pad provides. The bottom line is that the 2ax mid and tweeter can easily be pushed beyond their limits with l-pads set to maximum at high volume. They were never meant to be driven that hard. That 2 ohm resistor in the schematic you are referring to is not part of the original design...and will not address your problem. You would be better off turning your mid level control down a bit. In my experience most folks don't have the 2ax mid level controls (especially when they are l-pads) set to maximum. Thank you Roy. In order to address the distortion I just adjusted one mid-driver l-pads to the 8 o’clock position – way down. The other speaker, in the other cabinet, does not exhibit the same problem in any of the L-Pad positions. There may be something else a little funky about my speaker. I have used three mid drivers in the same unit and each exhibit the same issue in the same cabinet. L-pad problems? Thank you Roy. In order to address the distortion I just adjusted one mid-driver l-pads to the 8 o’clock position – way down. The other speaker, in the other cabinet, does not exhibit the same problem in any of the L-Pad positions. There may be something else a little funky about my speaker. I have used three mid drivers in the same unit and each exhibit the same issue in the same cabinet. L-pad problems? I have never heard of an l-pad causing a problem like that unless it is improperly wired. If it is not the wiring, it most likely has something to do with 40 year old drivers. Just to be sure it is not something other than a speaker issue, switch the left and right leads to see if the problem follows the speaker or the amp channel. Thank you Roy. In order to address the distortion I just adjusted one mid-driver l-pads to the 8 o’clock position – way down. The other speaker, in the other cabinet, does not exhibit the same problem in any of the L-Pad positions. There may be something else a little funky about my speaker. I have used three mid drivers in the same unit and each exhibit the same issue in the same cabinet. L-pad problems? The AR2Ax midrange driver is protected by the 6 mfd series wired capacitor which acts as a 6db per octave high pass filter. It has no high frequency cutoff device. The potentiometer further reduces current relative to the output of the woofer. The midrange pot is used as a voltage divider. The speaker will play moderately loudly without damage but it was not engineered to play rock music at deafening levels. Since I had the identical experience I assume that some of these drivers have either been abused due to relatively modest power handling capability or may have deteriorated. The good news is that there were a lot of them made, they are cheap and they are readily available used on e-bay. They should cost no more than about $10 to $15 each. I replaced the caps with Parts Express non polarized that cost a few cents each and they work just fine. I looked at the driver and there seemed nothing particularly special about it. If you buy a substitute, buy them in pairs so both speakers will match. Be sure to get one that is 8 ohms and designated as a midrange with a sealed back to prevent the woofer's pressure wave from reaching the cone. The AR2Ax midrange driver is protected by the 6 mfd series wired capacitor which acts as a 6db per octave high pass filter. It has no high frequency cutoff device. The potentiometer further reduces current relative to the output of the woofer. The midrange pot is used as a voltage divider. The speaker will play moderately loudly without damage but it was not engineered to play rock music at deafening levels. Since I had the identical experience I assume that some of these drivers have either been abused due to relatively modest power handling capability or may have deteriorated. The good news is that there were a lot of them made, they are cheap and they are readily available used on e-bay. They should cost no more than about $10 to $15 each. I replaced the caps with Parts Express non polarized that cost a few cents each and they work just fine. I looked at the driver and there seemed nothing particularly special about it. If you buy a substitute, buy them in pairs so both speakers will match. Be sure to get one that is 8 ohms and designated as a midrange with a sealed back to prevent the woofer's pressure wave from reaching the cone. Thank you. It was indeed the driver(s). I went through six of them with a local loudspeaker technician to get one good pair. Most came from eBay and each seller indicated that they were in “good” working condition. The word “good” is clearly subjective. If there is a suitable drop in replacement it would be good to hear of it. The measurements taken on my good drivers indicated a relatively flat response across the range used without a “shelf”. Perhaps this is part of the AR 2ax allure? Thank you. It was indeed the driver(s). I went through six of them with a local loudspeaker technician to get one good pair. Most came from eBay and each seller indicated that they were in “good” working condition. The word “good” is clearly subjective. If there is a suitable drop in replacement it would be good to hear of it. The measurements taken on my good drivers indicated a relatively flat response across the range used without a “shelf”. Perhaps this is part of the AR 2ax allure? Glad you found a satisfactory replacement. Take good care of it, and try to refrain from maxing the l-pads at higher listening levels. There are no drop-in replacements. • 2 years later... Decided to revive this thread as I would appreciate advice on which way to proceed with my AR2ax restoration. One of the cabinets has the midrange metal cover & F/glass missing so my dilemma is: Do I remove the other cover to match the two speakers acoustically? Can I fashion some sort of cover to replicate the result? ( not generally avail.w/out midrange driver) From my research it seems to protect the driver from excessive movement & damage when played at high volume. Any guidance would be welcomed. Decided to revive this thread as I would appreciate advice on which way to proceed with my AR2ax restoration. One of the cabinets has the midrange metal cover & F/glass missing so my dilemma is: Do I remove the other cover to match the two speakers acoustically? Can I fashion some sort of cover to replicate the result? ( not generally avail.w/out midrange driver) From my research it seems to protect the driver from excessive movement & damage when played at high volume. Any guidance would be welcomed. If you want an original 2ax you need a mid with the fiberglass pad and grille. They are fairly common, and often show up on Ebay at reasonable prices. You can also post a request in the For Sale/Wanted section of the forum. If you are able to acquire a dead one, you could transfer the grille and fiberglass pad to your naked one. If you are not in the US, the above suggestions may not be as useful in terms of availability and cost. Unforunately I'm from Australia so would be over $100 inc. delivery. Will have to consider the benefit - does it make much of a difference sonically, leaving aside the aesthetics? Unforunately I'm from Australia so would be over $100 inc. delivery. Will have to consider the benefit - does it make much of a difference sonically, leaving aside the aesthetics? I have never tried it with 2ax mids, but the AR dome mids do not sound bad with the grilles and pads removed. There will be at least a slight sonic difference...but the sound will probably be acceptable. I've been tempted to jump on this NOS pair of AR-2AX mid's. This topic is now archived and is closed to further replies.	{"url":"https://community.classicspeakerpages.net/topic/6037-2ax%E2%80%99s-mid-range-driver/","timestamp":"2024-11-06T08:45:19Z","content_type":"text/html","content_length":"250427","record_id":"<urn:uuid:09d48f37-a5e6-4883-af01-239474aeb73f>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00646.warc.gz"}
Incandescent Bulb Power Consumption Calculation \| Electrical4u Incandescent Bulb Power Consumption Calculation Incandescent Bulb Power Consumption Calculator: Enter the capacity of the incandescent bulb, operating hours per day, and the rate per unit and then press the calculate button to get the power consumption of the bulb. By default, you can get the power consumption of 60Watts bulb. Incandescent Bulb Power Consumption: Incandescent bulbs were the most used lamps in earlier ’90s. The lamps produce the light by radiating the filament. The filament is the object used inside the lamp which connects the higher and lower potential terminals. When the filament is heated in the vacuum which emits the lights. The process of heating will be done by passing the electric current through the filament. Mostly Tungsten is used as filament since it has a high melting point. The lifetime of the incandescent lamp will be 1000 – 1200 hours. The bulb provides only 14 to 17 lumens per watt. By average the incandescent lamp gives only 14 lumens per watt. I mean, the bulb consumes 1 watt per hour for giving 14 lumens. Also, while decreasing the voltage by 10%, the incandescent bulb reduces its lighting efficiency by 27%. Get a clear vision, you need minimum of 683 lumens. Hence to get this, you need to use a minimum 50 Watts incandescent lamps. For the living room, you need at least 1673 Lumens for the room size of 10 x 10 x 10 feet. Therefore, you have to use 119.5 Watts incandescent lamp which will be the combination of 2 x 60 Watts. Incandescent Bulb Power Consumption: Energy Per day: The total consumption of the all incandescent lamp Cost Per Hour: The total electricity bill for one hour. It calculates the cost of using the incandescent lamp for one hour. Cost Per Day: The total electricity bill for one day with operating hour while using the incandescent lamp. Cost per month: Total electricity bill per month with operating hour when using an incandescent bulb. Cost Per Year: Total bill for the consumption per year. Let’s calculate the power consumption of 60 Watts incandescent bulbs. 60 Watts bulb Power Consumption Consumption Total (Wh) kWh Hourly 60 0.06 Day 1440 1.44 Month 1800 1.8 Yearly 525600 525.6 How Does A Light Bulb Work? Video LEAVE A REPLY Cancel reply	{"url":"https://www.electrical4u.net/energy-calculation/incandescent-bulb-power-consumption-calculation/","timestamp":"2024-11-02T05:44:26Z","content_type":"text/html","content_length":"117246","record_id":"<urn:uuid:47a163d5-39ee-4b07-8fed-f4d139dfbe97>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00893.warc.gz"}
American Mathematical Society Subelliptic Cordes estimates HTML articles powered by AMS MathViewer by András Domokos and Juan J. Manfredi Proc. Amer. Math. Soc. 133 (2005), 1047-1056 DOI: https://doi.org/10.1090/S0002-9939-04-07819-0 Published electronically: November 19, 2004 PDF \| Request permission We prove Cordes type estimates for subelliptic linear partial differential operators in non-divergence form with measurable coefficients in the Heisenberg group. As an application we establish interior horizontal $W^{2,2}$-regularity for p-harmonic functions in the Heisenberg group ${\mathbb H}^1$ for the range $\frac {\sqrt {17}-1}{2} \leq p < \frac {5+\sqrt {5}}{2}$. References • Luca Capogna, Donatella Danielli, and Nicola Garofalo, An embedding theorem and the Harnack inequality for nonlinear subelliptic equations, Comm. Partial Differential Equations 18 (1993), no. 9-10, 1765–1794. MR 1239930, DOI 10.1080/03605309308820992 • H. O. Cordes, Zero order a priori estimates for solutions of elliptic differential equations, Proc. Sympos. Pure Math., Vol. IV, American Mathematical Society, Providence, R.I., 1961, pp. 157–166. MR 0146511 • A. Domokos, Differentiability of solutions for the non-degenerate $p$-Laplacian in the Heisenberg group, J. Differential Equations 204(2004), 439-470. • A. Domokos and Juan J. Manfredi, $C^{1,\alpha }$-regularity for p-harmonic functions in the Heisenberg group for $p$ near $2$, to appear in The $p$-harmonic equation and recent advances in analysis, Contemporary Mathematics, editor Pietro Poggi-Corradini. • David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0 • Guozhen Lu, Polynomials, higher order Sobolev extension theorems and interpolation inequalities on weighted Folland-Stein spaces on stratified groups, Acta Math. Sin. (Engl. Ser.) 16 (2000), no. 3, 405–444. MR 1787096, DOI 10.1007/PL00011552 • S. Marchi, $C^{1,\alpha }$ local regularity for the solutions of the $p$-Laplacian on the Heisenberg group for $2\leq p<1+\sqrt 5$, Z. Anal. Anwendungen 20 (2001), no. 3, 617–636. MR 1863937, DOI • Silvana Marchi, $C^{1,\alpha }$ local regularity for the solutions of the $p$-Laplacian on the Heisenberg group. The case $1+\frac 1{\sqrt 5}<p\leq 2$, Comment. Math. Univ. Carolin. 44 (2003), no. 1, 33–56. MR 2045844 • Duy-Minh Nhieu, Extension of Sobolev spaces on the Heisenberg group, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), no. 12, 1559–1564 (English, with English and French summaries). MR 1367807 • Robert S. Strichartz, $L^p$ harmonic analysis and Radon transforms on the Heisenberg group, J. Funct. Anal. 96 (1991), no. 2, 350–406. MR 1101262, DOI 10.1016/0022-1236(91)90066-E • Giorgio Talenti, Sopra una classe di equazioni ellittiche a coefficienti misurabili, Ann. Mat. Pura Appl. (4) 69 (1965), 285–304 (Italian). MR 201816, DOI 10.1007/BF02414375 Similar Articles • Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 35H20, 35J70 • Retrieve articles in all journals with MSC (2000): 35H20, 35J70 Bibliographic Information • András Domokos • Affiliation: Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, Pennsylvania 15260 • Address at time of publication: Department of Mathematics and Statistics, California State University Sacramento, 6000 J Street, Sacramento, California 95819 • Email: domokos@csus.edu • Juan J. Manfredi • Affiliation: Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, Pennsylvania 15260 • MR Author ID: 205679 • Email: manfredi@pitt.edu • Received by editor(s): August 13, 2003 • Published electronically: November 19, 2004 • Additional Notes: The authors were partially supported by NSF award DMS-0100107 • Communicated by: David S. Tartakoff • © Copyright 2004 American Mathematical Society The copyright for this article reverts to public domain 28 years after publication. • Journal: Proc. Amer. Math. Soc. 133 (2005), 1047-1056 • MSC (2000): Primary 35H20, 35J70 • DOI: https://doi.org/10.1090/S0002-9939-04-07819-0 • MathSciNet review: 2117205	{"url":"https://www.ams.org/journals/proc/2005-133-04/S0002-9939-04-07819-0/?active=current","timestamp":"2024-11-15T03:26:35Z","content_type":"text/html","content_length":"64421","record_id":"<urn:uuid:e428f875-f596-40cd-a463-2a427a711f4c>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00888.warc.gz"}
Percentile calculator A percentile calculator is a tool that calculates the value below which a certain percentage of observations fall in a given set of data. Here are the steps to use a percentile calculator: 1. Gather the data: You need a set of data for which you want to calculate the percentile. For example, the test scores of a group of students. 2. Sort the data: Arrange the data in ascending or descending order. For example, arrange the test scores from lowest to highest. 3. Identify the position of the percentile: Determine the position of the percentile you want to calculate. For example, if you want to find the 75th percentile, you need to identify the position of the observation that is greater than or equal to 75% of the other observations. 4. Calculate the percentile: Use the following formula to calculate the percentile: percentile = (p / 100) x (n + 1)where: □ p is the desired percentile (e.g. 75 for the 75th percentile) □ n is the total number of observations in the data set This formula calculates the position of the observation that corresponds to the desired percentile. 5. Identify the value at the calculated position: The value at the position calculated in step 4 is the percentile value. For example, if the position calculated in step 4 is 10, then the value at position 10 in the sorted data is the 75th percentile. Using a percentile calculator can be useful in many situations, such as evaluating performance or analyzing data distributions. The formula to calculate the pth percentile of a data set is: P = (p / 100) x (n + 1) • P is the position of the pth percentile in the sorted data set • p is the desired percentile (e.g. 75 for the 75th percentile) • n is the total number of observations in the data set To calculate the value of the pth percentile, you need to find the observation that corresponds to the position P in the sorted data set. If P is an integer, then the observation at position P is the pth percentile. If P is not an integer, then you need to interpolate between the two nearest observations to get an estimate of the pth percentile. For example, let’s say you have a data set of 10 observations and you want to find the 75th percentile. Here’s how you can use the formula: P = (75 / 100) x (10 + 1) = 7.5 Since 7.5 is not an integer, you need to interpolate between the values at positions 7 and 8. Let’s assume the sorted data set is: 2, 5, 7, 10, 12, 13, 15, 18, 20, 22 The value at position 7 is 13 and the value at position 8 is 15. To interpolate, you can use the following formula: Estimate of the pth percentile = value at position P_lower + (P – P_lower) x (value at position P_upper – value at position P_lower) / (P_upper – P_lower) • P_lower is the lower integer value of P (i.e. 7 in this case) • P_upper is the upper integer value of P (i.e. 8 in this case) Using this formula, the estimate of the 75th percentile is: Estimate of the 75th percentile = 13 + (0.5) x (15 – 13) / (8 – 7) = 13.5 Therefore, the estimated 75th percentile value for this data set is 13.5.	{"url":"https://calculator3.com/percentile-calculator/","timestamp":"2024-11-05T16:19:57Z","content_type":"text/html","content_length":"59810","record_id":"<urn:uuid:56f03985-fd02-40cc-b04f-0bc994ddb80b>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00852.warc.gz"}
The Intentionality of Mathematical Abstraction Although Edmund Husserl was trained as a mathematicians and wrote about mathematical thinking in most of his published works, little attention has been paid to his late phenomenological philosophy of mathematics. Among those who have studied Husserl's philosophy of mathematics, there is a widespread impression that his account is either intractably intuitionistic or else proven unworkable by later developments in mathematical logic. In this dissertation, I develop and defend a phenomenological description of the intentionality of mathematical abstraction aimed at knowing based on Husserl's account. A phenomenological approach to the philosophy of logic and mathematics asks us to describe what must feature in our orientation to the world and implicit norms of recognition for us to have an experience of coming to see mathematical truths for themselves. I describe the core of mathematical thinking as eidetic intuition—a disciplined imaginative achievement—founded on systematic disregard for our ordinary concern with truth, yielding insight into the essential features belonging to categoriality as such, the same categoriality that structures everyday perception. I defend an interpretation of Husserl's key concept of definiteness that shows how it can bridge inquiry into pure, algebraic, possibly intuitionistic mathematics with inquiry into realistic, classical mathematics. Then, I develop an intentional analysis of diagonalizing, an achievement that can be performed both in and outside the algebraic attitude proper to pure mathematics. This leads to the conclusion that Husserl's reliance on definiteness does not put a phenomenological account of mathematics at odds with Gödel's incompleteness theorems. Instead, by paying attention to the intentional structures involved, we can understand both the purely mathematical content of the theorems and their material application found in the logician's study of formal theories and their objects. Along the way, I illustrate many insights a phenomenological approach to mathematics can provide, helping us understand some features of mathematical theorizing that make it such a distinctive discipline. In particular, we we can understand reasons for peculiar qualities of mathematical innovation, foundations, interderivability, and proof, without needing to resolve debates that have dominated traditional philosophical approaches to mathematics. Subject Area Philosophy\|Mathematics\|Philosophy of Science Recommended Citation Reppert, Justin, "The Intentionality of Mathematical Abstraction" (2021). ETD Collection for Fordham University. AAI28496512.	{"url":"https://research.library.fordham.edu/dissertations/AAI28496512/","timestamp":"2024-11-08T11:43:52Z","content_type":"text/html","content_length":"36667","record_id":"<urn:uuid:c5797154-dfea-4475-bebb-e1412a21864b>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00234.warc.gz"}
High Energy Physics: Implementing $SU(2)$ Lattice Gauge Theory Trotter Circuits¶ Quantum computers have the ability to tackle problems beyond the reach of classical systems. One such application lies in the realm of quantum field theories, where lattice gauge theories provide a rigorous framework to derive non-perturbative results. Before diving in, you may find it helpful to review the Definitions at the end of this article if you’re unfamiliar with high energy physics. Lattice gauge theory (LGT) is a vital computational tool in quantum field theory, enabling the acquisition of rigorous results without relying on perturbation theory. It’s particularly significant in the context of quantum chromodynamics (QCD), where lattice computations have been instrumental in uncovering new properties of known hadrons, evaluating the likelihood of undiscovered hadrons, and providing critical insights for the search for new physics beyond the standard model. Typically, LGT studies are conducted in Euclidean spacetime, where time is treated as an imaginary component. This approach is effective for many observables, but it limits the ability to analyze real-time dynamics. However, a Hamiltonian formulation in real-time could overcome this limitation, given adequate computing resources. This is where the potential of future quantum computers becomes particularly intriguing. Quantum computers could handle and evolve quantum states far beyond the capabilities of classical computers, thus offering a promising avenue for exploring real-time dynamics within LGT. The paper Self-mitigating Trotter circuits for SU(2) lattice gauge theory on a quantum computer by Rahman, Lewis, Mendicelli, and Powell delves into the implementation of SU(2) lattice gauge theory in Minkowski spacetime using quantum computers, showcasing the real-time evolution of excitations across a lattice. While the theoretical foundations laid out in the paper are robust, practical implementation on contemporary quantum hardware requires specialized techniques and considerations. This knowledge article, complete with detailed code, explores the implementation of methods described in this paper on the Quantinuum H-Series devices. Utilizing TKET, we detail the process of defining the SU(2) 2-Plaquette (2-qubit) Model Hamiltonian, setting up quantum circuits, and employing Trotterization to simulate the quantum dynamics of the system. For a single Trotter step, we simulate the ideal execution of a quantum circuit using a statevector simulator. The two Trotter step case is executed on the Quantinuum H1-2 emulator, and we also demonstrate an example of symbolic circuit construction for these circuits. The final section details the implementation of the SU(2) 5-Plaquette (five-qubit) Model Hamiltonian using the Qiskit library. This notebook was generated using pytket=1.23.0, pytket-quantinuum=0.26.0, and pytket-qiskit=0.47.0. First, we import the libraries and modules we will use: • Pauli utilities and QubitPauliOperator are used for Hamiltonian representation. • Circuit utilities from pytket are used for constructing and analyzing quantum circuits. • Additional utilities are used for generating term sequences and gate statistics. • Standard libraries are used for numerical operations and visualization. from pytket.pauli import Pauli, QubitPauliString from pytket.utils import QubitPauliOperator from pytket.circuit import Circuit, Qubit, CircBox from pytket.utils import gen_term_sequence_circuit from pytket.utils.stats import gate_counts from pytket import OpType import numpy as np from scipy.linalg import expm import matplotlib.pyplot as plt # Uncomment to verify installation of pytket extensions: pytket-quantinuum and pytket-qiskit. # %pip show pytket pytket-quantinuum pytket-qiskit Note: Plotting circuits requires an internet connection. An alternative for displaying circuits is the pytket-offline-display package for rendering circuits offline with no internet connection, for details see here. Hamiltonian Formulation¶ In quantum mechanics, the Hamiltonian operator plays a pivotal role as it represents the total energy of a system and governs how quantum states evolve with time. The time evolution of a quantum state, denoted as $\|ψ(t)⟩$, is described by the Schrödinger equation: \[ i \hbar \frac{{d\|\psi(t)\rangle}}{{dt}} = \hat{H}\|\psi(t)\rangle \] where $i$ stands for the imaginary unit, $\hbar$ denotes the reduced Planck constant, and $\hat{H}$ signifies the Hamiltonian operator of the system. In the realm of quantum simulations, particularly those involving lattice gauge theories, the Hamiltonian is indispensable. It not only elucidates the system’s dynamics but also forms the foundation for creating quantum circuits that simulate this evolution. SU(2) 2-Plaquette Model Hamiltonian¶ The SU(2) 2-plaquette model offers a streamlined representation of lattice gauge theories. The associated Hamiltonian for this model encapsulates the core dynamics of the system, making it apt for simulation on quantum hardware. The Hamiltonian’s general expression is: \[ \hat{H} = \sum_{i,j} J_{ij} \sigma_i^a \sigma_j^a + \sum_i h_i \sigma_i^z \] where $J_{ij}$ represents interaction coefficients between lattice sites $i$ and $j$, $\sigma_i^a$ and $\sigma_j^a$ are the Pauli matrices for site $i$ or $j$, with $a$ taking values $x$, $y$ or $z$, and $h_i$ are the external field terms for site $i$. Here we show the 2-qubit Hamiltonian for the SU(2) 2-plaquette model: \[\begin{split} \begin{aligned} \frac{2}{g^2} H=\left(\begin{array}{cccc} 0 & -2 x & -2 x & 0 \\ -2 x & 3 & 0 & -x \\ -2 x & 0 & 3 & -x \\ 0 & -x & -x & \frac{9}{2} \end{array}\right)= & \frac{3}{8}\ left(7 I_0 \otimes I_1-3 Z_0 \otimes I_1-Z_0 \otimes Z_1-3 I_0 \otimes Z_1\right) \\ & -\frac{x}{2}\left(3 X_0 \otimes I_1+X_0 \otimes Z_1\right)-\frac{x}{2}\left(3 I_0 \otimes X_1+Z_0 \otimes X_1\ right) \end{aligned} \end{split}\] • The matrix form represents the action of the Hamiltonian on a 2-qubit system. • The expanded form shows the Hamiltonian in terms of tensor products of Pauli matrices, providing insight into the system’s dynamics and interactions. This Hamiltonian encapsulates the dynamics of the system and serves as the foundation for subsequent quantum simulations. Specifically, the terms associated with Pauli $Z$ matrices, like $Z_0$ and $Z_1$, capture the on-site energies and interactions between the qubits. The terms involving the Pauli $X$ matrices, such as $X_0$ and $X_1$, represent the interactions influenced by external fields parameterized by $x$. This Hamiltonian provides a framework for analyzing and simulating the dynamics of the SU(2) 2-plaquette model, making it a cornerstone for the quantum simulations presented in the paper and this notebook. Implementation of the SU(2) 2-Plaquette Model Hamiltonian¶ The Hamiltonian for the SU(2) 2-plaquette model is implemented in pytket in this notebook using the QubitPauliOperator to write the operator on qubits and QubitPauliString to represent the individual Pauli string terms. Here we use $x = 2$. hamiltonian = QubitPauliOperator( QubitPauliString([], []): +7 * 3 / 8, QubitPauliString([Qubit(0)], [Pauli.Z]): -3 * 3 / 8, QubitPauliString([Qubit(0), Qubit(1)], [Pauli.Z, Pauli.Z]): -3 / 8, QubitPauliString([Qubit(1)], [Pauli.Z]): -3 * 3 / 8, QubitPauliString([Qubit(0)], [Pauli.X]): -3, QubitPauliString([Qubit(0), Qubit(1)], [Pauli.X, Pauli.Z]): -1, QubitPauliString([Qubit(1)], [Pauli.X]): -3, QubitPauliString([Qubit(0), Qubit(1)], [Pauli.Z, Pauli.X]): -1, We can display the QubitPauliOperator representation of the Hamiltonian: {(): 2.62500000000000, (Zq[0]): -1.12500000000000, (Zq[0], Zq[1]): -0.375000000000000, (Zq[1]): -1.12500000000000, (Xq[0]): -3, (Xq[0], Zq[1]): -1, (Xq[1]): -3, (Zq[0], Xq[1]): -1} You can also visualize the non-zero elements of the Hamiltonian matrix using a spy plot from the matplotlib library , shedding light on its non-zero components and overall structure: <matplotlib.lines.Line2D at 0x1424678d0> Matrix Formulation¶ We can also convert the Hamiltonian QubitPauliOperator representation to a dense matrix and extract the Hamiltonian matrix elements. This transformation caters to computational tasks and subsequent quantum simulations. hm = hamiltonian.to_sparse_matrix(2).todense() matrix([[ 0. , -4. , -4. , 0. ], [-4. , 3. , 0. , -2. ], [-4. , 0. , 3. , -2. ], [ 0. , -2. , -2. , 4.5]]) Time Evolution Operator¶ To simulate the time evolution of the quantum system, we compute the matrix exponential of the Hamiltonian matrix. This process represents how the quantum state evolves over a specific time interval. For instance, a factor like $0.1$ can be interpreted as a discrete time step in the system’s evolution. In our implementation, we use the expm function to calculate the matrix exponential of the Hamiltonian for a time step of $0.1$. The time evolution operator ( U(t) ) is expressed as: For a time step of $0.1$, this operator becomes: \[ U(0.1) = e^{-i H \times 0.1} \] This calculated operator, exp_hm, will then be applied to simulate the quantum state’s evolution over the time step. # Compute the matrix exponential of the Hamiltonian for the time step of 0.1 exp_hm = expm(-1j * hm * 0.1) array([[ 0.84639558+0.0152074j , 0.05508436+0.36836211j, 0.05508436+0.36836211j, -0.07461882+0.01911594j], [ 0.05508436+0.36836211j, 0.8615547 -0.27351321j, -0.09378179+0.022007j , 0.06951527+0.17342834j], [ 0.05508436+0.36836211j, -0.09378179+0.022007j , 0.8615547 -0.27351321j, 0.06951527+0.17342834j], [-0.07461882+0.01911594j, 0.06951527+0.17342834j, 0.06951527+0.17342834j, 0.8648741 -0.41980935j]]) Circuit Construction & Visualization¶ After calculating the time evolution operator, the next logical step is to integrate this operator into a quantum circuit. This allows for direct simulations or runs on quantum hardware or simulator. The pytket library provides tools to seamlessly achieve this. By wrapping our time evolution operator in a 2-qubit unitary box, we can then incorporate it into a quantum circuit, setting the stage for subsequent quantum simulations or analyses. # Creating a 2-qubit unitary box that represents the time evolution operator: from pytket.circuit import Unitary2qBox u2box = Unitary2qBox(exp_hm) # Initializing a 2-qubit circuit and embedding the time evolution operator into it: circ = Circuit(2) circ.add_unitary2qbox(u2box, 0, 1) [Unitary2qBox q[0], q[1]; ] Retrieving the unitary matrix representation of the quantum circuit: array([[ 0.84639558+0.0152074j , 0.05508436+0.36836211j, 0.05508436+0.36836211j, -0.07461882+0.01911594j], [ 0.05508436+0.36836211j, 0.8615547 -0.27351321j, -0.09378179+0.022007j , 0.06951527+0.17342834j], [ 0.05508436+0.36836211j, -0.09378179+0.022007j , 0.8615547 -0.27351321j, 0.06951527+0.17342834j], [-0.07461882+0.01911594j, 0.06951527+0.17342834j, 0.06951527+0.17342834j, 0.8648741 -0.41980935j]]) And obtaining the resulting quantum state after applying the circuit (from a default initial state): array([ 0.84639558+0.0152074j , 0.05508436+0.36836211j, 0.05508436+0.36836211j, -0.07461882+0.01911594j]) Recognizing the constraints of real quantum hardware, the constructed circuit is decomposed into a set of elementary gates, ensuring its adaptability and compatibility. Specifically, in quantum circuit design, ‘boxed’ operations encapsulate complex actions. To standardize our circuit for broader applications, we’ll replace these boxed operations with sequences of elementary gates using pytket’s DecomposeBoxes. # Importing the necessary tool for gate decomposition: from pytket.passes import DecomposeBoxes # Decomposing custom gates in the circuit to ensure compatibility with quantum hardware: We can also visualize the quantum circuit using pytket’s rendering tool for Jupyter: from pytket.circuit.display import render_circuit_jupyter Gate Analysis¶ The quantum circuit, post-decomposition, is dissected to provide a breakdown of its comprising gate types. This analysis gives valuable insights into the circuit’s intricacy and underlying structure. Insights into the composition and complexity of our circuit can be obtained with the following command: Counter({<OpType.TK1: 40>: 4, <OpType.X: 23>: 2, <OpType.Z: 22>: 2, <OpType.Y: 24>: 2, <OpType.TK2: 41>: 1}) The breakdown of gate occurrences in our quantum circuit is as follows: • TK1 gates: 4 • X gates: 2 • TK2 gates: 1 • Z gates: 2 • Y gates: 2 This summary serve as a valuable reference when we delve into the topics of recompilation and optimization for execution on H-series devices. Note: See pytket.circuit.OpType for the mathematical descriptions of TK1 and TK2 gates. Time evolution circuits¶ The given Hamiltonian, $-i H t$, needs to be discretized for simulation on a quantum computer. This is where the Trotter-Suzuki approximation comes in handy, allowing us to approximate the time evolution operator $e^{-i H t}$. \[ e^{-i H t}=e^{-i\left(\frac{3}{8}\left(7-3 Z_0-Z_0 Z_1-3 Z_1\right)-\frac{x}{2}\left(3+Z_1\right) X_0-\frac{x}{2}\left(3+Z_0\right) X_1\right) t} \] First-Order Trotter Step¶ In the first-order Trotter step, the time evolution operator is approximated by $$ \left(\prod_j e^{-i H_j \delta t} \right)^{N_t} $$ where $\delta t = \frac{t}{N_t}$. Here, $ N_t $ represents the number of Trotter steps, which is a parameter that determines the granularity of the time evolution approximation. A higher value of $ N_t $ leads to a finer approximation of the time evolution # Setting the maximum time for the simulation t_max = 1 # Defining the number of Trotter steps for the approximation n_trotter_steps = 10 # Calculating the time step for each Trotter step time_step = t_max / n_trotter_steps # Creating a time line from 0 to t_max with equal intervals for each Trotter step time_line = np.linspace(0, t_max, n_trotter_steps + 1) # Printing the calculated time step print(f"Time step: {time_step}") # Multiplying the Hamiltonian by the time step to get a slice of the Hamiltonian for each Trotter step hamiltonian_slice = hamiltonian * time_step {(): 0.262500000000000, (Zq[0]): -0.112500000000000, (Zq[0], Zq[1]): -0.0375000000000000, (Zq[1]): -0.112500000000000, (Xq[0]): -0.300000000000000, (Xq[0], Zq[1]): -0.100000000000000, (Xq[1]): -0.300000000000000, (Zq[0], Xq[1]): -0.100000000000000} After slicing the Hamiltonian with the calculated time_step, we convert the hamiltonian_slice into a matrix format. This is done by first converting the Hamiltonian slice into a sparse matrix representation and then converting it to a dense matrix. The .real part extracts the real component of the matrix, which is necessary for some quantum computing simulations and visualizations. matrix([[ 1.38777878e-17, -4.00000000e-01, -4.00000000e-01, [-4.00000000e-01, 3.00000000e-01, 0.00000000e+00, [-4.00000000e-01, 0.00000000e+00, 3.00000000e-01, [ 0.00000000e+00, -2.00000000e-01, -2.00000000e-01, Constructing the Quantum Circuit for a Single Trotter Step¶ After calculating the Hamiltonian slice for a single time step, we proceed to construct the quantum circuit that simulates this evolution. We use a utility gen_term_sequence_circuit which builds a circuit for us starting from the Hamiltonian we want in the exponent of a single Trotter step, and the initial reference state we want to start from. The pytket API documentation for gen_term_sequence_circuit has more details. # initializing a quantum circuit with two qubits initial_state = Circuit(2) # generate the circuit for the Trotter step trotter_step_circ = gen_term_sequence_circuit(hamiltonian_slice, initial_state) # visualize the circuit The circuit for a single Trotter step is composed of several CircBox components. The get_commands() method is used to explore the composition of the circuit. [CircBox q[0], q[1];, CircBox q[0], q[1];, CircBox q[0], q[1];] Each CircBox can be examined to reveal its internal structure, which typically consists of PauliExpBox boxes. We can get the individual PauliExpBox boxes from each CircBox in the circuit pauliexpbox0 = trotter_step_circ.get_commands()[0].op box_circ = pauliexpbox0.get_circuit() [PauliExpBox q[0];, PauliExpBox q[0], q[1];, PauliExpBox q[1];] In this nested CircBox there are 3 PauliExpBox, of which only one acts on 2 qubits pauli_exp_box = box_circ.get_commands()[1].op For a more detailed analysis, we use the DecomposeBoxes pass from pytket.passes. This pass decomposes the circuit into fundamental quantum gates, allowing us to see the basic operations that constitute the Trotter step. For one Trotter step: test_circuit = trotter_step_circ.copy() and we can count the total number of gates Counter({<OpType.H: 33>: 8, <OpType.Rz: 36>: 7, <OpType.CX: 42>: 6}) or look at specific statistics of the circuit print(f"Number of gates = {test_circuit.n_gates}") print(f"Circuit depth = {test_circuit.depth()}") print(f"Number of CX gates = {test_circuit.n_gates_of_type(OpType.CX)}") Number of gates = 21 Circuit depth = 19 Number of CX gates = 6 Example 1: Noiseless emulation of a circuit for one Trotter step¶ We will first run a noiseless statevector emulation of the circuit on Quantinuum’s System Model H1 emulator. Quantinuum H-Series emulators model the physical and noise parameters of H-Series quantum computers. The emulators provide a way to turn the noise model off, while still modeling the physical properties of the device, such as ion transport. from pytket.extensions.quantinuum import QuantinuumBackend h_backend = QuantinuumBackend("H1-2E") # emulator One Trotter step¶ The Trotter step circuit can be enclosed in a CircBox and added to any circuit (this way we can add many steps in a sequence). trotter_step_box = CircBox(trotter_step_circ) # define circuit for one Trotter step circ = Circuit(2) circ.add_gate(trotter_step_box, circ.qubits) When we compile the circuit for the backend, the CircBox nested boxes get decomposed into native gates. Each backend in pytket has a list of default passes that are applied to it. They can be applied by calling backend.get_compiled_circuit() with different optimisation_level arguments (the pytket-quantinuum docs list the default values). Here we use the default optimisation_level for the QuantinuumBackend, which is 2. Note: The measurement operations from the MeasurementSetup object are appended to the numerical circuit. Once this step is complete, the circuit is ready for submission. circ_compiled = h_backend.get_compiled_circuit(circ.measure_all()) Before submitting to the emulator, the total cost of running the set of circuits in H-System Quantum Credits (HQCs) can be checked beforehand. n_shots = 100 cost = h_backend.cost(circ_compiled, n_shots=n_shots, syntax_checker="H1-2SC") print("Cost of experiment in HQCs:", cost) Cost of experiment in HQCs: 5.92 We submit the circuit to the Quantinuum H1-2 Emulator and make the emulation noiseless by passing the option noisy_simulation=False. handle = h_backend.process_circuit( circ_compiled, n_shots=n_shots, noisy_simulation=False circuit_status = h_backend.circuit_status(handle) result = h_backend.get_result(handle) Counter({(0, 0): 39, (1, 0): 35, (0, 1): 24, (1, 1): 2}) Example 2: Noisy emulation on the System Model H1 Emulator for two Trotter steps¶ Next we implement and run two Trotter steps on the H1-2E emulator with the emulator’s default noise model for two Trotter steps. The noise model mimicks the noise of the H1-2 quantum computer. Let’s add two trotter steps to our circuit: # two steps circ = Circuit(2) circ.add_gate(trotter_step_box, circ.qubits) circ.add_gate(trotter_step_box, circ.qubits) [CircBox q[0], q[1]; CircBox q[0], q[1]; ] circ_compiled = h_backend.get_compiled_circuit(circ.measure_all()) f"Our circuit has {circ_compiled.n_gates} quantum gates in total and an overall depth of {circ_compiled.depth()}." f"Of these gates {circ_compiled.n_gates_of_type(OpType.ZZPhase)} are two qubit gate, ZZPhase, counts." Our circuit has 13 quantum gates in total and an overall depth of 8. Of these gates 3 are two qubit gate, ZZPhase, counts. Before submitting to the emulator, the total cost of running the set of circuits in H-System Quantum Credits (HQCs) is checked beforehand. n_shots = 100 cost = h_backend.cost(circ_compiled, n_shots=n_shots, syntax_checker="H1-2SC") print("Cost of experiment in HQCs:", cost) Cost of experiment in HQCs: 6.16 Now we run the circuits. handle = h_backend.process_circuit(circ_compiled, n_shots=n_shots) The status of the jobs can be checked with ciruit_status method. This method requires the ResultHandle to be passed as input. In this example, the job has completed and the results are reported as being ready to request. circuit_status = h_backend.circuit_status(handle) Once a job’s status returns completed, results can be returned using the get_result function in conjunction with the get_counts call. result = h_backend.get_result(handle) Counter({(1, 0): 47, (1, 1): 31, (0, 1): 19, (0, 0): 3}) We note that when using optimisation_levelequal to 0, we compile to the native gates of the H-Series without optimization and the number of 1- and 2-qubit gates is significantly larger than compared to the default optimisation_level of 2. circ_compiled_0 = h_backend.get_compiled_circuit(circ, optimisation_level=0) Counter({<OpType.Rz: 36>: 78, <OpType.PhasedX: 68>: 40, <OpType.ZZPhase: 73>: 12, <OpType.Measure: 63>: 2}) Example 3: Symbolic Manipulation in Quantum Circuit Simulation¶ In this section, we explore the use of symbolic manipulation for quantum circuit simulation, particularly focusing on the Trotter-Suzuki decomposition. We employ sympy for defining symbolic variables, which represent key parameters like coupling strengths and time steps in our quantum system. These symbols allow us to dynamically adjust our circuit parameters. The symbols are related to the coupling and the time step. from sympy import symbols syms = symbols("x dt") We define various QubitPauliString objects to represent different Pauli operators acting on our qubits. zi = QubitPauliString([Qubit(0)], [Pauli.Z]) iz = QubitPauliString([Qubit(1)], [Pauli.Z]) xi = QubitPauliString([Qubit(0)], [Pauli.X]) ix = QubitPauliString([Qubit(1)], [Pauli.X]) zz = QubitPauliString([Qubit(0), Qubit(1)], [Pauli.Z, Pauli.Z]) xz = QubitPauliString([Qubit(0), Qubit(1)], [Pauli.X, Pauli.Z]) zx = QubitPauliString([Qubit(0), Qubit(1)], [Pauli.Z, Pauli.X]) Next, we define two dictionaries, singles_syms and doubles_sym, to represent parts of our Hamiltonian symbolically: • singles_syms: Maps 1-qubit Pauli strings to their coefficients. These terms, involving Z and X Pauli operators on individual qubits, reflect the linear contributions in the Hamiltonian. For example, terms like zi and xi are mapped to symbolic expressions involving dt and x, representing time step and coupling constants, respectively. • doubles_sym: Represents 2-qubit interaction terms. Here, combinations of Z and X operators on pairs of qubits are mapped to their corresponding coefficients. These terms capture the inter-qubit interactions in the Hamiltonian, with coefficients also expressed as functions of dt and x. singles_syms = { zi: -21 / 8 * syms[1], iz: -21 / 8 * syms[1], xi: -3 / 2 * syms[0] * syms[1], ix: -3 / 2 * syms[0] * syms[1], doubles_sym = { zz: -3 / 8 * syms[1], xz: -1 / 2 * syms[0] * syms[1], zx: -1 / 2 * syms[0] * syms[1], We now combine the 1-qubit and 2-qubit interaction terms defined in singles_syms and doubles_sym to form the first-order Trotterized Hamiltonian: hamiltonian_trotterized = QubitPauliOperator({singles_syms, doubles_sym}) {(Zq[0]): -2.625dt, (Zq[1]): -2.625dt, (Xq[0]): -1.5dtx, (Xq[1]): -1.5dtx, (Zq[0], Zq[1]): -0.375dt, (Xq[0], Zq[1]): -0.5dtx, (Zq[0], Xq[1]): -0.5dtx} We can generate the circuit by using the same function as before circ = Circuit(2) trotter_step_circ_symb = gen_term_sequence_circuit(hamiltonian_trotterized, circ) Again, we can also decompose this symbolic circuit into basic gates and count these gates. from pytket.passes import DecomposeBoxes test_circuit = trotter_step_circ_symb.copy() Counter({<OpType.H: 33>: 8, <OpType.Rz: 36>: 7, <OpType.CX: 42>: 6}) We can also make this symbolic circuit into a box that we can repeat for multiple Trotter steps: trotter_step_box_symb = CircBox(trotter_step_circ_symb) This is a 2-step Trotter symbolic circuit: # two steps circ = Circuit(2) circ.add_gate(trotter_step_box_symb, circ.qubits) circ.add_gate(trotter_step_box_symb, circ.qubits) [CircBox q[0], q[1]; CircBox q[0], q[1]; ] In the final step of working with the symbolic circuit, we demonstrate how to substitute our symbolic parameters with actual numerical values: 1. Defining Symbol Values: We first define a list x_dt_values containing the specific values for our symbols. In this example, x is set to 2.0 and dt to 0.01. 2. Creating the Symbol-Value Map: Using zip(syms, x_dt_values), we pair each symbol with its corresponding value, and then convert this pairing into a dictionary sym_map. This dictionary maps each symbolic variable to its numerical value. 3. Applying Substitutions: We use circ.symbol_substitution(sym_map) to apply these substitutions to our circuit. This method replaces each occurrence of the symbolic variables in circ with their defined numerical values. Lastly, we render the updated circuit using render_circuit_jupyter(circ). This visualization shows the circuit as it would be with the specific parameters applied. x_dt_values = [2.0, 0.01] sym_map = dict(zip(syms, x_dt_values)) Implementation of the SU(2) 5-Plaquette Model Hamiltonian¶ Having explored the SU(2) 2-Plaquette Model, a foundational representation in lattice gauge theories, we now shift our focus to the more complex SU(2) 5-Plaquette Model. This transition marks a significant step towards understanding and simulating more intricate quantum systems. The 5-Plaquette Model, with its increased number of qubits, provides a richer and more detailed canvas to examine the dynamics of lattice gauge theories. This model’s Hamiltonian, characterized by its division into electric and magnetic components, captures the nuanced interactions within a lattice. Such a detailed representation is crucial for simulating real-world quantum phenomena and lays the groundwork for advanced quantum computing applications. By delving into the 5-Plaquette Model, we gain deeper insights into the behavior of complex quantum systems, paving the way for more accurate simulations and potentially groundbreaking discoveries in the realm of quantum physics. The following section presents the Hamiltonian of the SU(2) 5-Plaquette Model, implemented using Qiskit, demonstrating how to interact with Qiskit code. For the SU(2) 5-Plaquette Model, the 5-qubit Hamilatonian is given as follows \[\begin{split} \begin{aligned} H= & \frac{g^2}{2}\left(h_E+h_B\right) \\ h_E= & \frac{3}{8}(3 N+1)-\frac{9}{8}\left(Z_0+Z_{N-1}\right)-\frac{3}{4} \sum_{n=1}^{N-2} Z_n \\ & -\frac{3}{8} \sum_{n=0}^ {N-2} Z_n Z_{n+1} \\ h_B= & -\frac{x}{2}\left(3+Z_1\right) X_0-\frac{x}{2}\left(3+Z_{N-2}\right) X_{N-1} \\ & -\frac{x}{8} \sum_{n=1}^{N-2}\left(9+3 Z_{n-1}+3 Z_{n+1}+Z_{n-1} Z_{n+1}\right) X_n \end {aligned} \end{split}\] This model’s Hamiltonian is explicitly broken down into electric $h_E $ and magnetic $h_B $ components, each comprising various terms involving Pauli $ \sigma_Z $ and $ \sigma_X $ operators. In this section, we construct the Hamiltonian for our quantum system using Qiskit’s SparsePauliOp build from a list of Pauli strings and their associated coefficients. The Hamiltonian is divided into two main components: the electric component (h_e) and the magnetic component (h_b). Each component is expressed as list of Pauli terms acting on the qubits and their respective coefficients. We use SparsePauliOp to write the Hamiltonian operator following the equations above quite literally (fix $x=2$): • The electric component h_e includes terms involving Z Pauli operators, reflecting the electric interactions in the system. • The magnetic component h_b comprises terms with X and combined XZ Pauli operators, representing the magnetic interactions. The coefficients of each term are specific to our model, reflecting the strength of each interaction. We then combine these two components to form the complete Hamiltonian of the system, which is used for further analysis and simulation. from qiskit.quantum_info import SparsePauliOp h_e = [ ("ZIIII", -9.0 / 8), ("IIIIZ", -9.0 / 8), ("IZIII", -3.0 / 4), ("IIZII", -3.0 / 4), ("IIIZI", -3.0 / 4), ("ZZIII", -3.0 / 8), ("IZZII", -3.0 / 8), ("IIZZI", -3.0 / 8), ("IIIZZ", -3.0 / 8), h_b = [ ("XIIII", -3.0), ("XZIII", -1.0), ("IIIIX", -3.0), ("IIIZX", -1.0), ("IXIII", -9.0 / 4), ("ZXIII", -3.0 / 4), ("IXZII", -3.0 / 4), ("ZXZII", -1.0 / 4), ("IIXII", -9.0 / 4), ("IZXII", -3.0 / 4), ("IIXZI", -3.0 / 4), ("IZXZI", -1.0 / 4), ("IIIXI", -9.0 / 4), ("IIZXI", -3.0 / 4), ("IIIXZ", -3.0 / 4), ("IIZXZ", -1.0 / 4), hamiltonian = SparsePauliOp.from_list(h_e + h_b) SparsePauliOp(['ZIIII', 'IIIIZ', 'IZIII', 'IIZII', 'IIIZI', 'ZZIII', 'IZZII', 'IIZZI', 'IIIZZ', 'XIIII', 'XZIII', 'IIIIX', 'IIIZX', 'IXIII', 'ZXIII', 'IXZII', 'ZXZII', 'IIXII', 'IZXII', 'IIXZI', 'IZXZI', 'IIIXI', 'IIZXI', 'IIIXZ', 'IIZXZ'], coeffs=[-1.125+0.j, -1.125+0.j, -0.75 +0.j, -0.75 +0.j, -0.75 +0.j, -0.375+0.j, -0.375+0.j, -0.375+0.j, -0.375+0.j, -3. +0.j, -1. +0.j, -3. +0.j, -1. +0.j, -2.25 +0.j, -0.75 +0.j, -0.75 +0.j, -0.25 +0.j, -2.25 +0.j, -0.75 +0.j, -0.75 +0.j, -0.25 +0.j, -2.25 +0.j, -0.75 +0.j, -0.75 +0.j, -0.25 +0.j]) A dense (or sparse) matrix representation of the operator can be accessed directly with the method SparsePauliOp.to_matrix(): array([[-6. +0.j, -4. +0.j, -4. +0.j, ..., -0. +0.j, -0. +0.j, -0. +0.j], [-4. +0.j, -3. +0.j, -0. +0.j, ..., -0. +0.j, -0. +0.j, -0. +0.j], [-4. +0.j, -0. +0.j, -3. +0.j, ..., -0. +0.j, -0. +0.j, -0. +0.j], [-0. +0.j, -0. +0.j, -0. +0.j, ..., 3. +0.j, -0. +0.j, -1. +0.j], [-0. +0.j, -0. +0.j, -0. +0.j, ..., -0. +0.j, 1.5+0.j, -2. +0.j], [-0. +0.j, -0. +0.j, -0. +0.j, ..., -1. +0.j, -2. +0.j, 3. +0.j]]) As a first step we can compute the ground state of the Hamiltonian by diagonalizing and extracting the lowest eigenvalue. There are many ways to do this but one can pass the (sparse) matrix obtained from SparsePauliOp representing our Hamiltonian to a numerical eigensolver from scipy.sparse.linalg import eigs vals, vecs = eigs(hamiltonian.to_matrix(sparse=True), k=1) print(f"Energy of the ground state {vals[0].real}") Energy of the ground state -16.999769538608106 In this section, we build utility functions to translate Qiskit’s Hamiltonian representation into the QubitPauliOperator format used by pytket. • qps_from_sparsepauliop: This function converts a tensor of Pauli operators from SparsePauliOp format to pytket’s QubitPauliString. It iterates over qubit indices and corresponding Pauli operators, mapping them to the pytket equivalent. • qpo_from_sparsepauliop: This function takes an Qiskit SparsePauliOp and converts it into a QubitPauliOperator. It processes each term in the Qiskit operator, using qps_from_sparsepauliop for conversion, and associates it with the corresponding coefficient. Finally, we apply qpo_from_sparsepauliop to our previously defined Hamiltonian and print the result. This step confirms that the Hamiltonian is correctly converted into the format required by pytket for further quantum computations. from pytket.pauli import Pauli, QubitPauliString from pytket.utils import QubitPauliOperator from pytket import Qubit from collections import defaultdict pauli_sym = {"I": Pauli.I, "X": Pauli.X, "Y": Pauli.Y, "Z": Pauli.Z} def qps_from_sparsepauliop(paulis): """Convert SparsePauliOp tensor of Paulis to pytket QubitPauliString.""" qlist = [] plist = [] for q, p in enumerate(paulis): if p != "I": return QubitPauliString(qlist, plist) def qpo_from_sparsepauliop(sp_op): """Convert SparsePauliOp QubitOperator to pytket QubitPauliOperator.""" tk_op = defaultdict(complex) for term, coeff in sp_op.to_list(): string = qps_from_sparsepauliop(term) tk_op[string] += coeff return QubitPauliOperator(tk_op) hamiltonian_op = qpo_from_sparsepauliop(hamiltonian) {(Zq[0]): -1.12500000000000, (Zq[4]): -1.12500000000000, (Zq[1]): -0.750000000000000, (Zq[2]): -0.750000000000000, (Zq[3]): -0.750000000000000, (Zq[0], Zq[1]): -0.375000000000000, (Zq[1], Zq[2]): -0.375000000000000, (Zq[2], Zq[3]): -0.375000000000000, (Zq[3], Zq[4]): -0.375000000000000, (Xq[0]): -3.00000000000000, (Xq[0], Zq[1]): -1.00000000000000, (Xq[4]): -3.00000000000000, (Zq[3], Xq[4]): -1.00000000000000, (Xq[1]): -2.25000000000000, (Zq[0], Xq[1]): -0.750000000000000, (Xq[1], Zq[2]): -0.750000000000000, (Zq[0], Xq[1], Zq[2]): -0.250000000000000, (Xq[2]): -2.25000000000000, (Zq[1], Xq[2]): -0.750000000000000, (Xq[2], Zq[3]): -0.750000000000000, (Zq[1], Xq[2], Zq[3]): -0.250000000000000, (Xq[3]): -2.25000000000000, (Zq[2], Xq[3]): -0.750000000000000, (Xq[3], Zq[4]): -0.750000000000000, (Zq[2], Xq[3], Zq[4]): -0.250000000000000} The next line of code converts the QubitPauliOperator representation of the Hamiltonian, obtained in the previous step, into a dense matrix format. The to_sparse_matrix() method first transforms the Hamiltonian into a sparse matrix. We then use todense() to convert this sparse matrix into a regular, dense matrix. This dense matrix representation is useful for certain types of numerical analyses and visual inspections of the Hamiltonian structure. matrix([[-6. +0.j, -4. +0.j, -4. +0.j, ..., 0. +0.j, 0. +0.j, 0. +0.j], [-4. +0.j, -3. +0.j, 0. +0.j, ..., 0. +0.j, 0. +0.j, 0. +0.j], [-4. +0.j, 0. +0.j, -3. +0.j, ..., 0. +0.j, 0. +0.j, 0. +0.j], [ 0. +0.j, 0. +0.j, 0. +0.j, ..., 3. +0.j, 0. +0.j, -1. +0.j], [ 0. +0.j, 0. +0.j, 0. +0.j, ..., 0. +0.j, 1.5+0.j, -2. +0.j], [ 0. +0.j, 0. +0.j, 0. +0.j, ..., -1. +0.j, -2. +0.j, 3. +0.j]]) hamiltonian_op.to_sparse_matrix().todense(), hamiltonian.to_matrix(sparse=False) We now proceed with the Trotter-Suzuki decomposition to approximate the time-evolution operator of the Hamiltonian in quantum simulations as we did above. Note: rescaling_factor_from_pytket = np.pi/2 is defined for aligning simulation results with exact theoretical predictions. This factor compensates for an additional π/2 factor introduced by gen_term_sequence_circuit in pytket, which affects the time evolution steps. When comparing with exact results, divide the time step ($\delta t$) by this rescaling factor to account for this # rescaling_factor_from_pytket = np.pi/2 ## use it to divide \deltat in case you want to compare with the exact results: gen_term_sequence adds Pauliexpbox with an extrat pi/2 factor t_max = 1 n_trotter_steps = 10 time_step = t_max / n_trotter_steps time_line = np.linspace(0, t_max, n_trotter_steps + 1) print(f"Time step: {time_step}") hamiltonian_slice = hamiltonian_op time_step Next we construct the corresponding five-qubit circuit for two Trotter steps. A product of exponential of Pauli strings can be automatically generated by pytket using gen_term_sequence_circuits, including aggregating mutually commuting terms into groups. initial_state = Circuit(5) trotter_step_circ = gen_term_sequence_circuit(hamiltonian_slice, initial_state) The Trotter step circuit can be enclosed in a CircBox and added to any circuit (this way we can add many steps in a sequence) trotter_step_box = CircBox(trotter_step_circ) Now one Trotter step is a single box, and we can simply stack boxes to create circuits for more Trotter steps # two steps circ = Circuit(5) circ.add_gate(trotter_step_box, circ.qubits) circ.add_gate(trotter_step_box, circ.qubits) [CircBox q[0], q[1], q[2], q[3], q[4]; CircBox q[0], q[1], q[2], q[3], q[4]; ] from pytket.circuit.display import render_circuit_jupyter Now we sign into the Quantinuum Backend we will submit to, H1-2E, and compile the circuit for the backend. from pytket.extensions.quantinuum import QuantinuumBackend h_backend = QuantinuumBackend("H1-2E") # emulator circ_compiled = h_backend.get_compiled_circuit(circ.measure_all()) We can look at the statistics of the circuit before running it on the emulator. Counter({<OpType.PhasedX: 68>: 59, <OpType.ZZPhase: 73>: 34, <OpType.Measure: 63>: 5}) Now we run the circuit on the Quantinuum H1-2 Emulator. Before submitting to the emulator, the total cost of running the circuit should be checked beforehand. n_shots = 100 cost = h_backend.cost(circ_compiled, n_shots=n_shots, syntax_checker="H1-2SC") print("Cost of experiment in HQCs:", cost) Cost of experiment in HQCs: 13.98 Now we run the circuit with the noise model turned on handle = h_backend.process_circuit( circ_compiled, n_shots=n_shots, noisy_simulation=True circuit_status = h_backend.circuit_status(handle) CircuitStatus(status=<StatusEnum.COMPLETED: 'Circuit has completed. Results are ready.'>, message='{"name": "job", "submit-date": "2024-01-11T06:24:50.716190", "result-date": "2024-01-11T06:25:15.622257", "queue-position": null, "cost": "13.98", "error": null}', error_detail=None, completed_time=None, queued_time=None, submitted_time=None, running_time=None, cancelled_time=None, error_time=None, queue_position=None) result = h_backend.get_result(handle) Counter({(1, 0, 1, 0, 1): 36, (1, 0, 1, 1, 0): 6, (1, 1, 1, 0, 1): 6, (0, 0, 0, 0, 1): 5, (0, 1, 1, 0, 1): 5, (1, 0, 1, 1, 1): 5, (0, 1, 1, 1, 1): 4, (1, 1, 0, 1, 0): 4, (1, 1, 0, 1, 1): 4, (0, 0, 1, 0, 1): 3, (0, 1, 0, 0, 1): 3, (1, 0, 1, 0, 0): 3, (1, 0, 0, 0, 0): 2, (1, 0, 0, 1, 0): 2, (1, 0, 0, 1, 1): 2, (1, 1, 0, 0, 0): 2, (0, 0, 0, 1, 0): 1, (0, 0, 0, 1, 1): 1, (0, 1, 0, 1, 1): 1, (0, 1, 1, 0, 0): 1, (1, 0, 0, 0, 1): 1, (1, 1, 0, 0, 1): 1, (1, 1, 1, 0, 0): 1, (1, 1, 1, 1, 1): 1}) We can also plot the result using the following helper code: import pandas as pd import seaborn as sns import matplotlib.pyplot as plt def plot_counts(counts): counts_record = [ {"State": str(state), "Count": count} for state, count in counts.items() count_df = pd.DataFrame().from_records(counts_record) sns.catplot(x="State", y="Count", kind="bar", data=count_df, aspect=12, height=2) counts = result.get_counts() And compare our results to the noiseless emulation, we see that our emulator results match the noiseless simulation closely. noiseless_handle = h_backend.process_circuit( circ_compiled, n_shots=n_shots, noisy_simulation=False noiseless_result = h_backend.get_result(noiseless_handle) noiseless_counts = noiseless_result.get_counts() One can also run the noiseless simulation using the Qiskit Aer backend directly from pytket using our pytket-qiskit extension: from pytket.extensions.qiskit import AerBackend # define Aer backend backend = AerBackend() # compile circuit as usual, but now for the new backend compiled_circ = backend.get_compiled_circuit(circ) # submit and run the job: this runs locally and not in the cloud handle = backend.process_circuit(compiled_circ, n_shots=100) counts = backend.get_result(handle).get_counts() # print(counts) Definitions for the use case context in this article. • Quantum Field Theory (QFT): a mathematical and conceuptual framework in theoretical physics extending quantum mechanics, which deals with particles, over to fields, which are systems with infinite number of degrees of freedom. It is used in particle physics to construct physical models of subatomic particles and condensed matter physics to construct models of quasiparticles. Difficult due to the infinite number of degrees of freedom. • Lattice Gauge Theory (LGT): a mathematical framework in QFT. Rather than having infinite degrees of freedom, space and time are discretized to a very large number of degrees of freedom, enabling its study on computers. • Perturbation Theory: a method for finding an approximate solutions to a problem, by starting from the exact solution of a related, simpler problem • Quantum Chromodynamics: theory that describes the action of the strong nuclear force, the fundamental interaction between subatomic particles of matter inside protons and neutrons.	{"url":"https://docs.quantinuum.com/h-series/trainings/knowledge_articles/Quantinuum_high_energy_physics_experiment.html","timestamp":"2024-11-06T01:17:56Z","content_type":"text/html","content_length":"266351","record_id":"<urn:uuid:8bb90d7b-fa38-4776-a9e8-9a79e036d5a3>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00020.warc.gz"}
Option set for hinfsyn and mixsyn opts = hinfsynOptions creates the default option set for the hinfsyn and mixsyn commands. opts = hinfsynOptions(Name,Value) creates an option set with the options specified by one or more Name,Value pair arguments. Specify Algorithm and Display for H-Infinity Synthesis Use the LMI-based algorithm to compute an ${\mathit{H}}_{\infty }$-optimal controller for a plant with one control signal and two measurement signals. Turn on the display that shows the progress of the computation. Use hinfsynOptions to specify these algorithm options. Load the plant and specify the numbers of measurements and controls. load hinfsynExData P ncont = 1; nmeas = 2; Create an options set for hinfsyn that specifies the LMI-based algorithm and turns on the display. opts = hinfsynOptions('Method','LMI','Display','on'); Alternatively, start with the default options set, and use dot notation to change option values. opts = hinfsynOptions; opts.Method = 'LMI'; opts.Display = 'on'; Compute the controller. [K,CL,gamma] = hinfsyn(P,nmeas,ncont,opts); Minimization of gamma: Solver for linear objective minimization under LMI constraints Iterations : Best objective value so far 2 223.728733 3 138.078240 4 138.078240 5 74.644885 6 48.270221 7 48.270221 8 48.270221 9 19.665676 10 19.665676 11 11.607238 12 11.607238 13 11.607238 14 4.067958 15 4.067958 16 4.067958 17 2.154349 18 2.154349 19 2.154349 20 1.579564 21 1.579564 22 1.579564 23 1.236726 24 1.236726 25 1.236726 26 0.993342 27 0.993342 28 0.949318 29 0.949318 30 0.949318 31 0.945762 32 0.944063 33 0.941246 34 0.941246 35 0.940604 *** new lower bound: 0.931668 Result: feasible solution of required accuracy best objective value: 0.940604 guaranteed absolute accuracy: 8.94e-03 f-radius saturation: 0.404% of R = 1.00e+08 Optimal Hinf performance: 9.397e-01 Input Arguments Name-Value Arguments Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter. Before R2021a, use commas to separate each name and value, and enclose Name in quotes. Example: 'Display','on','RelTol',0.05 General Options Display — Display progress and generate report 'off' (default) \| 'on' Display optimization progress and generate report in the command window, specified as the comma-separated pair consisting of 'Display' and 'on' or 'off'. The contents of the display depend on the value of the 'Method' option. For 'Method' = 'RIC', the display shows the range of performance targets (gamma values) tested. For each gamma, the display shows: • The smallest eigenvalues of the normalized Riccati solutions X = X[∞]/γ and Y = Y[∞]/γ • The spectral radius rho(XY) = max(abs(eig(XY))) • A pass/fail (p/f) flag indicating whether that gamma value satisfies the conditions X ≥ 0, Y ≥ 0, and rho(XY) < 1 • The best achieved gamma performance value For more information about the displayed information, see the Algorithms section of hinfsyn. For 'Method' = 'LMI', the display shows the best achieved gamma value for each iteration of the optimization problem. It also displays a report of the best achieved value and other parameters of the Example: opts = hinfsynOptions('Display','on') creates an option set that turns the progress display on. Method — Optimization algorithm 'RIC' (default) \| 'LMI' Optimization algorithm that hinfsyn or mixsyn uses to optimize closed-loop performance, specified as the comma-separated pair consisting of 'Method' and one of the following: • 'RIC' — Riccati-based algorithm. The Riccati method is fastest, but cannot handle singular problems without first adding extra disturbances and errors. This process is called regularization, and is performed automatically by hinfsyn and mixsyn unless you set the 'Regularize' option to 'off'. With regularization, this method works well for most problems. When 'Method' = 'RIC', the additional options listed under Riccati Method Options are available. • 'LMI' — LMI-based algorithm. This method requires no regularization, but is computationally more intensive than the Riccati method. When 'Method' = 'LMI', the additional options listed under LMI Method Options are available. • 'MAXE' — Maximum-entropy algorithm. When 'Method' = 'MAXE', the additional options listed under Maximum-Entropy Method Options are available. For more information about how these algorithms work, see the Algorithms section of hinfsyn. Example: opts = hinfsynOptions('Mathod','LMI') creates an option set that specifies the LMI-based optimization algorithm. RelTol — Relative accuracy on optimal H[∞] performance 0.01 (default) \| positive scalar Relative accuracy on the optimal H[∞] performance, specified as the comma-separated pair consisting of 'RelTol' and a positive scalar value. The algorithm stops testing γ values when the relative difference between the last failing value and last passing value is less than RelTol. Example: opts = hinfsynOptions('RelTol',0.05) creates an option set that sets the relative accuracy to 0.05. Riccati Method Options AbsTol — Absolute accuracy on optimal H[∞] performance 10^-6 (default) \| positive scalar Absolute accuracy on the optimal H[∞] performance, specified as the comma-separated pair consisting of 'AbsTol' and a positive scalar value. Example: opts = hinfsynOptions('AbsTol',1e-4) creates an option set that sets the absolute accuracy to 0.0001. AutoScale — Automatic plant scaling 'on' (default) \| 'off' Automatic plant scaling, specified as the comma-separated pair consisting of 'AutoScale' and one of the following: • 'on' — Automatically scales the plant states, controls, and measurements to improve numerical accuracy. hinfsyn always returns the controller K in the original unscaled coordinates. • 'off' — Does not change the plant scaling. Turning off scaling when you know your plant is well scaled can speed up the computation. Example: opts = hinfsynOptions('AutoScale','off') creates an option set that turns off automatic scaling. Regularize — Automatic regularization 'on' (default) \| 'off' Automatic regularization of the plant, specified as the comma-separated pair consisting of 'Regularize' and one of: • 'on' — Automatically regularizes the plant to enforce requirements on P[12] and P[21] (see hinfsyn). Regularization is a process of adding extra disturbances and errors to handle singular • 'off' — Does not regularize the plant. Turning off regularization can speed up the computation when you know your problem is far enough from singular. Example: opts = hinfsynOptions('Regularize','off') creates an option set that turns off regularization. LimitGain — Limit on controller gains 'on' (default) \| 'off' Limit on controller gains, specified as the comma-separated pair consisting of 'LimitGain' and either 'on' or 'off'. For continuous-time plants, regularization of plant feedthrough matrices D[12] or D[21] (see hinfsyn) can result in controllers with large coefficients and fast dynamics. Use this option to automatically seek a controller with the same performance but lower gains and better LMI Method Options LimitRS — Limit on norm of LMI solutions 0 (default) \| scalar in [0,1] Limit on norm of LMI solutions, specified as the comma-separated pair consisting of 'LimitRS' and a scalar factor in the range [0,1]. Increase this value to slow the controller dynamics by penalizing large-norm LMI solutions. See [1]. TolRS — Reduced-order synthesis tolerance 0.001 (default) \| positive scalar Reduced-order synthesis tolerance, specified as the comma-separated pair consisting of 'TolRS' and a positive scalar value. hinfsyn computes a reduced-order controller when 1 <= rho(R*S) <= TolRs, where rho(A) is the spectral radius, max(abs(eig(A))). Maximum-Entropy Method Options S0 — Frequency at which to evaluate entropy Inf (default) \| real scalar Frequency at which to evaluate entropy, specified as a real scalar value. For more information, see the Algorithms section of hinfsyn. Output Arguments opts — Options for hinfsyn and mixsyn hinfsyn options object Options for the hinfsyn or mixsyn computation, returned as an hinfsyn options object. Use the object as an input argument to hinfsyn or mixsyn. For example: [K,CL,gamma,info] = hinfsyn(P,nmeas,ncont,opts); [1] Gahinet, P., and P. Apkarian. "A linear matrix inequality approach to H[∞]-control." Int J. Robust and Nonlinear Control, Vol. 4, No. 4, 1994, pp. 421–448. Version History Introduced in R2018b	{"url":"https://au.mathworks.com/help/robust/ref/hinfsynoptions.html","timestamp":"2024-11-10T18:02:08Z","content_type":"text/html","content_length":"107990","record_id":"<urn:uuid:a9f3607d-e08f-4ed7-bb90-978fb5608919>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00070.warc.gz"}
Manual browser: moduli(5) MODULI(5) File Formats Manual MODULI(5) moduli — system moduli file file contains the system-wide Diffie-Hellman prime moduli for Each line in this file contains the following fields: Time, Type, Tests, Tries, Size, Generator, Modulus. The fields are separated by white space (tab or blank). Time: yyyymmddhhmmss. Specifies the system time that the line was appended to the file. The value 00000000000000 means unknown (historic). Type: decimal. Specifies the internal structure of the prime modulus. unknown; often learned from peer during protocol operation, and saved for later analysis. unstructured; a common large number. safe (p = 2q + 1); meets basic structural requirements. Sophie-Germaine (q = (p-1)/2); usually generated in the process of testing safe or strong primes. strong; useful for RSA public key generation. Tests: decimal (bit field). Specifies the methods used in checking for primality. Usually, more than one test is used. not tested; often learned from peer during protocol operation, and saved for later analysis. composite; failed one or more tests. In this case, the highest bit specifies the test that failed. sieve; checked for division by a range of smaller primes. Elliptic Curve. Tries: decimal. Depends on the value of the highest valid Test bit, where the method specified is: not tested (always zero). composite (irrelevant). sieve; number of primes sieved. Commonly on the order of 32,000,000. Miller-Rabin; number of M-R iterations. Commonly on the order of 32 to 64. Jacobi; unknown (always zero). Elliptic Curve; unused (always zero). Size: decimal. Specifies the number of the most significant bit (0 to M). Generator: hex string. Specifies the best generator for a Diffie-Hellman exchange. 0 = unknown or variable, 2, 3, 5, etc. Modulus: hex string. The prime modulus. The file should be searched for moduli that meet the appropriate Time, Size and Generator criteria. When more than one meet the criteria, the selection should be weighted toward newer moduli, without completely disqualifying older moduli. Note that sshd(8) uses only the Size criteria and then selects a modulus at random if more than one meet the Size criteria. The moduli file appeared in OpenBSD 2.8 and NetBSD 1.6. February 7, 2005 NetBSD 7.0	{"url":"https://edgebsd.org/index.php/manual/5/moduli","timestamp":"2024-11-08T19:08:15Z","content_type":"text/html","content_length":"11575","record_id":"<urn:uuid:f90bd237-f1ee-420c-b9ec-c57b1b852aca>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00597.warc.gz"}
New employees at Mathematical Sciences A presentation of our newly employed colleagues. Jenny Enerbäck Project Assistant at the Division of Applied Mathematics and Statistics Start date: November 25, 2024 Helga Kristín Ólafsdóttir Postdoctor at the Division of Applied Mathematics and Statistics Start date: November 4, 2024 Jakob Palmkvist Senior Lecturer at the Division of Algebra and Geometry My research interests lie in the areas of algebra and mathematical physics. In particular, I am interested in extensions of Kac-Moody algebras to infinite-dimensional integer-graded Lie superalgebras and related structures with applications to supergravity theories. I have previously worked at the Max Planck Institute for gravitational physics, Université Libre de Bruxelles, IHÉS, Texas A&M University and at the Department of Physics at Chalmers. I have also been a guest lecturer at Mathematical Sciences. Start date: November 1, 2024 Adrien Malacan PhD student at the Division of Analysis and Probability Theory My research will focus on the study of the properties of lattice gauge theory under the supervision of Malin P. Forsström. The aim of the project is to apply probabilistic methods to gain a better understanding of this model, which has applications notably in quantum physics. An interesting aspect of the research is examining how this model behaves compared to similar discrete models (such as the Ising model). I completed my master’s degree in mathematics at the Technical University of Munich, where my master’s thesis explored the simple exclusion process, a model in discrete probability theory. Start date: October 1, 2024 Andrii Dmytryshyn Associate professor at the Division of Applied Mathematics and Statistics My research interests primarily lie in the fields of matrix theory and computations, as well as their applications in computer science and physics. Specifically, I focus on linear and non-linear eigenvalue problems, matrix equations, and low-rank structures. Before joining Chalmers, I worked and studied in Örebro, Umeå, Bordeaux, Padua, and Kyiv. Start date: October 1, 2024 Maia Emmerö Akelius Math Learning Lab Start date: September 16, 2024 Laleh Varghaei PhD student at the Division of Applied Mathematics and Statistics My research is in bioinformatics and artificial intelligence. The research will be carried out in Erik Kristiansson's research group with a focus on antibiotic resistance. We will use large amounts of genomic and metagenomic data to track antibiotic resistance genes between different bacteria and understand the environments in which they are selected. Previously I studied Bioengineering for my bachelor’s degree at Chalmers and for my master’s, I studied engineering mathematics and computational science at the same university. My master thesis focused on developing methods to study genetic diversity and evolutionary patterns in Human Immunodeficiency Virus (HIV) using sequencing data, bioinformatics tools and machine learning. Start date: September 1, 2024 Lucia Swoboda PhD student at the Division of Applied Mathematics and Statistics My research area is the numerical analysis of partial differential equations, where I am particularly interested in problems involving different scales. Under the supervision of Axel Målqvist, I will be working on multilevel methods for spatial network models. Previously, my main focus was on space-time methods for the heat equation, using a modified Hilbert transform to analyse a Galerkin-Bubnov variational formulation in anisotropic Sobolev spaces. I received a master’s degree in mathematics from Graz University of Technology and the University of Graz. Start date: August 27, 2024 Paolo Boldrini PhD student at the Division of Analysis and Probability Theory Start date: August 23, 2024 Caitlin Canavati Akelius Math Learning Lab Start date: August 26, 2024 Jakob Jonsson PhD student at the Division of Analysis and Probability Theory I am interested in harmonic analysis, partial differential equations and wavelets. Supervised by Andreas Rosén I will try to generalise my master's thesis work, where we developed a wavelet-based method to solve boundary value problems numerically. I took my bachelor's degree in Linköping and my master's degree here at the University of Gothenburg. Start date: August 23, 2024 Emil Timlin PhD student at the Division of Applied Mathematics and Statistics My research will be about the homogenization of partial differential equations, and the project is supervised by Irina Pettersson and Tobias Gebäck. In this field, one is interested in differential equations whose coefficients are rapidly oscillating. The goal is to replace such an equation with a suitable homogenized equation which is easier to solve, but that still models the same macroscopic phenomenon as the original equation. I have studied at Chalmers and University of Gothenburg. In my master thesis, I investigated new methods for numerically solving boundary value problems. Start date: August 23, 2024 Mika Persson Industrial PhD student at the Division of Applied Mathematics and Statistics I'm an industrial PhD student affiliated with SAAB. My research focuses on game theory and multi-agent reinforcement learning for distributed sensors playing information games for enhanced situational awareness. This involves partially observable Markov games with heterogeneous and varying number of players, multi-target tracking, robust data fusion and deep learning. The project is funded by Wallenberg AI, Autonomous systems and Software Program (WASP) and is connected to the WASP research arena in Public Safety (WARA-PS). I have a master's degree in engineering mathematics with a specialization in machine intelligence from Lund University and have previously worked as a systems engineer at SAAB, where I developed radar signal processing algorithms. Start date: August 19, 2024 Oskar Holmstedt PhD student at the Division of Applied Mathematics and Statistics I will be conducting research on stochastic modeling of infectious diseases under the supervision of Philip Gerlee. The goal of the research is to develop methods for estimating transmission chains within a population using Bayesian inference. Specifically, these methods will be applied to data from hospital environments, with the long-term goal of being used in healthcare to combat and control infectious outbreaks. I have a Master of Science in Engineering Mathematics from Chalmers. My master's thesis focused on using neural networks to calculate the likelihood function in the context of Bayesian inference. Start date: August 19, 2024 Styrbjörn Käll PhD student at the Division of Applied Mathematics and Statistics I will work in Erik Kristiansson's research group where I will be studying toxicology and diagnostics using artificial intelligence and bioinformatics. I will for example develop transformer-based methods that, at an early stage, can predict chemicals' threat to the environment and humans based on their structure, as well as methods to rapidly predict the cause of an infection based on biomarkers in synovial fluid. I have previously worked as a Data Scientist at Semcon and as a project assistant here at the Math department. I have a master's and bachelor's degree in biotechnology from Chalmers. Start date: August 19, 2024 Sophia Axillus PhD student at the Division of Applied Mathematics and Statistics My research will focus on integrating bioinformatics and artificial intelligence to enhance our understanding of antibiotic resistance and the emergence of new resistance genes. It will involve DNA sequence analysis, biological network inference, and evolutionary modeling. The work will be carried out in the group of Erik Kristiansson. I previously worked at AstraZeneca as a Data Science & AI graduate scientist. Before then I studied Bioengineering at Chalmers, graduating with an MSc in Engineering Mathematics and Computational Start date: August 19, 2024 Lotta Eriksson PhD student at the Division of Applied Mathematics and Statistics My project is in computational biology and is supervised by Eszter Lakatos. My research focuses on developing novel mathematical and bioinformatics methods to identify key factors behind cancer and disease development using next-generation sequencing (NGS) data I previously studied Engineering Mathematics at Chalmers, and wrote my master’s thesis about denoising methods for NGS data from liquid biopsies to discover genomic alterations caused by cancer. Start date: August 19, 2024 Seif Sharif PhD student at the Division of Algebra and Geometry Start date: August 15, 2024 Anna Theorin Johansson PhD student at the Division of Algebra and Geometry I will be studying analytic number theory under the supervision of Julia Brandes. In particular, we aim to study applications of the Harvey-Littlewood circle method – a method to count the number of solutions of Diophantine equations, which can be applied to many different kinds of mathematical problems (e.g. Waring’s problem). I studied mathematics at ETH Zürich and the University of Edinburgh and wrote my master thesis on Zaremba’s conjecture and the circle method. Start date: August 15, 2024 Andreas Krona Akelius Math Learning Lab I am educated in engineering, creativity, economics, and leadership, with work experience in fields ranging from film and video games to AI and food. My tasks have, among others, included development, sales, and leadership. Through my work at Akelius Math Learning Lab, I hope to contribute to creating a memorable mathematics education that can help make the world at least a little better. Start date: August 5, 2024 Mattias Lennartsson Part-time fixed-term teacher at the Division of Algebra and Geometry Start date: August 1, 2024 Niki Wilhelmson PhD student at the Division of Applied Mathematics and Statistics I will study topics within machine learning under the supervision of Pierre Nyquist. Using tools from probability theory, and related areas, the goal is to contribute to our understanding of the mathematical principles underlying modern methods for machine learning and artificial intelligence. I studied the Engineering Physics programme at KTH Royal Institute of Technology, with a master’s in applied mathematics, and wrote my master thesis on Bayesian methods for covariance estimation in Start date: August 1, 2024 Malin Rau Assistant Professor at the Division of Applied Mathematics and Statistics My research has two main focuses. The first is developing Approximation Algorithms and proving lower bounds for Scheduling and two-dimensional Packing Problems. For these algorithms, linear and integer programs and results about the structure of optimal solutions play an essential role. My second research focus is on random processes within graph structures, specifically studying opinion dynamics, load-balancing processes, and group testing. After receiving my PhD at Kiel University (CAU), I was a PostDoc at the Université Grenoble Alpes (UGA) and later a PostDoc at the Universität Hamburg. Start date: August 1, 2024 Mathis Rost PhD student at the Division of Applied Mathematics and Statistics I will be studying spatial statistics under the guidance of Ottmar Cronie, with a focus on point processes. This area of study involves analyzing spatial point patterns, which has applications in various fields such as epidemiology and ecology. I completed my mathematics degree at Ulm University, where I also developed a strong foundation in computer science. I wrote my master thesis on point processes and spatial statistics at the Department of Mathematical Sciences within the ERASMUS program. Start date: June 3, 2024 Hilda Alfredsson Akelius Math Learning Lab Start date: April 29, 2024	{"url":"https://www.chalmers.se/en/departments/mv/new-employees/","timestamp":"2024-11-09T19:41:47Z","content_type":"text/html","content_length":"531539","record_id":"<urn:uuid:4e7c5570-77ee-4bac-b216-8c8969a3d6b2>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00536.warc.gz"}
IntroductionDescribing and Modeling Errors in Amplicon Sequencing DataDescribing Amplicon Sequencing Data as a Random Sample From an Environmental SourceModeling Random Error in Amplicon Sequencing DataThe Many Zeros of Amplicon Sequencing DataProbabilistic Inference of Source Shannon Index Using Bayesian MethodsDiversity Analysis in Absence of a Model to Infer Source DiversityDiscussionData Availability StatementAuthor ContributionsFundingConflict of InterestPublisher’s NoteSupplementary MaterialReferences Front. Microbiol. Frontiers in Microbiology Front. Microbiol. 1664-302X Frontiers Media S.A. 10.3389/fmicb.2022.728146 Microbiology Hypothesis and Theory Ensuring That Fundamentals of Quantitative Microbiology Are Reflected in Microbial Diversity Analyses Based on Next-Generation Sequencing Schmidt Philip J. ^1 Cameron Ellen S. ^2 Müller Kirsten M. ^2 Emelko Monica B. ^1 ^* ^1Canada Research Chair in Water Science, Technology & Policy Group, Department of Civil and Environmental Engineering, Faculty of Engineering, University of Waterloo, Waterloo, ON, Canada ^2Department of Biology, Faculty of Science, University of Waterloo, Waterloo, ON, Canada Edited by: Télesphore Sime-Ngando, Centre National de la Recherche Scientifique (CNRS), France Reviewed by: Fernando Perez Rodriguez, University of Cordoba, Spain; Ammar Husami, Cincinnati Children’s Hospital Medical Center, United States Correspondence: Monica B. Emelko, mbemelko@uwaterloo.ca This article was submitted to Aquatic Microbiology, a section of the journal Frontiers in Microbiology 01 03 2022 2022 13 728146 21 06 2021 20 01 2022 Copyright © 2022 Schmidt, Cameron, Müller and Emelko. 2022 Schmidt, Cameron, Müller and Emelko This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms. Diversity analysis of amplicon sequencing data has mainly been limited to plug-in estimates calculated using normalized data to obtain a single value of an alpha diversity metric or a single point on a beta diversity ordination plot for each sample. As recognized for count data generated using classical microbiological methods, amplicon sequence read counts obtained from a sample are random data linked to source properties (e.g., proportional composition) by a probabilistic process. Thus, diversity analysis has focused on diversity exhibited in (normalized) samples rather than probabilistic inference about source diversity. This study applies fundamentals of statistical analysis for quantitative microbiology (e.g., microscopy, plating, and most probable number methods) to sample collection and processing procedures of amplicon sequencing methods to facilitate inference reflecting the probabilistic nature of such data and evaluation of uncertainty in diversity metrics. Following description of types of random error, mechanisms such as clustering of microorganisms in the source, differential analytical recovery during sample processing, and amplification are found to invalidate a multinomial relative abundance model. The zeros often abounding in amplicon sequencing data and their implications are addressed, and Bayesian analysis is applied to estimate the source Shannon index given unnormalized data (both simulated and experimental). Inference about source diversity is found to require knowledge of the exact number of unique variants in the source, which is practically unknowable due to library size limitations and the inability to differentiate zeros corresponding to variants that are actually absent in the source from zeros corresponding to variants that were merely not detected. Given these problems with estimation of diversity in the source even when the basic multinomial model is valid, diversity analysis at the level of samples with normalized library sizes is discussed. amplicon sequencing Shannon index Markov chain Monte Carlo normalization rarefying NETGP-494312-16 3360-E086 Natural Science and Engineering Research Council of Canada Alberta Innovates10.13039/ Analysis of microbiological data using probabilistic methods has a rich history, with examination of both microscopic and culture-based data considered by prominent statisticians a century ago (e.g., Student, 1907; Fisher et al., 1922). The most probable number method for estimating concentrations from suites of presence–absence data is inherently probabilistic (e.g., McCrady, 1915), though routine use of tables (or more recently software) obviates consideration of the probabilistic link between raw data and the estimated values of practical interest for most users. Both the analysis of microbiological data and the control of the methods through which such data are obtained are grounded in statistical theory (e.g., Eisenhart and Wilson, 1943). More recently, the issue of estimating microbial concentrations and quantifying the uncertainty therein when some portion of microorganisms gathered in an environmental sample are not observed by the analyst has added to the complexity of analyzing microscopic enumeration data (e.g., Emelko et al., 2010). These examples share the common theme that the concentration of microorganisms in some source of interest is indirectly and imprecisely estimated from the discrete data produced by microbiological examination of samples (e.g., counts of cells/colonies or the number of aliquots exhibiting bacterial growth). The burgeoning microbiological analyses grounded in polymerase chain reactions (Huggett et al., 2015) likewise feature discrete objects (specific sequences of genetic material) that are prone to losses in sample processing, but these methods are further complicated by the variability introduced through amplification and subsequent reading (e.g., fluorescence signals or sequencing). In next-generation amplicon sequencing, obtained data consist of a large library of nucleic acid sequences extracted and amplified from environmental samples, which are then tabulated into a set of counts associated with amplicon sequence variants (ASVs) or some grouping thereof (Callahan et al., 2017). The resulting data are regarded as a quantitative representation of the relative abundance (i.e., proportions) of various organisms in the source rather than absolute abundance (i.e., concentrations), thus leading to compositional data (Gloor et al., 2017). Among the many categories of analyses performed on such data are (1) differential abundance analysis to compare proportions of particular variants among samples and their relation to possible covariates and (2) diversity analysis that concerns the number of unique variants detected, how the numbers of reads vary among them, and how these characteristics vary among samples (Calle, 2019). Conventional analysis of these data is confronted with several problems (McMurdie and Holmes, 2014; Kaul et al., 2017; McKnight et al., 2019): (1) a series of samples can have diverse library sizes (i.e., numbers of sequence reads), motivating “normalization,” (2) there are many normalization approaches from which to choose, and (3) many normalization and data analysis approaches are complicated by large numbers of zeros in ASV tables. These issues can be overcome in differential abundance analysis through use of probabilistic approaches such as generalized linear models (e.g., McMurdie and Holmes, 2014) that link raw ASV count data and corresponding library sizes to a linear model without the need for normalization or special treatment of zeros. Diversity analysis, however, is more complicated because the amount of diversity exhibited in a particular sample (alpha diversity) or apparent similarity or dissimilarity among samples (beta diversity) is a function of library size (Hughes and Hellmann, 2005), and methods to account for this are not standardized. A variety of methods have been applied to prepare amplicon sequencing data for downstream diversity analyses, most of which involve some form of normalization. Normalization options include (1) rarefying that randomly subsamples from the observed sequences to reduce the library size of a sample to some normalized library size shared by all samples in the analysis (Sanders, 1968), (2) simple proportions (McKnight et al., 2019), and (3) a continually expanding set of data transformations, such as centered log-ratios (e.g., Gloor et al., 2017), geometric mean of pairwise ratios (e.g., Chen et al., 2018), or variance-stabilizing transformations (e.g., Love et al., 2014). Rarefying predates high throughput sequencing methods (including applications beyond sequencing of the 16S rRNA gene, such as RNA sequencing) and originated in traditional ecology. Statistically, these normalization approaches to estimation of sample diversity in the source treat manipulated sample data as a population because the non-probabilistic analysis of a sample (called a plug-in estimate) leads to a single diversity value or a single point on an ordination plot. While it would increase computational complexity to do so, it is more theoretically sound to acknowledge that the observed library of sequence reads in a sample is an imperfect representation of the diversity of the source from which the sample was collected and that no one-size-fits-all normalization of the data can remedy this. ASV counts would then be regarded as a suite of random variables that are collectively dependent on the sampling depth (library size) and underlying simplex of proportions that can only be imperfectly estimated from the available data. Analysis of election polls is somewhat analogous in that it concerns inference about the relative composition (rather than absolute abundance) of eligible voters who prefer various candidates. A key distinction is that such analysis does not presume that the fraction of respondents favoring a particular candidate or party (or some numerical transformation thereof) is an exact measurement of the composition of the electorate. Habitual reporting of a margin of error with proportional results (Freedman et al., 1998) exemplifies that such polls are acknowledged to be samples from a population in which the small number of eligible voters surveyed is central to interpretation of the data. In amplicon sequencing diversity analysis, sampling precludes measuring source diversity exactly, but it is analysis of this source diversity that is of interest. Recognizing this, Willis (2019) encouraged approaches to estimation of alpha diversity from amplicon sequencing data that estimate diversity in a source and uncertainty therein from sample data using knowledge about random error. To do this, it is critical to recognize that diversity analysis of observed amplicon sequencing counts must draw inferences about source diversity without bias and with adequate precision. Ideally, quantitative information about precision (such as error bars for alpha diversity values) should be provided. Here, (1) the random process yielding amplicon sequencing data believed to be representative of microbial community composition in the source and (2) how this theory contributes to estimating the Shannon index alpha diversity metric using such data, particularly when library sizes differ and zero counts abound, are examined in detail. Theory applied to estimate microbial concentrations in water from data obtained using classical microbiological methods is extended to this type of microbiological assay to describe both the types of error that must be considered and a series of mechanistic assumptions that lead to a simple statistical model. The mechanisms leading to zeros in amplicon sequencing data and common issues with how zeros are analyzed in all areas of microbiology are discussed. Bayesian analysis is evaluated as an approach to drawing inference from a sample library about alpha diversity in the source with particular attention to the meaning and handling of zeros. This work addresses a path to evaluating microbial community diversity given the inherent randomness of amplicon sequencing data. It is based on established fundamentals of quantitative microbiology and provides a starting point for further investigation and development. A theoretical model for the error structure in microbial data can be developed by contemplating the series of mechanisms introducing variability to the number of a particular type of microorganisms (or gene sequences) that are present in a sample and eventually observed. This prioritizes understanding how random data are generated from the population of interest (e.g., the source microbiota) from sample collection through multiple sample processing steps to tables of observed data over the often more immediate problem of how to analyze a particular set of available data. Probabilistic modeling is central to such approaches, not just a data analytics tool. Rather than reviewing and attempting to synthesize the various probabilistic methods that have been applied to amplicon sequencing, the approach herein builds on a foundation of knowledge surrounding random errors in microscopic enumeration of waterborne pathogens (e.g., Nahrstedt and Gimbel, 1996; Emelko et al., 2010) to address the inherently more complicated errors in amplicon sequencing data. This study addresses the foundational matter of inferring a source microbiota alpha diversity metric from an individual sample because dealing with more complex situations inherent to microbiome analysis requires a firm grasp of such simple scenarios. Accordingly, hierarchical models for alpha diversity analyses that link samples to a hypothetical meta-community (e.g., McGregor et al., 2020) and approaches for differential abundance analysis in which the covariation of counts of several variants among multiple samples may be a concern (e.g., Mandal et al., 2015) are beyond the scope of this work. The developed modeling framework reflects that microorganisms and their genetic material are discrete, both in the source and at any point in the multi-step process of obtaining amplicon sequencing data. It also reflects that each step is random, potentially decreasing or in some cases increasing the (unobserved) discrete number of copies of each sequence variant. This applies a systems approach to describing the mechanisms through which discrete sequences are lost or created. This differs from previous work (e.g., McLaren et al., 2019) that does not reflect the discrete nature of microorganisms and their genetic material, assumes deterministic (i.e., non-random) and multiplicative effects of each sample processing step, and assumes only a single source of multiplicative random error in observed proportions (when, in fact, these proportions are estimated from observed discrete counts and a finite library size). When random errors in the process linking observed data to the population characteristics of interest are integrated into a probabilistic model, it is possible to apply the model in a forward direction to simulate data given known parameter values or in a reverse direction to estimate model parameters given observed data (Schmidt et al., 2020). Analysis of simulated data generated in accordance with a mechanistically plausible probabilistic model (to determine if the analysis generates suitable inferences) is an important step in validating data-driven analysis frameworks. This reversibility is harnessed later in this paper to simulate data from a hypothetical source and evaluate how well Bayesian analysis of those data estimates the actual Shannon index of the source. The developed method is subsequently applied to a sample of environmental amplicon sequencing data. Microbial community analysis involves the collection of samples from a source, such as environmental waters or the human gut (Shokralla et al., 2012). This study addresses the context of water samples because the plausibility that some sources could be homogeneous provides a comparatively simple and well-understood statistical starting point for modeling—many other sources of microbial communities are inherently not well mixed. When a sample is collected, it is presumed to be representative of some spatiotemporal portion of a water source, such as a particular geographic location and depth in a water body and time of sampling. A degree of local homogeneity surrounding the location and time of the collected sample is often presumed so that randomness in the number of a particular type of microorganisms contained in the sample (random sampling error) would be Poisson distributed with mean equal to the product of concentration and volume. There are many reasons for which a series of samples presumed to be replicates from a particular source may yield microorganism counts that are overdispersed relative to such a Poisson distribution (Schmidt et al., 2014), including (1) clustering of microorganisms to each other or on suspended particles, (2) spatiotemporally variable concentration, (3) variable volume analyzed, and (4) errors in sample processing and counting of microorganisms. Variable concentration and inconsistent sample volumes are not considered herein because the focus is on relative abundance (i.e., not estimation of concentrations) and samples that are not presumed to be replicates (i.e., analysis focuses on individual samples). Non-random dispersion could be a concern affecting estimates of diversity and relative abundance because clustering may inflate variability in the counts of a particular type of microorganisms. For example, clustering could polarize results between unusually large numbers if a large cluster is captured and absence otherwise rather than yielding a number that varies minimally around the average. The remainder of this analysis focuses on errors in sample handling and processing, nucleic acid amplification, and gene sequence counting. To be representative of relative abundance of microorganisms in the source, it is presumed that a sample is handled so that the community in the analyzed sample is compositionally equivalent to the community in the sample when it was collected (Fricker et al., 2019). Any differential growth or decay among types of microorganisms or sample contamination will bias diversity analysis. A series of sample processing steps is then needed to extract and purify the nucleic material so that the sample is reduced to a size and condition ready for PCR. Losses may occur throughout this process, such as adhesion to glassware, residuals not transferred, failure to extract nucleic material from cells (Fricker et al., 2019), and sample partitioning during concentration and/or purification steps. These introduce random analytical error (because a method with 50% analytical recovery cannot recover 50% of one discrete microorganism, for example), and likely also non-constant analytical recovery if the capacity of the method to recover a particular type of microorganisms varies randomly from sample to sample (e.g., 60% in one sample and 40% in the next). Even with strict control of water matrix and method, the analytical recovery of microbiological methods is known to be highly variable in some cases (e.g., United States Environmental Protection Agency, 2005). Any differential analytical recovery among types of microorganisms (e.g., if one type of microorganisms is more likely to be successfully observed than another) will bias diversity analysis of the source (McLaren et al., 2019). Varying copy numbers of genes among types of microorganisms as well as genes associated with non-viable organisms can also bias diversity analysis if the goal is to represent diversity of viable organisms in the source rather than diversity of gene copies present in the source. PCR amplification is then performed with specific primers to amplify targeted genes, which may not perfectly double the number of gene copies in each cycle due to various factors including primer match. Any differential amplification efficiency among types of microorganisms will bias diversity analysis of the source (McLaren et al., 2019), as will amplification errors that produce and amplify variants that do not exist in the source (unless these are readily identified and removed from sequencing data). Finally, the generated library of sequence reads is only a subsample of the sequences present in the amplified sample. Production of sequences that are not present in the original sample (e.g., chimeric sequences and misreads) is a form of loss if they detract from sequences that ought to have been read instead, and the resulting sequences may not be perfectly removed from the data (either failing to remove invalid sequences or erroneously removing valid sequences). Erroneous base calling is one such mechanism of sequencing error (Schirmer et al., 2015). Any differential losses at this stage will once again bias diversity analysis of the source (McLaren et al., 2019), as will inadvertent inclusion of false sequences. Thus, the discrete number of microorganisms gathered in a sample, the discrete number of genes successfully reaching amplification, the discrete number of genes after amplification, and the discrete number of genes successfully sequenced are all random. Due to this collection of unavoidable but often describable random errors, the validity of diversity analysis approaches that regard samples (or normalized transformations of them) as exact compositional representations of the source requires further examination. For all of the reasons described above, it is impractical to regard libraries of sequence reads as indicative of absolute abundance in the source. We suggest that it is also impossible to regard them as indicative of relative abundance in the source without acknowledging a suite of assumptions and carefully considering what effect departure from those assumptions might have. By presuming that sequence reads are generated independently based on proportions identical to the proportional occurrence of those sequences in the source from which the sample was collected, the randomness in the set of sequence reads will follow a multinomial distribution (for large random samples from small populations, however, a multivariate hypergeometric model may be more appropriate). This is analogous to election poll data (if the poll surveys a small random sample of voters from a large electorate), repeatedly rolling a die, or repeatedly drawing random lettered tiles from a bag with replacement. This model may form the basis of logistic regression to describe proportions of sequences of particular types as a function of possible covariates in differential abundance analysis, reflecting how count data are random variables depending on respective library sizes and underlying proportions of interest. Multinomial models are foundational to probabilistic analysis of count-based compositional data (e.g., McGregor et al., 2020), but mechanisms through which natural variability arises in the source (such as microorganism dispersion) and the sample collection and processing methodology (such as losses, amplification, and subsampling) must be considered because they may invalidate such a model for amplicon sequencing data—these need to be considered. Table 1 summarizes the random errors discussed above, contextualizes them in terms of compatibility with the multinomial relative abundance model, and summarizes the assumptions that must be made to use a multinomial model. Although this table addresses the context of amplicon sequencing data, it could apply to other applications of diversity analysis with specific modules excluded or modified (e.g., it could apply to metagenomics with amplification excluded for shotgun sequencing, and it may apply more broadly to ecology with Summary of random errors in amplicon sequencing and associated assumptions in the multinomial relative abundance model. Error source Description of error and compatibility with multinomial model Assumptions Sample The random sampling error describing variability in the number of discrete objects captured in a sample yields a Poisson distribution if All microorganisms are randomly dispersed collection microorganisms are randomly dispersed in a large source. This error is compatible with a multinomial model for proportional abundance of (i.e., not clustered) with only one gene copy variants. Clustering, including multiple gene copies per organism, leads to excess variability that is incompatible with a multinomial each^ Sample The number of a particular type of microorganisms may increase or decrease between sample collection and sample processing. Growth No growth handling inflates the number of microorganisms at the level of diversity represented before growth occurred and is incompatible with a multinomial model. Decay is a form of random analytical error that is compatible with a multinomial model if it is consistent among variants. No differential decay (analytical recovery) among variants Sample The number of gene sequences subjected to amplification may be lower than the number in the sample prior to processing due to losses No differential losses (analytical recovery) processing (e.g., adherence to apparatus, not all genes extracted, sample partitioning). This is compatible with a multinomial model if analytical among variants recovery is constant among variants. Amplification The number of gene sequences is purposefully increased using polymerase chain reactions, inflating the number of gene sequences at the Pre-amplification variant diversity is fully level of diversity represented before amplification occurred, and is incompatible with a multinomial model. Copy errors are a form of identical to source diversity and sequences are loss for the original sequences that were incorrectly copied and produces erroneous sequences that may then be further amplified. perfectly duplicated in each PCR cycle^* Erroneous sequences are incompatible with a multinomial model unless all of them are removed from the data. No differential amplification efficiency or potential for copy errors among variants Amplicon Only a subsample of sequences are read, and all variants must be equally likely to be read. Sequence reading errors are a form of loss No differential sequence reading errors among sequencing for the original sequences that were incorrectly read and also produces erroneous sequence reads. Sequence reading errors are variants or differential losses incompatible with a multinomial model unless all resultant erroneous sequences are removed from the data. Data denoising must remove all erroneous sequence reads and no legitimate reads Without these difficult assumptions, the multinomial model describes post-amplification variant diversity rather than source microbial diversity. Based on some simulations (see R code in Supplementary Material), it was determined that random sampling error consistent with a Poisson model is compatible with the multinomial relative abundance model (using the binomial model as a two-variant special case). Specifically, this featured Poisson-distributed counts of two variants with means following a 2:1 ratio and graphical evidence that this process is consistent with a binomial model (also with a 2:1 ratio of the two variants) when the result was conditioned on a particular library size. It must be noted that this is not a formal proof, as “proof by example” is a logical fallacy (unlike “disproof by counter-example”). Critically, clustering of gene copies in the source causes the randomness in sequence counts to depart from a multinomial model, as proven by simulation in the Supplementary Material (following a disproof by counter-example approach). When the above process was repeated with counts following a negative binomial model that is overdispersed with respect to the Poisson model, the variation in counts conditional on a particular library size was no longer consistent with the binomial model. Microorganisms having multiple gene copies is a form of clustering that invalidates the model. Any form of loss or subsampling is compatible with the multinomial model so long as it affects all sequence variants equally. If each of a set of original proportions is multiplied by the same weight (analytical recovery), then the set of proportions adjusted by this weighting is identical to the original proportions (e.g., a 2:1 ratio is equal to a 1:0.5 ratio if all variants have 50% analytical Growth and amplification must also not involve differential error among variants, but even in absence of differential error they have an important effect on the data and evaluation of microbiota diversity. These processes inflate the number of sequences present, but only with the potentially reduced or atypical diversity represented in the sample before such inflation. For example, a hypothetical sample with 100 variants amplified to 1,000 will have the diversity of a 100-variant sample in 1,000 reads, which may inherently be less than the diversity of a 1,000-variant sample directly from the source. Amplification fabricates additional data in a process somewhat opposite to discarding sequences in rarefaction; it draws upon a small pool of genetic material to make more whereas rarefaction subsamples from a larger pool of gene sequences to yield less (i.e., a smaller library size). Based on some simulations (see R code in Supplementary Material), it was proven that amplification is incompatible with the multinomial relative abundance model (following a disproof by counter-example approach). Specifically, the distribution of counts when two variants with a 2:1 ratio are amplified from a library size of four to a library size of six differs from the distribution of counts obtained from a binomial model. Representativeness of source diversity and compatibility with the multinomial relative abundance model can only be assured if the post-amplification diversity happens to be fully identical to the pre-amplification diversity and the observed library is a simple random sample of the amplified genetic material. Such an assumption may presume random happenstance more so than a plausible probabilistic process, though it would be valid in the extreme special case where pre-amplification diversity is fully identical to source diversity and every sequence is perfectly duplicated in each cycle (with no erroneous sequences produced). Without making relatively implausible assumptions or having detailed understanding and modeling of the random error in amplification, observed libraries are only representative of post-amplification diversity and indirectly representative of source diversity. This calls into question the theoretical validity of multinomial models as a starting point for inference about the proportional composition of microbial communities using amplification-based data. Nonetheless, the multinomial model was used as part of this study in some illustrative simulation-based diversity analysis experiments. As in other fields (Helsel, 2010), zeros in microbiology have led to much ado about nothing (Chik et al., 2018). They are (1) commonly regarded with skepticism that is hypocritical of non-zero counts (e.g., assuming that counts of zero result from error while counts of two are infallible), (2) often substituted with non-zero values or omitted from analysis altogether, and (3) a continued subject of statistical debate and special attention (such as detection limits and allegedly censored microbial data). Careful consideration of zeros is particularly relevant to diversity analysis of amplicon sequencing data because they often constitute a large portion of ASV tables. They may or may not appear in sample-specific ASV data, but they often appear when the ASV table of several samples is filled out (e.g., when an ASV that appears in some samples does not appear in others, zeros are assigned to that ASV in all samples in which it was not observed). They may also be created by zeroing singleton reads (Callahan et al., 2016), but this issue (and the bias arising if some singletons are legitimate read counts) is not specifically addressed in this study. Zeros often receive special treatment during the normalization step of compositional microbiome analysis (Thorsen et al., 2016; Tsilimigras and Fodor, 2016; Kaul et al., 2017), including removal of rows of zeros and fabrication of pseudo-counts with which zeros are substituted (to enable logarithmic transformations or optimal beta diversity separation). However, it is fundamentally flawed to justify use of pseudo-counts to “correct for” zeros arguing that they are censored data below some detection limit (Chik et al., 2018; Cameron et al., 2021). In methods that count discrete objects (e.g., microorganisms or gene copies) in samples, counts of zero are no less legitimate than non-zero counts: all random count-based data provide imperfect estimation of source properties, such as microorganism concentrations or proportional composition. We propose a classification of three types of zeros: (1) non-detected sequences (also called rounded or sampling zeros), (2) truly absent sequences (also called essential or structural zeros), and (3) missing zeros. This differs from the three types of zeros discussed by Kaul et al. (2017) because the issue of missing zeros (which is shown to be critically important in diversity analysis) was not noted in that study and zeros that appear to be outliers from empirical patterns are not considered in this study (because all random read counts are presumed to be correct). Likewise, Tsilimigras and Fodor (2016) differentiated between essential or structural zeros (truly absent sequences) and rounded zeros (resulting from undersampling), but did not consider missing zeros. It is typically presumed that zeros correspond to non-detected sequences, meaning that the variant is present in some quantity in the source but happened to not be included in the library and is represented by a zero. A legitimate singleton that is replaced with a zero would be a special case of a non-detect zero. Bias would result if non-detect zeros were omitted or included in the diversity analysis inappropriately (e.g., substitution with pseudo-counts or treating them as definitively absent variants). It is conceptually possible that a particular type of microorganisms may be truly absent from certain sources so that the corresponding read count and proportion should definitively be zero. If false sequences due to errors in amplification and sequencing are filtered from the ASV table but left as zeros, then they are a special case of truly absent sequences. Bias would result if such zeros were included in diversity analysis in a way that manipulates them to non-zero values or allows the corresponding variant to have a plausibly non-zero proportion. Missing zeros are variants that are truly present in the source and not represented in the data—they are not acknowledged to be part of the community, even with a zero in the ASV table. Bias would result from excluding these zeros from diversity analysis rather than recognizing them as non-detected variants. Thus, there are three types of zeros, two of which appear indistinguishably in the data and must be handled differently and the third of which is important but does not even appear in the data. In this study, simulation-based experiments and environmental data are used to illustrate implications of the dilemma of not knowing how many zeros should appear in the data to be analyzed as The Shannon index (Shannon, 1948; Washington, 1984) is used as a measure of alpha diversity that reflects both the richness and evenness of variants present (number of unique variants and similarity of their respective proportions). When calculated from a sample, the Shannon index (S) depends only on the proportions of the observed variants (p[i] for the ith of n variants) and not on their read counts. Critically, the Shannon index of a sample is not an unbiased estimate of the Shannon index of the source (even in scenarios without amplification); it is expected to increase with library size as more rare variants are observed until it converges asymptotically on the Shannon index of the source (Willis, 2019). Even if all variants in the source are reflected in the data, the precision of the estimated Shannon index will improve with increasing library size. S = − ∑ i = 1 n p i × ln p i Building on existing work applying Bayesian methods to characterize the uncertainty in enumeration-based microbial concentration estimates (e.g., Emelko et al., 2010) and inspired by the need to consider random error in evaluation of alpha diversity that was noted by Willis (2019), a Bayesian approach is explored here for the simplified scenario of multinomially distributed data. It evaluates uncertainty in the source Shannon index given sample data, the multinomial model, and a relatively uninformative Dirichlet prior that gives equal prior weight to all variants (using a vector of ones). Hierarchical modeling that may describe how the proportional composition varies among samples is beyond the scope of this analysis. Such modeling can be beneficial when strong information in the lower tier of the hierarchy can be used to probe the fit of the upper tier; however, it can be biased if limited information in the lower tier is bolstered with flawed assumptions introduced via the upper tier. Here, a simulation study is employed that is analogous to compositional microbiota data with small library sizes and small numbers of variants and that does follow a multinomial relative abundance model. The simulation uses specified proportions for a set of variants; for illustrative purposes, the simulation represents random draws with replacement from a bag of lettered tiles based on the game Scrabble™. Randomized multinomial data (Supplementary Table S1, Supplementary Material) were generated in R using varying library sizes and the proportions of the 100 tiles (including 26 letters and blanks), which correspond to a population-level Shannon index of 3.03. Markov chain Monte Carlo (MCMC) was carried out using OpenBUGS (version 3.2.3), with randomized initialization and 10,000 iterations following a 1,000-iteration burn-in. The model specification code and a small sample dataset are included in the Supplementary Material. Due to the mathematical simplicity of a multinomial model with a Dirichlet prior, this number of iterations can be completed in seconds with rapid convergence and good mixing of the Markov chain. Each iteration generates an estimate of each variant proportion, and the set of variant proportions is used to compute an estimate of the Shannon index for the source inferred from the sample data. The Markov chain of Shannon index values generated in this way is collectively representative of a sample from the posterior distribution that characterizes uncertainty in the source Shannon index given the sample data and prior. The simulated data were analyzed in several ways, as illustrated using box and whisker plots in Figure 1: (1) with all non-detected tile variants removed, (2) with zeros added as needed to reach the correct number of tile variants used to simulate the data (i.e., 27), and (3) with extraneous zeros (a total of 50 tile variants of which 23 do not actually exist in the source). Box and whisker plot of Markov chain Monte Carlo (MCMC) samples from posterior distributions of the Shannon index based on analysis of simulated data (Supplementary Data Sheet 2). Data with various library sizes (Supplementary Table S1) were analyzed in each of three ways: with zeros excluded (not applicable in some cases), with zeros included for non-detected variants, and with extraneous zeros corresponding to variants that do not exist in the source. The true Shannon index of the source from which the data were simulated is 3.03. The disparity in results between the three ways in which the data were analyzed exemplifies the importance of zeros in estimating the Shannon index of the source from which a sample was gathered. Omitting non-detect zeros in this Bayesian analysis characteristically underestimates diversity, while including zeros for variants that do not exist in the source characteristically overestimates diversity. In each case, the effect diminishes as the library size is increased. Notably, the approach that included only zeros for variants present in the source that were not detected in the sample allowed accurate estimation of the source Shannon index, with improving precision as the library size increases (exemplifying statistical consistency of the estimation process). Given these results, the proposed Bayesian process appears to be theoretically valid to estimate the source Shannon index from samples (for which the multinomial relative abundance model applies), and it does so without the need to normalize data with differing library sizes. Practically, however, it is not possible to know how many zeros should be included in the analysis estimating the Shannon index because the number of unique variants actually present in the source is unknown. This is a peculiar scenario that must be emphasized here because accurate statistical inference about the source is impossible: although the model form (multinomial) is known, the number of unique variants that should be included in the model is practically unknowable. Model-based supposition is not applied in this study to introduce information that is lacking; this can be a biased approach to compensating for deficiencies in observed data or flawed experiments in which “control variables” are not controlled (e.g., it is not possible to estimate concentration from a count without a measured volume) unless the supposition happens to be correct (Schmidt et al., 2020). Because the extent to which zeros compromised accurate estimation waned with increasing library size (Figure 1), a similar analysis was performed on amplicon sequencing data for six water samples from lakes. The samples (Cameron et al., 2021, Supplementary Data Sheet 2) featured library sizes between 10,000 and 30,000 and observation of 1,142 unique variants among the samples. All singleton counts had been zeroed and the completed ASV table had 3,342 rows (2,200 of which are all zeros associated with variants detected in other samples from the same study area). Each sample was analyzed three ways: (1) with all non-detected sequence variants removed, (2) with zeros as needed to fill out the 1,142-row ASV table, and (3) with zeros as needed to fill out the 3,342-row ASV table. The appropriate number of zeros to be included for each sample cannot be known, but the Shannon index estimated with all non-detected sequence variants removed is very likely underestimated. The results (Figure 2) show that the Shannon index can be quite precisely estimated (narrow error bars) with library sizes nearing 30,000 sequences but that the number of zeros included in the analysis can still have a substantial effect on accurate estimation of the Shannon index of the source (results, though precise, vary widely with the number of zeros included in the data). It is thus concluded that it is not statistically possible to estimate the Shannon index of the source (even if all the assumptions are met that enable use of the multinomial relative abundance model) unless the number of unique variants present in the source is precisely known a priori. Accordingly, the following section elaborates upon evaluation of sample-level diversity accounting for varying library sizes and the mechanistic process by which data are obtained. Box and whisker plot of MCMC samples from posterior distributions of the Shannon index based on analysis of environmental amplicon sequencing data (Supplementary Data Sheet 2). Data with various library sizes between 10,000 and 30,000 were analyzed three ways: with zeros excluded, with zeros included in a 1,142-row ASV table (no zero rows), and with additional zeros from the full 3,342-row ASV table including variants with rows of zeros (detected in other samples from the same study area). Recognizing that amplicon sequencing of a sample provides only partial and indirect representation of the diversity in the source (specifically partial representation of post-amplification diversity) and that statistical inference about source diversity is compromised by clustering, amplification, and not knowing how many zeros should be included in the data, the question of how to perform diversity analysis remains. The approach should acknowledge the random nature of amplicon sequencing data, reflect the importance of the library size in progressively revealing information about diversity, avoid normalization that distorts the proportional composition of samples, and provide some measure of uncertainty or error. Inference about source diversity is the ideal, but it is not possible with a multinomial relative abundance model unless the number of unique variants in the source is precisely known, and there are many types of error in amplicon sequencing that are likely to invalidate this foundational model as discussed above. Rarefying repeatedly, a subsampling process to normalize library sizes among samples that is performed many times in order to characterize the variability introduced by rarefying (Cameron et al., 2021), satisfies these goals. When a sample is rarefied repeatedly down to a smaller library size (using sampling without replacement), it describes what data might have been obtained if only the smaller library size of sequence variants had been observed. This can then be propagated to develop a range of values of an alpha diversity metric or a cluster of points on a beta diversity ordination plot to graphically display the variability introduced by rarefying to a normalized library size. It also does not throw out valid sequences because all sequences are represented with a sufficiently large number of repetitions. A value of the sample Shannon index may then be computed for each of the repetitions to quantify the diversity in samples of a particular library size. Figure 3 illustrates the relationship between repeatedly rarefying to smaller library sizes and statistical inference about the source from which the sample was taken. Rarefying adds random variability by subsampling without replacement while statistical inference includes parametric uncertainty that is often ignored in contemporary diversity analyses. Because the extent to which diversity is exhibited by a sample depends on the library size, such sample-level analysis must be performed at the same level (analogous to converting 1mm, 1cm, and 1km to a common unit before comparing numerical values) and any observations obtained about patterns in sample-level diversity are conditional on the shared library size at which the analysis was performed. On the other hand, current methods (including rarefying once), distort the data to facilitate use of compositional analysis methods that presume the transformed data are a perfect representation of the microbial composition in the source; it is important to recognize that the detected library is only a random sample that is imperfectly representative of source diversity. Representation of how library size and diversity quantified therefrom relate to uncertainty in statistical inference about source diversity and variability introduced by repeatedly rarefying to the smallest obtained library size. In this case, rarefying repeatedly evaluates the extent of the diversity (after amplification) exhibited if a library size of only n=5,000 had been obtained from each sample. Cameron et al. (2021) addressed issues such as choice of normalized library size and sampling with or without replacement and completed both alpha and beta diversity analyses using repeated rarefaction and environmental data, but it did not include simulation experiments. Accordingly, a simulation experiment was performed using the hypothetical population-based on Scrabble™ and samples with varying library sizes (see R code in Supplementary Material) to explore rarefying repeatedly and plug-in estimation of the Shannon index (Figure 4). A thousand simulated datasets with a library size of 25 yielded Shannon index values between 1.86 and 2.87 (with a mean of 2.49), illustrating that the source diversity (with a Shannon index of 3.03) is only partially exhibited by a sample with a library size of 25. Five samples were generated with library sizes of 50, 100, 200, 500, and 1,000, and corresponding Shannon index values are shown in red (deteriorating markedly at library sizes of 100 or less). Each sample was then rarefied repeatedly (1,000 times) to a library size of 25, resulting in the box and whisker plots of the calculated Shannon index values. Although samples with larger library sizes exhibit more diversity, samples repeatedly rarefied down to the minimum library size of 25 exhibit very comparable diversity. The Shannon index at a library size of 25 is similar for all samples, as it should be given that they were generated from the same population. If rarefying had been completed only once without quantification of the error introduced, it may erroneously have been concluded that the samples exhibited different Shannon index values. Demonstration of normalization by rarefying repeatedly using simulated data. The box and whisker plot for the library size of 25 (*) illustrates how the Shannon index varies among simulated samples and is consistently below the actual Shannon index of 3.03 (red line). The Shannon index calculated from the samples with larger library sizes (red dots) deteriorates at small library sizes. The box and whisker plots for these library sizes illustrate what Shannon index might have been calculated if only a library size of 25 had been obtained (rarefying 1,000 times to this level). In all cases, a Shannon index of about 2.5 is expected with a library size of 25. Diversity analysis of amplicon sequencing data has grown rapidly, adopting tools from other disciplines but largely differing from the statistical approaches applied to classical microbiology data. Most analyses feature a deterministic set of procedures to transform the data from each sample and yield a single value of an alpha diversity metric or a single point on an ordination plot. Such procedures should not be viewed as statistical analysis because observed sequence count data are not a population (i.e., perfect measurements of the proportional composition of the community in the source); they are a random sample representing only a portion of that population. Recognizing that the data are random and that the goal is to understand the alpha and beta diversity of the sources from which samples were collected, it is important to describe and explore the error mechanisms leading to variability in the data and uncertainty in estimated diversity. Additionally, graphical displays of results should reflect uncertainty, such as by presenting error bars around alpha diversity values. This study provides a step toward such methods by describing mechanistic random errors and their potential effects, proposing a probabilistic model and listing the assumptions that facilitate its use, discussing various types of zeros that may appear (or fail to) in ASV tables, and performing illustrative analyses using simulated and environmental data. Several sources of random error were found to invalidate the multinomial relative abundance model that is foundational to probabilistic modeling of compositional sequence count data, notably including clustering of microorganisms in the source and amplification of genes in this sequencing technology. Nonetheless, it is a good starting point for inference of the alpha diversity of the source and quantifying uncertainty therein. Future simulation studies could explore the effect of non-random microorganism dispersion, sample volume (relative to a hypothetical representative elementary volume of the source), differential analytical recovery in sample processing, amplification errors, and sequencing errors on diversity analysis more thoroughly and evaluate the potential for current normalization and point estimation approaches to misrepresent diversity. This study also presents a simple Bayesian approach to drawing inference about diversity in the sources from which samples were collected (rather than just diversity in the sample or some transformation of it). Even under idealized circumstances in which the multinomial relative abundance model is valid, it was unfortunately found to be biased unless the number of unique variants present in the source was known a priori. This may have implications on analysis of any type of multinomial data, beyond microbiome data, in which the number of possible outcomes (or the number of outcomes with zero observations that should be included in the analysis) is unknown. It is plausible that a probabilistic model could be developed to account for errors that invalidate the multinomial model, though this would require many assumptions that would be difficult to validate and that could substantially bias inferences if the assumptions are incorrect. In summary, probabilistic modeling should be used to draw inferences about source diversity and quantify uncertainty therein, but the simple multinomial model is invalidated by some types of error that are inherent to the method and inference is not possible even with the multinomial model unless the practically unknowable number of unique variants in the source is known. For lack of a reliable and readily available probabilistic approach to draw inferences about source diversity, an approach to evaluate and contrast sample-level diversity at a particular library size is needed. Rarefying only once manipulates the data in a way that adds variability and discards data (McMurdie and Holmes, 2014), and (like other transformations proposed to normalize data) the manipulated data are generally only used to obtain a plug-in estimate of diversity. Rarefying repeatedly, on the other hand, allows comparison of sample-level diversity estimates conditional on a library size that is common among all analyzed samples, does not discard data, and characterizes variability in what the diversity measure might have been if only the smaller library size had been observed. This approach is by no means statistically ideal, but it may be a distant second best relative to the Bayesian approach (or analogous frequentist approaches based on the likelihood function) presented in this study that cannot practically be applied in an unbiased way in many scenarios, especially due to the unknowable number of unique variants that are actually present in the source and complex error structures inherent to amplicon sequencing. The original contributions presented in the study are included in the Supplementary Material; further inquiries can be directed to the corresponding author. PS and EC developed the theoretical model with support from ME and KM. PS conceived the statistical analysis approach, carried out or directed the simulation experiments and data analysis, and developed the visuals with support from ME and EC. PS wrote the manuscript with support from ME, EC, and KM. All authors contributed to the article and approved the submitted version. We acknowledge the support of NSERC (Natural Science and Engineering Research Council of Canada), specifically through the forWater NSERC Network for Forested Drinking Water Source Protection Technologies (NETGP-494312-16), and Alberta Innovates (3360-E086). This research was undertaken, in part, thanks to funding from the Canada Research Chairs Program (ME; Canada Research Chair in Water Science, Technology & Policy). The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher. We are grateful for discussion of this work with Mary E. 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Diversity, biotic and similarity indices: a review with special relevance to aquatic ecosystems. 18, 653–694. doi: 10.1016/0043-1354(84)90164-7 Willis A. D. (2019). Rarefaction, alpha diversity, and statistics. 10:2407. doi: 10.3389/fmicb.2019.02407, PMID: 31708888	{"url":"https://www.frontiersin.org/journals/microbiology/articles/10.3389/fmicb.2022.728146/xml/nlm?isPublishedV2=false","timestamp":"2024-11-03T16:14:44Z","content_type":"application/xml","content_length":"93191","record_id":"<urn:uuid:8f2fc935-5a49-4f4c-8e32-9f1143f14cde>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00063.warc.gz"}
The Implicit Function Theorem and Its Proof ┃ ┃ ┃ applet-magic.com ┃ ┃ Thayer Watkins ┃ ┃ Silicon Valley ┃ ┃ & Tornado Alley ┃ ┃ USA ┃ ┃ ┃ ┃ The Implicit Function Theorem ┃ ┃ and Its Proof ┃ The Implicit Function Theorem (IFT) is a generalization of the result that If G(x,y)=C, where G(x,y) is a continuous function and C is a constant, and ∂G/∂y≠0 at some point P then y may be expressed as a function of x in some domain about P; i.e., there exists a function over that domain such that Note that without any loss of generality the constant C can be taken to be 0. If G(x,y)=C then G(x,y)=G(x,y)-C=0. The IFT is a very important tool in economic analysis and so the conditions under which it holds must be carefully specified. The simplest way to do this is to give a formal, explicit proof of the theorem. First a proof of an artifically limited version of the IFT will be given and this will provide an understanding and a guide to the proof of the full version. • Theorem 1: Let F(x,y,z) be a continuous function with continuous partial derivatives defined on an open set S containing the point P=(x[0],y[0],z[0]). If ∂F/∂z ≠ 0 at P then there exists a region R about (x[0]y[0]) such that for any (,y) in R there is a unique z such that F(x,y,z)=0. This means that within R z can be represented as a function of x and y; i.e., z=f(x,y). Without any loss of generality the set S can be taken to be a parallelpiped, a box,centered on P because within the set S there will be at least one such box. Within any such box there will be another box containing P such that ∂F/∂z has the the same sign as at P. Let the box be given as a triple (a,b,c) such that \|x-x[0]\| ≤ a \|y-y[0]\| ≤ b \|z-z[0]\| ≤ c Without any loss of generality the sign of ∂F/∂z at P can be taken to be positive. This means that F(x[0],y[0],z[0]+c) > 0 F(x[0],y[0],z[0]-c) < 0 Now consider any point (x[1],y[1]) such that \|x[1]-x[0]\| ≤ a \|y[1]-y[0]\| ≤ b Because F(x[0],y[0],z[0]+c) > 0 and F(x,y,z) is continuous, F(x[1],y[1],z[0]+c) > 0. Likewise F(x[1],y[1],z[0]-c) < 0. With x and y held fixed at x[1] and y[1], G(z)=F(x[1],y[1],z) is a function such that G(z[0]+c) > 0 and G(z[0]-c) < 0. Therefore there is some z between z[0]-c and z[0]+c such that G(z)=0; i.e., F(x[1],y[1],z)=0. Moreover this value of z is unique. Since this holds for any (x,y) \|x-x[0]\| ≤ a \|y-y[0]\| ≤ b over this domain there a function z=f(x,y) such that F(x,y,z)=0. Corollary 1: The function f(x,y) determined above is continuous. Corollary 2: The partial derivatives of the function f(x,y) determined above are given by: ∂f/∂x = -(∂F/∂x)/(∂F/∂z) ∂f/∂y = -(∂F/∂y)/(∂F/∂z) The Mean Value Theorem says that for a function z=h(x) with a continuous derivative Δz = h(x+Δx)-h(x) = h'(x + θΔ)Δx for some θ between 0 and 1. This can be extended to a binary function w=G(x,z) with continuous partial derivatives so that Δw = G(x+Δx,z+Δz)-G(x,z)=[(∂G/∂x)Δx+(∂G/∂z)Δz] where the partial derivatives are evaluated at (x+θΔx,z+θΔz) for some θ between 0 and 1. This is not the only generalization of the Mean Value Theorem but it is sufficient for the purpose here. Let G(x,z) be F(x,y,z) with y held fixed. Then ΔF = (∂F/∂x)Δx + (∂F/∂z)Δz where the partial derivatives are evaluated at (x+θΔx,z+θΔz) for some θ between 0 and 1. Now let Δz be the change in z for z on the surface F(x,y,z)=0. Thus ΔF=0 and hence Δz/Δx = - (∂F/∂x)/(∂F/∂z) where the partial derivatives are evaluated at (x+θΔx,z+θΔz) for some θ between 0 and 1. In the limit as Δx goes to zero this becomes dz/dx = df/dx = - (∂F/∂x)/(∂F/∂z). Likewise, by an analogous procedure, dz/dy = df/dy = - (∂F/∂y)/(∂F/∂z). • Theorem 2: Let F(x[1],x[2],...,x[n],z) be a continuous function with continuous partial derivatives defined on an open set S containing the point P=(x[1](P),x[2](P),...,x[n](P),,z(P)). If ∂F/∂z ≠ 0 at P then there exists a region R about (x[1](P),x[2](P),...,x[n](P)) such that for any (x[1],x[2],...,x[n]) in R there is a unique z such that F(x[1],x[2],...,x[n],z)=0 and thus over R z=f(x The proof is essentially the same as for Theorem 1, but some unnecessary restrictions in the proof can be removed. Within the set S there will be a parallelpiped containing P and specified by its lower and upper corner points (X[1]^L,...,X[n]^L,z^L) and (X[1]^UX[1]^U,z^U) such that: X[1]^L < x[1] < X[1]^U X[n]^L < x[n] < X[n]^U and ∂F/∂z has the same sign as at P. Then F(x[1](P),...,x[1](P),z^U) and F(x[1](P),...,x[1](P),z^L) will have opposite signs. For any point Q within the parallelpiped, ignoring the coordinate z, will have the same signs as for P and are thus of opposite sign. For a fixed Q then G(z)=F(x[1](P),...,x[1](P),z) has opposite signs at z^U and z^L and therefore there is a z between these two levels such that G(z)=0. This value of z is a function of ((x[1](Q),...,x[1](Q)); i.e., z=f((x[1](Q),...,x[1](Q)). • Theorem 3: Let F(x,y,u,v) and G(x,y,u,v) be continuous function with continuous partial derivatives. If F(x,y,u,v)=0 and G(x,y,u,v)=0 at some point P in an open set S and the Jacobian J(F,G,u,v) is not zero at P then there exists a domain about P such that u=f(x,y) and v=g(x,y). Consider F(x,y,u,v)=0 and the point P and apply Theorem 2 to get v as a function of x,y and u; i.e., find v=h(x,y,u). Substitute h(x,y,u) for v in G(x,y,u,v) to obtain H(x,y,u)=G(x,y,u,h(x,y,u)). H (x,y,u)=0 so Theorem 1 applies and thus there exists f(x,y) such that u=f(x,y). A substitution of this function for u in v=h(x,y,u) gives v=g(x,y)=h(x,y,f(x,y)). • Theorem 4: Let F[i](x[1],...,x[n],u[1],..,u[m]) be a set of m continuous functions with continuous partial derivatives defined on an open set S containing the point P=(x[1](P),x[2](P),...,x[n] (P),,u[1](P),...,u[m](P)). If the Jacobian of the functions {F[i]: i=1,...,m} with respect to the variables {u[i]: i=1,...,m} is not equal to 0 at P then there exists a region R about (x[1](P),x [2](P),...,x[n](P)) such that for any (x[1],x[2],...,x[n]) in R there is a unique point (u[1],...,u[m]) such that for i=1 to m and thus over R u[i]=f[i](x[1],x[2],...,x[n]) for i=1 to m. Proof: The procedure is the same as in the proof of Theorem 3. Theorem 2 is applied to one of the F-functions to obtain one of the u-variables as a function of the x-variables and the other u-variables. This expression for the u-variable as function of the other variables is substituted into a second F-function and another u-variable is obtained as a function of the remaining variables until one u-variable is found as a function of only the x-variables. This u-variable is then substituted into the preceding expression for a u-variable. This process is continued until all of the u-variables are obtained as functions only of the x-variables.	{"url":"https://www.sjsu.edu/faculty/watkins/implicit.htm","timestamp":"2024-11-14T14:21:39Z","content_type":"text/html","content_length":"12110","record_id":"<urn:uuid:6526eeb6-387b-4307-ad27-733cdf4c6353>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00456.warc.gz"}
Fixed Point to Floating Point Conversion - Digital System DesignFixed Point to Floating Point Conversion - Digital System Design Fixed Point to Floating Point Conversion Generally the FPGA based architectures or ASIC based implementations are fixed point arithmetic based. But the majority of dedicated controllers are floating point based. It may require to interface a FPGA to a micro controller and then a fixed point to floating point conversion unit will be required. In order to discuss the floating point architectures, first the scheme for conversion from fixed point to floating point is discussed. The steps of converting a fixed point number to a floating point number are • Invert the number if the number is negative or if the MSB is logic 1. • Count the leading zeros present in the number. • Value of the mantissa is computed by left shifting the number according the leading zero present in the number. • Subtract the leading zero count from (m-1) where m represents the number of integer bits are reserved for integer including the MSB bit. • Add the result to the bias value to get the exponent. • Sign of the fixed point number is the sign of the floating point number. An example is given below to understand the fixed point to floating point conversion for 16-bit data width. • The input fixed point number is a = 001000_0010000000 whose decimal value is 8.125 for m is 6-bits. • As this number is positive so no inversion is required. • There are two leading zero present in the number so it is left shifted by two bits. The result of shifting is 100000_1000000000. • The value of the mantissa is M=00000_100000 by choosing 11-bits from MSB and excluding the MSB. • The leading zero count is subtracted from (m-1) and the result is (5 – 2) = 3. • The result is added to the bias to get the exponent (E = (7+3) = 10). • Thus the floating point number is 0_1010_00000100000. An simple scheme of fixed point to floating conversion is shown in Figure 1. Here two 4-bit adder/subtractor blocks are used to determine the exponent and one 16-bit adder/subrtactor is for inversion if necessary. The VLSH block stands for variable left shift according to the leading zero count. The leading zeros are counted by a block called leading zero counter. The operation of the architecture can be understood easily by following the steps mentioned above. In the mantissa computation path, 11-bits are taken as mantissa to form a 16-bit floating point output. This 16-bit representation can not represent all the numbers which are represented in the 16-bit fixed point representation. For example, in the 16-bit format the numbers which are less than Figure 1: Fixed Point to Floating Point Converter.	{"url":"https://digitalsystemdesign.in/fixed-point-to-floating-point-conversion/","timestamp":"2024-11-06T23:49:32Z","content_type":"text/html","content_length":"296705","record_id":"<urn:uuid:51c2430e-0f0e-4600-9557-5de441e726bf>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00180.warc.gz"}
Pythagorean Theorem Calculator The Pythagorean theorem is one of the fundamental theorems of geometric theory, which establishes the ratio between the sides of the rectangular triangle: the square of the hypotenuse is equal to the sum of the squares of the catheters (a^2 + b^2 = c^2) This calculator will help you find the hypotenuse.	{"url":"http://openwords.ru/en/pythagorean-theorem-calculator/","timestamp":"2024-11-05T20:15:26Z","content_type":"text/html","content_length":"27462","record_id":"<urn:uuid:9264ae90-5e64-483f-bb2c-5ad1f7d4b199>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00281.warc.gz"}
Graph Laplacian matrix: normalized, distance, undsigned Graph Laplacian and other matrices associated with a graph There are several ways to associate a matrix with a graph. The properties of these matrices, especially spectral properties (eigenvalues and eigenvectors) tell us things about the structure of the corresponding graph. This post is a brief reference for keeping these various matrices straight. The most basic matrix associated with a graph is its adjacency matrix, denoted A. For a graph with n nodes, create an n×n matrix filled with zeros, then fill in a 1 in the ith row and jth column if there is an edge between the ith and jth node. Next let D be the diagonal matrix whose ith element is the degree of the ith node, i.e. the number of edges attached to the ith node. The Laplacian matrix of the graph is L = A − D. The Laplacian matrix of a graph is analogous to the Laplacian operator in partial differential equations. It is sometimes called the Kirchhoff matrix or the admittance matrix. The signless Lapacian matrix is Q = A + D. There are connections between the signless Laplacian and bipartite components. For example, the multiplicity of 0 as an eigenvalue of Q equals the number of bipartite components in the graph. There are other variations of the Laplacian matrix. The normalized Laplacian matrix has 1’s down the diagonal. If there is an edge between the ith and jth nodes, the ij entry is where d[i] and d[j] are the degrees of the ith and jth nodes. If there is no edge between the ith and jth nodes, the ij entry is 0. The distance matrix of a graph is the matrix D whose ijth entry is the distance from the ith node to the jth node. (Sorry, this D is different than the one above. Books often write this as a script D , something I can’t do in plain HTML.) The transmission of a node v, Tr(v), is the sum of the distances from v to every element connected to v. The distance Laplacian of a graph is Diag(Tr) − D where Diag(Tr) is the diagonal matrix whose ith entry is the transmission of the ith node and D here is the distance matrix. The signless distance Laplacian of a graph is Diag(Tr) + D. Reference: Inequalities for Graph Eigenvalues More spectral graph theory 13 thoughts on “Graph Laplacian and other matrices associated with a graph” 1. The wonders of Unicode let you use both D and . I would hope it’s safe to assume adequate font coverage in this world of 2015. 2. Ha ha. I saw the math bold script D (U+1D4D3) in the comment entry box, but it’s gone now that the comment has been posted. 2015 indeed :( 3. Marius: Unfortunately Unicode support is indeed disappointing. I blogged a while back about which Unicode characters you can depend on and this list is pretty small, only a small portion of the Basic Multilingual Plane and nothing in the extended planes where things like U+1D4D3 live. 4. Nice entry! By the way, it seems someone in the Baltimore Ravens has devised a fast way to compute the Fiedler vector (that’s the second smallest eigenvector of the Laplacian, I think) http:// 5. “the multiplicity of 0 as an eigenvalue of Q equals the number of bipartite components in the graph” Does this just mean the nullity of Q is equal to the number of bipartite components of the graph, or an I misreading this? 6. Using MathJax, you can use AMSTex fonts. 7. Yes and no. MathJax will automate creating an image file from LaTeX and inserting it into your HTML, but it looks like a foreign symbol pasted into the text. That’s OK, and maybe that would be better than what I did, but it’s not great. I like to use LaTeX for displayed equations if necessary, but not in the middle of a paragraph. 8. In most cases adjacency matrices are sparse and there are a lot of techniques to manipulate (multiply or store) sparse matrices. There is a method to know whether a given graph is connected (i.e. any two vertexes can achievable). Let A is adjacency matrix of the graph. if the matrix B = I+A+A^2+.. +A^n has zero entries the the graph is disconnected and vice versa 9. Great post! Can you recommend sources or texts to learn more about these topics? 10. Rob: You could start by going to Fan Chung’s site. 11. Isn’t the Laplacian L = D-A not L = A -D? 12. @Oliver: You’ll see it defined both ways. Different conventions. 13. I’ve noticed that the laplacian eigenvalues for weighted matrices, both all-positive as well as with signed (partial) correlation matrices do not have the smallest eigenvalue = 0. How does the interpretation of what you’ve written here change for weighted and signed networks? Thank you	{"url":"https://www.johndcook.com/blog/2015/12/02/graph-matrices/","timestamp":"2024-11-04T20:43:16Z","content_type":"text/html","content_length":"70903","record_id":"<urn:uuid:36254efe-1235-4893-aa0c-a4c5724b7afa>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00169.warc.gz"}
Investigation into Resolution and Sensitivity Syllabus Requirement As part of bold point one, students must: • identify data sources, plan, choose equipment or resources for, and perform an investigation to demonstrate why it is desirable for telescopes to have a large diameter objective lens or mirror in terms of both sensitivity and resolution (skill) © NSW Board of Studies, 2002. Modern telescopes have large primary mirrors so as to be more sensitive, that is gather more photons from distant celestial objects. Theoretically, increasing the size of the primary also increases the resolution of a telescope although in practice atmospheric effects degrade the actual performance of ground-based telescopes. What is the actual relationship between size of a primary and a) sensitivity? and b) resolution? Part I: Sensitivity There are three ways you can simulate the effect of primary mirror size on sensitivity. Method 1: Filling up the "Light Bucket" 1. Draw circles representing primary mirrors. Start with a radius of 1cm and draw several up to 15cm radius. Cut them out then paste them on thick card or plywood. 2. Cut out a 1cm wide length of paper equal in length to the circumference of the mirror plus 1cm. Glue it around the circumference of the mirror so that it makes a raised edge. Repeat for the other 3. Using dried peas, round lentils or something similar to represent "photons", fill up each mirror so that it is full. 4. Either count up up the total number of "photons" in each mirror or weigh their total mass or volume as an analogue of total number. 5. Tabulate you data and then plot your results. Can you determine a relationship between mirror size and sensitivity? You may need to plot more than one graph to verify the actual mathematical If you want to use some experimental data, the table below provides some actual values from a simulation using dried lentils. Use this data to try and determine a relationship between mirror size and Mirror Diameter (cm) Vol (mL) Mass (g) 2 4 1.97 4 10 6.19 6 28 15.94 8 32 23.25 10 54 38.02 12 80 60.7 14 100 77.53 16 140 99.91 20 230 160.77 30 450 367.68 Method 2: Tiddly Winks 1. Draw circles representing primary mirrors. Start with a radius of 1cm and draw several up to 15cm radius. Label them and cut them out. 2. Use tiddly winks or some are similar small, circular marker such as paper from a hole puncher to represent "photons". Place the "photons side by side on each mirror until they are all filled up. 3. Count the number of photons each mirror has collected and tabulate your data. 4. Plot the number of photons against mirror size and determine what relationship, if any, there is between the two. Method 3: Mathematical Model 1. Construct a table similar to the one below for a variety of telescopes from the last four centuries. Allocate a small number of "photons" as the relative number that a human eye can receive. In the example below, three have been selected. You can either perform your calculations manually or set up a spreadsheet to help with your calculations and plotting. Telescope Diameter of Radius Number of "photons" Sensitivity compared to eye Primary (m) (m) human eye 0.008 0.004 3 1 Galilean 0.03 0.015 4-inch refractor 0.1 0.05 8-inch Dobsonian 0.2 0.1 60-cm reflector 0.6 0.3 Yerkes Refractor 1 0.5 Herschel's reflector 1.2 0.6 Lord Rosse's 1.83 0.915 ANU 2.3m 2.3 1.15 AAT 3.9 1.95 Hale 5 2.5 Bolshoi Teleskop 6 3 Gemini 8.1 4.05 Keck 10 5 CELT 30 15 OWL 100 50 2. What other columns may you need to add to the table? (Hint: What effect does increasing the diameter of the primary have on its surface area?) 3. If you want to check your results with a sample solution, you can download an Excel spreadsheet file here (30KB). 4. What is the mathematical relationship between the size of a primary mirror and its sensitivity? Part II: Resolution Method 1: Mathematical Model A simple investigation for resolution involves a mathematical model using the resolution equation: 1. Using the information supplied for the same telescopes as in the sensitivity model above, calculate the theoretical resolution each telescope can achieve. Assume observations are at a wavelength of 550nm, ie peaking in yellow light. Remember, θ = resolution and D = diameter of primary. D and λ need to be in the same units. 2. Calculate the relative resolution of each compared to the human eye. 3. Plot your results and verify the mathematical relationship graphically. Diameter of Radius Theoretical Resolution Telescope Primary (m) (m) resolution milli- arcsec compared (arcsec) to eye human eye 0.008 0.004 1 Galilean 0.03 0.015 4-inch refractor 0.1 0.05 8-inch Dobsonian 0.2 0.1 60-cm reflector 0.6 0.3 Yerkes Refractor 1 0.5 Herschel's reflector 1.2 0.6 Lord Rosse's 1.83 0.915 ANU 2.3m 2.3 1.15 AAT 3.9 1.95 Hale 5 2.5 Bolshoi Teleskop 6 3 Gemini 8.1 4.05 Keck 10 5 CELT 30 15 OWL 100 50 To check your calculations you can download an Excel spreadsheet file with values calculated here (30KB)	{"url":"https://www.atnf.csiro.au/outreach/education/senior/astrophysics/resolution_investigation.html","timestamp":"2024-11-04T07:47:06Z","content_type":"application/xhtml+xml","content_length":"71639","record_id":"<urn:uuid:5117a080-4016-4cdd-a520-ecdd08dcf432>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00140.warc.gz"}
AP Board 9th Class Maths Notes Chapter 7 Triangles Students can go through AP Board 9th Class Maths Notes Chapter 7 Triangles to understand and remember the concepts easily. AP State Board Syllabus 9th Class Maths Notes Chapter 7 Triangles → Two line segments are congruent if they have equal length. → Two squares are congruent if they have same side. → Squares that have same measure of diagonals are also congruent. → Two triangles are congruent if they cover each other exactly. → If two triangles are congruent then Corresponding Parts of Congruent Triangles (CPCT) are equal. → Two triangles are congruent if two sides and the included angle of one triangle are equal to the two sides and included angle of the other triangle (S.A.S. congruence rule). → Two triangles are congruent, if two angles and the included side of one triangle are equal to two angles and the included side of the other triangle (A.S.A. congruence rule). → Two triangles are congruent if any two pairs of angles and one pair of corresponding sides are equal (A.A.S.). → Angles opposite equal sides of a triangle are equal. → The sides opposite to equal angles of a triangle are equal. → If three sides of one triangle are respectively equal to the corresponding three sides of another triangle, then the two triangles are congruent (SSS). → If in two right triangles the hypotenuse and one side of one triangle are equal to the hypotenuse and one side of the another triangle, then the two triangles are congruent (RHS). → In a triangle, of the two sides, side opposite to greater angle is greater. → Sum of any two sides of a triangle is greater than the third side.	{"url":"https://apboardsolutions.in/ap-board-9th-class-maths-notes-chapter-7/","timestamp":"2024-11-05T16:10:05Z","content_type":"text/html","content_length":"61464","record_id":"<urn:uuid:df5a43da-5f4d-4036-9574-378ffcf84194>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00677.warc.gz"}
Word Problems Worksheets ,Free Simple Printable - BYJU'S Frequently Asked Questions Students will get an opportunity to improve their math problem solving skills with these worksheets for word problems which cover several core math concepts. As a result, students can expect to get to build deeper learning about the various math concepts. The word problems worksheets are available for all grade levels. Students can choose the worksheets on concepts pertaining to their grade ensuring students to deepen their learning of the given math concepts in an interesting way. There are three levels of difficulty – easy, medium and hard for the word problems worksheets. This is done to ensure that the students gain confidence in their math skills with the different levels of difficulty. The word problems worksheets are ideal for students who want to gain an in-depth knowledge on different math concepts in a unique way. The word problems are designed as per real life situations thus giving students a context of real world problems and acquainting them with various real life situations where math can be a handy and effective problem solving tool. Yes, you can always solve as many worksheets as possible on word problems from the BYJU’S Math website for free to improve your understanding of the math concepts.	{"url":"https://byjus.com/us/math/word-problems-worksheets/","timestamp":"2024-11-03T13:06:03Z","content_type":"text/html","content_length":"158310","record_id":"<urn:uuid:c9509e7e-ace8-42e3-ab38-a7e1b7cc4716>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00399.warc.gz"}
The problem Answer provided by our tutors What you have type in is not an equation, but expression, so the solver simplifies it up to -24x - 60 thus, there is no solution for "x". However, is you type in the equation -8(6+6x)+4(-3+6x) = 0 you will get the following solution for "x": x = -5/2 x = -2 1/2	{"url":"https://www.doyourmath.com/algebra-help/in-this-problem-what-does-x-equal","timestamp":"2024-11-05T15:09:29Z","content_type":"text/html","content_length":"11226","record_id":"<urn:uuid:f041a272-ed8d-4a96-9199-11e0e539fb00>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00421.warc.gz"}
Pharmacokinetic ODE models in Stan – Notes from a data witch This is cmdstanr version 0.5.3 - CmdStanR documentation and vignettes: mc-stan.org/cmdstanr - CmdStan path: /home/danielle/.cmdstan/cmdstan-2.32.1 - CmdStan version: 2.32.1 A newer version of CmdStan is available. See ?install_cmdstan() to install it. To disable this check set option or environment variable CMDSTANR_NO_VER_CHECK=TRUE.	{"url":"https://blog.djnavarro.net/posts/2023-05-16_stan-ode/index.html","timestamp":"2024-11-09T07:13:47Z","content_type":"application/xhtml+xml","content_length":"193155","record_id":"<urn:uuid:7d06a2a2-01fa-4ba6-956a-1e6188f26e9c>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00158.warc.gz"}
Forces in MCAT Physics Tutorial Video Series As an MCAT tutor, I find that, when students master this in forces, they have an easier time connecting other topics. My goal with this video series is to help you make sense of Newton's Laws and all the wacky force equations. For extra guidance as you study these videos, be sure to download my MCAT Forces Cheat Sheet. This series requires a foundation in non-calculator math and kinematics. For a quick refresher, be sure to watch my MCAT Math Without A Calculator video series and my MCAT Kinematics video series. Every video series needs an introduction, and Forces in MCAT Physics is no different. This video introduces forces along with the proper way to derive units and ‘memorize' equations. This video also shows you how to relate the kinematic equations to forces to ensure you don't mentally isolate the different topics. Newton's First Law is more than just an explanation of inertia. For the MCAT you will be required to UNDERSTAND and apply inertia concepts to bigger picture questions and passages. This includes understanding inertia for objects at rest or in motion, and objects in motion changing their direction. Newton's Second Law talks about how the sum of forces results in an acceleration. This video breaks down the F=ma concept by looking at the sum in the x direction, y direction, and some component between the 2 axis. Newton's third law of motion is potentially the most difficult to grasp. Fa = -Fb. This video breaks it down to help you understand how a force can ‘magically' change to balance a counter-force. When it comes to static or dynamic equilibrium, it's important that you understand how to visualize forces in every direction. This video shows you how to set up free body diagrams and then to set up a simplified equation to help you minimize time wasted on potentially complex MCAT questions. This video explains the concept of Normal Force and shows you how to apply it to MCAT-style questions when objects are resting on horizontal surfaces, under a surface, and even on an inclined plane. We tend to take the force of gravity for granted and, on the MCAT, you might see questions that ask you to calculate the gravitational pull on an object under very specific circumstances. This video will show you how to calculate the force of gravity for objects near the surface of earth AND for objects far from earth, using both the simple and complex equations for gravity. This video defines the forces of static and kinetic friction using real-life examples and shows you how to draw free body diagrams that incorporate friction, for both objects in motion and objects that are stationary, on a flat surface. You’ll also learn about the coefficient of static and kinetic friction. Plus, I'll walk you through a full-length practice problem. Additional Force videos to follow include: • Tension • Centripetal Force • Torque (not a force but worth studying) Organic Chemistry Reference Material and Cheat Sheets Alkene Reactions Overview Cheat Sheet – Organic Chemistry The true key to successful mastery of alkene reactions lies in practice practice practice. However, … [Read More...] MCAT Tutorials Introduction To MCAT Math Without A Calculator While the pre-2015 MCAT only tests you on science and verbal, you are still required to perform … [Read More...] Organic Chemistry Tutorial Videos Keto Enol Tautomerization Reaction and Mechanism Keto Enol Tautomerization or KET, is an organic chemistry reaction in which ketone and enol … [Read More...]	{"url":"https://leah4sci.com/mcat/mcat-physics/forces-in-mcat-physics-tutorial-video-series/","timestamp":"2024-11-10T21:15:09Z","content_type":"text/html","content_length":"62766","record_id":"<urn:uuid:b14d0537-3a85-4d5f-8bc8-4162ae27b2b4>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00381.warc.gz"}
Volumetric Modelling - dbt Volumetric Modelling This research investigates a range of different mathematical models in computational design for the creation and modification of three-dimensional geometry. With additive manufacturing, especially binder-jet 3D printing, we have the most radical fabrication process at hand, one that is beyond tectonics. The logic at work in many CAD packages is that of assembling linear or planar individual elements. But within the build volume of a large-scale binder-jet 3D printer, any grain of sand can either be bound or not, with no geometric limitation. Hence, the main question is how the range of not only tools but also conceptual thought models can be extended to fully exploit the freedom offered by novel fabrication processes. The benefits of this research for architecture are manifold. Forms and shapes can be created and manipulated in a controlled manner conventional software would fail to handle. Customized materials can be designed and produced. Materiality can be graded within components to locally adapt to requirements. Intricate internal lattice structures allow to tailor the porosity, reduce weight or tune structural performance properties. Topologies of construction elements are not bound to the limits of predefined standard protocols. Thin shell surfaces can strategically be deformed by functional textures into ribs and ripples, to allow or prevent certain deformations. Technical Description There are different ways – modes – of how to encode geometry (primitives, forms, shapes, volumes) to be read, transmitted and modified by a computer. The most well-known, used by many CAD packages for its versatility and speed, is the so-called boundary representation, or BRep for short. Control points called vertices are placed in empty Cartesian space (2D or 3D) and connected with curves (2D) or surfaces (3D) describing – or enclosing – shapes and volumes. A pseudo-code notation for a rectangle with sides a and b could first include a list of coordinates, v(-a/2,-b/2), v(-a/2,b/2), v (a/2,-b/2) and v(a/2,b/2), followed by a list of index pairs l(1,2), l(2,3), l(3,4) and l(4,1), that indicate what vertices to connect with lines. The vertices can be moved around in the plane, it can be stretched, rotated, distorted or skewed but it always remains a quadrilateral polygon, as its topology – the connectivity of the vertices among each other – is explicitly predefined. Hence the term explicit modelling. Structure is primary and form is secondary. An alternative model to describe geometry is implicit modelling, distance fields or what we call volumetric modelling. Space is not empty to begin with but there exists a function f(x,y,z) that returns a value for every point in space (R^3 → R). It is sometimes also referred to as function representation, or FRep for short. This return value can for example be the distance to the surface enclosing the object to be modelled. It is then called a Signed Distance Function (SDF) – signed because negative values are inside the object, positive values outside and the surface of the object is where the function evaluates to 0 (isosurface). Space is described in pure mathematical terms, Boolean operations (addition, subtraction and intersection) as well as smooth blending between objects is done without the need to calculate intersection points and reconstructing the topology of how the vertices are connected. There is no geometry – vertices or lines or triangles – until it needs to be rendered on screen (not even then necessarily) or exported to some other CAD package or 3D printer. Form is primary and structure is secondary. The same rectangle as above in this notation would be max(abs(x)-a/2, abs(y)-b/2) = 0 where (x, y) denotes the point in question, any point in the plane. In the example below, drag around the objects, change the Boolean operation or the display mode with the buttons below. MODE: change display mode, FLIP: change order for subtraction, ADD,SUB,INT,SMOOTH: Boolean operations, SHELL: parallel offset thickening In volumetric modelling space, a sphere and a torus – the two prototypical primitives to illustrate the topological meaning of genus – coexist next to each other and one can crossfade between the two with ease. Continuous blending between a sphere and a torus mb \| dbt Any number of primitives can easily be combined, the zero level isosurface (e.g. marching cubes) will always create a watertight manifold mesh. Objects of any complexity can easily be turned into a hollow shell or filled with intricate micro lattice structures, hard to model with explicitly placing vertices and surfaces. Such lattices can significantly reduce weight while maintaining structural performance. Precisely designed porosity can also affect the physical properties of the material in many ways. The surface area to volume ratio can arbitrarily be set or many spatially separate void structures can tightly be interwoven. To understand space as a continuous field of values also enables a direct approach to efficient multi-material printing.	{"url":"https://dbt.arch.ethz.ch/research-stream/volumetric-modelling/","timestamp":"2024-11-09T04:14:46Z","content_type":"text/html","content_length":"59024","record_id":"<urn:uuid:a7b3c388-5710-4475-961a-270e81026459>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00205.warc.gz"}
Beyond Cauchy and Quasi-Cauchy Sequences In this paper, we investigate the concepts of downward continuity and upward continuity. A real valued function on a subset $E$ of $\mathbb{R}$, the set of real numbers is downward continuous if it preserves downward quasi Cauchy sequences; and is upward continuous if it preserves upward quasi Cauchy sequences, where a sequence $(x_{ k})$ of points in $\mathbb{R}$ is called downward quasi Cauchy if for every $\varepsilon>0$ there exists an $n_{0}\in{\mathbb{N}}$ such that $x_{n+1}-x_{n} <\varepsilon$ for $n \geq n_{0}$, and called upward quasi Cauchy if for every $\varepsilon>0$ there exists an $n_{1}\in{\mathbb{N}}$ such that $x_{n}-x_{n+1} <\varepsilon$ for $n \geq n_{1}$. We investigate the notions of downward compactness and upward compactness and prove that downward compactness coincides with above boundedness. It turns out that not only the set of downward continuous functions, but also the set of upward continuous functions is a proper subset of the set of continuous functions • There are currently no refbacks.	{"url":"https://journal.pmf.ni.ac.rs/filomat/index.php/filomat/article/view/4877","timestamp":"2024-11-11T14:30:52Z","content_type":"application/xhtml+xml","content_length":"16429","record_id":"<urn:uuid:cd42e48d-8144-4539-9b42-4678133b3ce7>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00683.warc.gz"}
No.011 Large-scale Distributed Computation \| SeminarsNII Shonan Meeting NO.011 Large-scale Distributed Computation January 12 - 15, 2012 (Check-in: January 11, 2012 ) • Graham Cormode • S. Muthukrishnan • Ke Yi □ Hong Kong University of Science and Technology, Hong Kong As the amount of data produced by large scale systems such as environmental monitoring, scientific experiments and communication networks grows rapidly, new approaches are needed to effectively process and analyze such data. There are many promising directions in the area of large-scale distributed computation, that is, where multiple computing entities work together over partitions of the (huge) data to perform complex computations. Two important paradigms in this realm are distributed continuous monitoring, which continually maintains an accurate estimate of a complex query, and MapReduce, a primarily batch approach to large cluster computation. The aim of this meeting is to bring together computer scientists with interests in this field to present recent innovations, find topics of common interest and to stimulate further development of new approaches to deal with massive data. Description of the Meeting If the Computer Science in the 20th Century was defined by the inception and astounding growth of computing power, in the 21st Century it will be defined by the huge growth in data. The ability to capture vast quantities of data (from network traffic, scientific experiments, online activity) is expanding rapidly, and at a rate far higher than the increase in our ability to process it. While Moore’s law has held for many decades, tracking an exponential growth in computational power, the corresponding law for data has recently shown an even faster growth in data production: see recent estimates such as http://wikibon.org/blog/cloud-storage/. Therefore, the new challenge for Computer Science in the 21st Century is how to deal with such a data deluge. Several promising new directions have emerged in recent years: Distributed Continuous Monitoring. In many settings the new data is observed locally?at a router in a network, at a sensor in a larger sensor network. The volume of observations is too large to move to a central location and process together; instead, it is necessary to perform distributed computation over the data. Since the new observations are continually arriving, we must produce a continual answer to complex monitoring queries, all while ensuring that the communication cost necessary to maintain the result, and the computational cost of the tracking, are minimized to meet the data throughput demands. In recent years, there have been several advances in this field: • The geometric monitoring approach, which views the value of the function being monitored as points in a metric space, covered by monotonic balls in the region. This enables arbitrary complex functions to be decomposed into local conditions that can be monitored efficiently [16, 17, 15]. • Use of sketches and randomized summaries to compactly summarize large complex distributions, allowing approximation of fundamental quantities such as order statistics, inner products, distinct items and frequency moments [7, 8]. • New ways to incorporate time decay and sliding windows, to allow queries to be based on only the more recent observations, and reduce or remove the contribution of outdated observations, while still communicating much fewer than the full set of observations [4, 9]. • Extension of monitoring techniques to more complex, non-linear functions such as the (empirical) entropy [16, 2]. • Advances in sampling technology to enable drawing uniform samples over multiple streams of arrivals, arriving at different rates, both with and without replacement [3, 9]. However, there remain many challenging questions to address in this area: • A more general theory of distributed continuous computation: what functions and computations can be performed in this model? Are there hardness results and separation theorems that can be proved? Can a stronger set of lower bounds be provided? • A stronger sense of the connection to other areas in computer science, such as communication complexity, network coding, coding theory, compressed sensing, and so on. Can techniques from these areas be extended to the distributed continuous setting? • Systems issues have so far received only minor consideration. Can general purpose systems be designed, in a comparable way to centralized databases, which can accept complex queries in a high-level language, and then produce and deploy query monitoring plans? Cluster Computing. As data sizes increase while the power of individual computing cores begin to saturate, there has been a move to adopt cluster computing: harnessing multiple computing nodes to work together to solve huge data processing tasks in parallel. The best known example of this is the MapReduce paradigm, and its open source Hadoop implementation. Computationally speaking, the approach is for each compute node to process data which is stored local to it in a distributed file system, and Map this data by deriving a new data set indexed by a key value. The system then collects all tuples with the same key together at a second node, which proceeds to Reduce these tuples to obtain the (partial) output. This paradigm has proved highly successful for many large scale computations, such as search engine log-analysis, building large machine learning models and data analysis such as clustering. Other key technical advances include: • A foundation for models of this style of computation, including the Massive Unordered Distributed (MUD) model, and the Karloff-Suri-Vassilivitskii model (KSV). These help to understand what computations can be performed effectively on clusters [11, 13]. • Development of Online versions of the protocols, which use pipelining of data between operators, to move beyond the batch view of cluster computing, and to quickly provide partial results to complex queries which are refined as computation progresses [6]. • Initial attempts to understand what class of algorithms can be effectively performed in these models, such as set cover, clustering, database, geometric and graph computations [5, 14, 10, 1, 12]. • Overlays, such as Facebooks Hive, which provides advanced data processing and abstract query language on top of Hadoop to enable data warehousing applications [18]. The main challenges in this area include: • Understanding the extent to which approximation and randomization can further advance efficiency of computation. • Richer models which help us understand the tradeoff between the different cost parameters: amount of communication, number of compute nodes, memory available at each node, number of rounds of Map-Reduce, etc. • More examples of non-trivial computations that can be performed in Map-Reduce, leading to a more general theory of cluster computation and data processing under Map-Reduce. • More efficient implementations and variations of the model allowing real-time and more interactive use of clusters. The aim of this workshop is to bring together researchers active in the areas of distributed data processing, to address these fundamental issues. We hope to encourage greater cooperation and interaction between currently separate communities, ultimately leading to new advances in these important developing areas.	{"url":"https://shonan.nii.ac.jp/seminars/011/","timestamp":"2024-11-10T23:46:59Z","content_type":"text/html","content_length":"19986","record_id":"<urn:uuid:e360df28-8514-4264-afad-83536f30fd72>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00438.warc.gz"}
RGB to HSV ConverterRGB to HSV Converter RGB to HSV Converter Here’s an RGB to HSV converter. Convert any RGB value to its HSV color code, along with corresponding HSV, Hex and CMYK values. Code Value HTML/CSS RGB Percentage Exact CMYK What is the HSV Color Model? HSV is a cylindrical color model based on three components: hue, saturation (or amount of gray), and value (or brightness). Created in the 1970s by graphic researchers, this model is an alternative representation to the RGB model. This model remaps the RGB colors into dimensions that are easier for people to understand. Unlike RGB and CMYK, the HSV color model is closer to how people perceive color. What is Hue in HSV? Hue represents the pure hue and is measured in degrees from 0 to 360, creating a color wheel: • 0° – 60°: Red • 61° – 120°: Yellow • 121° – 180°: Green • 181° – 240°: Cyan • 241° – 300°: Blue • 301° – 360°: Magenta What is Saturation in HSV? Saturation determines the intensity of a color, describing the amount of gray in a particular hue. A fully saturated color appears vivid and bright, while a desaturated color is more muted and pastel-like. Saturation is measured as a percentage, with 0% being gray and 100% being fully saturated. What is Value in HSV? Value represents the brightness or darkness of a color. In the HSV model, the value ranges from 0 to 100%, where 0% is black and 100% is the brightest. HSV vs. RGB HSV is an alternative representation of the RGB color model. Although it is more intuitive to human vision, RGB is the most preferred model by hardware display manufacturers. Both HSV and RGB are additive color models. Furthermore, HSV and RGB go hand in hand, with HSV remapping RGB colors into a more easily perceived dimension. If you’re wondering if HSV is better than RGB, the short answer is yes because it uses two additional dimensions than pure hue, namely saturation (S) and brightness (V). So, you can separate the pure hue (or color dimension) from luminance. It is also better than RGB when it comes to color detection. This model is more robust to lighting changes than RGB. For example, shadows affect HSV values less than RGB values. Thus, HSV transforms the RGB model into a more convenient representation. How to Convert RGB to HSV Using a Formula To convert RGB to HSV, first you need to divide the RGB values by 255 to get values between 0 and 1. Then you need to determine Cmax and Cmin, and then calculate Δ, specifically Cmax-Cmin. Calculate Hue Afterwards, you will calculate Hue according to Cmax. Afterwards, you will calculate Hue according to Cmax. Thus, Cmax can be R, G, or B. If Δ = 0, then hue = 0. Here are the formulas to calculate the Hue using your RGB values: Calculate Saturation To calculate saturation (H) from RGB, calculate the difference between Cmax (maximum value from R, G, and B) and Cmin (minimum value from R,G,B) and then divide it by Cmax. If Cmax = 0, then the saturation will also be 0. The simplest formula to calculate saturation from the RGB values is: S = 1 - (min(R, G, B) / V) if V != 0 else 0 , where V (value) = max(R, G, B). Value Calculation Value represents the brightness of the color. It is calculated as the maximum of the RGB values. The decimal RGB values must be divided by 255 to get values between 0 and 1. Other Color Conversion Tools Here are other color conversion tools: Leave a Comment	{"url":"https://thecolorsmeaning.com/rgb-to-hsv/","timestamp":"2024-11-14T21:40:57Z","content_type":"text/html","content_length":"123313","record_id":"<urn:uuid:e30ff51b-67c6-4587-ade9-c1ef684f8a7b>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00606.warc.gz"}
Circle Perimeter And Area Questions and Answers Tests the student's understanding of basic circle measurement concepts, and practices the calculation skills for perimeter and area of circles. • 1. The distance between two points on a circle, passing through the circle's centre, is the □ A. □ B. □ C. □ D. □ E. Correct Answer B. Diameter The distance between two points on a circle that passes through the circle's center is called the diameter. The diameter is a line segment that passes through the center of the circle and is twice the length of the radius. It divides the circle into two equal halves and is the longest chord in the circle. • 2. The distance from the centre of a circle to any point on the edge is the □ A. □ B. □ C. □ D. □ E. Correct Answer D. Radius The distance from the center of a circle to any point on the edge is called the radius. The radius is the straight line segment that connects the center of the circle to any point on its circumference. It is a fundamental measurement in geometry and is used to calculate various properties of the circle, such as its area and circumference. The radius is half the length of the diameter, which is the distance across the circle passing through the center. • 3. The distance around the outside edge of a circle is the □ A. □ B. □ C. □ D. □ E. Correct Answer A. Circumference The distance around the outside edge of a circle is known as the circumference. It is the total length of the circle's boundary. The circumference can be calculated using the formula C = 2πr, where r is the radius of the circle. The other options, such as radius, diameter, perimeter, and chord, are not applicable to measuring the distance around the outside edge of a circle. • 4. The perimeter of a circle divided by it's diameter is known as Correct Answer The perimeter of a circle divided by its diameter is known as pi. This is a fundamental concept in mathematics and is represented by the Greek letter π. Pi is an irrational number, approximately equal to 3.14159, which is the ratio of a circle's circumference to its diameter. It is used in various mathematical and scientific calculations involving circles and other curved shapes. • 5. Pi is an irrational number, meaning that it has an infinite number of decimal places with no recurring pattern. Correct Answer A. True Pi is indeed an irrational number, which means that it cannot be expressed as a fraction and has an infinite number of decimal places. Unlike rational numbers, which have a repeating pattern in their decimal representation, pi does not have a recurring pattern. This property of pi has been proven mathematically, and it is one of the fundamental characteristics of this mathematical constant. Therefore, the given statement is true. • 6. Pi can also be approximated by the fraction □ A. □ B. □ C. □ D. Correct Answer C. 22/7 The fraction 22/7 is often used as an approximation for the value of pi because it is a close approximation. Pi is an irrational number, meaning it cannot be expressed as a simple fraction or a finite decimal. However, the fraction 22/7 is a commonly used approximation that is close enough for many practical purposes. While it is not exactly equal to pi, it is a convenient and relatively accurate representation of the value. • 7. Taking pi = 3.14, what is the circumference of this circle? Correct Answer 12.56 metres The circumference of a circle can be calculated using the formula C = 2πr, where π is approximately 3.14 and r is the radius of the circle. Since the radius is not given in the question, we cannot calculate the exact circumference. Therefore, any of the given options (12.56, 12.56 m, 12.56 metres, 12.56 meters) could be the correct answer, depending on the unit of measurement for the radius. • 8. Taking pi = 3.14, what is the circumference of this circle? (Only write the numeric answer with decimal) Correct Answer • 9. Taking pi = 3.14, what is the circumference of this circle? □ A. 37.68 m 18.84 m , 31.41 m □ B. Correct Answer B. 37.68 m The circumference of a circle can be calculated using the formula C = 2πr, where π is approximately 3.14 and r is the radius of the circle. In this question, the radius is not given, so we cannot calculate the exact circumference. However, we can see that the closest option to the calculated value of 2πr is 37.68 m, which suggests that this is the correct answer. • 10. Taking pi = 3.14, what is the circumference of this circle? □ A. □ B. □ C. Correct Answer A. 31.4 m The circumference of a circle can be found by multiplying the diameter of the circle by pi. In this case, since the diameter is not given, we can assume that the radius of the circle is 31.4/2 = 15.7 m. Therefore, the circumference of the circle is 2 * pi * 15.7 m = 31.4 m. • 11. Taking pi = 3.14, what is the area of this circle (in square metres)? Correct Answer 78.5 square metres The area of a circle can be calculated using the formula A = πr^2, where A is the area and r is the radius of the circle. In this question, the radius of the circle is not provided, so it is not possible to calculate the exact area. However, since the answer choices all state that the area is 78.5 square metres, it can be assumed that the radius of the circle is 5 metres (approximately). Using this radius, the area of the circle would indeed be 78.5 square metres. • 12. Taking pi = 3.14, what is the area of this circle (in square metres)? □ A. □ B. □ C. □ D. Correct Answer D. 28.26 The area of a circle can be calculated using the formula A = πr^2, where A is the area and r is the radius of the circle. In this question, the radius is not given, so we cannot directly calculate the area. However, the answer choices are all numbers, suggesting that one of them might be the radius squared multiplied by π. By checking each answer choice, we find that 28.26 is equal to π multiplied by (3.14)^2, which is the correct formula for calculating the area of a circle. • 13. Taking pi = 3.14, what is the area of this circle (in square metres)? Correct Answer 314 square metres The area of a circle can be calculated using the formula A = πr², where A is the area and r is the radius. In this question, the radius is not given, so we cannot calculate the exact area. However, the given answer options all state the area as 314, which suggests that the radius is 10 (since 3.14 * 10² = 314). Therefore, the correct answer is 314 square meters. • 14. What is the diameter of a circle that has a circumference of 28 cm? □ A. □ B. □ C. Correct Answer A. 8.92 cm The diameter of a circle is the distance across the circle passing through the center. The formula to calculate the circumference of a circle is C = πd, where C is the circumference and d is the diameter. In this case, the circumference is given as 28 cm. Rearranging the formula, we can solve for the diameter: d = C/π. Plugging in the given circumference, we get d = 28/π. Evaluating this expression, we find that the diameter is approximately 8.92 cm. • 15. What is the radius of a circle that has a circumference of 45 km? □ A. □ B. □ C. Correct Answer B. 7.17 km The radius of a circle is the distance from the center of the circle to any point on its circumference. The formula to calculate the circumference of a circle is C = 2πr, where C is the circumference and r is the radius. In this case, we are given that the circumference is 45 km. Plugging this value into the formula, we can solve for the radius. Rearranging the formula, we get r = C / (2π). Substituting the given circumference, we find r = 45 km / (2π) ≈ 7.17 km. Therefore, the correct answer is 7.17 km. • 16. What is the radius of a circle that has an area of 78 cm^2? Correct Answer 4.98 cm The radius of a circle can be found using the formula A = πr^2, where A is the area and r is the radius. In this case, the area is given as 78 cm^2. By rearranging the formula, we can solve for the radius: r = √(A/π). Plugging in the given area, we get r = √(78/π) ≈ 4.98 cm. Therefore, the radius of the circle is approximately 4.98 cm. • 17. What is the diameter of a circle that has an area of 85 m^2? Correct Answer 10.4 m 10.4 metres 10.4 meters The diameter of a circle can be calculated using the formula D = √(4A/π), where A is the area of the circle. In this case, the area is given as 85 m2. Plugging this value into the formula, we get D = √(4*85/π) ≈ 10.4 m. Therefore, the correct answer is 10.4 m. The other options, 10.4, 10.4 metres, and 10.4 meters, are just different ways of expressing the same measurement. • 18. What is the perimeter of this semicircle? □ A. □ B. □ C. Correct Answer C. 33.42 m Half the whole-circle circumference, plus the diameter! • 19. To the nearest whole number, what is the area of the above shape (in square metres)? Correct Answer 1593 square meters Find the area of the whole circle, and then take three quarters of it...	{"url":"https://www.proprofs.com/quiz-school/story.php?title=circle-perimeter-area","timestamp":"2024-11-05T17:20:54Z","content_type":"text/html","content_length":"496516","record_id":"<urn:uuid:acab261e-9be3-4e76-887b-bb83b066934a>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00854.warc.gz"}
Hinge-loss & relationship with Support Vector Machines - GeeksforGeeks (2024) Here we will be discussing the role of Hinge loss in SVM hard margin and soft margin classifiers, understanding the optimization process, and kernel trick. Support Vector Machine(SVM) Support Vector Machine(SVM) is a supervised machine learning algorithm for classification and regression. Let us use the binary classification case to understand the Hinge loss. In the case of binary classification, the objective of SVM is to construct a hyperplane that divides the input data in such a way that all the data that belongs to class 1 lie on one side of the plane and all the data that belongs to class -1 lie on the other side of the plane. The objective function of the SVM classifier is (for derivation please refer to the link of the SVM article shared above) Minimize: [Tex]\frac{1}{2}* {\|\|w\|\|}^2[/Tex] Subject to: [Tex] yᵢ * (w·xᵢ + b) [/Tex] ≥ 1 for all training examples (xᵢ) • “w” represents the weight vector of the hyperplane, “b” is the bias term, • “xᵢ” are the training data points, and “yᵢ” is the class label of the data point (either -1 or 1 for a binary classification problem). • The objective is to minimize the L2-norm of the weight vector, which corresponds to maximizing the margin between the two classes. The inequality constraint [Tex](yᵢ * (w·xᵢ + b) ≥ 1)[/Tex] ensures that all data points are correctly classified, and those closest to the decision boundary (support vectors) satisfy the margin There are two types of SVM 1. Hard Margin: Here we aim to classify all the points correctly. We assume the data is linearly separable by a hyperplane. 2. Soft Margin: In the real world, the datasets are typically not linearly separable. In soft margin SVM we allow the model to misclassify some data points. We penalize the model for such misclassification . We will study hard margin and soft margin SVM in detail latter. Let us first understand hinge loss. Hinge Loss Hinge loss is used in binary classification problems where the objective is to separate the data points in two classes typically labeled as +1 and -1. Mathematically, Hinge loss for a data point can be represented as : [Tex]L(y, f(x)) = max(0, 1 – y * f(x))[/Tex] • y- the actual class (-1 or 1) • f(x) – the output of the classifier for the datapoint Lets understand it with the help of below graph Case 1 : Correct Classification and \|y\| ≥ 1 In this case the product t.y will always be positive and its value greater than 1 and therefore the value of 1-t.y will be negative. So the loss function value max(0,1-t.y) will always be zero. This is indicated by the green region in above graph. Here there is no penalty to the model as model correctly classifies the data point. Case 2 : Correct Classification and \|y\| < 1 In this case the product t.y will always be positive , but its value will be less than 1 and therefore the value of 1-t.y will be positive with value ranging between 0 to 1. Hence the loss function value will be the value of 1-t.y. This is indicated by the yellow region in above graph. Here though the model has correctly classified the data we are penalizing the model because it has not classified it with much confidence (\|y\| < 1) as the classification score is less than 1. We want the model to have a classification score of at least 1 for all the points. Case 3: Incorrect Classification In this case either of t or y will be negative. Therefore the product t.y will always be negative and the value of (1-t)y will be always positive and greater than 1. So the loss function value max (0,1-t.y) will always be the value given by (1-t)y . Here the loss value will increase linearly with increase in value of y. This is indicated by the red region in above graph. Relationship Between Hinge Loss and SVM Let us understand the relationship between hinge loss and svm mathematically . Hard Margin and Hinge Loss A hard margivn SVM is a type of SVM which aims to find a hyperpalne that perfectly separates the two classes without any misclassification. It assumes that the data is linearly separable, and the objective is to maximize the margin while ensuring that all training data points are correctly classified So mathematically speaking for hard margin we want our model to classify all the points in such a way that [Tex]y * (w.x +b) [/Tex]is at least 1 while minimizing the weight vector w. Thus a good classifier i.e. a good hyperplane will be one that will give a large positive value of [Tex]y(wx+b) [/Tex]for all the points. It encourages the SVM to find a hyperplane that not only separates the two classes but also maximizes the margin between them. Mathematically Minimize: [Tex]\frac{1}{2} {\|\|w\|\|}^2[/Tex] Subject to: [Tex] yᵢ * (w·xᵢ + b)[/Tex] ≥ 1 for all training examples (xᵢ) If we look at the mathematical formulation the hinge loss is effectively present in the constraints of a hard margin. This ensures that the decision boundary (the hyperplane) is positioned in such a way that it maximizes the margin without allowing any data points to be within or on the wrong side of the margin. Now solving a hardmargin equation becomes solving a constrained NLP (Non LInear Programming) problem. Here we need to find the lagrange function and KKT conditions to solve this. Lagrange Function and KKT Equation The Lagrange function incorporates the original objective function along with terms that account for the constraints. The Lagrange multipliers λ and μ are introduced to measure the impact of constraints on the objective function. Let us briefly look at generalized lagrange function. Given an objective function f(x) to be minimized or maximized, subject to a set of constraints [Tex]gi(x) = 0[/Tex] (for equality constraints) and [Tex]hk(x) ≤ 0[/Tex] (for inequality constraints), the Lagrange function L(x, λ, μ) is defined as: [Tex]L(x, λ, μ) = f(x) + Σ(λi * gi(x)) + Σ(μj * hj(x)) [/Tex] • L(x, λ, μ) is the Lagrange function. • x is the vector of decision variables. • λ (lambda) is a vector of Lagrange multipliers associated with the equality constraints g[i](x) = 0. • μ (mu) is a vector of Lagrange multipliers associated with the inequality constraints h[k](x) ≤ 0. To find the solution to the constrained optimization problem, we typically set the gradient of the Lagrange function with respect to the decision variables (x) and the Lagrange multipliers (λ and μ) to zero and solve the resulting system of equations. This approach is known as the Karush-Kuhn-Tucker (KKT) conditions, and it helps identify the optimal solution that satisfies the constraints. The above equation is called primal form of equation. When we cannot solve the lagrangian function in its primal form we tend to write the dual form and try to solve it. The dual form of lagrangian function is written by converting it to max min form [Tex]D(λ, μ) = max(λ, μ) min(x) L(x, λ, μ) [/Tex] where λ, μ ≥ 0 where D(λ, μ) is the dual equation Dual form removes the constraints of g(x) and h(x). In SVM we will use the dual form. Solving SVM using Lagrange and KKT Using the above method the SVM problem can be formulated as : Minimize: [Tex]L(w,b,λ) = \frac{1}{2} * {\|\|w\|\|}^2 – ∑λi[yᵢ * (w·xᵢ + b)-1] [/Tex] ———- Eq (1) • λ[i] are lagrange multipliers. • Here we have converted our greater than equality sign to less than i.e. [Tex]yᵢ * (w·xᵢ + b)[/Tex] ≥ 1 =>. – [Tex][yᵢ * (w·xᵢ + b) -1][/Tex] ≤ 0 to incorporate into the Lagrange equation. • Summation is over all data points Compute the partial derivatives of the Lagrange function (EQ(1)) with respect to w, b and λ[i] and set them to zero to find the critical points [Tex]\frac{dl}{dw} = w-∑λiyᵢ xᵢ = 0[/Tex] ——– Eq(2) [Tex]\frac{dl}{dλ} = ∑[yᵢ * (w·xᵢ + b)-1] = 0[/Tex] ——– Eq(3) [Tex]\frac{dl}{db} = ∑λiyi = 0[/Tex] ——– Eq(4) For Eq(2) we get [Tex]w = ∑λiyᵢ xᵢ[/Tex] —— Eq(5) Substituting the value of w in form Eq(5) into Eq(1) [Tex]L(w,b,λ) = \frac{1}{2} * ∑i\|\|λiyᵢ x\|\|2 – ∑jλj[yj * (∑iλiyᵢ xᵢ·xj + b) +∑jλj[/Tex] [Tex]L(w,b,λ) = \frac{1}{2} * (∑iλiyᵢx) (∑iλiyᵢ x – ∑i∑jλiyᵢλjyj xᵢ·xj + b∑iλiyᵢ +∑jλj [/Tex]—— Eq(6) The term b∑iλiyᵢ = 0 if we consider Eq(4). Thus Eq (6) can be rearranged to [Tex]L(w,b,λ) = ∑jλj – \frac{1}{2} * ∑i∑jλiyᵢλjyj xᵢ·xj [/Tex] Using the dual form we can write: [Tex]D(λ) = max(λ)min(w,b) L(w,b,λ)[/Tex] Sine we have obtaind L(w,b,)λ in form of λ only we can simply write dual as [Tex]D(λ) = max(λ) [ ∑jλj – 1/2 * ∑i∑jλiyᵢλjyj xᵢ·xj ][/Tex] ——— EQ(7) Thus we have converted our constrained optimization to unconstrained optimization . This is a special case of NON LINEAR PROGRAMMING known as QUADRATIC PROGRAMMING. Above maximization operation can be solved with the SMO ( sequential minimization optimization) algorithms designed to solve the quadratic programming problem that arises when training a SVM. Understanding the Kernel Trick It may be possible that the data is not linearly seperable in the given dimension however it could be linearly separable in higher dimension. To find a hyperplane in higher dimension , one approaach would be we would need to covnert our data in n higher dimension and proceed with our optimization process. However calculating features in higher dimension is computationally expensive. If we closely look at the equation 7 we observe that all we need is dot porduct of each observation with other observation and not the individual vector to solve our optimization problem. This fact is utilized in kernel trick. We use a kernel i.e. a function which takes two vector as input and gives us the inner product of the transformed feature vectors in the higher dimension space without explicitly calculating the individual features! Some commonly used kernls are • Linear Kernel • Polynomial Kernal • RBF Kernal The choice of kernel function depends on the nature of the data and the problem at hand. SVMs with different kernels can capture different types of decision boundaries. For example, the RBF kernel is commonly used for capturing complex, nonlinear decision boundaries, while the linear kernel is suitable for linearly separable data. A hard margin is an idealized form of SVM which assumes perfect separability. However most datasets are not perfectly linearly separable. This is where Soft margin SVM comes. Support Vector in SVM Most of the lagrange multipliers will be zero except for those data points which lies on the boundary of margin plane. These vectors are called as support vectors as they guide the optimization since the hyperplane has to maximize the distance margin from these vectors. Hence the name support vector margins. Soft Margin and Hinge Loss Since many real practical dataset contains points that are not linealy seprable we need a plane that allows minimum misclassification of data points. For this we have Soft margin SVM. A soft margin SVM is an extension of hard marginv SVM that allows for some misclassification of data points withing the margin. The soft margin SVM can be mathematically represented as Minimize: [Tex]\frac{1}{2} * {\|\|w\|\|}^2 + C * Σ ϵi[/Tex] where [Tex]yᵢ * (w·xᵢ + b) ≥ 1 -ϵi[/Tex] Here we introduce slack variable ϵ[i]. • if confidence score < 1, it means that classifier did not classify the point correctly and incurring a linear penalty of ϵ[i] • If 0<ϵ[i] <1 it means the point is correctly classified but lies between the hyperplane and margin plane • If ϵ[i]>1 if means the point is on wrong side of hyperplane • C is a regularization parameter that balances the trade-off between maximizing the margin and minimizing classification errors. Here the hinge loss component, is part of the objective function itself through slack variable. It encourages the SVM to correctly classify or have a margin of at least 1 for each training data point. The term, [Tex]\frac{1}{2} * \|\|w\|\|^2[/Tex], encourages a simple and effective separating hyperplane. The SVM aims to find the optimal values of w and b that minimize this combined objective.C, which controls the trade-off between maximizing the margin and tolerating misclassification. If value of C = 0 we effectively train hard margin classifier. The lagrange equation becomes: [Tex]L(w,b,λ) = \frac{1}{2} * {\|\|w\|\|}^2 – ∑λi[yᵢ * (w·xᵢ + b)-1 + ϵi][/Tex] This can be similarly solved using the method discussed in hard margin classifier. Implementing Hinge Loss Functions in Python We will use iris dataset to construct a SVM classifier using Hinge loss. The notebook is available at SVM Hinge Loss 1. Import Necessary Libraries from sklearn import datasetsfrom sklearn.model_selection import train_test_splitfrom sklearn.metrics import confusion_matrix, precision_score, recall_score 2. Load the IRIS Dataset iris = datasets.load_iris() 3. Split in Train and Test Set X_train, X_test, y_train, y_test = train_test_split( iris['data'], iris['target'], test_size=0.33, random_state=42) 4. Model training using Hinge Loss • We have imported SGD Classifier from scikit-learn and specified the loss function as ‘hinge’. • Model uses the training data and corresponding labels to classify data based on hinge loss function. # Using hinge loss from sklearn.linear_model import SGDClassifier clf_hinge = SGDClassifier(loss="hinge", max_iter=1000) clf_hinge.fit(X_train, y_train) y_test_pred_hinge = clf_hinge.predict(X_test) 5. Model Evaluation when, loss = ‘hinge’ print('\033[1m' + "Hinge Loss" + '\033[0m') print( f"Precision score : {precision_score(y_test_pred_hinge,y_test,average='weighted')}") print( f"Recall score : {recall_score(y_test_pred_hinge,y_test, average='weighted')}") print("Confusion Matrix") confusion_matrix(y_test_pred_hinge, y_test) Output : Hinge Loss Precision score : 0.95125 Recall score : 0.94 Confusion Matrix array([[19, 0, 0], [ 0, 15, 3], [ 0, 0, 13]]) Advantages of using hinge loss for SVMs There are several advantages to using hinge loss for SVMs: • Hinge loss is a simple and efficient loss function to optimize. • Hinge loss is robust to noise in the data. • Hinge loss encourages SVMs to find hyperplanes with a large margin. Disadvantages of using hinge loss for SVMs There are a few disadvantages to using hinge loss for SVMs: • Hinge loss is not differentiable at zero.This can make it difficult to optimize using some gradient-based methods. • Hinge loss can be sensitive to outliers. Hinge loss is a popular loss function for training SVMs. It is simple, efficient, and robust to noise in the data. However, it is not differentiable at zero and can be sensitive to outliers. Overall, hinge loss is a good choice for training SVMs for classification and regression tasks. Next Article Optimal feature selection for Support Vector Machines	{"url":"https://ballarattechnologypark.com/article/hinge-loss-relationship-with-support-vector-machines-geeksforgeeks","timestamp":"2024-11-03T13:07:50Z","content_type":"text/html","content_length":"86184","record_id":"<urn:uuid:74b028eb-fdf2-479d-aa37-3f735f294e78>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00646.warc.gz"}
If m is chosen in the quadratic equation (m2+1)x2−3x+(m2+1)2=0 ... \| Filo If is chosen in the quadratic equation such that the sum of its roots is greatest, then the absolute difference of the cubes of its roots is Not the question you're searching for? + Ask your question (b) Given quadratic equation is Let the roots of quadratic Eq. (i) are and , so According to the question, the sum of roots is greatest and it is possible only when " is and "min value of , when . and , as Now, the absolute difference of the cubes of roots Was this solution helpful? Video solutions (4) Learn from their 1-to-1 discussion with Filo tutors. 6 mins Uploaded on: 1/23/2023 Was this solution helpful? 25 mins Uploaded on: 5/3/2023 Was this solution helpful? Found 4 tutors discussing this question Discuss this question LIVE for FREE 13 mins ago One destination to cover all your homework and assignment needs Learn Practice Revision Succeed Instant 1:1 help, 24x7 60, 000+ Expert tutors Textbook solutions Big idea maths, McGraw-Hill Education etc Essay review Get expert feedback on your essay Schedule classes High dosage tutoring from Dedicated 3 experts Practice questions from Arihant Mathematics Master Resource Book (Arihant) View more Practice more questions from Complex Number and Quadratic Equations 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 If is chosen in the quadratic equation such that the sum of its roots is greatest, then the absolute difference of the cubes of its roots is Updated On May 3, 2023 Topic Complex Number and Quadratic Equations Subject Mathematics Class Class 11 Answer Type Text solution:1 Video solution: 4 Upvotes 499 Avg. Video Duration 10 min	{"url":"https://askfilo.com/math-question-answers/if-m-is-chosen-in-the-quadratic-equation-leftm21right-x2-3-xleftm21right20-such-187137","timestamp":"2024-11-07T09:59:48Z","content_type":"text/html","content_length":"525093","record_id":"<urn:uuid:654a0255-1f26-471c-9bc9-cf6d08cd1fc3>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00577.warc.gz"}
The unlist() Function in R \| R-bloggersThe unlist() Function in RThe unlist() Function in R The unlist() Function in R [This article was first published on Steve's Data Tips and Tricks , and kindly contributed to ]. (You can report issue about the content on this page ) Want to share your content on R-bloggers? if you have a blog, or if you don't. Hey fellow R enthusiasts! Today, we’re diving deep into the incredible world of R programming to explore the often-overlooked but extremely handy unlist() function. If you’ve ever found yourself dealing with complex nested lists or vectors, this little gem can be a lifesaver. The unlist() function is like a magician that simplifies your data structures, making them more manageable and easier to work with. Let’s unlock its magic together! What is the unlist() function? The unlist() function in R does exactly what its name suggests: it “un-lists” nested lists or vectors and converts them into a simple atomic vector. In other words, it takes a list that contains other lists, vectors, or atomic elements and flattens it into a single vector. This can be useful for a variety of tasks, such as: • Simplifying the structure of a data object • Passing a list to a function that only accepts vectors • Combining the elements of a list into a single vector Syntax of unlist() The syntax for the unlist() function is straightforward: unlist(list, recursive = TRUE, use.names = TRUE) • list: This is the input list that you want to flatten. • recursive: A logical value that determines whether to flatten the list recursively or not. If TRUE, it will flatten nested lists; if FALSE, it will only flatten one level. • use.names: A logical value that specifies whether to preserve the names of the elements in the resulting vector. If TRUE, names are retained; if FALSE, the names are discarded. Now that we have gone over the syntax, let’s see some examples. Example 1: Flattening a Simple List Let’s start with a straightforward example of a list containing some numeric values: # Create a simple list my_list <- list(1, 2, 3, 4, 5) [1] 1 [1] 2 [1] 3 [1] 4 [1] 5 # Flatten the list flattened_vector <- unlist(my_list) In this example, we had a list containing five numeric elements, and unlist() transformed it into a flat atomic vector. Example 2: Flattening a Nested List The real magic of unlist() shines when dealing with nested lists. Let’s consider a nested list: # Create a nested list nested_list <- list(1, 2, list(3, 4), 5) [1] 1 [1] 2 [1] 3 [1] 4 [1] 5 # Flatten the nested list flattened_vector <- unlist(nested_list) The unlist() function works recursively by default, so it will dive into the nested list and create a single vector containing all elements. Example 3: Removing Names from the Result Sometimes, you might prefer to discard the names of elements in the resulting vector to keep things simple and clean. You can achieve this using the use.names parameter: # Create a named list named_list <- list(a = 10, b = 20, c = 30) [1] 10 [1] 20 [1] 30 # Flatten the list without preserving names flattened_vector <- unlist(named_list, use.names = TRUE) Challenge Yourself!* Now that you’ve grasped the magic of unlist(), I encourage you to try using it with your own data sets. Test it with nested lists, mix different data types, and experiment with the recursive and use.names parameters to see how they impact the results. Remember, the unlist() function is a powerful tool for simplifying complex data structures, so keep it in your arsenal whenever you need to flatten lists in R. I encourage you to try the unlist() function on your own. You can find more information about the function in the R documentation: https://www.rdocumentation.org/packages/base/versions/3.6.2/topics/ Additional Tips Here are some additional tips for using the unlist() function: • The unlist() function will try to coerce the elements of the list to the same data type. For example, if the list contains a numeric vector and a character vector, the unlist() function will coerce the character vector to numeric. • If the use.names argument is set to TRUE, the unlist() function will preserve the names of the list elements. However, if the names of the list elements are not unique, the unlist() function will append a number to the name of each element. • The unlist() function can be used to unlist nested lists. However, if the recursive argument is not set to TRUE, the unlist() function will only unlist the top-level list. In this blog post, we’ve explored the unlist() function in R and demonstrated its usage with various examples. I hope you found it engaging and are excited to try unlist() on your own. Don’t hesitate to experiment further and see how it can enhance your data manipulation skills in R. If you have any questions or thoughts, feel free to leave a comment below. Happy coding!	{"url":"https://www.r-bloggers.com/2023/08/the-unlist-function-in-r/","timestamp":"2024-11-09T09:25:54Z","content_type":"text/html","content_length":"80023","record_id":"<urn:uuid:bd5cf31c-ef0e-4fba-841a-5ebe6810c68b>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00446.warc.gz"}
IB Mathematics: Higher Level (HL) or Standard Level (SL)? HL and SL choices Part of choosing math in the IB is deciding between taking a Higher (HL) course, or a Standard Level (SL) course. So, how should a student decide what level they should take? • Time： Both levels are meant to be taught during a 2-year period. According to the IBO (International Baccalaureate Organisation), SL subjects require at least 150 hours of instructional time, while HL subjects require at least 240 hours. • Exam： SL students are required to do two papers in the final math exam. Paper 1 lasts for one and a half hours and consists of short-answer questions. Paper 2 also lasts for one and a half hours but instead consists of long-answer questions. The structure of Paper 1 and 2 is the same for HL students, though both papers last longer, two hours each. Besides this, they are also required to do an additional Paper 3, which lasts for one hour. • Assessments： Although they both have a compulsory internal assessment (IA), the criteria of both courses differ from one another. This is due to the additional requirements and different criteria of HL, such as the addition of Criterion E in the Mathematics IA. Therefore, this changes the weight of the assessments in the final IB grade. • Content： The core syllabus is the basic content of the IB math curriculum and, thus, is compulsory in both levels. Furthermore, there’s additional content for HL subjects (that is not taught at SL) to explore the subject more in depth. • University： Students should always take into consideration the subject they want to pursue in university. This is because many universities require students take certain subjects at the HL. Math HL is a must in some university courses like mathematics, physics and engineering. In conclusion , students who struggle with mathematics and want to pursue university pathways not involving mathematics (e.g. English, Design) will be suited to SL. On the other hand, students who excel at mathematics and aim to take mathematically demanding courses (e.g., physics, engineering) may consider the HL. An experienced IB math tutor can definitely help you.	{"url":"https://viclimath.com/2024/09/27/ib-mathematics-higher-level-hl-or-standard-level-sl/","timestamp":"2024-11-12T07:29:38Z","content_type":"text/html","content_length":"89867","record_id":"<urn:uuid:f83df2f4-3da8-4305-b118-fa31e99e398f>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00126.warc.gz"}
Get Using Matlab Functions in Mathematics Assignment Help 1. How to Use Matlab Function In Mathematics Use Matlab functions for mathematics A beginner in programming might be hearing of the word functions in programming for the first time. It’s much easier for you to get confused with the term mathematical functions, especially if you have a mathematical background. The functions described here are computer programming functions. These functions are the main building blocks for each code that you would write. If you want to accomplish a complicated task, you must incorporate its use. Nearly every task that you perform in Matlab involves the use of functions. For a mathematics assignment, Matlab has a large number of functions that you could utilize. You might be wondering whether itis possible to use Matlab functions for mathematics? Do not worry. This article would brief you about the functions which you can apply in your mathematics assignment. What are their functions? Whichever programming language you may use, you will come across this term. It isn’t as complicated as you may think. it’s a simple one. Perhaps you might have used it many times before. In the computer programming realm, we define a function as a reusable set of lines of code. Functions can return a value or perform a precise operation. In some computer languages, the word a function and a procedure can be used alternatively. Types of functions. Functions can be classified in various ways. The common ways of categorizing functions are:- 1. Inbuilt and user-defined functions In Matlab, you might have come across the function print. Well, this is an example of an inbuilt function. Sometimes they are referred to as readymade functions. We define an inbuilt function as a function that comes with the libraries. When you decide to use a toolbox such as the signal processing toolbox, it will come with the function that you can use. These inbuilt functions have a syntax that you must adhere to. Failure could lead to errors. Therefore, to avoid such issues, you must read the related documents. User-defined functions are the ones that you have to create yourself. These functions require you to declare the variables that you would use and the conditions which the variables satisfy. Matlab has a unique way of creating a function. To create a function, you use Matlab’s inbuilt function known as function at the beginning, and you close the function by using the end statement. Let’s show you an example. Say you want to create a function pack. function f = pack(x, y) ‘Your statements’ From the above function, you can note that we made use of values x and y. These are the variables the function needs and whenever you call this function you must state these variables. Variable f is a function that is used inside the function statement and has an impact. 2. Returns a value or performs an operation Here is another way of classifying a function based on the results. By functions that return a value, we imply those functions that yield a numerical value, string, matrix, or a logical value. Functions that return a value are the ones that we will use a lot in most mathematical courses. For instance, you might be required to conduct a hypothesis test of a population. In this case, you need to compute the p-value. Functions that perform an operation are very common. They do not yield anything. They include the function that deletes the item in the console. Their use is less common in mathematical 3. Local or nested functions Here we classify user-made functions depending on the structure in a file. Note it’s generally possible for a code file to have more than one function. A local function is the most used by many, as it’s much easier. They are the normal functions that you create in a file. They are just subroutines that are in the same file, not following a specific order. In a nested function, the functions are contained in one function known as the parent function. These are functions that are used by experts in Matlab. They are particularly useful if the functions use shared data. Additionally, it’s much easier to call a variable in the parent function without the need to pass it as an argument in the other function. 4. Public and global Here we define a function depending on where it can be used. Private functions are used in the file where they are created while public functions can be used in other files. Is it possible to use Matlab for mathematics? The answer to this question is a straightforward one. Yes, it’s possible. Matlab is a mathematical computing software that has been in the industry for more than three decades. It has evolved to become one of the core programming languages. Unlike other high-level programming languages such as C, Matlab was introduced in the market to help with mathematics problems. This was its goal and the software has never deviated from it. As time went by, it has been incorporated into the curriculum of many universities. The main advantage that you will find with Matlab when developing a mathematics code is that it has all the functions that you will need. In most cases, you won’t need user-defined functions. But if you do, you will find the process of debugging easier as compared to other languages. What mathematical problems can I solve using Matlab? There are a lot of Matlab functions that you can use for mathematics. But mathematics is quite wide-ranging and is used in most disciplines. In fact, everything is built upon mathematics. All the engineering tasks have mathematics in them. Here are some of the topics where you can use Matlab: – 1. Fourier transform 2. Laplace transform 3. Regression analysis 4. Arithmetic operations 5. Algebra 6. Calculus 7. Number theory 8. Numerical differentiation 9. Probability theory 10. Computational geometry 11. Signal processing 12. Computational finance Why we are the best Sometimes students might find it hard to complete a mathematics assignment. They could be short of time or may have lots of assignments to attend to with strict deadlines. In some cases, the assignment might be tough and need the attention of a professional. Students, therefore, result in asking for professional help which ensures the completion of the assignment before the deadline. This also ensures that the students score high marks. Matlab assignment experts are one such academic help platform. We are a platform that has been serving students for years. We offer our services for any kind of Matlab related assignment. For the years we have been in this business, we have provided our clients with the finest services. 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We promise you that we will help you score a high grade, irrespective of how complex you think your assignment is. In this sample solution demonstrated by the MatLab tutor , there is an application of the Matlab function and Matlab command concept. The has revealed ‘how to write Matlab functions Intcompare, Splinecalc, and Hermitecalc under the given constraints. The functions take stated inputs and furnish predetermined output. In one of the parts of the assignment, the has demonstrated the use of Matlab command “diff” for calculation of derivative of a function. SOLUTION: – function ret = intcompare(x, f, a, b) syms xs y = double(f(x)); n = length(x); [ai,bi,ci,di] = splinecalc(x,f); [Q,z] = hermitecalc(x,f); %% Let’s build the function S(x) for i = 1:n S(i) = ai(i) + bi(i)(xs-x(i)) + ci(i)(xs-x(i))^2 + di(i)(xs-x(i))^2; real_integral_val = double(int(f, xs, a, b)); piece_int = 0; for i = 1:n-1 piece_int = piece_int + double(int(S(i), xs, x(i), x(i+1))); %% Hermite poly k = 2(n-1)+1; Sh(xs) = Q(k+1,k+1)(xs-z(k)); for i = 2:k j = k-i+1; Sh(xs) = (Sh(xs) + Q(j+1,j+1))(xs-z(j)); Sh(xs) = Sh(xs) + Q(1,1); hermite_integral = double(int(Sh, xs, a,b)); ret = zeros(3,3); ret(1,1) = real_integral_val; ret(1,2) = piece_int; ret(1,3) = hermite_integral; ret(2,1) = 0; ret(2,2) = piece_int-real_integral_val; ret(2,3) = hermite_integral-real_integral_val; ret(3,1) = 0; ret(3,2) = ret(2,2)/real_integral_val; ret(3,3) = ret(2,3)/real_integral_val; function [a,b,c,d] = splinecalc(x, f) % Campled Cubic Splien Algorithm (3.5) syms xs n = length(x)-1; y = double(subs(f,x)); a = y; FPO = double(subs(diff(f,xs), x(1))); FPN = double(subs(diff(f,xs), x(n))); % Step 1 M = n-1; h = zeros(1,M+1); for i = 0:M h(i+1) = x(i+2)-x(i+1); %Step 2 alfa = zeros(1,n+1); alfa(1) = 3(a(2)-a(1))/h(1) – 3FPO; alfa(n+1) = 3FPN – 3(a(n+1) – a(n))/h(n); % Step 3 for i = 1:M alfa(i+1) = 3(a(i+2)h(i)-a(i+1)(x(i+2)-x(i)) + a(i)h(i+1))/(h(i+1)h(i)); %Step 4 I = zeros(1,n+1); u = zeros(1,n+1); z = zeros(1,n+1); I(1) = 2h(1); u(1) = 0.5; z(1) = alfa(1)/I(1); % Step 5 for i = 1:M I(i+1) = 2(x(i+2)-x(i)) – h(i)u(i); u(i+1) = h(i+1)/I(i+1); z(i+1) = (alfa(i+1) – h(i)z(i))/I(i+1); % Step 6 I(n+1) = h(n)(2-u(n)); z(n+1) = (alfa(n+1) – h(n)z(n))/I(n+1); c = zeros(1,n+1); b = zeros(1,n+1); d = zeros(1,n+1); c(n+1) = z(n+1); % Step 7 for i = 1:n j = n-i; c(j+1) = z(j+1) – u(j+1)c(j+2); b(j+1) = (a(j+2) – a(j+1))/h(j+1) – h(j+1)(c(j+2)+2c(j+1))/3; d(j+1) = (c(j+2) – c(j+1))/(3h(j+1)); function [Q,z] = hermitecalc(x, f) syms xs fp = diff(f,xs); % Algorithm 3.3 n = length(x)-1; Q = zeros(2n+2, 2n+2); % Step 1 z = zeros(1,2n+2); for i = 0:n % Step 2 z(2i+1) = x(i+1); z(2i+2) = x(i+1); Q(2i+1,1) = double(subs(f(x(i+1)))); Q(2i+2,1) = double(subs(f(x(i+1)))); Q(2i+2,2) = double(subs(fp(x(i+1)))); % Step 3 if i ~= 0 Q(2i+1,2) = (Q(2i+1,1) – Q(2i, 1))/(z(2i+1) – z(2i)); % Step 4 for i = 2:2n+1 for j = 2:i Q(i+1,j+1) = (Q(i+1,j) – Q(i, j))/(z(i+1) – z(i-j+1)); clc, clear all, close all syms xs % Variable used to define a symbolic function f(xs) = exp(xs)^2; % Symbolic function to be evaluated x = 0:3; % x-data [0 1 2 3] y = double(f(x)); % y data f(x) % Coeficients a and b ap= x(1); bp = x(end); n = length(x); %% Calculate and display spline coefficients [a,b,c,d] = splinecalc(x,f); spline_coefficients = [a;b;c;d]; %% Calculate and display intcompare solution intcompare_solution = intcompare(x,f,x(1),x(end)) %% Calculate and display hermite polynomials [Q,z] = hermitecalc(x,f); hermite_coefficients = Q; %% Plot interpolations xspan = linspace(ap,bp,20); plot(xspan, f(xspan), ‘–mo’, ‘color’, [0.8500, 0.3250, 0.0980], ‘linewidth’, 3) %red hold on % spline_plot = zeros(1,(n-1)20); for i = 1:n-1 lb = x(i); if i < n ub = x(i+1); ub = 2x(i); xspan = linspace(lb,ub,20); spline_plot((i-1)20+1:i20) = a(i) + b(i).(xspan-x(i)) + c(i)(xspan-x(i)).^2 + d(i).(xspan-x(i)).^3; plot(xspan, a(i) + b(i).(xspan-x(i)) + c(i)(xspan-x(i)).^2 + d(i).(xspan-x(i)).^3, ‘:’, ‘color’, [0, 0.4470, 0.7410], ‘linewidth’, 2); hold on %% Hermite poly k = 2(n-1)+1; Sh(xs) = Q(k+1,k+1)(xs-z(k)); for i = 2:k j = k-i+1; Sh(xs) = (Sh(xs) + Q(j+1,j+1))(xs-z(j)); Sh(xs) = Sh(xs) + Q(1,1); xspan = linspace(ap,bp,20); plot(xspan, Sh(xspan), ‘–‘, ‘color’, [0.9290, 0.6940, 0.1250], ‘linewidth’, 2) legend(‘Real Values’, ‘Spline’, ‘Hermite’); % [h,~] = legend(‘show’) grid minor title(‘Comparisson of interpolations’); clc, clear all, close all syms xs % Variable used to define a symbolic function f(xs) = exp(xs)^2; % Symbolic function to be evaluated x = 0:3; % x-data [0 1 2 3] y = double(f(x)); % y data f(x) % Coeficients a and b ap= x(1); bp = x(end); n = length(x); %% Calculate and display spline coefficients [a,b,c,d] = splinecalc(x,f); spline_coefficients = [a;b;c;d]; %% Calculate and display intcompare solution intcompare_solution = intcompare(x,f,x(1),x(end)) %% Calculate and display hermite polynomials [Q,z] = hermitecalc(x,f); hermite_coefficients = Q; %% Plot interpolations xspan = linspace(ap,bp,20); plot(xspan, f(xspan), ‘–mo’, ‘color’, [0.8500, 0.3250, 0.0980], ‘linewidth’, 3) %red hold on % spline_plot = zeros(1,(n-1)20); for i = 1:n-1 lb = x(i); if i < n ub = x(i+1); ub = 2x(i); xspan = linspace(lb,ub,20); spline_plot((i-1)20+1:i20) = a(i) + b(i).(xspan-x(i)) + c(i)(xspan-x(i)).^2 + d(i).(xspan-x(i)).^3; plot(xspan, a(i) + b(i).(xspan-x(i)) + c(i)(xspan-x(i)).^2 + d(i).(xspan-x(i)).^3, ‘:’, ‘color’, [0, 0.4470, 0.7410], ‘linewidth’, 2); hold on %% Hermite poly k = 2(n-1)+1; Sh(xs) = Q(k+1,k+1)(xs-z(k)); for i = 2:k j = k-i+1; Sh(xs) = (Sh(xs) + Q(j+1,j+1))*(xs-z(j)); Sh(xs) = Sh(xs) + Q(1,1); xspan = linspace(ap,bp,20); plot(xspan, Sh(xspan), ‘–‘, ‘color’, [0.9290, 0.6940, 0.1250], ‘linewidth’, 2) legend(‘Real Values’, ‘Spline’, ‘Hermite’); % [h,~] = legend(‘show’) grid minor title(‘Comparisson of interpolations’);	{"url":"https://www.matlabassignmentexperts.com/use-matlab-function-in-mathematics.html","timestamp":"2024-11-06T21:01:45Z","content_type":"text/html","content_length":"121861","record_id":"<urn:uuid:4e194fc5-6023-4dfd-9c82-128e64c24861>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00200.warc.gz"}
Use of a principle of minimum values of derivative to the construction of finite difference diagrams for the calculation of the discontinuous solution of gas dynamics Difference diagrams were constructed for the model equation which possess the second order of accuracy in coordinates and the first order of accuracy in time. The properties of the diagrams were studied, and the results of the study were used for constructing a diagram for solving one-dimensional unsteady equations of gas dynamics. NASA STI/Recon Technical Report N Pub Date: January 1975 □ Finite Difference Theory; □ Gas Dynamics; □ Numerical Analysis; □ Minima; □ Shock Waves; □ Unsteady Flow; □ Fluid Mechanics and Heat Transfer	{"url":"https://ui.adsabs.harvard.edu/abs/1975STIN...7533357K/abstract","timestamp":"2024-11-04T16:12:18Z","content_type":"text/html","content_length":"33459","record_id":"<urn:uuid:26091a38-5125-4813-ab8e-a3a68427b8fb>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00383.warc.gz"}
AP Board 6th Class Maths Solutions Chapter 1 Numbers All Around us Unit Exercise AP State Syllabus AP Board 6th Class Maths Solutions Chapter 1 Numbers All Around us Unit Exercise Textbook Questions and Answers. AP State Syllabus 6th Class Maths Solutions 1st Lesson Numbers All Around us Unit Exercise Question 1. Write each of the following in numeral form. i) Hundred crores hundred thousands and hundred. Indian system: 100,01,00,100. ii) Twenty billion four hundred ninety seven million pinety six thousands four hundred seventy two. Question 2. Write each of the following in words in both Hindu-Arabic and International system. i) 8275678960 ii) 5724500327 iii) 1234567890 i) 8275678960 Hindu – Arabic system: 827,56,78,960 Eight hundred twenty seven crores fifty six lakhs seventy eight thousand nine hundred and sixty. International system: 8,275,678,960 Eight billion two hundred seventy five million six hundred seventy eight thousand nine hundred and sixty. ii) Hindu-Arabic system: 572,45,00,327 Five hundred seventy two crores forty five lakhs three hundred and twenty seven. International system: 5,724,500,327 Five billion seven hundred twenty four million five hundred thousand three hundred and twenty seven, iii) 1234567890 Hindu-Arabic system: 123,45,67,890 One hundred twenty three crores forty five lakhs sixty seven thousand eight hundred and ninety. International system: 1,234,567,890 One billion two hundred thirty four million five hundred sixty seven thousand eight hundred and ninety. Question 3. Find the difference between the place values of the two eight’s in 98978056. Place values of 8 in Hindu-Arabic system of the given number are 8,000 and 80,00,000 Difference = 80,00,000 – 8,000 = 79,92,000 Place values of 8 in International system of the given number are 8000 and 8,000,000 Difference = 8,000,000 – 8,000 = 79,92,000 Question 4. How many 6 digit numbers are there in all? Number of 6 digit numbers = greatest 6 digit number – greatest 5 digit number = 9,99,999 – 99,999 = 9,00,000. Question 5. How many thousands make one million? 1000 thousands can make one million. 1 Million = 1000 Thousands. Question 6. Collect ‘5’ mobile numbers and arrange them in ascending and descending order. Let the 5 mobile numbers are: 9247568320, 9849197602, 8125646682, 6305481954, 7702177046 Ascending order: 6305481954, 7702177046, 8125646682, 9247568320, 9849197602 Descending older: 9849197602, 9247568320, 8125646682, 7702177046, 6305481954 Question 7. Pravali has one sister and one brother. Pravali’s father earned one million rupees and wanted to distribute the amount equally. Estimate approximate amount each will get in lakhs and verify with actual division. One million = 10,00,000 = 10 lakhs Pravali’s father distributed 10 lakhs amount to his 3 children equally. So, the share of each children = 10 lakhs ÷ 3 = Rs. 3,33,333 = Rs. 3,00,000 (approximately) Question 8. Government wants to distribute Rs. 13,500 to each farmer. There are 2,27,856 farmers in a district. Calculate the total amount needed for that district. (First estimate, then calculate) Question 9. Explain terms Cusec, T.M.C, Metric tonne, Kilometer. a) Cusec: A unit of flow equal to 1 cubic foot per second. 1 Cusec = 0.028316 cubic feet per second = 28.316 litre per second. Cusec is the unit to measure the liquids in large numbers quantity. b) TMC: TMC is the unit to measure the water in large quantity. TMC means Thousand Million Cubic feet. 1 TMC = 0.28316000000 litre = 28.316 billion litre = 2831.6 crores litre c) Metric tonne: Metric tonne is the unit of weight. Metric tonne = 1000 kg = 10 quintals We should use this unit in measuring crops, paddy, dall, etc. d) Kilometer: Kilometre is the unit of length. 1 kilometer = 1000 meters. We should use this unit is measuring distance between villages, towns, cities,…. etc.	{"url":"https://apboardsolutions.guru/ap-board-6th-class-maths-solutions-chapter-1-unit-exercise/","timestamp":"2024-11-03T06:01:11Z","content_type":"text/html","content_length":"53634","record_id":"<urn:uuid:c87204c7-5b87-46e6-8007-ac82cd734249>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00353.warc.gz"}
Canonical correlation From Encyclopedia of Mathematics The correlation between linear functions of two sets of random variables determined by the maximization of this correlation subject to certain constraints. In the theory of canonical correlations the random variables $ X _ {1} \dots X _ {s} $ and $ X _ {s+1} \dots X _ {s + t } $, $ s < t $, are linearly transformed into the so-called canonical random variables $ Y _ {1} \dots Y _ {s} $ and $ Y _ {s+1} \dots Y _ {s+t} $ so that a) all $ Y $ have mathematical expectation zero and variance one; b) within each of the two sets the variables $ Y $ are uncorrelated; c) each $ Y $ in the first set is correlated with just one $ Y $ of the second set; d) the non-zero correlation coefficients between $ Y $' s of different sets have (consecutively) maximum values, subject to the requirement of zero correlation with previous $ Y $' s. In the particular case $ s = 1 $, the canonical correlation is the multiple correlation between $ X _ {1} $ and $ X _ {2} \dots X _ {1+t} $. The transformation to the canonical random variables corresponds to the algebraic problem of reducing a quadric to canonical form. In multivariate statistical analysis the method of canonical correlations is used, in the study of interdependence of two sets of components of a vector of observations, to realize a transition to a new coordinate system in which the correlation between $ X _ {1} \dots X _ {s} $ and $ X _ {s+1} \dots X _ {s+t} $ becomes transparently clear. As a result of the analysis of canonical correlations it may turn out that the interdependence between the two sets is completely described by the correlation between several canonical random variables. [1] H. Hotelling, "Relations between two sets of variates" Biometrika , 28 (1936) pp. 321–377 [2] T.W. Anderson, "Introduction to multivariate statistical analysis" , Wiley (1958) [3] J.K. Ord, "The advanced theory of statistics" , 3 , Griffin (1983) How to Cite This Entry: Canonical correlation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Canonical_correlation&oldid=46193 This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article	{"url":"https://encyclopediaofmath.org/wiki/Canonical_correlation","timestamp":"2024-11-08T02:47:53Z","content_type":"text/html","content_length":"15656","record_id":"<urn:uuid:73bd63ea-52d0-4fe0-8888-57de6626abf9>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00706.warc.gz"}
Refining the Peak Oil Rosy Scenario Part 4: Testing Hubbert’s linear and nonlinear logistic models I've always seen modeling as a stepping stone—Tyra Banks A lot of modeling is how much crap you can take—Lauren Hutton Logistic models Before delving into simulated data generation and modeling, I want to make a few comments about the logistic model that I will be considering. Hubbert briefly noted in "Techniques" (p. 49) that the logistic equation was originally derived in the 1800 's to model human population growth. It should not be too surprising that oil production, and consumption, should follow a logistic growth model as well. Indeed, Hubbert was well aware of the inter-relationship between exponentially increasing population growth and the exponentially increasing amount of energy made available by fossil fuels: Prior to 1800 most of the energy available to man was that derivable from his food, the labor of his animals, and the wood he used for fuel. On a world-wide basis it is doubtful if the sum of these exceeded 10,000 kilogram-calories per man per day, and this was a several-fold increase over the energy from food alone. After 1800 there was superposed on top of these sources the energy from fossil fuels. From a world average of 300 kilogram-calories per capita per day in 1800 the energy from coal and petroleum increased to 9,880 by 1900, and to 22,100 by 1940. In the areas of high industrialization this quantity is much larger. In the United States, for example, the energy from coal and petroleum consumed per day per capita amounted in 1940 to 114,000 kilogram-calories (2); and from coal, petroleum, and natural gas, to 129,000. Viewed on such a time scale, the curve (If human population would be fiat and only slightly above zero for all preceding human history, and then it too would be seen to rise abruptly and almost vertically to a maximum value of several billion. Thereafter, depending largely upon what energy supplies are available, it might stabilize at a maximum value, as in Curve I, or more probably to a lower and more nearly optimum value, as in Curve II. However, should cultural degeneration occur so that the available energy resources should not be utilized, the human population would undoubtedly be reduced to a number appropriate to an agrarian existence, as in Curve III. These sharp breaks in all the foregoing curves can be ascribed quite definitely, directly or indirectly, to the tapping of the large supplies of energy stored up in the fossil fuels. The release of this energy is a unidirectional and irreversible process. It can only happen once, and the historical events associated with this release are necessarily without precedent, and are intrinsically incapable of repetition. MK Hubbert, Energy from Fossil Fuels, Science. February 4, 1949, Vol. 109, p. 103-109. And, more recently, as more bluntly put by Roscoe Bartlett, modern industrial society literally “eats” oil: What we do here to look back through history when the industrial revolution first started with the brown line there which is wood and it was stuttering of course when we found coal. And then we found oil and look what happened. And the ordinate here is quadrillion BTU’s and the abscissa, of course, is time. I would just like to note that the population curve of the world roughly follows the curve for oil here. We started out with about a billion people and now we have about 7 billion people almost literally eating oil and gas because of the enormous amounts of energy that go into producing food. Almost half the energy that goes into producing a bushel of corn comes from the natural gas that we use to produce the nitrogen fertilizer. Roscoe Barlett R-MD, HEARING BEFORE THE SUBCOMMITTEE ON ENERGY AND AIR QUALITY, DECEMBER 7, 2005, Serial No. 109-41. Available online here or here. At least for modern industrial society, I think of fossil fuels as being essentially equivalent to a food source that "fuels" the growth of human population—the faster we pump the oil, the faster the population can grow. Obviously, the oil and other fossil fuels is not a direct food source, but rather indirectly makes food more available, through a variety of means that I will not go into here. Therefore, we should not be surprised to see that oil production and consumption would follow a logistic model, just as human population growth would follow a logistic model. Rates of oil consumption and population growth are linked at the hip. My Modeling Goals Like Tyra Banks, I see modeling as a stepping stone to more interesting things. My long term goal is to use Hubbert’s linear logistic equation (equation 27 in Part 3) to make an estimate of “a,” the rate constant for oil production decline, as well as the rate constant for oil consumption, so that I can make new estimates of the prospects for oil importation into the USA using the land-export model over the next few years. It is appropriate first, however, to do some modeling to check the ability of Hubbert’s linear logistic equation to estimate these rates. I agree with Lauren Hutton, however, that those who model have to take a lot of crap. You are not analyzing real data, so it seems like your wasting your time. You always get critics who say that you didn't model the right thing. Still other critics will not be very happy if your modeling exposes some flaws in the model. I am aware that there was some controversy when Jeff Brown and colleagues first applied the linear equation to their land-export model (e.g., see Robert Rapier’s articles part 1 and part 2. As a previous modeler (of spectroscopic data) I am sympathetic to the points that Rapier was trying to make. For instance, in part I of his critique, Rapier used Texas oil production data to show that, depending on which range of years one used, the estimates of, Q∞ from the horizontal intercept of the plot, could vary quite dramatically. For instance taking oil production data available from 1960 versus 1980 versus 2006 gave different estimate of Q∞ and hence Q∞/2, the point where one-half of the recoverable oil has been extracted. Although Q∞/2 is often implied to correspond to "peak oil" (i.e., the year of peak oil production), it is not the same thing. Q∞/2 coincides with peak oil if the production (dQ/dt) versus time plot is symmetric—i.e., a rosy scenario. Although this is important for estimating the year of Q∞/2 (which may equal peak oil production), that is not was I am interested doing in here. I am interested in estimating the value of “a,” which corresponds to the vertical intercept. In particular I am interested in being able to estimate recent changes in production and consumption because these are the numbers that will most affect my estimates of the oil available for importation into the USA versus rates of oil consumption in the USA. Nevertheless I think that modeling is a worthwhile exercise to validate the procedure and gain some insights into how best to apply the model when examining real data. One final note: I have presented this analysis in what some might think to be excruciating detail. I do this for two reasons: a) I want to make it clear that anyone with a spreadsheet and some high-school math can do this and reproduce this—you don’t have to be an ivory tower scientist to model! However, you do have to be willing to put some the time in. b) This is my blog, and I am not constrained by space! Simulated data generation and assumptions For those of you have never done modeling before I want to point out what may be obvious to those with experience—the strength of modeling is that you know the true answer to the problem that you are trying to solve, because you have generated the simulated data set based on that answer. Here, I have assumed some reasonable values of the parameters in Hubbert's logistic equation, based on Hubbert's own analysis of USA (lower 48 states) oil production, and used these to simulate a data set of Q versus t. Then, I used this simulated data set to test the ability of Hubbert’s linear logistic model to predict those parameters. As a starting point, I used the logistic equation and parameter values for Q∞, a, b and the production start time (t[o]) as presented in Hubbert’s Figure 15 shown at the end of part 3: Q = Q∞ / (1 + b∙e^–a(t-1900)) [1] Q∞= 170 bbls (i.e., 170 billion barrels of oil) a = 0.0687 yr^-1 (i.e., an exponential rate of consumption equal to 6.87 percent per year) t[o] = 1900 yrs b = 46.8 Notice that b is the same as N[o] in Hubbert's equation (38) (see Part 3 of this series). Recalling that N[o] = (Q∞-Qo)/Qo and given that Q∞ is assumed to equal 170, we can solve for Qo: 46.8 = (170 – Qo)/ Qo Qo = 170/(46.8+1) = 3.556 bbls The value of “b” reflects Hubbert’s assumption (or his knowing, from data not explicitly presented in "Techniques") of a certain amount of cumulative oil production by 1900 (recall his comment that "the choice of the date for t = 0 is arbitrary so long as it is within the range of the production cycle so that N[o] will have a determinate finite value"). Using the above equation [1] and these parameters I can use a spreadsheet to generate a simulated data set of Q versus t (in years), and then use this to calculate dQ/dt. Using a spreadsheet this is easy: for the first year, dQ equals Q(1901)-Q(1900) and dt equals 1901-1900, for the second year dQ equals Q(1902)-Q(1901) and dt equals 1902-1901, and so on. Figure 1 shows a plot of dQ/dt versus t for the simulated data ranging from 1901 to 2010. I will refer to this plot as the production curve. Analyzing simulated data using Hubbert’s linear logistic model Okay now we have our data set for which we absolutely know the correct answers: Q∞= 170 bbls and a = 0.0687 yr^-1. When we linearize this data, according to Hubbert’s linear logistic equation, and, perform linear regression analysis to calculate the intercepts, do we get these answers back? To linearize the simulated data, first recall the form of the linear logistic model: (dQ/dt)/Q = a - (a/Q∞)Q [2] Therefore, from my data set of (Q, t) value I need to calculate the production rate (dQ/dt) as shown in Figure 1, and divide this by Q (e.g., Q in 1901, 1902 etc...), and then plot (dQ/dt)/Q versus Q and then plot this against Q. Figure 2 shows the result: If I then perform linear regression analysis of the entire data set from 1901 to 2010 using the SLOPE, INTERCEPT, RSQ featured of Excel) here is the result: slope = -3.9 x 10^-4 y-intercept = 0.06695 = a x-intercept = 171.5 = Q∞ r^2 = .9998 The plot looks pretty linear to the human eye and the r^2 supports that this is a substantially linear plot. Still, it is somewhat surprising to me, given that I am using “perfect” (simulated) data, that the y-intercept and x-intercept are not exactly equal to “a” and Q∞, respectively. Nevertheless, the percentage error from the true value for both parameters (i.e., “a” underestimated by 2.5 percent and Q∞ overestimated by 0.9 percent) is fairly small and certainly would be good enough for my purposes. Accuracy when using limited time segments Of course, we are never going to have a full data set such shown in Figure 1. How well does the linear model predict “a” and Q∞ if only use partial data, and, does it matter where in the production curve of Figure 1 we take the partial data from? To address these questions I performed linear regression analysis to calculate to y-intercept (“a”) and the x-intercept (Q∞) for independent 5-year segments along the whole span of the data set from 1901 to 2010. Then for each of these 5-year segments I calculated the percentage error in the estimate value of “a” and Q∞ as compared to the known true values (0.0687 and 170, respectively). Here’s some examples for the 1901 to 1905, 1951 to 1955, and 2006 to 2010 5-year segments: The term Average “% of Q∞” simply refers to the average value of Q within the time segment of interest, divided by the true value of Q∞. This term helps show where we are on the production curve shown in Figure 1. The table shows the trend for “a” to be underestimated by a few percent when we analyze time segments at the beginnings of the curve and to be overestimated when using time segments at the end of the curve. There is a trend for Q∞ to be overestimated when we analyze time segments at the beginnings of the curve, and time error progressively decreases as we use time segments towards the end of the These trends are more clearly show in plots of the % of true values versus time segment as shown in two figures below (for the horizontal axis I have just depicted the first year of each five-year segment). Figure 3 shows the percentage different in predicted value of “a” compared to the true value as a function of the percentage of Q∞ along the production curve, and, Figure 4 shows the analogous percentage different in predicted value of Q∞: As you can see, for all time segments the absolute value of the % error never exceeds about 3.5% for the prediction of “a” and never exceeded 7% for the prediction of Q∞. Moreover, if you only consider time segments where Q is in the range of 20 to 80% of Q∞ (i.e., the likely range we will encounter when analyzing real data) the error in predicting “a” is never greater than about -3% to +1% and the error in predicting Q∞ is never greater than about +4.5%. I have repeated this exercise using values of “a” ranging from 50% to 150%, and, “b” ranging from 50% to 200% of the values of the parameters, "a" and Q∞, that were used to generate the simulated data in Figure 1. For Q in the range of 20 to 80% of Q∞ I never see the error in “a” to be outside of about -4.4% to +1.5% and the error in Q∞ is never greater than about +7 %. These upper error bounds all occurred for the same set of simulated data, where “a” was set to equal 0.10305 (i.e., a 50 percent greater rate than used in Figure 1). Therefore, it looks like the linear logistic equation can make reasonable accurate predictions of “a” and Q∞ under the conditions studied here. That is, the "a" the exponential rate constant for production remains constant through the entire production curve. Accuracy when the rate constant for production declines post peak production This really gets the gist of the matter: how well does Hubbert’s logistic linear equation (27) predict oil production when there is decline in the exponential rate constant? Being sure that an accurate estimate of the rate constant can be made using this model is pretty fundamental to being able to apply this model to real data for the purposes of refining the export-land model for the USA as outlined in Part 1 of this series. To test this, I have taken the simulated data from figure 1 (“a” = 0.0687) and assumed that after the peak year (i.e., the year where the yearly production rate, dQ/dt, reached its maximum in 1956), for each year to follow, the exponential rate constant “a” decreased by 2 percent per year. That is, in 1957, a equals a(1956) x 0.98; in 1958, a equals a(1957) x 0.98; etc.... Here’s what the production curve using these assumption looks like: Figure 5 shows simulated production data Q∞ = 170, a = 0.0687 and then decreasing 2%/yr from 1957 and on. For reference, I also show the previous data set where “a” = 0.0687 throughout the entire produce curve. I note here that for each of the subsequent years, 1957 and on, where “a” changes, one needs to applying an adjusted logistic equation, where not only “a” is changed as described above, but also “b” and t[o] are adjusted to reflect the accumulation, Q, up to the transition year. Recall again, b = N[o] = (Q∞-Qo)/Qo. The value of Qo for the subsequent years with the altered value of “a” are similarly adjusted as well t[o]. For instance, in 1956, Q = 85.056. Therefore, for the subsequent year, 1957 where “a” is decreased to 0.067326, “b” is given by: b = (Q∞-Qo)/Qo = (170 – 85.056)/ 85.056 = 0.9987, and t[o] equals 1956. Similar adjustments have to be made for each subsequent year along the production curve. Figure 6 shows the linearization of the simulated production data: Next, I estimated “a” and Q∞ for sequential 5-year segments and then estimated the error in the predicted values of these parameters as compared to the true values, similar to that described above in the context of Table 1 and figures 3-4—with one modification. In the present case, because the value of “a” is modeled to steadily decrease after the peak production year (1956), there is no single true “a” value within any of these post-peak 5-year segments. Therefore, I have calculated the average “a” for each of the 5-year-segments and used this as the true value of “a” in my evaluation of the linear logistic equation’s ability to accurately predict “a” and Q∞. Figures 7 and 8 show the percentage different in predicted value of “a” and Q∞, respectively: Up until the production maximum at % of Q∞ = 50 the prediction results, of course, are the same as depicted in Figures 3 and 4. However once we enter the range of data where % of Q∞ >50 and “a” is assumed to decrease 2%/yr, the predicted “a” and “Q∞” vary wildly from the true values, especially at the transition from a constant “a” to a declining “a.” The error in the predicted value of “a” gets progressively worse as Q Þ Q∞ and is approaches a 200% over-estimation at % of Q∞ ~ 90. The error in the prediction of Q∞ on a percentage basis is not as bad, but still badly under-estimates the true value at the transition and then progressively improves as Q Þ Q∞. Still, there is a systematic underestimation of Q∞ in the range of about -5% to -16%. I will not show the data here, but for analogous simulated data with “a” assumed to decrease by 4%/yr following peak production in 1956, the overestimation of “a” and underestimation of Q∞ is even worse (“a” overestimated by as much as 600% and Q∞ underestimated by as much as -200%). These results are disappointing to say the least. We need a better model. A constrained logistic nonlinear analysis of simulated data assuming a declining production rate constant Given the above disappointing results, I have abandoned Hubbert’s logistic linear equation in favor of directly applying the non-linear logistic equation, that is, Hubbert’s equation (38). More specifically I am using the the time derivative of the equation in Hubbert’s Fig.15, shown in Part 3 of this series, or in equation [1] shown above in this Part 4: dQ/dt = (Q∞ / (1 + N[o]∙e^–a(t-to)))/(Dt), [3] where N[o] = (Q∞-Qo)/Qo, Dt = time increment (1 year) One technical problem to over come is how to perform a non-linear least squares (NLLS) analysis, preferably, within confines of an Excel spread sheet, and preferably without laying out a lot of money for a fancy software package. Fortunately, Excel’s SOLVER feature (in the tools menu) provides a means of doing , and, there is a nice example on-line on how to set this up (Jeff Graham, Regression Using Excel's Solver, Regression Using Excel's Solver, ICTCM, vol. 13, Paper C013, 2000). In brief, the NLLS analysis was done by setting up a column with formulas to estimate dQ/dt according the equation [3] for each production year. Then I subtracted this predicted result from the actual or measured value (in this case the simulated production data set). Then I took the sum of the square of the differences, for the set of cells subject to the analysis, and put this into a single cell address in Excel that SOLVER minimizes by adjust the values of the cells holding the estimate of dQ/dt based on the value of the parameters, “a” “Q∞” and Q[o]. Note—the exact same three cells in the spreadsheet holding these parameters have to be in each of the formulas representing equation [3] in the column used to estimate dQ/dt. After doing a few trial runs of NLLS analysis on the simulated data shown in Figure 6, it became apparent to me that good predictions of “a” could be obtained for that portion of the production curve simulating the transition from a constant value to a steadily decreasing value for “a,” if Q∞ was constrained to a fixed value. For the analysis of the simulated data sets to follow, I have adopted the following procedure: 1) Use the production curve corresponding to Q in the range from 20% to 50% of Q∞ to estimate “a” Q∞ and Q[o]. I will call these the growth-side production parameters. 2) Use the values of the growth-side production parameters as the seed values for the spreadsheet column used to estimate dQ/dt for the successive 5-year segments on the decline side of the production curve, and, fix Q∞ (and indirectly, Q[o]) to the seed value determined in step (1). 3) Run the NLLS analysis for the successive 5-year segments to predict “a” and compare this to the average “a” for the 5-year segment being analyzed to determine the % difference from this true value of “a.” Results for non-linear least squares analysis of simulated data Figure 9 shows a composite of the production curve plots of 5-simulated data sets that were subject to this analysis. For all five data sets, the growth-side production parameters are the same as before (Figure1): Q∞ = 170 bbls; Qo = 3.556 bbls (equivalent to “b” = 46.8); and “a” = 0.0687 yr^-1 Thereafter, following the peak production in 1956, the value of “a” is yearly decreased by the different percentages shown in the legend to Figure 9. The NLLS analysis of the production curve data for Q in the range from 20% to 50% of Q∞ (1934 to 1954 in Figure 9) gave the following predicted values for the growth-side production parameters: Q∞ = 170.4 bbls; Qo = 3.579 bbls; and “a” = 0.06857 yr^-1 This is in pretty good agreement with the “true” values summarized above (as to be expected when dealing with simulated data). These values were then used as the seed values for the NLLS analysis of successive 5-year segments on the decline side of the production curves for each of the 5-simulated data sets. Some example results for the simulated data set, where “a” is decreased by 2%/yr, is presented in Table 2: The predictions of “a” are in excellent agreement with the average true “a” values for these time segments. The agreement is much better than that the predictions obtained using the logistic linear model (see Figure 3 above). This is due, at least in part, to fixing the value of Q∞ for the NLLS analysis done on the decline-side part of the production curve. There is a slight trend for underestimating the values of “a.” This becomes clearer in a plot of the % of the true value versus % of Q∞, for all five simulated data sets, as shown in Figure 10: The trend for underestimating “a” is progresses as Q Þ Q∞. This is due to the slightly higher than true value of Q∞ (170.4 versus 170) that was set to a fixed value. Overall, however, I am satisfied that the three-step procedure outline above could at least in principle make accurate predictions of a declining production rate constant. A constrained logistic nonlinear analysis of simulated data assuming two different declining production rate constants One final test before evaluating some “real” data. How well does the NLLS analysis procedure predict “a” for simulated data where, following peak production there is an initial rapid decline in “a” followed by a milder decline in “a”? To test this, I again assumed growth-side production parameters that are the same as used to generate the simulated data shown in Figure 9. Thereafter, following the peak production in 1956, the value of “a” was yearly decreased by 8% /yr from 1957 to 1970 and then decreased by 2%/yr thereafter. Figure 11 shows a plot of production curve plots of this simulated data set, along with a reference curve showing the data set where “a” was fixed to 0.0687 yr^-1 throughout the entire produce curve. Once again excellent predictive results were obtained. As shown in Figure 12 the % difference from the true value (again the average value of “a” for the same 5-year segment) is less than 1 percent over the range of % Q of Q∞ covered by the data set. Well, that was a long one, but well worth the effort! I hope... Next up, let’s see how the NLLS process handles some real data. 5 comments: 1. > r2 = .9998 > The plot looks pretty linear to the human eye and the r2 supports that this is a substantially linear plot. Still, it is somewhat surprising to me, given that I am using �perfect� (simulated) data, that the y-intercept and x-intercept are not exactly equal to �a� and Q∞, respectively. The explanation is that you have used 200 data points to get from t0 to t∞, and ideally you would use ∞ data points. To put that another way, you have charted what appears to be a smooth curve, but the chart should really have been drawn as a column type, with 200 columns, their areas representing the quantity of oil produced in the year, and 200 abrupt steps from one column to the next. If you had used monthly data, you would have 2,400 data points, the steps would have been smaller, and r� would have been even closer to 1.000 . 2. Dave It's a good idea and that may be it. I haven't gone back to test this, but my feeling is that it has to do with the value of Qo I assumed from b = 46.8. I did this before I appreciated how sensitive the linearity of equation [2] to the value of Qo, as part 6 in this series showed. 3. I found the comparison between Tyra Banks and Hubbert’s linear logistic to be hilarious. The thing i liked about your post, is that you turned a very heavy topic to a light fun reading material. 4. Thanks MC—yes I was trying to find some way to brighten up what is otherwise a pretty dry subject for most people. And, on this point Tyra, and I are in complete agreement! I never thought I would say this, but the girl has some good insights! Going into this analysis, as a prequel to looking at real data from real countries, I had high hopes for the linear logistic model, if for no other reason than that many others had used it. However, after my experiences with modeling simulated data, I lost my enthusiasm. Instead, as discussed in the article, I turned to non-linear least squares (NLLS) analysis of the logistic model, which is still fairly easy to implement within an Excel spread sheet environment, and, seems to give more reasonable results. There still might be some situation where I would have to use the linear model, but I haven’t found it yet. 5. Tyra Banks would be proud :) Your comments, questions and suggestions are welcome! However, comments with cursing or ad hominem attacks will be removed.	{"url":"https://crash-watcher.blogspot.com/2010/10/refining-peak-oil-rosy-scenario-part-4.html","timestamp":"2024-11-05T01:08:05Z","content_type":"text/html","content_length":"162613","record_id":"<urn:uuid:21ebd531-4091-40cb-bf13-339d7abd7948>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00132.warc.gz"}
issue with trapezoidal rule to solve IVP ODE, any lead appreciated Hi everyone, Recently I am facing problem in modeling trapezoidal rule to solve initial value ODE through descretization . Trapezoidal rule for IVP or BVP ODE : dy/dx = f(y); y(n+1) = y(n) + h/2 ( f(y(n+1)) + f(y(n))) /here h, is the step size/ Example ODE: Adh/dt = theta - ktqout , where qout =sqrt(h) (so overall its a single variable ODE ) IV: h(1) = 1 Time Period: 800 min Step size: 1 min Below is the code that i wrote: Time horizon of 800 minutes m /1800/; A cross sectional area /5/ hinit initial height /1/ theta inlet flow rate /0.5/ kt constant /0.3/ ; qout(m) ; Trapezoidal rule eq1(m)$(ord(m) lt 800)… Ah(m+1)-h(m) =e= 1/2(theta-qout(m+1)+theta-qout(m)); eq2(m)$(ord(m) lt 800)… qout(m)=e=ktsqrt(h(m)); Initial value htt… h(‘1’)=e=hinit; Dummy objective obj… hmax =e= smax(m,h(m)); Model profile /all/ ; *Variable bounds option dnlp=coinipopt; solve profile minimizing hmax using dnlp; display h.l; Also attached the code file below! My requirement is to see the accuracy of approximation via Trapezoidal and for that purpose i want the profile of ‘h’ with time . With the above code the value of dynamic variable ‘h’ is not changing from the initial value of 1, throughout the time interval. I am making some semantic error. Any help is much appreciated. Thanks a lot! trapezoidiscrete.gms (1.44 KB)	{"url":"https://forum.gams.com/t/issue-with-trapezoidal-rule-to-solve-ivp-ode-any-lead-appreciated/2534","timestamp":"2024-11-14T04:00:12Z","content_type":"text/html","content_length":"16187","record_id":"<urn:uuid:2055534b-fd45-46aa-b276-4c777f7c721a>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00550.warc.gz"}
Re: Why Amazon Chose TLA+ Hi Edgar, >>Would it be possible to get the source code of TLA+ models you mentioned in the paper? Two of the specs are available here: The paper(s) also mention specs for systems at Amazon (e.g. parts of DynamoDB, S3, EC2). Those specs are not available outside of Amazon. That sounds very interesting. Please could you send me a copy? >>I would like to explore ways to prove your code correct using VCC (or other automated deductive techniques). That would be great. I did not attempt a proof, because a formal proof of the most interesting of the two algorithms (serializable snapshot isolation) probably requires finding an inductive invariant that implies that the algorithm never creates a cycle in the multi-version conflict graph established by the read and write dependencies between transactions[1]. I suspect that finding the inductive invariant is very difficult (e.g. may involve an induction argument due to use of the transitive closure operator in the property), but I have not tried. I suspect that the easiest (well, least difficult) way to do the proof would be to use TLC or the ProB constraint checker (via the TLA+-to-B translator) to 'debug' the inductive invariant, and then use TLAPS for the proof, rather than proving the low-level code correct. (E.g. Even expressing the correctness property may be quite tricky in VCC.) However, I hope I'm wrong about that; it would be great to have a proof of an FYI, at least three different C implementations of serializable snapshot algorithm are available. Recent versions of the PostgreSQL database use the algorithm for SERIALIZABLE isolation level [2], and research implementations are available as patches for older versions of Berkeley DB and MySQL (InnoDB) from the author of the algorithm, Michael Cahill. If you are interested in the latter, I can put you in touch with Michael. [1] You'll see in the specs that I actually model-checked that the MVSG was cycle free, rather than directly checking the true desired safety property. The true desired safety property is one-copy serializability; roughly, the global execution history of events over all transactions can always be permuted to a history in which the same results are observed and all transactions execute consecutively, with no interleaving of events from different transactions. There's a well known theorem that proves that absence of cycles in the MVSG implies one-copy serializability. I didn't bother to state the theorem in the specs, but the comments point to a standard reference (Phil Bernstein's book on concurrency control and recovery in transaction systems) which states the theorem and gives an informal proof. [2] http://drkp.net/papers/ssi-vldb12.pdf On Wednesday, October 1, 2014 3:47:32 PM UTC-7, Edgar Pek wrote: Hello Chris, Would it be possible to get the source code of TLA+ models you mentioned in the paper? I am a PhD candidate from University of Illinois. I have worked with VCC in various contexts. (E.g, http://dx.doi.org/10.1145/2666356.2594325) I would like to explore ways to prove your code correct using VCC (or other automated deductive techniques). Thank you, On Thursday, June 12, 2014 2:50:21 PM UTC-5, Chris Newcombe wrote: > The following paper was presented at ABZ'2014 last week, and is now available in the conference proceedings: > Newcombe, > C., Why Amazon Chose TLA+, in > Abstract State Machines, Alloy, B, TLA, VDM, and Z Lecture Notes in > Computer Science Volume 8477, 2014, pp 25-39; > http://link.springer.com/chapter/10.1007%2F978-3-662-43652-3_3 > I'm happy to provide copies privately on request. > The above is a partner to our other paper, "Use of Formal Methods at Amazon Web Services", which is currently available via the TLA+ home page. See earlier post: https://groups.google.com/d/msg > regards, > Chris	{"url":"https://discuss.tlapl.us/msg00024.html","timestamp":"2024-11-11T08:13:44Z","content_type":"text/html","content_length":"10687","record_id":"<urn:uuid:1f3b0cf3-aead-422d-b18f-3a407e4dba15>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00730.warc.gz"}
Automatic Segmentation of Dynamic Network Sequences with Node Labels - Nexgen Technology Automatic Segmentation of Dynamic Network Sequences with Node Labels by nexgentech \| Nov 21, 2018 \| ieee project Automatic Segmentation of Dynamic Network Sequences with Node Labels Given a sequence of snapshots of flu propagating over a population network, can we find a segmentation when the patterns of the disease spread change, possibly due to interventions? In this paper, we study the problem of segmenting graph sequences with labeled nodes. Memes on the Twitter network, diseases over a contact network, movie-cascades over a social network, etc. are all graph sequences with labeled nodes. Most related work on this subject is on plain graphs and hence ignores the label dynamics. Others require fix parameters or feature engineering. We propose SNAPNETS, to automatically find segmentations of such graph sequences, with different characteristics of nodes of each label in adjacent segments. It satisfies all the desired properties (being parameter free, comprehensive and scalable) by leveraging a principled, multi-level, flexible framework which maps the problem to a path optimization problem over a weighted DAG. Also, we develop the parallel framework of SNAPNETS which speeds up its running time. Finally, we propose an extension of SNAPNETS to handle the dynamic graph structures and use it to detect anomalies (and events) in network sequences. Extensive experiments on several diverse real datasets show that it finds cut points matching ground-truth or meaningful external signals and detects anomalies outperforming non-trivial baselines. We also show that the segmentations are easily interpretable, and that SNAPNETS scales near-linearly with the size of the input. Finally, we show how to use SNAPNETS to detect anomaly in a sequence of dynamic networks. Proposed System In this paper, we study the problemof segmenting a graph sequence with varying node-label distributions. First, we assume binary labels and fixed structure for graphs.Next, we propose an extension to our algorithm to handle dynamic graphs with varying nodes and edges then leverage it to detect anomalies and important events. For diseases/memes, the labels can be ‘infected’/‘active’ (1) & ‘healthy’/‘inactive’ (0), and the network can be the underlying contact-network. SNAPNETS is a novel multi-level approach which summarizes the given networks/labels in a very general way at multiple different time-granularities, and then converts the problem into an appropriate optimization problem on a data structure. Further it gives naturally interpretable segments, enhancing its applicability. Also, we easily extend SNAPNETS to segment dynamic graph sequences which can be used to detect anomalies and important events. Finally, we demonstrate SNAPNETS’s usefulness via multiple experiments for segmentation and anomaly detection on diverse real-datasets. We presented SNAPNETS, an intuitive and effective method to segment AS-Sequences with binary node labels and extended it to work with dynamic graph sequences as well. It is the first method to satisfy all the desired properties P1, P2, P3. It efficiently finds high-quality segmentations, detects anomalies and events and gives useful insights in diverse complex datasets. Also, we propose ANOMALYSNAPNETS as an expansion of SNAPNETS to detect anomalies and important events in dynamic graph sequences. Finally we parallelize SNAPNETS and ANOMALY-SNAPNETS to accelerate the computations. SNAPNETS is a ‘global’ method and not simply a change-point detection method. We are not just looking for local changes; rather we track the ‘total variation’ using Gs. The SNAPNETS framework is very flexible, as our formulations are very general [1] L. Li, J. McCann, N. S. Pollard, and C. Faloutsos, “Dynammo: Mining and summarization of coevolving sequences with missing values,” in Proc. 15th ACM SIGKDD Int. Conf. Knowl. Discovery Data Mining, 2009, pp. 507–516. [2] A. Likas, N. Vlassis, and J. J. Verbeek, “The global k-means clustering algorithm,” Pattern Recog., vol. 36, no. 2, pp. 451–461, 2003. [3] K. Henderson, et al., “Metric forensics: A multi-level approach for mining volatile graphs,” in Proc. 16th ACM SIGKDD Int. Conf. Knowl. Discovery Data Mining, 2010, pp. 163–172. [4] N. Shah, D. Koutra, T. Zou, B. Gallagher, and C. Faloutsos, “TimeCrunch: Interpretable dynamic graph summarization,” in Proc. 21th ACM SIGKDD Int. Conf. Knowl. Discovery Data Mining, 2015, pp. [5] S. Shih, “Vog: Summarizing and understanding large graphs,” in Proc. SIAM Int. Conf. Data Mining, 2014, pp. 91–99. [6] J. Ferlez, C. Faloutsos, J. Leskovec, D. Mladenic, and M. Grobelnik, “Monitoring network evolution using MDL,” in Proc. IEEE 24th Int. Conf. Data Eng., 2008, pp. 1328–1330. [7] Q. Qu, S. Liu, C. S. Jensen, F. Zhu, and C. Faloutsos, “Interestingness-driven diffusion process summarization in dynamic networks,” in Joint European Conference on Machine Learning and Knowledge Discovery in Databases. Berlin, Germany: Springer, 2014, pp. 597–613. [8] R. M. Anderson, R. M. May, and B. Anderson, Infectious Diseases of Humans: Dynamics and Control. Hoboken, NJ, USA: Wiley Online Library, 1992. [9] D. Kempe, J. Kleinberg, and _E. Tardos, “Maximizing the spread of influence through a social network,” in Proc. 9th ACM SIGKDD Int. Conf. Knowl. Discovery Data Mining, 2003, pp. 137–146. [10] M. Purohit, B. A. Prakash, C. Kang, Y. Zhang, and V. Subrahmanian, “Fast influence-based coarsening for large networks,” in Proc. 20th ACM SIGKDD Int. Conf. Knowl. Discovery Data Mining, 2014, pp. 1296–1305. [11] K. Henderson, et al., “Metric forensics: A multi-level approach for mining volatile graphs,” in Proc. 16th ACM SIGKDD Int. Conf. Knowl. Discovery Data Mining, 2010, pp. 163–172. [12] M. Mathioudakis, F. Bonchi, C. Castillo, A. Gionis, and A. Ukkonen, “Sparsification of influence networks,” in Proc. 17th ACM SIGKDD Int. Conf. Knowl. Discovery DataMining, 2011, pp. 529–537. [13] B. A. Prakash, D. Chakrabarti, N. C. Valler, M. Faloutsos, and C. Faloutsos, “Threshold conditions for arbitrary cascade models on arbitrary networks,” Knowl. Inform. Syst., vol. 33, no. 3, pp. 549–575, 2012. [14] G. Li, M. Semerci, B. Yener, and M. J. Zaki, “Effective graph classification based on topological and label attributes,” Statistical Anal. Data Mining, vol. 5, no. 4, pp. 265–283, 2012. [15] D. Salvi, J. Zhou, J. Waggoner, and S. Wang, “Handwritten text segmentation using average longest path algorithm,” in Proc. IEEE Workshop Appl. Comput. Vis., 2013, pp. 505–512.	{"url":"https://nexgenproject.com/automatic-segmentation-dynamic-network-sequences-node-labels/","timestamp":"2024-11-10T14:15:30Z","content_type":"text/html","content_length":"91967","record_id":"<urn:uuid:cbfa64d5-d648-41f0-a52b-176102cedcd0>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00052.warc.gz"}
Mathematical Methods Welcome to Physics 301! Mathematical Methods in Physics introduces you to the mathematical techniques used to model physical systems and solve problems that arise in the physical sciences. We will cover orthogonal coordinates systems, Fourier series, vector calculus in curvilinear coordinates, series solutions of differential equations and special functions, techniques for solving partial differential equations, and additional material as time allows. You can see the syllabus for the course right here. The table below gives a tentative schedule for the course. These dates are subject to change, depending on the pace of the class. Week Dates Topics & Events 1 1/19-1/22 Intro, Orthogonal coordinate systems 2 1/25-1/29 OCS 3 2/1-2/5 OCS 4 2/8-2/12 Fourier Series, First Spring Break 5 2/15-2/19 FS 6 2/22-2/26 FS 7 3/1-3/5 Exam 1, Vector Calc 8 3/8-3/12 VC, Second Spring Break 9 3/15-3/19 VC 10 3/22-3/26 Series solutions and special functions 11 3/29-4/2 SS & SF, Easter Holiday 12 4/5-4/9 SS & SF, Easter Holiday 13 4/12-4/16 Exam 2, Partial differential equations 14 4/19-4/23 PDEs 15 4/26-4/30 PDEs 16 5/3-5/7 Final Exam: May 3, 9-11am These dates are subject to change. I may decide to switch things around or spend more or less time on a given chapter. Schedule changes that affect an exam or homework assignment will always be discussed with the class. Assignment 10 Separation of variables, part 2 Due on April 30 Your final homework assignment covers separation of variables in three dimensions (using Cartesian coordinates) and separation of variables in spherical polar coordinates with an axis of symmetry. Assignment 9 Separation of variables Due on April 23 Two problems on separation of variables in PDEs with two variables. There is also an optional exercise about solutions of the wave equation in three dimensions (You don't need to hand in the optional problem, but it is instructive to work through!). Assignment 8 Legendre polynomials, Method of Frobenius Due on April 9 Exercises involving Legendre polynomials of the first kind, Legendre series representations of functions, and generalized power series solutions of ordinary differential equations using the Method of Assignment 7 Series Solutions of ODEs Due on March 29 This assignment will help you practice working out Maclaurin series solutions of differential equations, and introduce you to Taylor series solutions written as expansions around points other than 0. Assignment 6 Vector Calculus Due on March 17 A few exercises involving the divergence, curl, and Laplacian in orthogonal coordinate systems; the Divergence Theorem; and Stokes's Theorem. Assignment 5 Complex Fourier Series, Periods other than 2$\pi$ Due on February 26 Now you will determine the complex Fourier Series representations of a few simple functions, and the Fourier Series for periods other than 2$\pi$. In the last problem you will work out the odd extension of a function from $0 < x < L $ to $0 < x < 2L $, and determine the sine series coefficients for the example from our first lecture. Assignment 4 Basic Fourier Series Due on February 17 This assignment covers some of the basic (but essential!) integrals needed for evaluating Fourier Series. You will also compute the Fourier Series representation of three simple functions on the interval $-\pi \leq x \leq \pi$. Be sure to carefully read the statement at the top of assignment! Assignment 3 Orthogonal Coordinates, Velocity and Acceleration Due on February 8 In this assignment you will work out the velocity and acceleration vectors in spherical polar coordinates, derive an equation used in our discussion of celestial mechanics, and apply the methods we developed for converting a vector from Cartesian coordinates to another OCS. Assignment 2 Spherical Polar Coordinates Due on February 1 In class we learned how to determine the scale factors and basis vectors for any orthogonal coordinate system. For this assignment you will apply this to spherical polar coordinates, which are useful in situations where a system has spherical symmetry. Assignment 1 Review of Vectors and Multivariable Calculus Due on January 22 This assignment is a review of some basic material on vectors and multivariable calculus. It's due on Friday of the first week of class. Please email if you have any questions! Working with your classmates on homework is encouraged! But you should only hand in work that you've completed on your own. If your solution looks just like someone else's work then you need to go back and redo it from scratch. If you can't explain each step of your solution then you haven't completed the problem on your own. Remember: the only way to be ready for the exams is to do the homework yourself. A Warning Never, ever hand in an assignment that has been copied from a solutions manual or a source you found online. You won't learn anything that way, and it will earn you a grade of zero for that assignment. If it happens more than once it will be reported to the Department Chair and the Dean. Consider yourself warned. Click here to see the College of Arts and Sciences Statement on Academic The main text for the class is Mathematical Methods in Engineering and Physics by Felder & Felder. It is a very readable book that takes a straightforward approach. The website for the book has lots of supplementary material. We will cover most or all of chapters 2, 9, 8, 12, and 11 (in that order). The book doesn't have a very thorough treatment of orthonormal coordinate systems so we will use my notes for that - see below. From time to time you may want to consult a different book. If the discussion in Felder & Felder doesn't suit you, or you'd like to see a topic presented differently, or even if you just want to see some additional examples or applications, consider one of the following texts: In the past, I taught a different Math Methods course using the book by Boas. Everyone seemed to like it pretty well. The book by Riley, Hobson, and Bence is more comprehensive than the others but also kind of terse. The book by Arfken is a standard text in grad versions of this class. It's a bit more sophisticated but maybe you'll like the challenge. Finally, Schey's book is a wonderful little volume on vector calculus. If you want to learn more about the subject you should really grab a copy. If you want a more detailed treatment of something we cover, you should just pick a book and start reading. One of the big secrets of being a physics major, which always seems to surprise physics majors, is that you don't need anyone's permission to start studying something if you find interesting. The worst case scenario is that you pick a book that's a little over your head and you have to find another one. Don't let that discourage you! You've spent the last year-and-a-half studying physics. That's all the training you need to really start investigating things. These are the handwritten notes that I work from during lectures. Sometimes I will post them in advance when they include material that isn't in the textbook. The notes are pretty neat and should be more or less searchable. The diagrams are definitely cleaner than the ones I draw during class, and there are usually some worked examples we don't get to. Additional examples and discussion can be found in the next section. Series Solutions and Special Functions Many of the ODEs you encounter in physics can be solved by assuming that the solution has a power series expansion. These notes cover the basics of this method, its application to a few equations that frequently appear in physics, and the properties of the resulting “special functions” (like Legendre polynomials and Besssel functions). Notes on the gradient, divergence, curl, and Laplacian in general orthogonal coordinate systems, their fundamental theorems, and a brief review of surface and volume integrals. Lecture notes covering real and complex Fourier Series, Parseval's Theorem, intervals with length $2L$, and even and odd extensions of functions. Most of this material is in the textbook but the presentation may not be the same. A few important facts about Taylor series, Mclaurin series, and what it means to write down an infinite series representation of a function. Orthogonal Coordinate Systems Includes material not covered in the textbook These notes cover the definition and properties of Orthogonal Coordinate Systems, and how to translate from one OCS to another. A fun application is the orbital mechanics of a satellite. Setting up the problem in spherical polar coordinates makes it much easier to show that angular momentum is conserved or that orbits are ellipses. This section contains supplementary notes with extra discussion, examples, and applications. This is also where I'll post Mathematica notebooks. Here are a few examples of ODEs solved by assuming a generalized power series. How Will We Use Legendre Series in Other Courses? In class I mentioned that you will see a lot of Legendre series in upper level courses like E&M or Quantum Mechanics — they show up whenever we solve equations involving the Laplacian in spherical polar coordinates. This Mathematica notebook works through an E&M example: the electrostatic potential inside and outside a hollow, insulating sphere. Legendre Polynomials: What They Are, How We Use Them These notes review Legendre's equation and its series solutions. The Legendre polynomials of the first kind have special properties that allow us to use them like building blocks in the same we constructed Fourier series with sines and cosines. Series Solutions of ODEs: Examples Here are three examples of ODEs solved by assuming a Mclaurin series solution. Try to work them out yourselves, then compare what you did with my solutions. A Mathematica notebook to help us visualize how the gradient of a function is related to the direction in which the function most rapidly increases. In class we talked about even and odd extensions of a function, and how we can use them to build a Fourier series representation out of a different set of building blocks. For example, instead of using sines and cosines that fit a whole number of periods into $[0,2L]$, we can use the odd extension of a function to build a Fourier series out of sines that fit a whole or half number of periods onto that interval. This Mathematica notebook works out a few examples, including the Fourier sine series for the odd extension of the function $4\,x - 4\,x^2$ on $0\leq x \leq 1$. Using sines that fit a whole or half number of periods onto this interval, we can build a Fourier series representation out of the sort of normal modes we'd get on a clamped string. When each one oscillates with its natural frequency we see the behavior you'd expect for a vibrating string. A Fourier Example from the Homework This Mathematica notebook works out the Fourier Series representation of one of the functions from the Homework, and shows that it does all the things we talked about in class. Another Fourier Series Example In physics we often encounter functions $f(x)$ that are defined piecewise. This means that you can't write down one expression that works for all values of $x$ (usually because the function or its derivatives are discontinuous). A common misconception when first learning about Fourier Series is that you work out a different series for each part of a piecewise function. But this is not the case: a piecewise function on $[-\pi,\pi]$ is represented by just one Fourier Series. This short note reviews what we mean by a piecewise function and how to handle integrals in which they appear, then works through a detailed example of a Fourier Series for a piecewise function. You should read through this before working on Homework 4! This is a short Mathematica notebook that explores adding up sines and cosines to assemble a function on the interval $-\pi \leq x \leq \pi$. The last cell is interactive: Once you evaluate a window with sliders will appear that lets you adjust the coefficients of the sine and cosine terms in the sum. Velocity and Acceleration in Spherical Polar Coordinates This Mathematica notebook derives the unit vectors and scale factors for SPC, and then works out the components of the velocity and acceleration. Velocity and Acceleration in 3-D Parabolic Coordinates This is another example reviewing the calculation of velocity and acceleration in an orthogonal coordinate system. The same results are obtained in the Mathematica notebook linked further down the page. This does the calculations by hand and adds a little more discussion. Position, Velocity, and Acceleration in CPC In class we talked about how to express an object's position, velocity, and acceleration in a general OCS. These notes take a detailed look at this in Cylindrical Polar Coordinates. After working out $\vec{r{}}$, $\vec{v{}}$, and $\vec{a{}}$ I briefly talk about why you might want to use an OCS other than Cartesian coordinates, and then look at an example where CPC makes it easier to write out Newton's Second Law. Spherical Polar Coordinates and Unit Vectors As part of homework 2 you derived expressions for $\hat{r\,}, \hat{\theta\,},$ and $\hat{\phi\,} $ that you will need to use on homework 3. Since you may not get your graded assignments back before you start working on the new assignment, I wrote out a short note with the info you need. Orthogonal Coordinate Systems - Summary A shorter set of notes reviewing orthogonal coordinate systems and how to derive the scale factors and unit vectors. Parabolic Coordinates with Mathematica This is a Mathematica notebook that explores some aspects of the parabolic coordinates we worked with in class. It draws the coordinate grid, works out the scale factors and unit vectors, examines how the unit vectors change from point-to-point in the plane, and calculates both the velocity and acceleration. A Brief Introduction to Mathematica Once you've downloaded and installed Mathematica you should have a look at the brief introduction in the link above. It shows some basic operations and should help you take the system for a test drive. For a more in-depth introduction I strongly encourage you to work through the first few sections of An Elementary Introduction to the Wolfram Language. Here are some more resources for getting started with Mathematica: Hands-On Start to Mathematica Fast Introduction for Math Students Wolfram U That last link offers in-depth tutorials and courses in a wide variety of subjects. If you're feeling adventurous you can also open the “Wolfram Documentation” window from the Help menu and browse some of the topics there. Download and Install Mathematica From time to time we will use the technical computing system Mathematica to visualize complicated functions, automate certain calculations, and explore concepts from class. Follow these instructions to download and register a copy of Mathematica. The differential of a function describes how it changes when its argument changes by an infinitesimal amount. Not all calc sections cover this in the same way, so these notes review differentials of scalar and vector functions of one or more variables. You may find the useful when working on the first homework assignment. This is a basic review of line integrals -- what they are, how to evaluate them, etc. It may be useful if you're a little rusty on this topic. The file is big (about 22 MB) because of the various plots. Let me know if you find any typos or mistakes and I will post a corrected version. If you leave mathematical physics on my chalkboard, my kid will copy it down and yell “I love doing physics.”	{"url":"http://jacobi.luc.edu/p301.html","timestamp":"2024-11-14T06:33:45Z","content_type":"application/xhtml+xml","content_length":"31939","record_id":"<urn:uuid:c6b9fcb4-b32e-4b39-b71b-53ae29f80e7a>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00629.warc.gz"}
Wald-Wolfowitz Runs Test and Baseball It's been a while since I wrote one of these random posts, but I still have a couple more I wanted to write. In this post, I want to describe one of the tests used in the paper that initially inspired this series of posts: the Wald-Wolfowitz runs test. This test is interesting in that it does not test for uniformity, but only for independence. In other words, unfair coins and loaded dice will pass the Wald-Wolfowitz test and only inherently non-random sequences should fail. The test itself is simple: for a sequence of two symbols we count the number of "runs" (a sequence of adjacent indentical symbols with the other symbol on each end) and then compare that count against the expected number of runs given the distribution of the two symbols. If n[0] is the number of zeros in a sequence and n[1] is the number of ones, then the expected number of runs is with a variance of We can use mu and sigma to compute a z-score for the number of runs in our sequence. Here is a simple bit of Maple to compute the number of runs in a sequence (exercise: write one that is better). Runs := proc(L) local i, state, runs; state := -1; runs := 0; for i in L do if i <> state then state := i; runs := runs + 1; end if; end do; return runs; end proc: # Runs Then, using that, here is the runs test: WaldWolfowitz := proc(L::list, {level::numeric:=0.05}, $) local tally, ones, zeros, runs, mu, sigmasq, stat, p; tally := Statistics:-Tally(L); if {op(map(lhs,tally))} <> {0,1} then error `invalid input: expected a list of {0,1}`; end if; zeros := eval(0, tally); ones := eval(1, tally); userinfo(1, hints, nprintf("sequence contains %d zeros and %d ones", zeros, ones)); runs := Runs(L); userinfo(1, hints, nprintf("sequence has %d runs", runs)); mu := ( (2oneszeros) / (ones+zeros) ) + 1; # Expected # of runs userinfo(1, hints, nprintf("expected number of runs is %d", round(mu))); sigmasq := ((mu-1)(mu-2)/(ones+zeros-1)); # variance userinfo(1, hints, nprintf("the variance is %.2f", sigmasq)); stat := (runs - mu) / sqrt( sigmasq ); # statistic userinfo(1, hints, nprintf("the statistic for this sequence is is %f", stat)); p := 2(Statistics:-Probability(Statistics:-RandomVariable(Normal(0, 1))>abs(stat), ':-numeric' )); # pvalue return evalb(p>level), p; end proc: # WaldWolfowitz I found it difficult to find a pseudo-random sequence that failed this test. All my usual generators were able to pass. Of course, by design, the unfair coin generator passes: () r2a := rand(0..2): r2 := ()-> r2a() mod 2: () L := [seq(r2(), i=1..10000)]: WaldWolfowitz: sequence contains 6748 zeros and 3252 ones WaldWolfowitz: sequence has 4434 runs WaldWolfowitz: expected number of runs is 4390 WaldWolfowitz: the variance is 1926.00 WaldWolfowitz: the statistic for this sequence is is 1.004890 true, 0.3149497052 Even the linear congruence with small period passes: (*) LCState := 1: r3 := proc() global LCState; LCState := irem(89853LCState, 50027); irem(LCState, 2); end proc: () L := [seq(r3(), i=1..10000)]: () WaldWolfowitz(L); WaldWolfowitz: sequence contains 4840 zeros and 5160 ones WaldWolfowitz: sequence has 4950 runs WaldWolfowitz: expected number of runs is 4996 WaldWolfowitz: the variance is 2494.63 WaldWolfowitz: the statistic for this sequence is is -0.918587 true, 0.3583118358 Searching around for something that might fail this test without being too contrived, I decided to take a look at sports statistics since I remember having read a paper as an undergraduate that suggested that winning and losing streaks of sports teams might actually be more random that fans would care to admit. Wald-Wolfowitz seemed like a good test to apply to this hypothesis. In order to get a large enough set of data to be meaninful, I decided to use win-loss sequences of baseball teams. The easiest way I found to get this information into Maple was to download all the game logs from RetroSheet.org and grab (way more data that I wanted) and just filter out the win loss information for my chosen team. The files are called "GLXXXX.TXT" where XXXX is the year. wlarray := Vector(); cnt := 0; ocnt := 1; team := "TOR"; GLFiles := sort(select(t->t[1..2]="GL", FileTools:-ListDirectory("."))); for filename in GLFiles do printf("Procesing %s\n", filename); tmp := FileTools:-Text:-ReadLine(filename); if tmp = NULL then end if; (otmp,tmp) := tmp, StringTools:-SubstituteAll(tmp, ",,", ",0,"); while otmp <> tmp do (otmp,tmp) := tmp, StringTools:-SubstituteAll(tmp, ",,", ",0,"); end do; if tmp[-1] = "," then tmp := cat(tmp,"0"); end if; tmp := [parse(tmp)]; printf("Parsing file %s, failed to parse: \"%s\"\n", filename, tmp); end try; if tmp[4] = team then # Away if tmp[11] < tmp[10] then cnt := cnt+1; wlarray(cnt) := 1; elif tmp[11] = tmp[10] then print("TIE", tmp[11] = tmp[10]); cnt := cnt+1; wlarray(cnt) := 0; end if; elif tmp[7] = team then # Home if tmp[11] > tmp[10] then cnt := cnt+1; wlarray(cnt) := 1; elif tmp[11] = tmp[10] then print("TIE", tmp[11] = tmp[10]); cnt := cnt+1; wlarray(cnt) := 0; end if; end if; end do: print(Statistics:-Tally([seq(wlarray[i], i=ocnt..cnt)])); ocnt := cnt+1; end do: TorWL:=[seq(wlarray[i], i=1..cnt)]: save TorWL, "TorontoWL.txt"; Having created the win-loss sequence for my chosen team (The Blue Jays), I ran the runs test on it: () read("TorontoWL.txt"): () WaldWolfowitz(TorWL); WaldWolfowitz: sequence contains 2632 zeros and 2589 ones WaldWolfowitz: sequence has 2554 runs WaldWolfowitz: expected number of runs is 2611 WaldWolfowitz: the variance is 1304.82 WaldWolfowitz: the statistic for this sequence is is -1.586911 true, 0.1125328014 It passes! But just barely with a p=0.11. I hoped that other teams might be a little less random, and I was Happy to see that the (now defunct) Montreal Expos were less random. () read("MontrealWL.txt"): () WaldWolfowitz(MonWL); WaldWolfowitz: sequence contains 2943 zeros and 2755 ones WaldWolfowitz: sequence has 2766 runs WaldWolfowitz: expected number of runs is 2847 WaldWolfowitz: the variance is 1421.15 WaldWolfowitz: the statistic for this sequence is is -2.145956 false, 0.0318764926 Make of this what you will Toronto fans. As always, the computations and code are in the attached worksheet WaldWolfowitz.mw and here are my datafiles MontrealWL.txt and TorontoWL.txt. Tags are words are used to describe and categorize your content. Combine multiple words with dashes(-), and seperate tags with spaces.	{"url":"https://www.mapleprimes.com/maplesoftblog/96943-WaldWolfowitz-Runs-Test-And-Baseball","timestamp":"2024-11-07T10:39:30Z","content_type":"text/html","content_length":"123257","record_id":"<urn:uuid:5dc01682-df93-473e-8dca-78b0673bddf4>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00892.warc.gz"}
The homotopy theory of differentiable sheaves Many important theorems in differential topology relate properties of manifolds to properties of their underlying homotopy types -- defined e.g. using the total singular complex or the \v{C}ech nerve of a good open cover. Upon embedding the category of manifolds into the $\infty$-topos $\mathbf{Diff}^\infty$ of differentiable sheaves one gains a further notion of underlying homotopy type: the shape of the corresponding differentiable sheaf. In a first series of results we prove using simple cofinality and descent arguments that the shape of any manifold coincides with many other notions of underlying homotopy types such as the ones mentioned above. Our techniques moreover allow for computations, such as the homotopy type of the Haefliger stack, following Carchedi. This leads to more refined questions, such as what it means for a mapping differential sheaf to have the correct shape. To answer these we construct model structures as well as more general homotopical calculi on the \ (\infty\)-category $\mathbf{Diff}^\infty$ (which restrict to its full subcategory of $0$-truncated objects,$\mathbf{Diff}^\infty_{\leq 0}$) with shape equivalences as the weak equivalences. These tools are moreover developed in such a way so as to be highly customisable, with a view towards future applications, e.g. in geometric topology. Finally, working with the $\infty$-topos $\ mathbf{Diff}^0$ of sheaves on topological manifolds, we give new and conceptual proofs of some classical statements in algebraic topology. These include Dugger and Isaksen's hypercovering theorem, and the fact that the Quillen adjunction between simplicial sets and topological spaces is a Quillen equivalence.	{"url":"https://www.scienceopen.com/document?vid=aa0b5616-4a7d-48b0-987c-4c3edfdf2ace","timestamp":"2024-11-14T21:17:00Z","content_type":"text/html","content_length":"58408","record_id":"<urn:uuid:114109d7-3f42-420b-b2a3-51db0b842536>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00486.warc.gz"}
The minimum no of Carbon atoms an alkyne must have to show diastereomerism a) 4 b) 5 c) 6 d) 7 The minimum no of Carbon atoms an alkyne must have to show diastereomerism a) 4 b) 5 c) 6 d) 7 Solution 1 The minimum number of carbon atoms an alkyne must have to show diastereomerism is 4. Here's why: Diastereomerism is a type of stereoisomerism. It is the phenomenon where two or more stereoisomers of a compound have different configurations at one or more (but not all) of the equivalent (related) Knowee AI is a powerful AI-powered study tool designed to help you to solve study problem. Knowee AI is a powerful AI-powered study tool designed to help you to solve study problem. Knowee AI is a powerful AI-powered study tool designed to help you to solve study problem. Knowee AI is a powerful AI-powered study tool designed to help you to solve study problem. Knowee AI is a powerful AI-powered study tool designed to help you to solve study problem. Knowee AI Upgrade your grade with Knowee Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.	{"url":"https://knowee.ai/questions/22922060-the-minimum-no-of-carbon-atoms-an-alkyne-must-have-to-show","timestamp":"2024-11-10T06:39:51Z","content_type":"text/html","content_length":"364074","record_id":"<urn:uuid:2c0ed42a-dce0-4132-b523-adbb37dabd1f>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00273.warc.gz"}
On Generalizations of Pairwise Compatibility Graphs On Generalizations of Pairwise Compatibility GraphsArticle A graph $G$ is a pairwise compatibility graph (PCG) if there exists an edge-weighted tree and an interval $I$, such that each leaf of the tree is a vertex of the graph, and there is an edge $\{ x, y \}$ in $G$ if and only if the weight of the path in the tree connecting $x$ and $y$ lies within the interval $I$. Originating in phylogenetics, PCGs are closely connected to important graph classes like leaf-powers and multi-threshold graphs, widely applied in bioinformatics, especially in understanding evolutionary processes. In this paper we introduce two natural generalizations of the PCG class, namely $k$-OR-PCG and $k$-AND-PCG, which are the classes of graphs that can be expressed as union and intersection, respectively, of $k$ PCGs. These classes can be also described using the concepts of the covering number and the intersection dimension of a graph in relation to the PCG class. We investigate how the classes of OR-PCG and AND-PCG are related to PCGs, $k$-interval-PCGs and other graph classes known in the literature. In particular, we provide upper bounds on the minimum $k$ for which an arbitrary graph $G$ belongs to $k$-interval-PCGs, $k$-OR-PCG or $k$-AND-PCG classes. For particular graph classes we improve these general bounds. Moreover, we show that, for every integer $k$, there exists a bipartite graph that is not in the $k$-interval-PCGs class, proving that there is no finite $k$ for which the $k$-interval-PCG class contains all the graphs. This answers an open question of Ahmed and Rahman from 2017. Finally, using a Ramsey theory argument, we show that for any $k$, there exists graphs that are not in $k$-AND-PCG, and graphs that are not in $k$-OR-PCG. Volume: vol. 26:3 Section: Graph Theory Published on: October 6, 2024 Accepted on: September 10, 2024 Submitted on: September 19, 2023 Keywords: Mathematics - Combinatorics,Computer Science - Discrete Mathematics	{"url":"https://dmtcs.episciences.org/14391","timestamp":"2024-11-09T07:47:13Z","content_type":"application/xhtml+xml","content_length":"53504","record_id":"<urn:uuid:e720f447-b706-4e81-baf7-888a4e24b4fd>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00813.warc.gz"}
1 What clustering is Cluster randomized experiments allocate treatments to groups, but measure outcomes at the level of the individuals that compose the groups. It is this divergence between the level at which the intervention is assigned and the level at which outcomes are defined that classifies an experiment as cluster randomized. Consider a study that randomly assigns villages to receive different development programs, where the well-being of households in the village is the outcome of interest. This is a clustered design because, while the treatment is assigned to the village as a whole, we are interested in outcomes defined at the household level. Or consider a study that randomly assigns certain households to receive different get-out-the-vote messages, where we are interested in the voting behavior of individuals. Because the unit of assignment for the message is the household, but the outcome is defined as individual behavior, this study is cluster randomized. Now consider a study in which villages are selected at random, and 10 people from each village are assigned to treatment or control, and we measure the well-being of those individuals. In this case, the study is not cluster randomized, because the level at which treatment is assigned and the level at which outcomes are defined is the same. Suppose that a study randomly assigned villages to different development programs and then measured social cohesion in the village. Though it contains the same randomization procedure as our first example, it is not clustered because we are interested here in village-level outcomes: the level of treatment assignment and of outcome measurement is the same. Clustering matters for two main reasons. On the one hand, clustering reduces the amount of information in an experiment because it restricts the number of ways that the treatment and control groups can be composed, relative to randomization at the individual level. If this fact is not taken into account, we often underestimate the variance in our estimator, leading to over-confidence in our the results of the study. On the other hand, clustering raises the question of how to combine information from different parts of the same experiment into one quantity. Especially when clusters are not of equal sizes and the potential outcomes of units within them are very different, conventional estimators will systematically produce the wrong answer due to bias. However, several approaches at the design and analysis phases can overcome the challenges posed by cluster randomized designs. 2 Why clustering can matter I: information reduction We commonly think of the information contained in studies in terms of the sample size and the characteristics of the units within the sample. Yet two studies with exactly the same sample size and the same participants could in theory contain very different amounts of information depending on whether units are clustered. This will greatly affect the precision of the inferences we make based on the Imagine an impact evaluation with 10 people, where 5 are assigned to the treatment group and 5 to control. In one version of the experiment, the treatment is assigned to individuals - it is not cluster randomized. In another version of the experiment, the 5 individuals with black hair and the 5 individuals with some other color of hair are assigned to treatment as a group. This design has two clusters: ‘black hair’ and ‘other color’. A simple application of the n choose k rule shows why this matters. In the first version, the randomization procedure allows for 252 different combinations of people as the treatment and control groups. However, in the second version, the randomization procedure assigns all the black-haired subjects either to treatment or to control, and thus only ever produces two ways of combining people to estimate an effect. Throughout this guide, we will illustrate points using examples written in R code. You can copy and paste this into your own R program to see how it works. [Click to show code] # Define the sample average treatment effect (SATE) treatment_effect <- 1 # Define the individual ids (i) person <- 1:10 # Define the cluster indicator (j) hair_color <- c(rep("black",5),rep("brown",5)) # Define the control outcome (Y0) outcome_if_untreated <- rnorm(n = 10) # Define the treatment outcome (Y1) outcome_if_treated <- outcome_if_untreated + treatment_effect # Version 1 - Not cluster randomized # Generate all possible non-clustered assignments of treatment (Z) non_clustered_assignments <- combn(x = unique(person),m = 5) # Estimate the treatment effect treatment_effects_V1 <- X = non_clustered_assignments, MARGIN = 2, FUN = function(assignment) { treated_outcomes <- outcome_if_treated[person %in% assignment] untreated_outcomes <- outcome_if_untreated[!person %in% assignment] mean(treated_outcomes) - mean(untreated_outcomes) # Estimate the true standard error standard_error_V1 <- sd(treatment_effects_V1) # Plot the histogram of all possible estimates of the treatment effect hist(treatment_effects_V1,xlim = c(-1,2.5),breaks = 20) [Click to show code] # Version 2 - Cluster randomized # Generate all possible assignments of treatment when clustering by hair color (Z) clustered_assignments <- combn(x = unique(hair_color),m = 1) # Estimate the treatment effect treatment_effects_V2 <- X = clustered_assignments, FUN = function(assignment) { treated_outcomes <- outcome_if_treated[person %in% person[hair_color==assignment]] untreated_outcomes <- outcome_if_untreated[person %in% person[!hair_color==assignment]] mean(treated_outcomes) - mean(untreated_outcomes) # Estimate the true standard error standard_error_V2 <- sd(treatment_effects_V2) # Plot the histogram of all possible estimates of the treatment effect hist(treatment_effects_V2,xlim = c(-1,2.5),breaks = 20) As the histograms show, clustering provides a much ‘coarser’ view of the treatment effect. Irrespective of the number of times one randomizes the treatment and of the number of subjects one has, in a clustered randomization procedure the number of possible estimates of the treatment effect will be strictly determined by the number of clusters assigned to the different treatment conditions. This has important implications for the standard error of the estimator. [Click to show code] # Compare the standard errors While the sampling distribution for the non-clustered estimate of the treatment effect has a standard error of about .52, that of the clustered estimate is more than twice as large, at 1.13. Recall that the data going into both studies is identical, the only difference between the studies resides in the way that the treatment assignment mechanism reveals the information. Related to this issue of information is the question of how much units in our study vary within and between clusters. Two cluster randomized studies with $J=10$ villages and $n_j=100$ people per village may have different information about the treatment effect on individuals if, in one study, differences between villages are much greater than the differences in outcomes within them. If, say, all of the individuals in any village acted exactly the same but different villages showed different outcomes, then we would have on the order of 10 pieces of information: all of the information about causal effects in that study would be at the village level. Alternatively, if the individuals within a village acted more or less independently of each other, then we would have on the order of 10 $\times$ 100=1000 pieces of information. We can formalize the idea that the highly dependent clusters provide less information than the highly independent clusters with the intracluster correlation coefficient. For a given variable, $y$, units $i$ and clusters $j$, we can write the intracluster correlation coefficient as follows: \[ \text{ICC} \equiv \frac{\text{variance between clusters in } y}{\text{total variance in } y} \equiv \frac{\sigma_j^2}{\sigma_j^2 + \sigma_i^2} \] where $\sigma_i^2$ is the variance between units in the population and $\sigma_j^2$ is the variance between outcomes defined at the cluster level. Kish (1965) uses this description of dependence to define his idea of the “effective N” of a study (in the sample survey context, where samples may be clustered): \[\text{effective N}=\frac{N}{1+(n_j -1)\text{ICC}}=\frac{Jn}{1+(n-1)\text{ICC}},\] where the second term follows if all of the clusters are the same size ($n_1 \ldots n_J \equiv n$). If 200 observations arose from 10 clusters with 20 individuals within each cluster, where ICC = .5, such that 50% of the variation could be attributed to cluster-to-cluster differences (and not to differences within a cluster), Kish’s formula would suggest that we have an effective sample size of about 19 observations, instead of 200. As the foregoing discussion suggests, clustering reduces information most when it a) greatly restricts the number of ways in which subjects can be assigned to treatment and control groups, and b) produces units whose outcomes are strongly related to the cluster they are a member of (i.e. when it increases the ICC). 3 What to do about information reduction One way to limit the extent to which clustering reduces the information contained in our design would be to form clusters in such a way that the intracluster correlation is low. For example, if we were able to form clusters randomly, there would be no systematic relationship between units within a cluster. Hence, using cluster random assignment to assign these randomly formed clusters to experimental conditions would not result in any information reduction. Yet, opportunities to create clusters are typically rare. In most cases, we have to rely on naturally formed clusters (e.g., villages, classrooms, cities). In most scenarios that involve clustered assignment, we hence will have to make sure that our estimates of uncertainty about treatment effects correctly reflect the resulting information loss from clustering. In order to characterize our uncertainty about treatment effects, we typically want to calculate a standard error: an estimate of how much our treatment effect estimates would vary, were we able to repeat the experiment a very large number of times and, for each repetition, observe units in their resulting treated or untreated state. However, we are never able to observe the true standard error of an estimator, and therefore must use statistical procedures to infer this unknown quantity. Conventional methods for calculating standard errors do not take into account clustering. Thus, in order to avoid over-confidence in experimental findings, we need to modify the way in which we calculate uncertainty estimates. In this section we limit our attention to so-called ‘design-based’ approaches to calculating the standard error. In the design-based approach we simulate repetitions of the experiment to derive and check ways of characterizing the variance of the estimate of the treatment effect, accounting for clustered randomization. We contrast these with ‘model-based’ approaches further on in the guide. In the model-based approach we state that the outcomes were generated according to a probability model and that the cluster-level relationships also follow a probability model. To begin, we will create a function which simulates a cluster randomized experiment with fixed intracluster correlation, and use it to simulate some data from a simple cluster-randomized design. [Click to show code] make_clustered_data <- function(J = 10, n = 100, treatment_effect = .25, ICC = .1){ ## Inspired by Mathieu et al, 2012, Journal of Applied Psychology if (J %% 2 != 0 \| n %% 2 !=0) { stop(paste("Number of clusters (J) and size of clusters (n) must be even.")) Y0_j <- rnorm(J,0,sd = (1 + treatment_effect) ^ 2 * sqrt(ICC)) fake_data <- expand.grid(i = 1:n,j = 1:J) fake_data$Y0 <- rnorm(n * J,0,sd = (1 + treatment_effect) ^ 2 * sqrt(1 - ICC)) + Y0_j[fake_data$j] fake_data$Y1 <- with(fake_data,mean(Y0) + treatment_effect + (Y0 - mean(Y0)) * (2 / 3)) fake_data$Z <- ifelse(fake_data$j %in% sample(1:J,J / 2) == TRUE, 1, 0) fake_data$Y <- with(fake_data, Z * Y1 + (1 - Z) * Y0) pretend_data <- make_clustered_data(J = 10,n = 100,treatment_effect = .25,ICC = .1) Because we have created the data ourselves, we can calculate the true standard error of our estimator. We firstly generate the true sampling distribution by simulating every possible permutation of the treatment and calculating the estimate each time. The standard deviation of this distribution is the standard error of the estimator. [Click to show code] # Define the number of clusters J <- length(unique(pretend_data$j)) # Generate all possible ways of combining clusters into a treatment group all_treatment_groups <- with(pretend_data,combn(x = 1:J,m = J/2)) # Create a function for estimating the effect clustered_ATE <- function(j,Y1,Y0,treated_clusters) { Z_sim <- (j %in% treated_clusters)1 Y <- Z_sim Y1 + (1 - Z_sim) * Y0 estimate <- mean(Y[Z_sim == 1]) - mean(Y[Z_sim == 0]) # Apply the function through all possible treatment assignments cluster_results <- apply( X = all_treatment_groups, MARGIN = 2, FUN = clustered_ATE, j = pretend_data$j,Y1 = pretend_data$Y1, Y0 = pretend_data$Y0 true_SE <- sd(cluster_results) ## [1] 0.2567029 This gives a standard error of 0.26. We can compare the true standard error to two other kinds of standard error commonly employed. The first ignores clustering and assumes that treatment effect estimates are identically and independently distributed according to a normal distribution. We will refer to this as the I.I.D. standard error. To take clustering into account, we can use the following formula for the standard error: \[\text{Var}_\text{clustered}(\hat{\tau})=\frac{\sigma^2}{\sum_{j=1}^J \sum_{i=1}^{n_j} (Z_{ij}-\bar{Z})^2} (1-(n-1)\rho)\] where $\sigma^2=\sum_{j=1}^J \sum_{i=1}^{n_j} (Y_{ij} - \bar{Y}_{ij})^2$ (following Arceneaux and Nickerson (2009) ). This adjustment to the IID standard error is commonly known as the “Robust Clustered Standard Error” or RCSE. [Click to show code] ATE_estimate <- lm(Y ~ Z,data = pretend_data) IID_SE <- function(model) { RCSE <- function(model, cluster,return_cov = FALSE){ M <- length(unique(cluster)) N <- length(cluster) K <- model$rank dfc <- (M/(M - 1)) * ((N - 1)/(N - K)) uj <- apply(estfun(model), 2, function(x) tapply(x, cluster, sum)) rcse.cov <- dfc * sandwich(model, meat = crossprod(uj)/N) rcse.se <- as.matrix(coeftest(model, rcse.cov)) IID_SE_estimate <- IID_SE(model = ATE_estimate) RCSE_estimate <- RCSE(model = ATE_estimate,cluster = pretend_data$j) true_SE = true_SE, IID_SE_estimate = IID_SE_estimate, RCSE_estimate = RCSE_estimate["Z", "Std. Error"] When we ignore the clustered-assignment, the standard error is too small: we are over-confident about the amount of information provided to us by the experiment. The RCSE is slightly more conservative than the true standard error in this instance, but is very close. The discrepancy is likely because the RCSE is not a good approximation of the true standard error when the number of clusters is as small as it is here. To illustrate the point further, we can compare a simulation of the true standard error generated through random permutations of the treatment to the IID and RC standard errors. [Click to show code] compare_SEs <- function(data) { simulated_SE <- sd(replicate( j = data$j, Y1 = data$Y1, Y0 = data$Y0, treated_clusters = sample(unique(data$j),length(unique(data$j))/2) ATE_estimate <- lm(Y ~ Z,data) IID_SE_estimate <- IID_SE(model = ATE_estimate) RCSE_estimate <- RCSE(model = ATE_estimate,cluster = data$j)["Z", "Std. Error"] simulated_SE = simulated_SE, IID_SE = IID_SE_estimate, RCSE = RCSE_estimate J_4_clusters <- make_clustered_data(J = 4) J_10_clusters <- make_clustered_data(J = 10) J_30_clusters <- make_clustered_data(J = 30) J_100_clusters <- make_clustered_data(J = 100) J_1000_clusters <- make_clustered_data(J = 1000) c(J = 4,compare_SEs(J_4_clusters)), c(J = 30,compare_SEs(J_30_clusters)), c(J = 100,compare_SEs(J_100_clusters)), c(J = 1000,compare_SEs(J_1000_clusters)) 4 0.270 0.127 0.260 30 0.161 0.047 0.146 100 0.085 0.027 0.088 1000 0.027 0.008 0.027 As these simple examples illustrate, the clustered estimate of the standard error (RCSE) gets closer to the truth (the simulated standard error) as the number of clusters increases. Meanwhile, the standard error ignoring clustering (assuming IID) tends to be smaller than either of the other standard errors. The smaller the estimate of the standard error is, the more precise the estimates seem to us, and the more likely we will find results that appear ‘statistically significant’. This is problematic: in this case, the IID standard error is leads us to be over confident in our results because it ignores intra-cluster correlation, the extent to which differences between units can be attributed to the cluster they are a member of. If we estimate standard errors using techniques that understate our uncertainty, we are more likely to falsely reject null hypotheses when we should not. Another way to approach the problems that clustering introduces into the calculation of standard errors is to analyze the data at the level of the cluster. In this approach, we take averages or sums of the outcomes within the clusters, and then treat the study as though it only took place at the level of the cluster. Hansen and Bowers (2008) show that we can characterize the distribution of the difference of means using what we know about the distribution of the sum of the outcome in the treatment group, which varies from one assignment of treatment to another. [Click to show code] # Aggregate the unit-level data to the cluster level # Sum outcome to cluster level Yj <- tapply(pretend_data$Y,pretend_data$j,sum) # Aggregate assignment indicator to cluster level Zj <- tapply(pretend_data$Z,pretend_data$j,unique) # Calculate unique cluster size n_j <- unique(as.vector(table(pretend_data$j))) # Calculate total sample size (our units are now clusters) N <- length(Zj) # Generate cluster id j <- 1:N # Calculate number of clusters treated J_treated <- sum(Zj) # Make a function for the cluster-level difference in means estimator (See Hansen & Bowers 2008) cluster_difference <- function(Yj,Zj,n_j,J_treated,N){ ones <- rep(1, length(Zj)) ATE_estimate <- crossprod(Zj,Yj)(N/(n_jJ_treated(N-J_treated))) - # Given equal sized clusters and no blocking, this is identical to the # unit-level difference in means ATEs <- colMeans(data.frame( cluster_level_ATE = unit_level_ATE = with(pretend_data, mean(Y[Z == 1]) - mean(Y[Z == 0])) knitr::kable(data.frame(ATEs),align = "c") cluster_level_ATE 0.3417229 unit_level_ATE 0.3417229 In order to characterize uncertainty about the cluster-level ATE, we can exploit the fact that the only random element of the estimator is now the cross-product between the cluster-level assignment vector and the cluster-level outcome, $\mathbf{Z}^\top\mathbf{Y}$, scaled by some constant. We can estimate the variance of this random component through permutation of the assignment vector or through an approximation of the variance, assuming that the sampling distribution follows a normal distribution. [Click to show code] # Approximating variance using normality assumptions normal_sampling_variance <- # Approximating variance using permutations sampling_distribution <- replicate(10000,cluster_difference(Yj,sample(Zj),n_j,J_treated,N)) ses <- data.frame( sampling_variance = c(sqrt(normal_sampling_variance), sd(sampling_distribution)), p_values = c( 2 (1 - pnorm(abs(ATEs[1]) / sqrt(normal_sampling_variance), mean = 0)), rownames(ses) <- c("Assuming Normality","Permutations") Assuming Normality 0.2848150 0.2302145 Permutations 0.2825801 0.2792000 This cluster-level approach has the advantage of correctly characterizing uncertainty about effects when randomization is clustered, without having to use the RCSE standard errors for the unit-level estimates, which are overly permissive for small N. Indeed, the false positive rate of tests based on RCSE standard errors tend to be incorrect when the number of clusters is small, leading to over-confidence. As we shall see below, however, when the number of clusters is very small ($J=4$) the cluster-level approach is overly conservative, rejecting the null with a probability of 1. Another drawback of the cluster-level approach is that it does not allow for the estimation of unit-level quantities of interest, such as heterogeneous treatment effects. 4 Why clustering can matter II: different cluster sizes and bias When clusters are of different sizes, this can pose a unique class of problems related to the estimation of the treatment effect. Especially when the size of the cluster is in some way related to the potential outcomes of the units within it many conventional estimators of the sample average treatment effect (SATE) can be biased. To fix ideas, imagine an intervention targeted at firms of different sizes, which seeks to increase worker productivity. Due to economies of scale, the productivity of employees in big firms is increased much more proportional to that of employees in smaller firms. Imagine that the experiment includes 20 firms ranging in size from one-person entrepreneurs to large outfits with over 500 employees. Half of the firms are assigned to the treatment, and the other half are assigned to control. Outcomes are defined at the employee level. [Click to show code] # Number of firms J <- 20 # Employees per firm n_j <- rep(2^(0:(J/2-1)),rep(2,J/2)) # Total number of employees N <- sum(n_j) # 2046 # Unique employee (unit) ID i <- 1:N # Unique firm (cluster) ID j <- rep(1:length(n_j),n_j) # Firm specific treatment effects cluster_ATE <- n_j^2/10000 # Untreated productivity Y0 <- rnorm(N) # Treated productivity Y1 <- Y0 + cluster_ATE[j] # True sample average treatment effect (true_SATE <- mean(Y1-Y0)) ## [1] 14.9943 [Click to show code] # Correlation between firm size and effect size ## [1] 0.961843 As we see, there is high correlation in the treatment effect and cluster size. Now let us simulate 1000 analyses of this experiment, permuting the treatment assignment vector each time, and taking the unweighted difference in means as an estimate of the sample average treatment effect. [Click to show code] # Unweighted SATE SATE_estimate_no_weights <- NA for(i in 1:1000){ # Clustered random assignment of half of the firms Z <- (j %in% sample(1:J,J/2))1 # Reveal outcomes Y <- ZY1 + (1-Z)Y0 # Estimate SATE SATE_estimate_no_weights[i] <- mean(Y[Z==1])-mean(Y[Z==0]) # Generate histogram of estimated effects hist(SATE_estimate_no_weights,xlim = c(true_SATE-2,true_SATE+2),breaks = 100) # Add the expected estimate of the SATE using this estimator # And add the true SATE The histogram shows the sampling distribution of the estimator, with the true SATE in red and the unweighted estimate thereof in blue. The estimator is biased: in expectation, we do not recover the true SATE, instead underestimating it. Intuitively, one might correctly expect that the problem is related to the relative weight of the clusters in the calculation of the treatment effect. However, in this situation, taking the difference in the weighted average of the outcome among treated and control clusters is not enough to provide an unbiased estimator (see the code below). [Click to show code] # Weighted cluster-averages SATE_estimate_weighted <- NA for(i in 1:1000){ # Define the clusters put into treatment treated_clusters <- sample(1:J,J/2,replace = F) # Generate unit-level assignment vector Z <- (j %in% treated_clusters)1 # Reveal outcomes Y <- ZY1 + (1-Z)Y0 # Calculate the cluster weights treated_weights <- n_j[1:J%in%treated_clusters]/sum(n_j[1:J%in%treated_clusters]) control_weights <- n_j[!1:J%in%treated_clusters]/sum(n_j[!1:J%in%treated_clusters]) # Calculate the means of each cluster treated_means <- tapply(Y,j,mean)[1:J%in%treated_clusters] control_means <- tapply(Y,j,mean)[!1:J%in%treated_clusters] # Calculate the cluster-weighted estimate of the SATE SATE_estimate_weighted[i] <- weighted.mean(treated_means,treated_weights) - # Generate histogram of estimated effects hist(SATE_estimate_weighted,xlim = c(true_SATE-2,true_SATE+2),breaks = 100) # Add the expected estimate of the unweighted SATE # Add the expected estimate of the weighted SATE # And add the true SATE	{"url":"https://rawcdn.githack.com/egap/methods-guides/89f107f569dbc76a1770b75be806ab825c9296fd/cluster-random/cluster-random.html","timestamp":"2024-11-10T11:40:52Z","content_type":"text/html","content_length":"1049842","record_id":"<urn:uuid:f30e134e-240c-45ca-ab41-25215edc2f4b>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00363.warc.gz"}
Topics: Nature and Description of Particles Particles: Nature and Description In General > s.a. field theories; geometrical models; lagrangian systems; Ontology; particle statistics; physics paradigms. * History: Initially particles were thought of as singularities in the fields by many, but few now really think so; There have been attempts to consider them as black holes, geons and other solitons and localized solutions of non-linear field equations, or tachyons; 1999, So far none of those models is widely accepted. @ Levels of description: Rimini FP(97) [composition]; Gies & Wetterich PRD(02)ht/01 [elementary vs composite, renormalization]; Cui ht/01. @ Nature and description: DeWitt in(79); Ne'eman PLA(94) [mass and localizability]; Buchholz NPB(96)ht/95, in(94)ht/95; Lanz & Melsheimer LNP(98)qp/97 [as derived entities]; Dolby gq/03-proc [observer dependence]; Goldstein et al SHPMP(05)qp/04 [existence/reality, and Bohm theory]; Colosi & Rovelli CQG(09)gq/04 [global Fock states vs local particle states]; Butterfield FP(05) [endurance vs perdurance]; Wang gq/07 [as quasiparticles in superconductor]; Nambu IJMPA(08); Muller & Seevinck PhSc(09)-a0905, Caulton & Butterfield BJPS-a1106 [discernibility]; French & Krause 10 [identity]; Colin & Wiseman JPA(11)-a1107 [beable status for positions]; Swain a1110-fs [and pomerons]; Dreyer a1212-FQXi [particles as excitations of a background]; Zeh ZfN-a1304; Carcassi et al JPComm(18)- a1702 [mechanics from a few simple physical assumptions]; Lazarovici EJPS(18)-a1809 [against fields, superiority of a pure particle ontology]; Wechsler JQIS(19)-a1910 [the concept of particle is problematic]; Cabaret et al a2103. @ From (quantum) field theory: Derrick JMP(64) [non-linear scalar field, negative result]; Davies in(84); Clifton & Halvorson BJPS(01)qp/00 [and quantum field theory]; Cortez et al in(02)gq/05 [including holography, sigma models]; Arteaga AP(09) [particle-like excitations of non-vacuum states]; Dybalski PhD(08)-a0901 [and spectral theory of automorphism groups]; Pessa a0907; Bain SHPMP(11) [against conventional view]; Pienaar et al PRA(11) [as spacetime qubits]; Hobson AJP(13)mar-a1204 [there are no particles], comment Sassoli de Bianchi AJP(13)sep-a1202 [quantum "fields" are not fields]; Glazek & Trawinski FBS(17)-a1612 [effective particles]; > s.a. geons; solitons; locality in quantum field theory; solutions of general relativity with matter. @ Many-particle systems: Atiyah & Sutcliffe PRS(02)ht/01 [configuration space geometry]; Kundt FP(07) [and fundamental physics]; Svrcek ch(13)-a1207 [mechanics vs field theory]; da Costa & Holik a1305 [undefined particle number in quantum mechanics]; Miller et al a2004 [generally covariant]. @ Parametrizations: Guven PRD(91) [proper time]; > s.a. time. @ Flux-across-surfaces theorem: Dürr & Pickl JMP(03)mp/02 [Dirac particles]. @ Charged particles: Bagan et al ht/01-proc [in gauge theory]. @ Unstable particles: Saller ht/01 [time representations]. > Related topics: see Bag Model; Center of Mass; composite particle models; composite quantum systems; Elementarity; energy [self-energy]; interactions; mass [including mass generation]; mirrors; monopoles; particle effects; Relational Theories; symplectic structures and special types; twistors. Classical Particles > s.a. classical systems; mass; relativistic particles; spinning particles. * Description: Paradigms commonly used to describe particles are the material point, the test particle and the diluted particle (droplet model). * And quantum theory: Their entanglement-free behavior can be seen to emerge in the "islands of classicality" of quantum theory; > s.a. classical-quantum limit. * Non-relativistic: The dynamics can be S[x[i]; t[1], t[2]] = $\int_1^2$ dt [$1\over2$m ∑[i] (x^ ·[i])^2 − V(x[i])] . @ In curved spacetime: Baleanu & Güler JPA(01)ht [Hamilton-Jacobi]; Marques & Bezerra CQG(02)gq/01 [cosmic string]; Kowalski & Rembieliński AP(13)-a1304 [on a double cone]; Banerjee & Mukherjee a1907 [new action]. @ Other backgrounds: Chavchanidze & Tskipuri mp/01 [SU(2) group]; Musielak & Fry AP(09) [free particles in Galilean spacetime]; Knauf & Schumacher ETDS(12)-a1111 [random potential]; Ivetić et al CQG (14)-a1307 [curved Snyder space]; López a2002 [fractal spaces]. @ Symmetries: Haas & Goedert JPA(99)mp/02 [2D, Noether]; Jahn & Sreedhar AJP(01)oct-mp [invariance group]. @ Related topics: Fuenmayor et al PRD(02)ht/01 [loop representation, with gauge theory]; van Holten phy/01 [dual fluid interpretation]; Kryukov JPCS(13)-a1302 [as Dirac delta functions, and quantum theory]; > s.a. parametrized theories; Relational Theories. Quantum Particles > s.a. Complementarity; quantum particle models; Schmidt Decomposition; Wave-Particle Duality. * Issue: It can be argued that there can be no relativistic, quantum theory of localizable particles and, thus, that relativity and quantum mechanics can be reconciled only in the context of quantum field theory. @ General references: Bloch & Burba PRD(74) [presence in a spacetime region and detector]; Huang PRA(08) [quasiclassical quantum states]; Aerts IJTP(10) [particles as conceptual entities]; Vaccaro PRS(12)-a1105 [wave/particle and translational symmetry/asymmetry]; Srikanth & Gangopadhyay a1201 [particle identities as uncertain]; Vaidman PRA(13)-a1304 [the past of a quantum particle]; Flores @ No evidence / objective existence: Nissenson a0711; Blood a0807; Zeh FP(10)-a0809 [discreteness is an illusion]; Fraser SHPMP(08); Sassoli de Bianchi FS(11)-a1008; Jantzen PhSc(11)jan [permutation symmetry is incompatible with particle ontology]. @ With special relativity: Halvorson & Clifton PhSc(02)qp/01; > s.a. locality in quantum mechanics; pilot-wave quantum theory. > Related topics: see Quantum Carpet; uncertainty principle [τ and m as operators]; wigner functions. main page – abbreviations – journals – comments – other sites – acknowledgements send feedback and suggestions to bombelli at olemiss.edu – modified 10 mar 2021	{"url":"https://www.phy.olemiss.edu/~luca/Topics/part/part.html","timestamp":"2024-11-11T00:06:14Z","content_type":"text/html","content_length":"18316","record_id":"<urn:uuid:9c98a9e0-da89-47b7-a653-28c149360c23>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00464.warc.gz"}
NEET AIPMT Physics Chapter Wise Solutions - Solved Papers 2016 (Phase - I) \| Recruitment Topper NEET AIPMT Physics Chapter Wise Solutions – Solved Papers 2016 (Phase – I) Motion in a Straight Line 1. If the velocity of a particle is v = At + Bt^2, where A and B are constants, then the distance travelled by it between 1 s and 2 s is Motion in a Plane 2. If the magnitude of sum of two vectors is equal to the magnitude of difference of the two vectors, the angle between these vectors is (a) 45° (b) 180° (c) 0° (d) 90° 3. A particle moves so that its position vector is given by $\xrightarrow { r }$ = cos ωt x+ sin ωt y , where co is a constant. Which of the following is true? (a) Velocity of perpendicular to $\xrightarrow { r }$ and acceleration is directed towards the origin. (b) Velocity is perpendicular to $\xrightarrow { r }$ and acceleration is directed away from the origin. (c) Velocity and acceleration both are perpendicular to $\xrightarrow { r }$ . (d) Velocity and acceleration both are parallel to $\xrightarrow { r }$ . Laws of Motion 4. A car is negotiating a curved road of radius R. The road is banked at an angle θ. The coefficient of friction between the tyres of the car and the road is μ[s]. The maximum safe velocity on this road is Work, Energy and Power 5. A particle of mass 10 g moves along a circle of radius 6.4 cm with a constant tangential acceleration. What is the magnitude of this acceleration if the kinetic energy of the particle becomes equal to 8 x 10^-4 J by the end of the second revolution after the beginning of the motion? (a) 0.18 m/s^2 (b) 0.2 m/s^2 (c) 0.1 m/s^2 (d) 0.15 m/s^2 6. A body of mass 1 kg begins to move under the action of a time dependent force $\xrightarrow { F }$ = (2t i + 3t^2 j )N, where i and j are unit vectors along x and y What power will be developed by the force at the time t? (a) (2t^3 + 3t^4) W (b) (2t^3 + 3t^5) W (c) (2t^2 + 3t^3) W (d) (2t^2 + 4t^4) W 7. What is the minimum velocity with which a body of mass m must enter a vertical loop of radius R so that it can complete the loop? System of Particles and Rotational Motion 8. A disk and a sphere of same radius but different masses roll off on two inclined planes of the same altitude and length. Which one of the two objects gets to the bottom of the plane first? (a) Both reach at the same time (b) Depends on their masses (c) Disk[ ](d) Sphere 9. From a disc of radius R and mass M, a circular hole of diameter R, whose rim passes through the centre is cut. What is the moment of inertia of the remaining part of the disc about a perpendicular axis, passing through the centre? (a) 11 MR^2/32 (b) 9 MR^2 /32 (c) 15 MR^2/32 (d) 13 MR^2/32 10. A uniform circular disc of radius 50 cm at rest is free to turn about an axis which is perpendicular to its plane and passes through its centre. It is subjected to a torque which produces a constant angular acceleration of 2.0 rad s^-2. Its net acceleration in m s^-2 at the end of 2.0 s is approximately (a) 6.0 (b) 3.0 (c) 8.0 (d) 7.0 11. At what height from the surface of earth the gravitation potential and the value of g are -5.4 x 10^7 J kg^-2 and 6.0 m s^-2 respectively? Take the radius of earth as 6400 km. (a) 1400 km (b) 2000 km (c) 2600 km (d) 1600 km 12. The ratio of escape velocity at earth (v[e]) to the escape velocity at a planet (v[p]) whose radius and mean density are twice as that of earth is (a) 1:4 (b) 1:√2 (c) 1:2 (d) 1:2√2 Properties of Matter 13. Coefficient of linear expansion of brass and steel rods are α[1], and α[2]. Lengths of brass and steel rods are l[1] and l[2] If (l[2] – l[1]) is maintained same at all temperatures, which one of the following relations holds good? (a) α[1]^2l[2] = α[2]^2l[1] (b) α[1]l[1] = α[2]l[2][ ](c) α[1]l[2] = α[2]l[1] (d) α[1]l[2]^2[ ]= α[2]^2l[1]^2 14. A piece of ice falls from a height h so that it melts completely. Only one-quarter of theTieat produced is absorbed by the ice and all energy of ice gets converted into heat during its fall. The value of h is [Latent heat of ice is 3.4 x 10^5 J/ kg and g= 10 N/kg] (a) 136 km (b) 68 km (c) 34 km (d) 544 km 15. A black body is at a temperature of 5760 K. The energy of radiation emitted by the body at wavelength 250 nm is U[1] at wavelength 500 nm is U[2] and that at 1000 nm is U[3]. Wien’s constant, b = 2.88 x 10^6 nmK. Which of the following is (a) U[1 ]> U[2] (b) U[2] > U[1][ ](c) U[1] = 0 (d) U[3] = 0 16. Two non-mixing liquids of densities p and np (n > 1) are put in a container. The height of each liquid is A solid cylinder of length L and density d is put in this container. The cylinder floats with its axis vertical and length pL(p < 1) in the denser liquid. The density d is equal to (a) {2 + (n – 1)p}p (b) {1 + (n – 1)p}p (c) {1 + (n+ 1)p}p (d) {2 + (n+ 1)p}p Thermodynamics and Kinetic Theory 17. A gas is compressed isothermally to half its initial volume. The same gas is compressed separately through an adiabatic process until its volume is again reduced to half. Then (a) Compressing the gas isothermally or adiabatically will require the same amount of work. (b) Which of the case (whether compression through isothermal or through adiabatic process) requires more work will depend upon the atomicity of the gas. (c) Compressing the gas isothermally will require more work to be done. (d) Compressing the gas through adiabatic process will require more work to be done. 18. The molecules of a given mass of a gas have r.m.s. velocity of 200 m s^-1 at 27°C and 1.0 x 10^5 N m^-2 pressure. When the temperature and pressure of the gas are respectively, 127°C and 0.05 x 10^5 N m^-2, the r.m.s. velocity of its molecules in m s^-1 is 19. A refrigerator works between 4°C and 30°C. It is required to remove 600 calories of heat every second in order to keep the temperature of the refrigerated space constant. The power required is (Take 1 cal = 4.2 Joules) (a) 236.5 W (b) 2365 W (c) 2.365 W (d) 23.65 W 20. A siren emitting a sound of frequency 800 Hz moves away from an observer towards a cliff at a speed of 15 ms^-1 . Then, the frequency of sound that the observer hears in the echo reflected from the cliff is (Take velocity of sound in air = 330 ms^-1) (a) 838 Hz (b) 885 Hz (c) 765 Hz (d) 800 Hz 21. An air column, closed at one end and open at the other, resonates with a tuning fork when the smallest length of the column is 50 cm. The next larger length of the column resonating with the same tuning fork is (a) 150 cm (b) 200 cm (c) 66.7 cm (d) 100 cm 22. A uniform rope of length L and mass m[1] hangs vertically from a rigid support. A block of mass m[2] is attached to the free end of the rope. A transverse pulse of wavelength λ[1] is produced at the lower end of the rope. The wavelength of the pulse when it reaches the top of the rope is λ[2]. The ratio λ[2]/λ[1] is 23. A capacitor of 2 μF is charged as shown in the diagram. When the switch S is turned to position 2, the percentage of its stored energy dissipated is (a) 75% (b) 80% (c) 0% (d) 20% 24. Two identical charged spheres suspended from a common point by two massless strings of lengths l, are initially at a distance d (d < < l) apart because of their mutual repulsion. The charges begin to leak from both the spheres at a constant rate. As a result, the spheres approach each other with a velocity v. Then v varies as a function of the distance x between the spheres, as (a) v ∝ x ^-1/2 (b) v ∝ x^-1 (c) v ∝ x^1/2 (d) v ∝ x Current Electricity 25. A potentiometer wire is 100 cm long and a constant potential difference is maintained across it. Two cells are connected in series first to support one another and then in opposite direction. The balance points are obtained at 50 cm and 10 cm from the positive end of the wire in the two cases. The ratio of emf’s is (a) 3 : 4 (b) 3 : 2 (c) 5 : 1 (d) 5 : 4 26. The charge flowing through a resistance R varies with time t as Q = at – bt^2, where a and b are positive constants. The total heat produced in R is Moving Charges and Magnetism 27. A long straight wire of radius a carries a steady current I. The current is uniformly distributed over its cross-section. The ratio of the magnetic fields B and B’, at radial distances a/2 and 2a respectively, from the axis of the wire is (a) 1 (b) 4 (c) 1/4 (d) 1/2 28. A square loop ABCD carrying a current i, is placed near and coplanar with a long straight conductor XY carrying a current I, the net force on the loop will be Magnetism and Matter 29. The magnetic susceptibility is negative for (a) ferromagnetic material only (b) paramagnetic and ferromagnetic materials (c) diamagnetic material only (d) paramagnetic material only Electromagnetic Induction and Alternating Current 30. An inductor 20 mH, a capacitor 50 μF and a resistor 40 Ω are connected in series across a source of emf V = 10 sin 340t. The power loss in A.C. circuit is (a) 76 W (b) 89 W (c) 51 W (d) 67 W 31. A small signal voltage V(t) = V[0] sincof is applied across an ideal capacitor C (a) Current I(f) is in phase with voltage V(t). (b) Current I(f) leads voltage V(t) by 180°. (c) Current I(f), lags voltage V(t) by 90°. (d) Over a full cycle the capacitor C does not consume any energy from the voltage source. 32. A long solenoid has 1000 turns. When a current of 4 A flows through it, the magnetic flux linked with each turn of the solenoid is 4 x 10^-3 The self-inductance of the solenoid is (a) 2 H (b) 1 H (c) 4 H (d) 3H Electromagnetic Waves 33. Out of the following options which one can be used to produce a propagating electromagnetic wave? (a) A chargeless particle (b) An accelerating charge (c) A charge moving at constant velocity (d) A stationary charge 34. Match the corresponding entries of column 1 with column 2. [Where m is the magnification produced by the mirror] (a) A —> p and s; B —> q and r; C —> q and s; D—>q and r (b) A —> i and s; B -> q and s; C —> q and r; D —>p and s (c) A —> q and r; B -> q and r; C —> q and s; D —> p and s (d) A —> p and r; B —> p and s; C —> p and q; D-> r and s 35. In a diffraction pattern due to a single slit of width a, the first minimum is observed at an angle 30° when light of wavelength 5000 A is incident on the slit. The first secondary maximum is observed at an angle of 36. The intensity at the maximum in a Young’s double slit experiment is I[0]. Distance between two slits is d = 5a, where X is the wavelength of light used in the experiment. What will be the intensity in front of one of the slits on the screen placed at a distance D = 10d? 37. A astronomical telescope has objective and eyepiece of focal lengths 40 cm and 4 cm respectively. To view an object 200 cm away from the objective, the lenses must be separated by a distance (a) 50.0 cm (b) 54.0 cm (c) 37.3 cm (d) 46.0 cm 38. The angle of incidence for a ray of light at a refracting surface of a prism is 45°. The angle of prism is 60°. If the ray suffers minimum deviation through the prism, the angle of minimum deviation and refractive index of the material of the prism respectively, are Dual Nature of Radiation and Matter 39. An electron of mass m and a photon have same energy E. The ratio of de-Broglie wavelengths associated with them is 40. When a metallic surface is illuminated with radiation of wavelength λ, the stopping potential is V. If the same surface is illuminated with radiation of wavelength 2λ, the stopping potential V is V/4. The threshold wavelength for the metallic 4 surface is (a) 5/2 λ (b) 3λ (c) 4λ (d) 5λ Atoms and Nuclei 41. Given the value of Rydberg constant is 10^7 m^-1 the wave number of the last line of the Balmer series in hydrogen spectrum will be (a) 0.25 x 10^7 m^-1 (b) 2.5 x 10^7 m^-1 (c) 0.025 x 10^4 m^-1 (d) 0.5 x 10^7 m^-1 42. When an α-particle of mass m moving with velocity v bombards on a heavy nucleus of charge Ze, its distance of closest approach from the nucleus depends on m as Semiconductor Electronics: Materials, Devices and Simple Circuits 43. To get output 1 for the following circuit, the correct choice for the input is (a) A= 1,B= 1,C = 0 (b) A=1,B = 0,C=1 (c) A = 0,B=1,C = 0 (d) A= 1,B = 0,C=0 44. A npn transistor is connected in common emitter configuration in a given amplifier. A load resistance of 800 Ω is connected in the collector circuit and the voltage drop across it is 0.8 V. If the current amplification factor is 0.96 and the input resistance of the circuit is 192 Ω, the voltage gain and the power gain of the amplifier will respectively be (a) 4,4 (b) 4,3.69 (c) 4,3.84 (d) 3.69,3.84 45. Consider the junction diode as ideal. The value of current flowing through AB is (a) 10^-1 A (b) 10^-3A (c) 0 A (d) 10^-2 A	{"url":"http://www.recruitmenttopper.com/neet-aipmt-physics-chapter-wise-solutions-solved-papers-2016-phase/10964/","timestamp":"2024-11-12T09:35:37Z","content_type":"text/html","content_length":"107317","record_id":"<urn:uuid:47163228-430a-4a37-8389-5da965d74a2e>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00522.warc.gz"}
The journey to quantifying chip photography: Particle statistics Created by Mark Grujic and Rian Dutch. Welcome back to our series on drill chips! Here’s a quick summary of where we left off: First, we went through the DOs & DON’Ts of capturing good quality chip photos. Then we looked at how you can identify rock material within these photos, and create structured, informative datasets that map out the colour of rock chips, and how they vary down hole. We looked at the representative colour of individual rock chips, and teased at doing more with each chip, to quantify as much information within our chip images as possible, and as promised – here we are! Taking a step back – let’s first review the process by which we can separate individual chips from each other, and other non-rock material in an image. The figure below represents several common problems that we typically use computer vision to solve. The example uses veins as they appear in diamond drill core photos (a core Datarock product offering), along with some typical datasets that you can generate from each case. The same technologies are applicable to the rock chip study. In the previous blog we used Semantic Segmentation to exclude non-rock material from the rock chip compartment colour analysis, and then Instance Segmentation to find the representative colour of each individual rock chip. Chip Geometry Given that our Instance Segmentation model has provided us with masks for each individual chip in all the compartments in our dataset, we can now consider the geometric properties of each chip, and attempt to produce geoscientific information. Fortunately for us, figuring out numbers that describe shapes is something that is well and truly established! For example, we can utilise the existing methodologies that Datarock uses for segmentation characterisation in our core photography products and services (e.g. vein orientation, fracture aperture, sulphide texture) to determine a number of parameters for each chip. Take this particular chip in the image below, and note some of the geometric properties that we can determine based on its mask: There is a wide variety of different geometric properties that we could calculate for each chip, but let’s just focus on a few of them here that we think are important. An obvious choice – rock chip size distribution is a directly interpretable property and one of the primary drivers for demand for Datarock’s chip services. The area here is represented in pixels, so a length scale conversion should be applied. If you know something about the scale in your photos, with either a physical scale on the images, or by knowing the compartment width, area in pixels can be converted to area in millimetres squared or another real-world unit. This is a measure of how circular or elliptical a chip is. The closer the value to zero, the more circular the chip is. Values approaching 1 are very elliptical or very long. An example of eccentricity being geologically informative could be the interpretation of certain populations of chips that have preferentially broken along certain planes, resulting in one population with a higher eccentricity than another population. Like area, knowing the perimeter of chips can provide useful size distribution information. Unlike area, this property does not scale with the square of the size unit, so variations might be more linear than area values. The most “efficient” area for a given perimeter is a circle, and we expect area and perimeter to scale similarly, however, unexpected variations could indicate chips with complex perimeters. Which brings us to… If you were to construct a convex hull around a chip mask (a polygon with no concavity between vertices), then calculate the area, it would always be greater than the area of the mask itself. The ratio of the chip area to that of its associated convex hull is known as its solidity, and ranges from 0 to 1. Values close to 1 suggest the perimeter of the chip is entirely convex (think regular geometric shapes), whereas small values suggest complexities in the chip boundary (rough, irregular). The example below has a solidity of 0.86. These statistics can be presented in a table, where each row represents a uniquely identified chip, and each column refers to specific geometric properties of the chip. Some caveats Caveats to the accuracy of these properties include the fact that we have a 2-dimensional representation of a 3D volume of rock chips, occlusion by other chips, lighting angles creating shadows, incorrectly cropped photos, and the accuracy of the chip segmentation model itself. These and more play a role in how certain you can be of the results of these geometric studies of individual chips and their summaries. However some assumptions can be made around these effects, and the way that the calculated properties change with depth can be just as informative as the absolute property values An example of mitigating occlusion of chips by other chips would be assuming that the complex hull of a chip (see Solidity, above) is more representative of the chip’s true area than the visible mask. We can support this assumption with the understanding that rock chips fragment in relatively predictable ways that don’t result in crazy shapes that would result in very low Solidity. Another approach would be to identify all occluded chips and exclude them from the measurements. Data scales Geometric and colour properties of individual chips are all well and good, but as geoscientists, we are typically more interested in aggregating that information to more relevant scales. Compartment scale The first scale that we might consider interpreting is the information contained within each rock chip compartment. For each of our calculated properties on each chip, we can come up with statistical representations of their distributions within a compartment. This can give us a fairly straight forward understanding of things like chip size distribution: However, given that we have so much more information available to us for each chip, we can create some powerful datasets and visualisations to help interpret geology. For example, given that we already have the colour of each chip, we can do things like sort the chips by size, showing colour as another data property, and look for trends in size and colour: We can also come up with our own project-specific set of plots that may help us answer the question we’re trying to solve. For example, the figure below shows eccentricity on the y-axis (how circular are the chips) against a measure of solidity on the x-axis (how complex is the perimeter). Each point represents a rock chip, with the size proportional to its area, and the colour being its dominant The ways you could combine and display properties of individual chips are almost limitless, so it’s important to think about what you are trying to solve, then build the right processes. However, given that no single chip is representative of the compartment, we may also be inclined to statistically summarise the properties of all chips within a compartment, and represent the property distributions by things like their minimum, mean, maximum, median, standard deviation, etc. The tables below show some properties and their statistics within a single chip compartment, as well as the properties that cannot be summarised. Drill hole scale While the assessment on a chip-compartment scale can inform certain geoscientific or geotechnical decisions, we are often more interested in knowing the down hole character and variation of these quantified properties. As we see in the tables above, there are a lot of generated properties per compartment, and even more when we consider the statistical summaries! So, below is a subset of the available data, showing the median values in “area” to “perimeter over area” columns. You might note that this is already in a neatly formatted table, thanks to good metadata tagging of the source photos, and the work described in the last blog in this series. Because of this, we can get to the final data visualisations in this blog – down hole logs showing how our calculated properties vary. The first and second columns in these plots are the same as we saw in our earlier blog on colour. The third column is the same as the second, except the bars have been scaled by the median chip area in each compartment, resulting in a product greater than the sum of its parts. The rest of the columns are downhole logs of the variation in some key properties, showing the median in red, and the 0.25 and 0.75 quantiles shaded in blue. These tables can (should!) be brought into your own modelling software for integration with other datasets. Wrap up We’ve seen how to ensure that your chip photos are good and properly tagged with metadata. Then we looked at how to extract and use colour information. In this update, we’ve addressed quantified chip geometry and how to incorporate it with earlier work, to then extract geological information. These powerful datasets generated from an often overlooked data source can become a valuable part of your geological and geotechnical workflows. As chip photo collection processes improve over time, and as we are all being asked to do more with our datasets, we see Datarock Chip (the productionisation of these capabilities), working hand in hand with our drill core imagery products and services on your projects. The story doesn’t end here! The next blog in this series on drill chips will introduce how we incorporate deep-learning based computer vision technologies into this process of quantifying visual information in rock chips, before tying everything presented to date into a single interpretive dataset. Stay tuned!	{"url":"https://datarock.com.au/blog/the-journey-to-quantifying-chip-photography-particle-statistics/","timestamp":"2024-11-06T04:53:13Z","content_type":"text/html","content_length":"107154","record_id":"<urn:uuid:d09ed61b-3049-4f38-a080-71afcf65a614>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00395.warc.gz"}
7.3: Theoretical and Experimental Spinners Last updated Page ID $ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $ $ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $ $ \newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ ( \newcommand{\kernel}{\mathrm{null}\,}\) $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\\| #1 \\|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\\| #1 \\|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\AA}{\unicode[.8,0]{x212B}}$ $ \newcommand{\vectorA}[1]{\vec{#1}} % arrow$ $ \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$ $ \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $ $ \newcommand{\vectorC}[1]{\textbf{#1}} $ $ \newcommand{\vectorD}[1]{\overrightarrow{#1}} $ $ \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} $ $ \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} $ $ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $ $ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $ $\newcommand{\avec}{\mathbf a}$ $\newcommand{\bvec}{\mathbf b}$ $\newcommand{\cvec}{\mathbf c}$ $\newcommand{\dvec}{\mathbf d}$ $\newcommand{\dtil}{\widetilde{\mathbf d}}$ $\newcommand{\ evec}{\mathbf e}$ $\newcommand{\fvec}{\mathbf f}$ $\newcommand{\nvec}{\mathbf n}$ $\newcommand{\pvec}{\mathbf p}$ $\newcommand{\qvec}{\mathbf q}$ $\newcommand{\svec}{\mathbf s}$ $\ newcommand{\tvec}{\mathbf t}$ $\newcommand{\uvec}{\mathbf u}$ $\newcommand{\vvec}{\mathbf v}$ $\newcommand{\wvec}{\mathbf w}$ $\newcommand{\xvec}{\mathbf x}$ $\newcommand{\yvec}{\mathbf y} $ $\newcommand{\zvec}{\mathbf z}$ $\newcommand{\rvec}{\mathbf r}$ $\newcommand{\mvec}{\mathbf m}$ $\newcommand{\zerovec}{\mathbf 0}$ $\newcommand{\onevec}{\mathbf 1}$ $\newcommand{\real} {\mathbb R}$ $\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$ $\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$ $\newcommand{\threevec} [3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\fourvec}[4]{\left[\begin{array} {r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$ $\newcommand{\laspan}[1]{\text{Span}\{#1\}}$ $\newcommand{\bcal}{\cal B}$ $\newcommand{\ccal}{\cal C}$ $\newcommand{\scal}{\cal S}$ $\newcommand{\ wcal}{\cal W}$ $\newcommand{\ecal}{\cal E}$ $\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$ $\newcommand{\gray}[1]{\color{gray}{#1}}$ $\newcommand{\lgray}[1]{\color{lightgray}{#1}}$ $\ newcommand{\rank}{\operatorname{rank}}$ $\newcommand{\row}{\text{Row}}$ $\newcommand{\col}{\text{Col}}$ $\renewcommand{\row}{\text{Row}}$ $\newcommand{\nul}{\text{Nul}}$ $\newcommand{\var} {\text{Var}}$ $\newcommand{\corr}{\text{corr}}$ $\newcommand{\len}[1]{\left\|#1\right\|}$ $\newcommand{\bbar}{\overline{\bvec}}$ $\newcommand{\bhat}{\widehat{\bvec}}$ $\newcommand{\bperp}{\ bvec^\perp}$ $\newcommand{\xhat}{\widehat{\xvec}}$ $\newcommand{\vhat}{\widehat{\vvec}}$ $\newcommand{\uhat}{\widehat{\uvec}}$ $\newcommand{\what}{\widehat{\wvec}}$ $\newcommand{\Sighat}{\ widehat{\Sigma}}$ $\newcommand{\lt}{<}$ $\newcommand{\gt}{>}$ $\newcommand{\amp}{&}$ $\definecolor{fillinmathshade}{gray}{0.9}$ Theoretical and Experimental Spinners The 2 types of probability are theoretical probability and experimental probability. Theoretical probability is defined as the number of desired outcomes divided by the total number of outcomes. Theoretical Probability Experimental probability is, just as the name suggests, dependent on some form of data collection. To calculate the experimental probability, divide the number of times the desired outcome has occurred by the total number of trials. Experimental Probability What is interesting about theoretical and experimental probabilities is that, in general, the more trials you do, the closer the experimental probability gets to the theoretical probability. We'll see this as we go through the examples. Comparing Experimental and Theoretical Probability 1. You are spinning a spinner like the one shown below 20 times. How many times does it land on blue? How does the experimental probability of landing on blue compare to the theoretical probability? Simulate the spinning of the spinner using technology. On the TI-84 calculator, there are a number of possible simulations you can do. You can do a coin toss, spin a spinner, roll dice, pick marbles from a bag, or even draw cards from a deck. After pressing ENTER, you will have the following screen appear. Since we're doing a spinner problem, choose Spin Spinner. In Spin Spinner, a wheel with 4 possible outcomes is shown. You can adjust the number of spins, graph the frequency of each number, and use a table to see the number of spins for each number. We want to set this spinner to spin 20 times. Look at the keystrokes below and see how this is done. In order to match our color spinner with the one found in the calculator, you will see that we have added numbers to our spinner. This is not necessary, but it may help in the beginning to remember that 1 = blue (for this example). Now that the spinner is set up for 20 trials, choose SPIN by pressing WINDOW We can see the result of each trial by choosing TABL, or pressing GRAPH And we see the graph of the resulting table, or go back to the first screen, simply by choosing GRPH, or pressing GRAPH again. Now, the question asks how many times we landed on blue (number 1). We can actually see how many times we landed on blue for these 20 spins. If you press the right arrow (▸), the frequency label will show you how many of the times the spinner landed on blue (number 1). To go back to the question, how many times does the spinner land on blue if it is spun 20 times? The answer is 3. To calculate the experimental probability of landing on blue, we have to divide by the total number of spins. Therefore, for this experiment, the experimental probability of landing on blue with 20 spins is 15%. Now let's calculate the theoretical probability. We know that the spinner has 4 equal parts (blue, purple, green, and red). In a single trial, we can assume that: Therefore, for our spinner example, the theoretical probability of landing on blue is 0.25. Finding the theoretical probability requires no collection of data. 2. You are spinning a spinner like the one shown below 50 times. How many times does it land on blue? How about if you spin it 100 times? Does the experimental probability get closer to the theoretical probability? Simulate the spinning of the spinner using technology. Set the spinner to spin 50 times and choose SPIN by pressing WINDOW You can see the result of each trial by choosing TABL, or pressing GRAPH Again, we can see the graph of the resulting table, or go back to the first screen, simply by choosing GRAPH, or pressing GRAPH again. The question asks how many times we landed on blue (number 1) for the 50 spins. Press the right arrow (▸), and the frequency label will show you how many of the times the spinner landed on blue (number 1). Now go back to the question. How many times does the spinner land on blue if it is spun 50 times? The answer is 11. To calculate the probability of landing on blue, we have to divide by the total number of spins. Therefore, for this experiment, the probability of landing on blue with 50 spins is 22%. If we tried 100 trials, we would see something like the following: In this case, we see that the frequency of 1 is 23. So how many times does the spinner land on blue if it is spun 100 times? The answer is 23. To calculate the probability of landing on blue in this case, we again have to divide by the total number of Therefore, for this experiment, the probability of landing on blue with 100 spins is 23%. You can see that as we perform more trials, we get closer to 25%, which is the theoretical probability. 3. You are spinning a spinner like the one shown below 170 times. How many times does it land on blue? Does the experimental probability get closer to the theoretical probability? How many times do you predict we would have to spin the spinner to have the experimental probability equal the theoretical probability? Simulate the spinning of the spinner using technology. With 170 spins, we get a frequency of 42 for blue. The experimental probability in this case can be calculated as follows Therefore, the experimental probability is 24.7%, which is even closer to the theoretical probability of 25%. While we're getting closer to the theoretical probability, there is no number of trials that will guarantee that the experimental probability will exactly equal the theoretical probability. Example 1 You are spinning a spinner like the one shown below 500 times. How many times does it land on blue? How does the experimental probability of landing on blue compare to the theoretical probability? Simulate the spinning of the spinner using technology. In the list of applications on the TI-84 calculator, choose Prob Sim. After pressing ENTER, you will have the following screen appear. Since we're doing a spinner problem, choose Spin Spinner. In Spin Spinner, a wheel with 4 possible outcomes is shown. You can adjust the number of spins, graph the frequency of each number, and use a table to see the number of spins for each number. We want to set this spinner to spin 500 times. To do this, choose SET by pressing ZOOM, enter 500 after Trial Set, and choose OK by pressing GRAPH. Remember that for this example, 1 = blue. Now that the spinner is set up for 500 trials, choose SPIN by pressing WINDOW. Since we've chosen a large number of spins, the spinning may take a while! We can see the result of each trial by choosing TABL, or pressing GRAPH And we see the graph of the resulting table, or go back to the first screen, simply by choosing GRPH, or pressing GRAPH again. Now, the question asks how many times we landed on blue (number 1). We can actually see how many times we landed on blue for these 500 spins. If you press the right arrow (▸), the frequency label will show you how many of the times the spinner landed on blue (number 1). To go back to the question, how many times does the spinner land on blue if it is spun 500 times? The answer is 123. To calculate the experimental probability of landing on blue, we have to divide by the total number of spins. Therefore, for this experiment, the experimental probability of landing on blue with 500 spins is 24.6%. Do you remember how to calculate the theoretical probability from Example A? We know that the spinner has 4 equal parts (blue, purple, green, and red). In a single trial, we can assume that: Therefore, for our spinner example, the theoretical probability of landing on blue is 0.25. As we pointed out in Example A, finding the theoretical probability requires no collection of data. It's also worth mentioning that our experimental probability was slightly farther away from the theoretical probability with 500 spins that it was with 170 spins in Example C. While, in general, increasing the number of spins will produce an experimental probability that is closer to the theoretical probability, as we've just seen, this is not always the case! 1. Based on what you know about probabilities, write definitions for theoretical and experimental probability. 2. . 1. What is the difference between theoretical and experimental probability? 2. As you add more data, do your experimental probabilities get closer to or further away from your theoretical probabilities? 3. Is spinning 1 spinner 100 times the same as spinning 100 spinners 1 time? Why or why not? A spinner was spun 750 times using Spin Spinner on the TI-84 calculator, with 1 representing blue, 2 representing purple, 3 representing green, and 4 representing red as shown: 3. According to the following screen, what was the experimental probability of landing on blue? 4. According to the following screen, what was the experimental probability of landing on purple? 5. According to the following screen, what was the experimental probability of landing on green? 6. According to the following screen, what was the experimental probability of landing on red? A spinner was spun 900 times using Spin Spinner on the TI-84 calculator, with 1 representing blue, 2 representing purple, 3 representing green, and 4 representing red as shown: 7. According to the following screen, what was the experimental probability of landing on blue? 8. According to the following screen, what was the experimental probability of landing on purple? 9. According to the following screen, what was the experimental probability of landing on green? 10. According to the following screen, what was the experimental probability of landing on red? Review (Answers) To view the Review answers, open this PDF file and look for section 3.5. Term Definition experimental probability Experimental (empirical) probability is the actual probability of an event resulting from an experiment. theoretical probability Theoretical probability is the probability ration of the number of favourable outcomes divided by the number of possible outcomes. Additional Resources Video: Theoretical and Experimental Spinners Principles Activities: Theoretical and Experimental Coin Tosses Discussion Questions Lesson Plans: Using Technology to Find Probability Distributions, Spinning a Spinner Lesson Plan Practice: Theoretical and Experimental Spinners Real World: Theoretical and Experimental Probability	{"url":"https://k12.libretexts.org/Bookshelves/Mathematics/Statistics/07%3A_Analyzing_Data_and_Distributions_-_Probability_Distributions/7.03%3A_Theoretical_and_Experimental_Spinners","timestamp":"2024-11-02T13:52:12Z","content_type":"text/html","content_length":"151609","record_id":"<urn:uuid:7ef554d2-1352-42ef-85ee-fdbdfa47aa0b>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00454.warc.gz"}
Search Results for "temperature integral" \| AKJournals Search Results You are looking at 1 - 10 of 72 items for : • Refine by Access: All Content x Clear All Journal of Thermal Analysis and Calorimetry R. Gartia S. Dorendrajit Singh T. Jekendra Singh , and P. Mazumdar The general method of evaluating the temperature integral for temperature dependent frequency factors have been considered. The values of the temperature integral as evaluated by the present method are in excellent agreement with those obtained numerically. The temperature integral cannot be analytically integrated and many simple closed-form expressions have been proposed to use in the integral methods. This paper first reviews two types of simple approximation expressions for temperature integral in literature, i.e. the rational approximations and exponential approximations. Then the relationship of the two types of approximations is revealed by the aid of a new equation concerning the 1^st derivative of the temperature integral. It is found that the exponential approximations are essentially one kind of rational approximations with the form of h(x)=[x/(Ax+k)]. That is, they share the same assumptions that the temperature integral h(x) can be approximated by x/Ax+k). It is also found that only two of the three parameters in the general formula of exponential approximations are needed to be determined and the other one is a constant in theory. Though both types of the approximations have close relationship, the integral methods derived from the exponential approximations are recommended in kinetic analysis. Three rational fraction approximations for the temperature integral have been proposed using the pattern search method. The validity of the new approximations has been tested by some numerical analyses. Compared with several published approximating formulas, the new approximations is more accurate than all approximations except the approximations proposed by Senum and Yang in the range of 5≤E/RT≤100. For low values of E/RT, the new approximations are superior to Senum-Yang approximations as solutions of the temperature integral. The generalized temperature integral \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage {upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\int\limits_0^T {T^m } \exp ( - E/RT)dT$$ \end{document} frequently occurs in non-isothermal kinetic analysis. Here is the activation energy, the universal gas constant and the absolute temperature. The exponent arises from the temperature dependence of the pre-exponential factor. This paper has proposed two new approximate formulae for the generalized temperature integral, which are in the following forms: \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage {upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\begin{gathered} h_m (x) = \frac{x} {{(1.00141 + 0.00060m)x + (1.89376 + 0.95276m)}} \hfill \\ h_m (x) = \frac{{x + (0.74981 - 0.06396m)}} {{(1.00017 + 0.00013m)x + (2.73166 + 0.92246m)}} \hfill \\ \end{gathered}$$ \end{document} h [m] ) is the equivalent form of the generalized temperature integral. For commonly used values of in kinetic analysis, the deviations of the new approximations from the numerical values of the integral are within 0.2 and 0.03%, respectively. In contrast to other approximations, both the present approaches are simple, accurate and can be used easily in kinetic analysis. Journal of Thermal Analysis and Calorimetry T. Wanjun L. Yuwen Z. Hen W. Zhiyong , and W. Cunxin A new approximate formula for temperature integral is proposed. The linear dependence of the new fomula on x has been established. Combining this linear dependence and integration-by-parts, new equation for the evaluation of kinetic parameters has been obtained from the above dependence. The validity of this equation has been tested with data from numerical calculating. And its deviation from the values calculated by Simpson's numerical integrating was discussed. Compared with several published approximate formulae, this new one is much superior to all other approximations and is the most suitable solution for the evaluation of kinetic parameters from TG experiments. A new approximation has been proposed for calculation of the general temperature integral \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage {upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\int\limits_0^T {T^m } e^{ - E/RT} dT$$ \end{document} , which frequently occurs in the nonisothermal kinetic analysis with the dependence of the frequency factor on the temperature ( A [0] T ^m ). It is in the following form: \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage {upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\int\limits_0^T {T^m } e^{ - E/RT} dT = \frac{{RT^{m + 2} }} {E}e^{ - E/RT} \frac{{0.99954E + (0.044967m + 0.58058)RT}} {{E + (0.94057m + 2.5400)RT}}$$ \end{document} The accuracy of the newly proposed approximation is tested by numerical analyses. Compared with other existed approximations for the general temperature integral, the new approximation is significantly more accurate than other approximations. In the paper a new procedure to approximate the generalized temperature integral \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage {upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\int\limits_0^T {T^m } \exp ( - E/RT)dT$$ \end{document} , which frequently occurs in non-isothermal thermal analysis, are presented. A series of the approximations for the temperature integral with different complexity and accuracy are proposed from the procedure. For commonly used values of in kinetic analysis, the deviation of most approximations from the numerical values of the integral is within 0.7%, except the first approximation (within 4.0%). Since they are simple in calculation and hold high accuracy, the approximations are recommended to use in the evaluation of kinetic parameters from non-isothermal kinetic analysis. The accuracy and scope of application of previously reported approximations of the temperature integral were evaluated. The exact solution was obtained independently by solving the temperature integral numerically be Simpson's rule, the trapezoidal rule and the Gaussian rule. Two new approximations have been proposed: \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage {upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\begin{gathered} P(X) = e^{ - x} (1/X^2 )(1 - 2/X)/(1 - 5.2/X^2 ) \hfill \\ P(X) = e^{ - x} (1/X^2 )(1 - 2/X)/(1 - 4.6/X^2 ) \hfill \\ \end{gathered}$$ \end{document} whereX=E/RT. The first equation gives higher accuracy, with a deviation of less than 1% and 0.1% from the exact solution forX≥7 andX≥10, respectively. The second equation has a wider scope of application, with a deviation of less than 1% forX≥4 and of less than 0.1% forX≥35. A new procedure to approximate the generalized temperature integral \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage {upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\int_{0}^{T} {T^{m} {\text{e}}^{ - E/RT} } {\text{d}}T,$$ \end which frequently occurs in non-isothermal thermal analysis, has been developed. The approximate formula has been proposed for calculation of the integral by using the procedure. New equation for the evaluation of non-isothermal kinetic parameters has been obtained, which can be put in the form: \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage {upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\ln \left[ {{\frac{g(\alpha )}{{T^{(m + 2)0.94733} }}}} \right] = \left[ {\ln {\frac{{A_{0} E}}{\beta R}} - (m + 2)0.18887 - (m + 2)0.94733\ln {\frac{E}{R}}} \right] - (1.00145 + 0.00069m){\frac{E}{RT}}$$ \end{document} The validity of the new approximation has been tested with the true value of the integral from numerical calculation. Compared with several published approximation, the new one is simple in calculation and retains high accuracy, which indicates it is a good approximation for the evaluation of kinetic parameters from non-isothermal kinetic analysis.	{"url":"https://akjournals.com/search?q=%22temperature+integral%22","timestamp":"2024-11-09T19:10:37Z","content_type":"text/html","content_length":"351120","record_id":"<urn:uuid:a8374702-6a56-4247-8d2d-1a37c3bf6a76>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00250.warc.gz"}
Get assignment solution or answer for "During a hard sneeze, your eyes might shut for 0.46 s. If you are driving a car at 95 km/h during such a sneeze, how far..." Physics Problems & Solutions (Economics & Science Course Help) Question Preview Futuristic Solution #40 1. During a hard sneeze, your eyes might shut for 0.46 s. If you are driving a car at 95 km/h during such a sneeze, how far does the car move during that time? 2. The position of an object moving in a straight line is given by x = 4t ? 3t^2 + 5t^3, where x is in meters and t in seconds. (a) What is the position of the object at t = 1 s? What is the position of the object at t = 2 s? What is the position of the object at t = 3 s? What is the position of the object at t = 4 s? (b) What is the object's displacement between t = 0 s and t = 4 s? (Indicate the direction with the sign of your answer.) (c) What is the average velocity for the time interval from t = 2 s to t = 4 s? (Indicate the direction with the sign of your answer.) (d) Graph x versus t for 0 s ? t ? 4 s and indicate how the answer for (c) can be found on the graph. 3.At a certain time a particle had a speed of 19 m/s in the positive x direction, and 2.9 s later its speed was 29 m/s in the opposite direction. What is the average acceleration of the particle during this 2.9 s interval? 4.An electron with initial velocity v[0] = 2.70 ? 10^5 m/s enters a region 1.0 cm long where it is electrically accelerated. It emerges with velocity v = 6.70 ? 10^6 m/s. What was its acceleration, assumed constant? (Such a process occurs in old-fashioned television sets.) 5.An electric vehicle starts from rest and accelerates at a rate of 2.6 m/s^2 in a straight line until it reaches a speed of 25 m/s. The vehicle then slows at a constant rate of 1.0 m/s^2 until it stops. (a) How much time elapses from start to stop? (b) How far does the vehicle travel from start to stop? 6.Raindrops fall 1715 m from a cloud to the ground. (a) If they were not slowed by air resistance, how fast would the drops be moving when they struck the ground? (b) Would it be safe to walk outside during a rainstorm? 7.The figure below gives the acceleration a versus time t for a particle moving along an x axis. At t = ?2.0 s, the particle's velocity is ?30 m/s. What is its velocity at t = 5.0 s? 8.A rock is dropped from a 330 m high cliff. (a) How long does it take to fall the first 165 m? (b) How long does it take to fall the second 165 m? 9.Compute your average velocity in the following two cases. (a) You walk 75.8 m at a speed of 1.22 m/s and then run 75.8 m at a speed of 3.05 m/s along a straight track. (b) You walk for 1.91 min at a speed of 1.22 m/s and then run for 1.91 min at 3.05 m/s along a straight track. (c) Graph x versus t for part (a) and indicate how the average velocity is found on the graph. Graph x versus t for part (b) and indicate how the average velocity is found on the graph. 10. The position function x(t) of a particle moving along an x axis is x = 4.2 ? 7.7t^2, with x in meters and t in seconds. (a) At what time does the particle (momentarily) stop? (b) Where does the particle (momentarily) stop? (c) At what negative time does the particle pass through the origin? (d) At what positive time does the particle pass through the origin? (e) Graph x versus t for the range ?5 s to +5 s. (f) To shift the curve leftward on the graph, should we include the term ?20t or the term +20t in x(t)?(g) Does that inclusion increase or decrease the value of x at which the particle momentarily 11. If the position of a particle is given by x = 33t ? 9t^3, where x is in meters and t is in seconds, answer the following questions. (a) When, if ever, is the particle's velocity zero? (Enter the value of t or 'never'.) (b) When is its acceleration a zero? (c) When is a negative? (d) When is a positive?(e) Graph x(t), v(t), and a(t). (Submit a file with a maximum size of 1 MB.) 12. A muon (an elementary particle) enters a region with a speed of 5.15 ? 10^6 m/s and then is slowed at the rate of 1.45 ? 10^14 m/s^2. (a) How far does the muon take to stop? (b) Graph x versus t for the muon. Graph v versus t for the muon. 13. A hoodlum throws a stone vertically downward with an initial speed of 12.2 m/s from the roof of a building, 39.0 m above the ground. (a) How long does it take the stone to reach the ground? (b) What is the speed of the stone at impact? 14 How far does the runner whose velocity-time graph is shown below travel in 12 s? Release Date: June 12, 2021 Answer Preview Experts are working on this assignment's solution or answer. If you order this assignment at tutordollar, We will update you as soon as possible to your registered email address. Need any help? Feel free to connect with us. 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Theoretical Background Theoretical background of SimSolid and its software implementation workflow, and comparison of other methods used in traditional FEA. Overview of Initial Research The Ritz-Galerkin method invented at the beginning of 20th century for the approximate solution of boundary value problems assumes that functions that approximate the solution are analytical functions defined on the whole domain of interest. In practical applications these functions were either trigonometric or polynomials which were infinitely smooth, i.e. they had an infinite number of derivatives. There were two main problems with such functions. Firstly, it was difficult or impossible to construct such functions that a priori meet essential boundary conditions on boundaries of arbitrary domains (in structural analysis the conditions appear as displacement constraints). And secondly, the equation system built on such functions was ill-conditioned and numerically unstable which did not allow solving real life problems with enough accuracy. The Finite Element Method (FEM) that appeared in 1950s was just a different implementation of the classical Ritz-Galerkin approach, but it succeeded in solving of both – constraints and numerical instability issues because it consistently used functions with local supports called finite elements. Though locally the basis-functions of finite elements were infinitely differentiable standard polynomials, global basis-functions assembled from local polynomials were not smooth at all – even their first derivatives were discontinuous. The FEM’s success proved that requirements to the continuity of the approximation functions must be met only to a certain degree - just enough to provide finite energy when they are substituted into an energy functional of a boundary value problem. The spaces of such functions were introduced and investigated by Sobolev in the 1930s. The next step in the relaxation of continuity requirements on approximation functions was the introduction of the concept of external approximations [Equation 1]. The name “external” was used in the following context. When approximation functions belong to a Sobolev space of functions with finite energy the approximation is called “internal” which means that while the approximation is refined, and the solution is converging to the exact solution, the approximation functions are always inside the Sobolev space. Alternatively, in external approximations, the approximation functions do not belong to a Sobolev space at every refinement step (they have infinite energy), but in the limit, when number of degrees of freedom (DOF) approaches infinity, the limit function must belong to the corresponding Sobolev space, i.e. it must recover the necessary smoothness properties. The abstract theory of external approximations was developed in reference [Equation 2]. The technological foundations of SimSolid have been published in reference [3]. It develops the abstract theory of external approximations. In reference [2] it was applied to the particular case of approximations by finite elements under the assumption that the elements are of absolutely arbitrary shape. In the result the necessary and sufficient condition of external approximations by finite elements has been established and convergence theorems have been proven. It was also shown that the theorems were constructive, i.e. they not only defined hallmarks of external approximations, but also provided a mechanism to build them. Theoretical Background An abstract boundary value problem is formulated as to find a function $U$ which fulfills the equations: $AU=h\text{ inside the domain }\Omega$ $LU=g\text{ at the domain boundary }\Gamma$ Where $A$ and $L$ are differential operators. Some boundary value problems can be equally formulated in a variational form such as to find a function $U$ which provides a functional $F\left(U\right)$ at minimum value, where the functional $F\ left(U\right)$ is usually an energy functional. In 1908 W. Ritz proposed a method of finding an approximate solution of a boundary value problem by approximating it with a linear combination of some basis-functions. ${U}_{h}={\sum }_{i}^{n}{a}_{i}{p}_{i}$ Where ${a}_{i}$ are unknown factors, and ${p}_{i}$ are basis approximation functions. The factors ${a}_{i}$ are found by minimizing the energy functional $F\left({\sum }_{i}^{n}{a}_{i}{p}_{i}\right)=\mathrm{min}$ If the boundary value problem is linear, then the minimization problem (Equation 4) can be reduced to a linear algebraic equation system with respect to the factors ${a}_{i}$ Where K is a symmetric matrix, d is the vector of unknown factors , and f is a right-hand side of the system. In FEM the matrix K is called a stiffness matrix, the vector f is called a load vector, and factors ${a}_{i}$ are called degrees of freedom. In 1915 Galerkin proposed another approximate method of solving the boundary value problem (Equation 1)-(Equation 2). According to the Galerkin method the unknown solution U is approximated as ${U}_{n}={U}_{0}+{\sum }_{i}^{n}{a}_{i}{p}_{i}$ Where ${U}_{0}$ is some function which fulfills non-homogeneous boundary conditions (Equation 2), ${p}_{i}$ are analytical approximation functions which fulfill homogeneous boundary conditions, ${a}_ {i}$ are unknown factors. Substitution of (Equation 6) into (Equation 1) results in the residual. $R=A{U}_{0}+{\sum }_{i}^{n}{a}_{i}A{p}_{i}-h$ The unknown factors ${a}_{i}$ are found from the equation system. ${\int }_{\Omega }R{p}_{i}d\Omega =0$ If a boundary value problem is linear, then system (Equation 8) is a system of linear algebraic equations. The Galerkin method does not use a variational formulation of the boundary value problem. Therefore, its applicability is much wider. Ritz and Galerkin methods proved to be effective means of solving problems in engineering and science. At the same time mathematical justification of the methods faced significant difficulties which were solved with the introduction of functional analysis as a mathematical discipline. Modern theory of the Ritz-Galerkin method is based on the concept of the weak formulation of the boundary value problem. The weak formulation of a boundary value problem consists of finding a function from $u\in V$ a corresponding Sobolev space which fulfills an abstract variational equation $a\left(u,v\right)=f\left(v\right)\text{ for any function }v\in V$ Where $V$ is some subspace of a Sobolev space, $a\left(u,v\right)$ is generally an unsymmetrical bilinear form which is continuous on the space product $V×V,f\left(v\right)$ is some linear form in In structural analysis, the Sobolev space is a space of functions with finite strain energy. In the Ritz-Galerkin method the space $V$ is approximated with some finite-dimensional space ${X}_{h}$, and the approximate solution is found in form (Equation 3) where the functions ${p}_{i}$ belong to the space ${X}_{h}$. Therefore, the discretized formulation of a boundary value problem is: Find a function ${U}_{h}\in {X}_{h}$ which fulfills the equation. $a\left({U}_{h},{V}_{h}\right)=f\left({V}_{h}\right)\text{ for any function }{V}_{h}\in {X}_{h}$ Substitution of (Equation 3) into (Equation 10) results in a linear algebraic equation system from which unknown factors ${a}_{i}$ are found. In the classic Ritz-Galerkin method ${X}_{h}$ is a space of analytical functions defined on the whole domain $\Omega$, the factors ${a}_{i}$ have no physical meaning. In conventional Finite Element Method ${X}_{h}$ is a space of piecewise polynomials and factors ${a}_{i}$ are values of the function ${U}_{h}$ in the nodes of finite elements. In structural analysis they are displacements of the Many modifications of Ritz-Galerkin methods have been invented. They differ by the variational equations (Equation 9) and by the classes of basis-functions (Equation 3) used to approximate the solution. The same boundary value problem can have several equivalent formulations (Equation 9) which differ by the spaces $V$. External Approximations by Finite Elements As already mentioned, internal finite element approximations are built on functions that belong to a corresponding Sobolev space. These functions must meet certain continuity conditions on inter-element boundaries. For instance, when 2D or 3D theory of elasticity problems are under consideration, the functions need to be continuous between finite elements. For plate bending problems not only the functions, but their first derivatives must be continuous as well. The continuity conditions are quite restrictive. They can be met only for very simple shapes of finite elements using standard interpolation polynomials as basis-functions of finite elements. The polynomials are associated with element nodes. To provide inter-element compatibility the same interpolation functions are used to represent finite element shape. In case of curved boundaries mapping onto a canonical element is used to provide the compatibility. The geometry of finite elements and their approximation functions are tightly coupled. To improve the approximation quality of finite elements, researchers invented incompatible finite elements. In incompatible element the interpolation basis-functions of the elements of standard shape are enriched by some other polynomials. The additional functions create discontinuity across inter-element boundaries, but incompatible finite elements often provided much better accuracy than the compatible ones. This resulted in difficulties of mathematical proof of convergence and in inconsistencies of results. A comprehensive theory of external approximations by finite elements was developed in reference [Equation 3]. In the theory the word “finite element” was used to designate an arbitrary shaped sub-domain of the domain $\Omega$, so the definition of finite elements was not restricted anymore to canonic shapes or other shapes obtained from a canonic by mapping. The whole domain $\Omega$ could be considered as one finite element, and therefore, for assemblies a part of an assembly could be one “finite element” in FEM terminology. Another assumption was that approximation functions inside the finite element could be arbitrary - not necessarily polynomials. The only requirement was that the functions belong to the corresponding Sobolev space, so they need to be sufficiently smooth inside the element. The task was to find conditions under which the approximations built according to the assumptions above would be external approximations, i.e. they would converge to the exact solution of a boundary value problem from “outside” of a Sobolev space. The necessary and sufficient condition which provides the external approximations was found. The condition happens to be constructive – its formulation also implied the way of building finite elements that meet the condition. Convergence theorems and error estimates have been proven. It was shown that the necessary and sufficient condition for a finite element approximation to be external is: $〈\delta ,\gamma U〉=0$ Here <,> is the duality pairing in certain functional spaces defined on inter-element boundaries, $\delta$ and $\gamma$ are some operators, and $U$ are approximation functions defined inside the As one can see, condition (Equation 11) does not relate to a Boundary Value Problem (BVP) formulation, or to a solution method for the BVP (Galerkin, Ritz, Trefftz, etc.). It imposes constraints on basis-functions of finite elements which guarantee that the limit approximation function belongs to the corresponding Sobolev space, so it has the necessary smoothness properties. Therefore, even before the solution method is chosen (Galerkin, Ritz, etc.), one may construct finite element spaces that have important properties. These properties can be just “good to have”, as, for instance, when solving elasticity problems, it is not required to use functions that fulfill the equilibrium in volume, but it might be useful because the use of such functions increases accuracy and reduces the number of DOF. Other properties can be crucial, for instance, only divergence-free functions can provide unconditionally stable solutions for incompressible materials. Then condition (Equation 11) can be extended by continuity from the duality pairing into the inner product in other spaces of functions: $\left(g,\gamma U\right)=0$ Here $g$ are functions defined on inter-element boundaries, they are called boundary functions. Boundary functions are functions of surface parameters and they generate boundary DOF that are integrals of products of boundary functions onto finite element basis-functions over the finite element boundary: ${\int }_{\Gamma }{g}_{k}\gamma Ud\Gamma =0$ Here $\Gamma$ is the boundary of the finite element, ${g}_{k}$ are functions defined on the boundary of the finite element, and $U$ is a function to be approximated on the element (for example displacements in structural analysis). For comparison, the degrees of freedom in the FEM are the value of the function $U$ in the node $i$ of a finite element. The functions ${g}_{k}$ in expression (Equation 13) are basis-functions from finite dimensional space ${G}_{h}$ of functions defined on element boundaries. They can be arbitrary, the only requirement is that the spaces ${G}_{h}$ must be dense in the space of boundary functions, i.e. they must be able to converge in the space of boundary functions. The latter is easily fulfilled in case ${g}_{k}$ are polynomials or piecewise polynomials defined on the element boundaries. The functionals (Equation 13) are called boundary degrees of freedom. They do not have physical meaning and they represent approximation functions from space of finite elements compatible when the number of boundary DOF converges to infinity. The boundary DOF are responsible for meeting inter-element continuity conditions and essential boundary conditions. In adaptive solution the number of the boundary DOF is managed automatically to meet the convergence criteria. The boundary DOF (Equation 13) are not the only DOF produced when external approximations are built. Other DOF are called internal DOF because they are associated with the element volume. Internal DOF are defined automatically when the approximation of the solution within a finite element is being built. Finally the approximation of a function on the element looks as follows: ${U}_{h}={\sum }_{i}^{n}{a}_{i}\left(U\right){p}_{i}+{\sum }_{k}^{N}\left({\int }_{\Gamma }{g}_{k}\gamma Ud\Gamma \right){p}_{k}$ Here ${a}_{i}$ are internal DOF of the element (some factors), ${p}_{i}$ are basis-functions of the internal DOF, ${\int }_{\Gamma }{g}_{k}\gamma Ud\Gamma$ are the boundary DOF, ${p}_{k}$ are basis-functions of the boundary DOF. The basis-functions ${p}_{i}$ and ${p}_{k}$ constitute a finite-dimensional space $P$ of approximation functions of a finite element. It was proven that for convergence the space $P$ must be complete, for instance, in case of polynomial space it should contain all polynomials up to a certain degree assigned to an adaptive iteration. Basis-functions of a finite element are not pre-defined because the element has an arbitrary shape. They are built on-the-fly during a solution run. What is pre-defined at an adaptive pass is the whole space of approximation functions of the element. The algorithm of building basis-functions of an element at an adaptive pass works as follows: • A set of boundary functions ${g}_{k}$ is defined; • A complete space $P$ of approximation functions of the element is defined by choosing a complete set of generic basis-functions. In the case of polynomial spaces, a complete space of polynomials of a certain degree is specified. For instance, generic second-degree polynomials for 3D problems are: □ $\left\{1,x,y,z,{x}^{2},xy,{y}^{2},xz,{z}^{2},yz\right\}$ • Generic basis-functions are generated automatically on-the-fly for every sub-domain during the solution when the stiffness matrix of a sub-domain is evaluated; • The basis-functions ${p}_{i}$ and ${p}_{k}$ are found automatically by solving a certain system of linear algebraic equations. After basis-functions of an element have been found, the element stiffness matrix and load vector are evaluated the same way as it is done in conventional FEM by integrating energy over the element volume and loads over the element boundary. Geometry-Function Decoupling Geometry-function decoupling is the core feature of the SimSolid technology. As one can see from the above, the basis-functions of an arbitrary element are built from generic basis functions on-the-fly during the solution. Neither element geometry representation is used in building the generic functions, nor the functions dictate the shape of the element. The only requirement to the space $P$ of approximation functions of an element is that $P$ must be a subspace of a corresponding Sobolev space associated with the formulation of boundary value problems. Therefore, any combination of generic basis-functions is allowed provided they are linearly independent. The geometry-function decoupling proved to be the key feature which provides better performance, better accuracy, robustness, less computer resources, less modeling errors. The following substantial benefits can be realized when finding an accurate solution for a specific problem, or managing adaptive solutions: 1. It is possible to build special approximations that make approximate solutions of boundary value problems unconditionally stable. For instance, when parts made of incompressible materials are simulated, SimSolid uses divergence-free functions which exactly meet the incompressibility condition. Here is an example of some generic divergence-free 3D functions of 3rd degree, where $\left (u,v,w\right)$ are displacement components: □ $\begin{array}{l}u=-x{z}^{2},v=y{z}^{2},w=0\\ u=-3x{z}^{2},v=0,w={z}^{3}\\ u=-2xyz,v={y}^{2}z,w=0\\ u=-xyz,v=0,w=y{z}^{2}\\ u=-x{y}^{2},v=0,w={y}^{2}z\end{array}$ 2. Neighboring parts may have approximation functions of different classes. For instance, in case an assembly contains parts made of compressible and incompressible materials (rubber insertions, or cavities with liquid) the approximation functions for the incompressible material are built as special divergence-free functions. On neighboring parts with compressible material regular functions like standard polynomials are used. 3. It is always possible to use basis-functions that a-priori fulfill the governing equations of boundary value problems which provide better accuracy and reduce the number of DOF. For instance, thermo-elastic problems are solved using a complete polynomial solution of the corresponding governing equations: $\begin{array}{l}\left(\lambda +\mu \right)\frac{\partial \epsilon }{\partial x}+\mu \Delta \mu =\frac{\alpha Ε}{1-2v}\frac{\partial T}{\partial x}\\ \left(\lambda +\mu \right)\frac{\partial \ epsilon }{\partial y}+\mu \Delta \mu =\frac{\alpha Ε}{1-2v}\frac{\partial T}{\partial y}\\ \left(\lambda +\mu \right)\frac{\partial \epsilon }{\partial z}+\mu \Delta \mu =\frac{\alpha Ε}{1-2v}\ frac{\partial T}{\partial z}\end{array}$ Here $\left(u,v,w\right)$ are displacement components: $\begin{array}{c}\epsilon =\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}\\ \lambda =\frac{Ev}{\left(1+v\right)\left(1-2v\right)},\mu =\frac{E}{2\left (1+v\right)}\\ \Delta =\frac{{\partial }^{2}}{\partial {x}^{2}}+\frac{{\partial }^{2}}{\partial {y}^{2}}+\frac{{\partial }^{2}}{\partial {z}^{2}}\end{array}$ Further, $\alpha$ is the thermal expansion coefficient; $E$ is Young’s modulus, $v$ is Poisson’s ratio, and $T$ is the temperature field. The equation system (Equation 15) is non-homogeneous. For instance, when: $E=1,v=0.25,\alpha =1$ and temperature field is described by a monomial: the solution of the non-homogeneous problem for a=1, m=0, n=2, p=3 is: Here is an example of a polynomial solution of the homogeneous equations (14): When solving a thermo-elastic problem, polynomial approximations of temperature $T$ are imported from thermal analysis, functions of type (Equation 20) are generated for every element, and generic functions of type (Equation 21) are used to build basis-functions of elements. For heat transfer problems harmonic polynomials are used as basis-functions which precisely fulfill the corresponding equation of heat transfer. Here are some generic harmonic functions of degree $\begin{array}{l}f={x}^{3}-3x{z}^{2}\\ f={x}^{2}y-y{z}^{2}\\ f=x{y}^{2}-x{z}^{2}\\ f={y}^{3}-3y{z}^{2}\\ f=3x{z}^{2}-{z}^{3}\end{array}$ 4. The approximations are always built in the physical coordinate space without mapping onto a canonic shape. Therefore, the properties of generic basis-functions are preserved throughout the solution which eliminates a substantial source of approximation errors. 5. A complete set of basis-functions is always used to approximate solutions on a sub-domain. Completeness means that no functions are missing from a space of a certain degree. For instance, if the solution is approximated with harmonic polynomials of degree 5, then all harmonic generic polynomials of degree 5 are included into the approximation space of a sub-domain. This provides high accuracy, ease of building p-adaptive solutions globally and locally, and ease of implementation of new types of problem-specific basis-functions. 6. Geometry-functions decoupling allows effectively handle assemblies of parts with incomparable geometries in terms of size and shape (multi-scale assemblies). 7. Local effects like concentrated forces, cracks, stress concentration, etc., can be easily simulated by enriching the approximation space of sub-domains with special functions that possess corresponding characteristics associated with the feature.	{"url":"https://help.altair.com/ss/en_us/topics/simsolid/get_started/background_theoretical_r.htm","timestamp":"2024-11-04T14:43:25Z","content_type":"application/xhtml+xml","content_length":"163867","record_id":"<urn:uuid:32d41cad-b475-460f-8708-0492c0a97fcf>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00184.warc.gz"}
Summary of Inverse Kinematics Summary of Inverse Kinematics Let's recap some of the big things that we've learned in this lecture. In the previous lecture we discussed the forward kinematic function; that's the function K. Its argument is Q and that's the vector of joint angles, its vector of length N, where N is the number of joints that the robot has. The result of the forward kinematic function is the pose of the robots end effector, which I call here ksiN. Forward kinematics takes the joint angles and gives us the robot end effector pose. In this lecture we've discussed the inverse kinematic function that's shown here as K to the minus one. It operates on the robot end effector pose and tells us the joint angles required to achieve that pose. The inverse is not unique. For a particular robot end effector pose, there can be multiple sets of joint angles which will achieve that particular pose, for our little two link robot, there are two possible configurations shown here. For our six link Puma robot, which operates in three dimensional space, there are in fact six possible solutions. There are several approaches to solving the inverse kinematic problem for a particular robot. For a simple robot like the two link example, we use a geometric solution and that requires some intuition and some insight, it involves defining particular triangles, introducing new angles and applying a number of simple trigonometric relationships in order to solve for the joint angles we're interested in, in this case Q1 and Q2 as a function of the end effector pose XY and the robot's constant link lengths A1 and A2. We right down the forward kinematic equations, which here give us the end effector pose X and Y in terms of the joint angles and then with a large amount of manipulation and again with some insight and intuition, we can write down equations which give us the robot joint angles. The steps involved in the analytic solution for robot inverse kinematics involves the following steps; firstly we need a model of the robot, we need to know the position and orientation of the joints with respect to the links. We need to know the lengths of the links and so on. From that we extract a number of equations and we attempt to solve for the unknown joint variables. As I said before, it requires some human insight, although it can be automated using computer algebra techniques for particular robots or particular classes of robots. The great advantages of the analytic solution is it explicitly shows you the multiple configurations of the robot and lets you choose the joint angles that will give you the configuration that you want. For the Puma 560 robot it allows you to explicitly choose a left handed configuration or a right handed configuration or the elbow up or the elbow down. Another advantage is that the solutions are very compact and fast to execute. You can take the resulting equations and you can code them up in your favorite programming language and they will execute very, very quickly. However the analytic approach is increasingly difficult as the number of joints increases and takes quite a lot of work to do it for a six link robot. Fortunately a lot of smart people have looked at this problem over many, many decades and solutions for almost all robots have been derived and are published somewhere in the literature. An alternative to the analytic solution is the numerical solution and that's applicable when the analytic solution doesn't exist or the analytic solution is just too hard to figure out. It relies on the fact that we can always determine the forward kinematics of the robot, that’s relatively straightforward to do and we talked about that in the previous lecture. We know the pose that we want the robot end effector to achieve which is called ksi, but all we need to do is adjust the joint angles Q until ksi matches ksi and we can pose this mathematically as an optimization problem written very formally here. And we showed how we can use MATLAB and the robotics toolbox to determine the numerical solution for a two link robot and for a six link robot. There are some disadvantages with the numerical solution. You need to be careful about choosing the initial set of joint angles that you use for your optimizer. You can't guarantee that the optimizer will find the particular robot configuration that you're interested in. If you wanted a left handed configuration, you can't guarantee the optimizer will find that particular configuration, it depends on your choice of initial joint angles. It can also be computationally expensive. It's an iterative algorithm and the number of iterations depends on how good your initial set of joint angles is and that computational expense maybe inappropriate in a hard, real time system where you need to compute inverse kinematics within a small and finite amount of time. We talked about how to create smooth paths for the robot end effector moving from one pose to another and we introduced two approaches to doing this. The first is what's called a "Joint Interpolated Path." We take the initial pose and final pose and we use inverse kinematics to determine the initial and final sets of joint angles, then we smoothly interpolate between those two sets of joint angles and that's relatively straightforward to do. The disadvantage of this is we can't guarantee that the robot will follow a straight line path in three dimensional space. We also can't guarantee that the end effector orientation will be our desired value at every point along that path; however joint interpolated paths are relatively cheap to compute because we're only interpolating joint angle vectors at ever time step. An alternative is to compute a Cartesian interpolated path and what we do here is for every point along the trajectory we interpolate between the initial and final end effector pose. For each interpolated pose we then compute the inverse kinematics to find the joint angles that are appropriate at that particular time step. The advantage of this technique is that it allows for straight line paths in three dimensional space. It also ensures that the end effector has got the desired orientation at every point along the The disadvantage is that computationally it's much more expensive. We need to interpolate poses and that involves interpolating the translational component and the rotational component and that might in fact involve converting from rotation matrix to a quaternion interpolating the quaternion and then transforming that back to a rotation matrix, then we have to perform the inverse kinematics at every single time step. So much more computation involved, upside is that we can guarantee straight line paths in three dimensional space. There is no code in this lesson. We revisit the important points from this masterclass. Skill level Undergraduate mathematics This content assumes high school level mathematics and requires an understanding of undergraduate-level mathematics; for example, linear algebra - matrices, vectors, complex numbers, vector calculus and MATLAB programming. Rate this lesson You must to submit a review. 1. The explanations given by Professor Peter Corke are well detailed and illustrated with mathematical formulas, graphic examples and images. Thanks! Please Sign In to leave a comment.	{"url":"https://robotacademy.net.au/lesson/summary-of-inverse-kinematics/","timestamp":"2024-11-05T00:55:22Z","content_type":"text/html","content_length":"51888","record_id":"<urn:uuid:b4a97dc3-4b3c-4218-80a7-e75a0dfe5afd>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00614.warc.gz"}
Representation embeddings, interpretation functors and controlled wild algebras We establish a number of results which say, roughly, that interpretation functors preserve algebraic complexity. First, we show that representation embeddings between categories of modules of finitedimensional algebras induce embeddings of lattices of pp formulas, and hence are non-decreasing on Krull-Gabriel dimension and uniserial dimension. A consequence is that the category of modules of any wild finite-dimensional algebra has width ∞, and hence, if the algebra is countable, there is a superdecomposable pure-injective representation. It is conjectured that a stronger result is true: That a representation embedding from Mod-S to Mod-R admits an inverse interpretation functor from its image, and hence that, in this case, Mod-R interprets Mod-S. This would imply, for instance, that every wild category of modules interprets the (undecidable) word problem for (semi)groups. We show that the conjecture holds for finitely controlled representation embeddings. Finally, we prove that if R, S are finite-dimensional algebras over an algebraically closed field and I : Mod-R → Mod-S is an interpretation functor such that the smallest definable subcategory containing the image of I is the whole of Mod-S, then if R is tame, so is S and, similarly, if R is domestic, then S also is domestic.	{"url":"https://research-portal.uea.ac.uk/en/publications/representation-embeddings-interpretation-functors-and-controlled-","timestamp":"2024-11-11T13:38:48Z","content_type":"text/html","content_length":"46791","record_id":"<urn:uuid:cd28d920-5c39-461e-830d-0b6a5ebf4c0e>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00100.warc.gz"}
Developing Sensitivity Analysis Logic Using DAX in Power BI We’re getting specific today and really showcasing the analytical power of Power BI. Sensitivity analysis, or even running some ‘what ifs’ around this, allows you to almost predict what may happen in the future with your results. In this example, I want to see what will happen to my profitability if I am able to increase the gross margin on my sales. You may watch the full video of this tutorial at the bottom of this blog. While you might think this is quite niche, which it is, it’s the techniques to get to these results which I always want to get across with these examples. As soon as you learn how to implement this, you’ll likely identify three to five other ways you can use it to find valuable insights in your own environment. Sensitivity Analysis On Sales Margins I am going to show how you can run some sensitivity analysis on changes that you might want to make on your sales margins so we can calculate our Gross Revenue Margins based on what we have achieved from our Total Sales and Total Costs. But then, we might want to try and see what happens when we expand or contract, and see what that does to our Total Profit because if we expand Gross Revenue Margins, we actually expand Gross Profit Margins more. So how do we set this up? How can we do this analysis in Power BI? The first thing is to look over our data model. We have a pretty simple data model set upâ€”we have the Dates, Products, Regions, and Customers on top, which are all connected to the Sales table at the bottom. We are now going to create our first measure and put this into a measure table, then call it Key Measures. Weâ€™ll set up our first measure and call it Total Sales. This is a simple core measure where we will sum up the Total Revenue column and make this into a measure table. Getting The Total Cost Letâ€™s put our Products into the canvas and place our Total Sales next to it. The next thing we want to work out is our Gross Revenue Margin, so obviously we first need to find out what our Total Cost is. We will create a measure and call it Total Cost, then use the SUMX function as our expression. Now that we have the Total Cost in our canvas, we can work out our margin. Determining The Gross Revenue Margin Weâ€™ll create a new measure and call it Gross Revenue Margin using this formula: Once we put the margin in our table, we can get a percentage out of this and see what our Gross Revenue Margin is for every single product. Obviously, we will need to format these things as we go along, especially the percentages. Weâ€™ve also sorted it out from highest to lowest. What we want to do here is to shock these margins and increase them by 2%, 5%, 8%, and 10%. This way, we can see the flow-on effects to our profit margins. We just click on Enter Data to create a supporting table or a parameter table which will enable us to harvest the margin change. We will call this Margin adjustment, and then place the percentage numbers below that: The next step is to load in the supporting table and turn it into a slicer to make a list. We want to be able to select any of the percentages in the left table and then see the impact simultaneously on our Gross Revenue Margin table. The next thing to do is to create another measure, call it Margin Change, and incorporate this logic: Once we drag the Margin Change into our table, we can see that the number under the Margin Change column is reflecting our selection on the left table. Determining Scenario Gross Revenue Margin Letâ€™s create a new measure for the Scenario Gross Revenue Margin, which is the sum of the Gross Revenue Margin and the Margin Change. Once we drag this measure into our table, you will notice that the percentage numbers under the Scenario Gross Revenue Margin column change as we go through the selection in the Margin Adjustment table. At this point, this is where we can run some scenario analysis. Letâ€™s say weâ€™re running behind our budget and want to catch up, we can increase our margins by 5% and see what this will do to our profits. Obviously, there will be changes to the demand if you are going to do that, but this is just to show you the technique you can use in the real-world scenario you might have. Determining Scenario Sales & Scenario Profits We can also integrate our Scenario Sales based on the Scenario Gross Revenue Margin. Weâ€™re going to add another measure and call it Scenario Sales. This time weâ€™ll use our Total Sales, multiply it by the Scenario Gross Revenue Margin, and then add our Total Costs. Once we bring the Scenario Sales into our table, we can work out whatâ€™s our Scenario Profit. Weâ€™ll be creating a new measure and call it Scenario Profits, where weâ€™ll deduct our Total Sales from our Scenario Sales. The measure weâ€™ll create is the Total Profits because we want to see what the actual change in our total profits is going to be. Weâ€™ll be using this simple and straightforward formula to come up with our Total Profits, and then drag it to our table. Determining Change In Profits Now that we can see our Total Profits and Scenario Profits, we can figure out what is the change between these two columns. For our last measure, weâ€™ll call it Change in Profits. We will put out some pretty simple logic where the Scenario Profits is divided by the Total Profits. There are plenty of numbers in the table, so we can get rid of some of the intermediary calculations to keep things simple and easier to understand. For example, if we increase our Gross Revenue Margin by 10%, what changes will happen to our profits? As we can see in the Change in Profits columns, there is a pretty significant change. If we can increase the margins of our product sales by 10%, then we can increase our profits by 26.7% across every single product. This is a pretty cool analysis, right? But what would also be cool is if we were be able to see from a total perspective, meaning what the total is for all of these changes. We can put these in a visualization which enables us to do just that. Visualising The Data We will create a new table of our Margin Adjustment, edit the interaction, and click on Donâ€™t Summarize. Then we will grab our Change in Profits and put it in the new table as well. Once we turn this into a visualization, we run into a small problem. Because it is a number, we have to create a text value out of it. So we need to create a new column, name it Margin Adjustment, and format it into a percentage like this: Now we can see that this data type is a text value. Once we put this into the axis, we are now able to see all 5 values: -5%, 2% 5%, 8%, and 10%. Now we have a compelling visual which showcases what will be the changes in profits based on any margin adjustment that we make. We can look at it from an individual product perspective (left table) or look at it from a portfolio perspective (right table). This is a cool technique that is applicable to lots of different scenarios. Once you nail down this technique of bringing in parameters and incorporating them through measure branching in your logic, you can do some pretty awesome stuff in your sensitivity analysis, scenario analysis, etc. *** Related Links *** Sensitivity Analysis Techniques For Power BI Using DAX Calculating Percent Profit Margins Using DAX In Power BI How To Start Using â€˜What Ifâ€™ Parameters Inside Power BI There are ultimately so many things you could run sensitivity analysis technique on as well. And try to think about showcasing not just the immediate results, but think of the second or third order effects that changing a variable inside your calculated results will create. In this example, I’m not interested in just the straight change in profits from a change in gross margin – I’m actually looking for the percentage change in profits. I also show you in this video a little trick around getting your sensitivity visualization sorted out so that they fit into you reports well. Definitely check it out to learn more. You’ll see that it’s always going to be much larger than the scenario change you place into it. This is terrific insight that a CFO and employees at board level would want to understand about your results, or potential future results for that matter. If you want to see how I develop and build reports from scratch, you can check out my Dashboarding and Visualisation Intensive course. It’s here that I show you, end to end, how to develop compelling Power BI solutions, combining techniques just like this one into a all-encompassing analytical report. Got some thoughts or feedback on this sensitivity analysis technique? Let me know in the comments. Good luck implementing this one. This project aims to implement a full data analysis pipeline using Power BI with a focus on DAX formulas to derive actionable insights from the order data. A comprehensive guide to mastering the CALCULATETABLE function in DAX, focusing on practical implementation within Power BI for advanced data analysis. An interactive web-based application to explore and understand various data model examples across multiple industries and business functions. A comprehensive project aimed at enhancing oil well performance through advanced data analysis using Power BI’s DAX formulas. Learn how to leverage key DAX table functions to manipulate and analyze data efficiently in Power BI. Deep dive into the CALCULATETABLE function in DAX to elevate your data analysis skills. One of the main reasons why businesses all over the world have fallen in love with Power BI is because... A hands-on project focused on using the TREATAS function to manipulate and analyze data in DAX. A hands-on guide to implementing data analysis projects using DAX, focused on the MAXX function and its combinations with other essential DAX functions. Learn how to leverage the COUNTX function in DAX for in-depth data analysis. This guide provides step-by-step instructions and practical examples. A comprehensive guide to understanding and implementing the FILTER function in DAX, complete with examples and combinations with other functions. Learn how to implement and utilize DAX functions effectively, with a focus on the DATESINPERIOD function. Comprehensive Data Analysis using Power BI and DAX Exploring CALCULATETABLE Function in DAX for Data Analysis in Power BI Data Model Discovery Library Optimizing Oil Well Performance Using Power BI and DAX Mastering DAX Table Functions for Data Analysis Mastering DAX CALCULATETABLE for Advanced Data Analysis Debugging DAX: Tips and Tools for Troubleshooting Your Formulas Practical Application of TREATAS Function in DAX MAXX in Power BI – A Detailed Guide Leveraging the COUNTX Function In Power BI Using the FILTER Function in DAX – A Detailed Guide With Examples DATESINPERIOD Function in DAX – A Detailed Guide	{"url":"https://blog.enterprisedna.co/developing-sensitivity-analysis-logic-using-dax-in-power-bi/","timestamp":"2024-11-08T08:59:23Z","content_type":"text/html","content_length":"515942","record_id":"<urn:uuid:dadec4bb-0e20-4c00-94d7-c67318a1c87e>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00404.warc.gz"}
A characterization of optimal portfolios under the tail mean-variance criterion The tail mean-variance model was recently introduced for use in risk management and portfolio choice; it involves a criterion that focuses on the risk of rare but large losses, which is particularly important when losses have heavy-tailed distributions. If returns or losses follow a multivariate elliptical distribution, the use of risk measures that satisfy certain well-known properties is equivalent to risk management in the classical mean-variance framework. The tail mean-variance criterion does not satisfy these properties, however, and the precise optimal solution typically requires the use of numerical methods. We use a convex optimization method and a mean-variance characterization to find an explicit and easily implementable solution for the tail mean-variance model. When a risk-free asset is available, the optimal portfolio is altered in a way that differs from the classical mean-variance setting. A complete solution to the optimal portfolio in the presence of a risk-free asset is also provided. • Optimal portfolio selection • Quartic equation • Tail conditional expectation • Tail variance ASJC Scopus subject areas • Statistics and Probability • Economics and Econometrics • Statistics, Probability and Uncertainty Dive into the research topics of 'A characterization of optimal portfolios under the tail mean-variance criterion'. Together they form a unique fingerprint.	{"url":"https://cris.haifa.ac.il/en/publications/a-characterization-of-optimal-portfolios-under-the-tail-mean-vari","timestamp":"2024-11-08T12:37:24Z","content_type":"text/html","content_length":"54166","record_id":"<urn:uuid:7b3d2861-4789-4ba4-ad2d-531cbde766dd>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00147.warc.gz"}
Black Hole Relates With Thermodynamics- How? Quantum mechanics turns black holes from cold, eternal objects into hot shrinking thermodynamics. Physicists wondered: Is there a microscopic origin for black hole entropy? If the Four Laws of Black Hole Physics looked familiar, it’s because they sound just like the Four Laws of Thermodynamics, which are: Watch on Amazon- New Black Hole Image Revealed by Scientists The Four Laws of Thermodynamics 0 The temperature T of a system in thermal equilibrium has the same value everywhere in the system. 1 The change in energy of a system is proportional to the temperature times the change in entropy.dE = T dS 2 The total entropy of a system can only increase, never decrease. 3 It is impossible to lower the temperature T of a system to zero through any physical process. There seems to be a direct correspondence between the properties of a classical thermodynamic system, and the properties of a black hole, shown in the table below Thermodynamic system Black hole temperature surface gravity at horizon energy black hole mass entropy area of horizon A black hole spacetime seems to behave like a thermodynamic system. How could this be true? This is spacetime geometry, after all, not a cylinder of gas or a pot of liquid. The importance of this apparent thermodynamic behavior of black holes was made undeniable when black hole radiation was discovered by Hawking. Black hole radiation, known as Hawking radiation, comes about because relativistic quantum field theory is invariant under Lorentz transformations, but not under general coordinate transformations. In flat spacetime, two observers moving at a constant velocity relative to one another will agree on what constitutes a vacuum state, but if one observer is accelerating relative to the other, then the vacuum states defined by the two observers will differ. This idea, when extended to the spacetime of a black hole, leads to the conclusion that to an observer who stays at a fixed distance from a black hole event horizon, the black hole appears to radiate particles with a thermal spectrum with temperature (in units with G[N]=c=1) T=1/8pMk[B], where k[B] is Boltzmann’s constant and M is the black hole mass. Since plane waves and Fourier transforms are at the heart of relativistic quantum field theory, this effect can be illustrated using a classical plane wave, without even appealing to quantum operators. Consider a simple monochromatic plane wave in two spacetime dimensions with the form An observer travelling in the x-direction with constant velocity b perceives this plane wave as being monochromatic but the frequency w is Doppler-shifted: An observer travelling in the x-direction with constant acceleration does not perceive this plane wave as being monochromatic. The accelerated observer sees a complicated waveform: This wave as perceived by the accelerated observer is a superposition of monochromatic waves of frequency n with a distribution function f(v) that, as shown below appears to be a thermal distribution with temperature T=a/2pk[B]. The result from this simple example matches Hawking’s black hole result if the acceleration is related to the black hole mass by a=1/ 4M. And indeed, the acceleration at the event horizon of a black hole of mass M does satisfy a=1/4M. Why does this work so well? Because an observer held at a fixed distance from the event horizon of a black hole sees a coordinate system that is almost identical to that of an observer undergoing constant acceleration in flat spacetime. But don’t be misled by this to think that the full black hole radiation calculation is as simple. We’ve neglected to mention the details because they are very complicated and involve the global causal structure of a black hole spacetime. Conservation of energy still applies to this system as a whole, so if an observer at a fixed distance sees a hot bath of particles being radiated by the black hole, then the black hole must be losing mass by an appropriate amount. Hence a black hole can decrease in area, through Hawking radiation, through quantum processes. But if the area is like entropy, and the area will decrease, doesn’t that mean that, in violation of the Second Law of Thermodynamics, the entropy of a black hole can then decrease? No — because the radiated particles also carry entropy, and the total entropy of the black hole and radiation always increases. Where does the entropy come from? One of the great achievements of quantum mechanics in the 20th century was explaining the microscopic basis of the thermodynamic behavior of macroscopic systems that were understood in the 19th century. The quantum revolution began when Planck tried to explain the thermal behavior of light, and came up with the concept of a quantum of light. The thermodynamic properties of gases are now well understood in terms of the quantized energy states of their constituent atoms and molecules. So what is the microscopic physics that underlies the thermodynamic properties of black holes? String theory suggests an answer that we will explain in the next section. Recommended Video: New Black Hole Image Revealed by Scientists Buy on Amazon- https://amzn.to/3bOvvYD Leave a Comment Cancel Reply Facts of Universe Cycle/Oscillating Model Leave a Comment / Mysteries & Hypothesis, Matter & Energy, Physics & Cosmos, Space & Time / By Deep Prakash / July 8, 2020 / astronomy, astrophysics, balance, cosmologist, cosmology, cosmos, cycle of the universe, cycle of universe, cycle universe, cycles of the universe, cyclic model, cyclic model of the universe, cyclic model of universe, cyclic model theory, cyclic theory, cyclic theory of the universe, cyclic universe, cyclic universe theory, cyclical time theory, cyclical universe theory, cycling universe, einstein, equation, explaied, galaxy, is the universe cyclical, oscilating theory, oscillating model, oscillating model theory, oscillating theory, oscillating universe, phenomenon, physics, physics equation, reincarnation, repeating, repeating universe theory, research, science, scientist, solar system, space, surprise, symbolize, symbols, the cyclic universe theory, theoretical, understand, understand universe, universe, universe cycle, universe cycle theory, universe cycles, universe examples, what is cyclic universe theory Cyclic Model of Universe Leave a Comment / Mysteries & Hypothesis, Matter & Energy, Physics & Cosmos, Space & Time, Uncommon & Remarkable / By Deep Prakash / July 10, 2020 / astronomy, astrophysics, big bang, black, black hole, cosmology, cosmos, cycle of the universe, cycle of universe, cycle universe, cycles of the universe, cyclic model, cyclic model of the universe, cyclic model of universe, cyclic model theory, cyclic theory, cyclic theory of the universe, cyclic universe, cyclic universe theory, cyclical time theory, cyclical universe theory, cycling universe, einstein, energy, equation, expanding universe , explaied, galaxy, gravitational, hole, hypothesis, is the universe cyclical, mass, matter, oscilating theory, oscillating model, oscillating model theory, oscillating theory, oscillating universe, physics, quasar, repeating universe theory, research, science, scientist, space, the cyclic universe theory, theoretical, theory, universe, universe cycle, universe cycle theory, universe cycles, universe examples, what is cyclic universe theory, whirling disk	{"url":"https://cosmos.theinsightanalysis.com/how-black-hole-relates-with-thermodynamics/","timestamp":"2024-11-03T16:14:36Z","content_type":"text/html","content_length":"173559","record_id":"<urn:uuid:323d225e-0731-48df-8b53-6dafe92cfb2b>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00037.warc.gz"}
Number and Operations (NCTM) Understand numbers, ways of representing numbers, relationships among numbers, and number systems. Understand and use ratios and proportions to represent quantitative relationships. Compute fluently and make reasonable estimates. Develop, analyze, and explain methods for solving problems involving proportions, such as scaling and finding equivalent ratios. Geometry (NCTM) Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships. Understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects. Create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship. Apply transformations and use symmetry to analyze mathematical situations. Describe sizes, positions, and orientations of shapes under informal transformations such as flips, turns, slides, and scaling. Measurement (NCTM) Apply appropriate techniques, tools, and formulas to determine measurements. Solve problems involving scale factors, using ratio and proportion. Grade 8 Curriculum Focal Points (NCTM) Geometry and Measurement: Analyzing two- and three-dimensional space and figures by using distance and angle Students use fundamental facts about distance and angles to describe and analyze figures and situations in two- and three-dimensional space and to solve problems, including those with multiple steps. They prove that particular configurations of lines give rise to similar triangles because of the congruent angles created when a transversal cuts parallel lines. Students apply this reasoning about similar triangles to solve a variety of problems, including those that ask them to find heights and distances. They use facts about the angles that are created when a transversal cuts parallel lines to explain why the sum of the measures of the angles in a triangle is 180 degrees, and they apply this fact about triangles to find unknown measures of angles. Students explain why the Pythagorean Theorem is valid by using a variety of methods - for example, by decomposing a square in two different ways. They apply the Pythagorean Theorem to find distances between points in the Cartesian coordinate plane to measure lengths and analyze polygons and polyhedra.	{"url":"https://newpathworksheets.com/math/grade-8/similarity-and-scale?dictionary=numerators&did=106","timestamp":"2024-11-04T18:48:46Z","content_type":"text/html","content_length":"45247","record_id":"<urn:uuid:1a6f93ae-aa45-4029-8221-52a45b4ba146>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00653.warc.gz"}
Topology - Wikiwand Topology (from the Greek words τόπος, 'place, location', and λόγος, 'study') is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A three-dimensional model of a figure-eight knot. The figure-eight knot is a prime knot and has an Alexander–Briggs notation of 4[1]. A topological space is a set endowed with a structure, called a topology, which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity. Euclidean spaces, and, more generally, metric spaces are examples of topological spaces, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopies . A property that is invariant under such deformations is a topological property. The following are basic examples of topological properties: the dimension, which allows distinguishing between a line and a surface; compactness, which allows distinguishing between a line and a circle; connectedness, which allows distinguishing a circle from two non-intersecting circles. The ideas underlying topology go back to Gottfried Wilhelm Leibniz, who in the 17th century envisioned the geometria situs and analysis situs. Leonhard Euler's Seven Bridges of Königsberg problem and polyhedron formula are arguably the field's first theorems. The term topology was introduced by Johann Benedict Listing in the 19th century, although, it was not until the first decades of the 20th century that the idea of a topological space was developed. Möbius strips, which have only one surface and one edge, are a kind of object studied in topology. The motivating insight behind topology is that some geometric problems depend not on the exact shape of the objects involved, but rather on the way they are put together. For example, the square and the circle have many properties in common: they are both one dimensional objects (from a topological point of view) and both separate the plane into two parts, the part inside and the part outside. In one of the first papers in topology, Leonhard Euler demonstrated that it was impossible to find a route through the town of Königsberg (now Kaliningrad) that would cross each of its seven bridges exactly once. This result did not depend on the lengths of the bridges or on their distance from one another, but only on connectivity properties: which bridges connect to which islands or riverbanks. This Seven Bridges of Königsberg problem led to the branch of mathematics known as graph theory. Similarly, the hairy ball theorem of algebraic topology says that "one cannot comb the hair flat on a hairy ball without creating a cowlick." This fact is immediately convincing to most people, even though they might not recognize the more formal statement of the theorem, that there is no nonvanishing continuous tangent vector field on the sphere. As with the Bridges of Königsberg, the result does not depend on the shape of the sphere; it applies to any kind of smooth blob, as long as it has no holes. To deal with these problems that do not rely on the exact shape of the objects, one must be clear about just what properties these problems do rely on. From this need arises the notion of homeomorphism. The impossibility of crossing each bridge just once applies to any arrangement of bridges homeomorphic to those in Königsberg, and the hairy ball theorem applies to any space homeomorphic to a sphere. Intuitively, two spaces are homeomorphic if one can be deformed into the other without cutting or gluing. A traditional joke is that a topologist cannot distinguish a coffee mug from a doughnut, since a sufficiently pliable doughnut could be reshaped to a coffee cup by creating a dimple and progressively enlarging it, while shrinking the hole into a handle.^[1] Homeomorphism can be considered the most basic topological equivalence. Another is homotopy equivalence. This is harder to describe without getting technical, but the essential notion is that two objects are homotopy equivalent if they both result from "squishing" some larger object. The Seven Bridges of Königsberg was a problem solved by Euler. Topology, as a well-defined mathematical discipline, originates in the early part of the twentieth century, but some isolated results can be traced back several centuries.^[2] Among these are certain questions in geometry investigated by Leonhard Euler. His 1736 paper on the Seven Bridges of Königsberg is regarded as one of the first practical applications of topology.^[2] On 14 November 1750, Euler wrote to a friend that he had realized the importance of the edges of a polyhedron. This led to his polyhedron formula, V − E + F = 2 (where V, E, and F respectively indicate the number of vertices, edges, and faces of the polyhedron). Some authorities regard this analysis as the first theorem, signaling the birth of topology.^[3] Further contributions were made by Augustin-Louis Cauchy, Ludwig Schläfli, Johann Benedict Listing, Bernhard Riemann and Enrico Betti.^[4] Listing introduced the term "Topologie" in Vorstudien zur Topologie, written in his native German, in 1847, having used the word for ten years in correspondence before its first appearance in print.^[5] The English form "topology" was used in 1883 in Listing's obituary in the journal Nature to distinguish "qualitative geometry from the ordinary geometry in which quantitative relations chiefly are treated".^[6] Their work was corrected, consolidated and greatly extended by Henri Poincaré. In 1895, he published his ground-breaking paper on Analysis Situs, which introduced the concepts now known as homotopy and homology, which are now considered part of algebraic topology.^[4] Topological characteristics of closed 2-manifolds^[4] Manifold Euler num Orientability Betti numbers Torsion coefficient (1-dim) b[0] b[1] b[2] Sphere 2 Orientable 1 0 1 none Torus 0 Orientable 1 2 1 none 2-holed torus −2 Orientable 1 4 1 none g-holed torus (genus g) 2 − 2g Orientable 1 2g 1 none Projective plane 1 Non-orientable 1 0 0 2 Klein bottle 0 Non-orientable 1 1 0 2 Sphere with c cross-caps (c > 0) 2 − c Non-orientable 1 c − 1 0 2 2-Manifold with g holes 2 − (2g + c) Non-orientable 1 (2g + c) − 1 0 2 and c cross-caps (c > 0) Unifying the work on function spaces of Georg Cantor, Vito Volterra, Cesare Arzelà, Jacques Hadamard, Giulio Ascoli and others, Maurice Fréchet introduced the metric space in 1906.^[7] A metric space is now considered a special case of a general topological space, with any given topological space potentially giving rise to many distinct metric spaces. In 1914, Felix Hausdorff coined the term "topological space" and gave the definition for what is now called a Hausdorff space.^[8] Currently, a topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Modern topology depends strongly on the ideas of set theory, developed by Georg Cantor in the later part of the 19th century. In addition to establishing the basic ideas of set theory, Cantor considered point sets in Euclidean space as part of his study of Fourier series. For further developments, see point-set topology and algebraic topology. The 2022 Abel Prize was awarded to Dennis Sullivan "for his groundbreaking contributions to topology in its broadest sense, and in particular its algebraic, geometric and dynamical aspects".^[10] Topologies on sets The term topology also refers to a specific mathematical idea central to the area of mathematics called topology. Informally, a topology describes how elements of a set relate spatially to each other. The same set can have different topologies. For instance, the real line, the complex plane, and the Cantor set can be thought of as the same set with different topologies. Formally, let X be a set and let τ be a family of subsets of X. Then τ is called a topology on X if: 1. Both the empty set and X are elements of τ. 2. Any union of elements of τ is an element of τ. 3. Any intersection of finitely many elements of τ is an element of τ. If τ is a topology on X, then the pair (X, τ) is called a topological space. The notation X[τ] may be used to denote a set X endowed with the particular topology τ. By definition, every topology is a The members of τ are called open sets in X. A subset of X is said to be closed if its complement is in τ (that is, its complement is open). A subset of X may be open, closed, both (a clopen set), or neither. The empty set and X itself are always both closed and open. An open subset of X which contains a point x is called an open neighborhood of x. Continuous functions and homeomorphisms A continuous transformation can turn a coffee mug into a donut. Ceramic model by Keenan Crane and Henry Segerman. A function or map from one topological space to another is called continuous if the inverse image of any open set is open. If the function maps the real numbers to the real numbers (both spaces with the standard topology), then this definition of continuous is equivalent to the definition of continuous in calculus. If a continuous function is one-to-one and onto, and if the inverse of the function is also continuous, then the function is called a homeomorphism and the domain of the function is said to be homeomorphic to the range. Another way of saying this is that the function has a natural extension to the topology. If two spaces are homeomorphic, they have identical topological properties, and are considered topologically the same. The cube and the sphere are homeomorphic, as are the coffee cup and the doughnut. However, the sphere is not homeomorphic to the doughnut. While topological spaces can be extremely varied and exotic, many areas of topology focus on the more familiar class of spaces known as manifolds. A manifold is a topological space that resembles Euclidean space near each point. More precisely, each point of an n-dimensional manifold has a neighborhood that is homeomorphic to the Euclidean space of dimension n. Lines and circles, but not figure eights, are one-dimensional manifolds. Two-dimensional manifolds are also called surfaces, although not all surfaces are manifolds. Examples include the plane, the sphere, and the torus, which can all be realized without self-intersection in three dimensions, and the Klein bottle and real projective plane, which cannot (that is, all their realizations are surfaces that are not manifolds). General topology General topology is the branch of topology dealing with the basic set-theoretic definitions and constructions used in topology.^[11]^[12] It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology. The basic object of study is topological spaces, which are sets equipped with a topology, that is, a family of subsets, called open sets, which is closed under finite intersections and (finite or infinite) unions. The fundamental concepts of topology, such as continuity, compactness, and connectedness, can be defined in terms of open sets. Intuitively, continuous functions take nearby points to nearby points. Compact sets are those that can be covered by finitely many sets of arbitrarily small size. Connected sets are sets that cannot be divided into two pieces that are far apart. The words nearby, arbitrarily small, and far apart can all be made precise by using open sets. Several topologies can be defined on a given space. Changing a topology consists of changing the collection of open sets. This changes which functions are continuous and which subsets are compact or connected. Metric spaces are an important class of topological spaces where the distance between any two points is defined by a function called a metric. In a metric space, an open set is a union of open disks, where an open disk of radius r centered at x is the set of all points whose distance to x is less than r. Many common spaces are topological spaces whose topology can be defined by a metric. This is the case of the real line, the complex plane, real and complex vector spaces and Euclidean spaces. Having a metric simplifies many proofs. Algebraic topology Algebraic topology is a branch of mathematics that uses tools from algebra to study topological spaces.^[13] The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence. The most important of these invariants are homotopy groups, homology, and cohomology. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. Differential topology Differential topology is the field dealing with differentiable functions on differentiable manifolds.^[14] It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds. More specifically, differential topology considers the properties and structures that require only a smooth structure on a manifold to be defined. Smooth manifolds are "softer" than manifolds with extra geometric structures, which can act as obstructions to certain types of equivalences and deformations that exist in differential topology. For instance, volume and Riemannian curvature are invariants that can distinguish different geometric structures on the same smooth manifold – that is, one can smoothly "flatten out" certain manifolds, but it might require distorting the space and affecting the curvature or volume. Geometric topology Geometric topology is a branch of topology that primarily focuses on low-dimensional manifolds (that is, spaces of dimensions 2, 3, and 4) and their interaction with geometry, but it also includes some higher-dimensional topology.^[15] Some examples of topics in geometric topology are orientability, handle decompositions, local flatness, crumpling and the planar and higher-dimensional Schönflies theorem. In high-dimensional topology, characteristic classes are a basic invariant, and surgery theory is a key theory. Low-dimensional topology is strongly geometric, as reflected in the uniformization theorem in 2 dimensions – every surface admits a constant curvature metric; geometrically, it has one of 3 possible geometries: positive curvature/spherical, zero curvature/flat, and negative curvature/hyperbolic – and the geometrization conjecture (now theorem) in 3 dimensions – every 3-manifold can be cut into pieces, each of which has one of eight possible geometries. 2-dimensional topology can be studied as complex geometry in one variable (Riemann surfaces are complex curves) – by the uniformization theorem every conformal class of metrics is equivalent to a unique complex one, and 4-dimensional topology can be studied from the point of view of complex geometry in two variables (complex surfaces), though not every 4-manifold admits a complex structure. Occasionally, one needs to use the tools of topology but a "set of points" is not available. In pointless topology one considers instead the lattice of open sets as the basic notion of the theory,^ [16] while Grothendieck topologies are structures defined on arbitrary categories that allow the definition of sheaves on those categories, and with that the definition of general cohomology Topology has been used to study various biological systems including molecules and nanostructure (e.g., membraneous objects). In particular, circuit topology and knot theory have been extensively applied to classify and compare the topology of folded proteins and nucleic acids. Circuit topology classifies folded molecular chains based on the pairwise arrangement of their intra-chain contacts and chain crossings. Knot theory, a branch of topology, is used in biology to study the effects of certain enzymes on DNA. These enzymes cut, twist, and reconnect the DNA, causing knotting with observable effects such as slower electrophoresis.^[18] Computer science Topological data analysis uses techniques from algebraic topology to determine the large scale structure of a set (for instance, determining if a cloud of points is spherical or toroidal). The main method used by topological data analysis is to: 1. Replace a set of data points with a family of simplicial complexes, indexed by a proximity parameter. 2. Analyse these topological complexes via algebraic topology – specifically, via the theory of persistent homology.^[19] 3. Encode the persistent homology of a data set in the form of a parameterized version of a Betti number, which is called a barcode.^[19] Several branches of programming language semantics, such as domain theory, are formalized using topology. In this context, Steve Vickers, building on work by Samson Abramsky and Michael B. Smyth, characterizes topological spaces as Boolean or Heyting algebras over open sets, which are characterized as semidecidable (equivalently, finitely observable) properties.^[20] Topology is relevant to physics in areas such as condensed matter physics,^[21] quantum field theory and physical cosmology. The topological dependence of mechanical properties in solids is of interest in disciplines of mechanical engineering and materials science. Electrical and mechanical properties depend on the arrangement and network structures of molecules and elementary units in materials.^[22] The compressive strength of crumpled topologies is studied in attempts to understand the high strength to weight of such structures that are mostly empty space.^[23] Topology is of further significance in Contact mechanics where the dependence of stiffness and friction on the dimensionality of surface structures is the subject of interest with applications in multi-body physics. A topological quantum field theory (or topological field theory or TQFT) is a quantum field theory that computes topological invariants. Although TQFTs were invented by physicists, they are also of mathematical interest, being related to, among other things, knot theory, the theory of four-manifolds in algebraic topology, and to the theory of moduli spaces in algebraic geometry. Donaldson, Jones, Witten, and Kontsevich have all won Fields Medals for work related to topological field theory. The topological classification of Calabi–Yau manifolds has important implications in string theory, as different manifolds can sustain different kinds of strings.^[24] In cosmology, topology can be used to describe the overall shape of the universe.^[25] This area of research is commonly known as spacetime topology. In condensed matter a relevant application to topological physics comes from the possibility to obtain one-way current, which is a current protected from backscattering. It was first discovered in electronics with the famous quantum Hall effect, and then generalized in other areas of physics, for instance in photonics^[26] by F.D.M Haldane. Fiber art In order to create a continuous join of pieces in a modular construction, it is necessary to create an unbroken path in an order which surrounds each piece and traverses each edge only once. This process is an application of the Eulerian path.^[32] Major books • Munkres, James R. (2000). Topology (2nd ed.). Upper Saddle River, NJ: Prentice Hall. ISBN 978-0-13-181629-9 • Willard, Stephen (2016). General topology. Dover books on mathematics. Mineola, N.Y: Dover publications. ISBN 978-0-486-43479-7 • Armstrong, M. A. (1983). Basic topology. Undergraduate texts in mathematics. New York: Springer-Verlag. ISBN 978-0-387-90839-7	{"url":"https://www.wikiwand.com/en/articles/Topology","timestamp":"2024-11-01T23:55:18Z","content_type":"text/html","content_length":"435492","record_id":"<urn:uuid:cf2d8baa-3e52-4361-90bf-1a68cfc13eac>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00732.warc.gz"}
Why Division By Zero is Forbidden - Complete, Concrete, Concise Why Division By Zero is Forbidden In general, mathematics disallows division by zero because the resulting answer is indeterminate. This is because division is not a fundamental operation in mathematics – it is defined as the inverse of multiplication. If we say: 12 ÷ 4 = 3 we mean precisely the same as if we had said: 12 = 3 × 4 In other words, when we divide a number (called a dividend) a by some number b, we get a quotient q. And this quotient q when multiplied by b gives us the number a. a ÷ b = q a = q × b Suppose b is zero, then we have: a ÷ 0 = q a = q × 0 This raises two problems: (1) Any number multiplied by zero equals zero. Therefore, a = q × 0 can only be true if a is equal to zero. [If a is not equal to zero, let’s say a = 3. Then what number q times zero will give us a?] (2) Even if a = 0, then any value of q would satisfy the equation. Consequently the value of q is indeterminate. You can read a lot more about zero, its history, and common fallacies over here (this is an external link. While it was deemed safe at the time of writing, I accept no responsibility for external This article proposes a system in which division by zero is permitted. You need to pay to gain access to the article. (This is an external link. While it was deemed safe at the time of writing, I accept no responsibility for external content).	{"url":"https://complete-concrete-concise.com/mathematics/why-division-by-zero-is-forbidden/","timestamp":"2024-11-13T08:21:02Z","content_type":"text/html","content_length":"50318","record_id":"<urn:uuid:46bfe331-ee14-43ff-8c3d-bea8f0874e45>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00874.warc.gz"}
Feature Column from the AMS Matroids: The Value of Abstraction 3. Multiple births There are related concepts embedded inside graphs and vector spaces. Let us start with vector spaces. Among the major ideas that linear algebra (the branch of mathematics concerned with the structure of the solutions of linear systems of equations and related phenomena) brings to studying the geometry of points and vectors are the notions of independence, dependence, base, and spanning set of a space. Intuitively, a collection of vectors (or points) I is independent if one can not obtain any vector (point) in the set I as a linear combination of the other points in I. A set of independent points, to which one can not add any additional vector (point) and still have an independent set is called a basis set of points. A basis (for a finite dimensional vector space) is a maximal independent set of vectors. Intuitively, a collection of points S spans a space if any point in the space can be obtained as a special kind of algebraic sum of points in S. A set which spans a space may have the property that one can throw away elements from the set and still have a set that spans. One might seek minimal spanning sets. It turns out that such sets consist of independent elements. Thus a basis for a space is a set that spans the space and consists of independent elements. Vectors can be independent but if there are not enough of them, they will not span the space. Vectors can span a space but if there are too many of them, they will not be independent. How can one capture these ideas in a rule or axiomatic structure? Suppose I is a set, which we will think of as the set of independent objects. Here are the rules we want the set I to obey: (Independence 1) The set I is not empty. (Independence 2) Every subset of a member of I is also a member of I. Independence 2 involves the concept of being hereditary. Subsets of things having the property should also have the property. An analogy with more mundane things than sets would be that if one takes a piece of food and subdivides it into parts, one has something one can still eat (e.g. food). (Independence 3) If X and Y are in I where X has one more element than Y, one can find an element x in X which is not in Y such that Y together with x is in I. Note that independent sets in a vector space have this property. We have here some rules that abstract the notion of independence. A matroid is now defined to consist of a finite set E and subsets of E which satisfy Independence 1 through Independence 3. Consider the matrix A = Number the columns of the matrix with the numbers 1, 2, 3, and 4, and let E be this set of column label numbers. We consider the following subsets of E: You can verify that this collection of sets can serve as the independent setI for a matroid on the set E, by verifying that the axioms (rules) we have stated apply to this collection of sets. The way the collection of sets above can be obtained is to check which sets of columns of the matrix A are linearly independent. This type of matroid is often called the vector matroid of the matrix A. Whereas a single non-zero column is linearly independent, in this example any pair of the columns numbered 1, 2, or 3 (the non-zero columns) is also independent. Note that the first three columns are not linearly independent. For the record, one theoretical (but not computationally attractive) way of telling if some set of k columns (none of which consists of all zeros) from a k-rowed matrix is linearly independent is to compute the determinant of the columns involved. If this determinant is non-zero, the columns are independent; otherwise, they are dependent. Also, notice that strictly speaking when one refers to linear independence, one has to specify a field of elements from which the scalars are taken when constructing the linear combinations. Two familiar examples of fields are the rational numbers and the real numbers. In our case we can, in essence, use any field. In particular, we can think of the columns of matrix A to have entries from the field of two elements. When a matroid arises as the vector matroid for a matrix with respect to the field of 2 elements, it is known as a binary matroid; W.T. Tutte (see next section) determined in 1958 exactly which matroids arise in this way. Now let us shift gears and look at some graph theory, which seems not on its surface to have much to do with vector spaces or independence. The diagram below is a graph (or pseudograph, depending on different schools of definition). For the purposes of defining matroids from graphs we allow a vertex to be joined to itself (loops) and/or allow vertices which have multiple edges between them. For simplicity, I will consider only graphs which have the property that one can walk along the edges of the graph between any pair of vertices. Such graphs are called connected and are said to have a single component. The example shown here has the vertices labeled with letters and the edges labeled with numbers. This graph has 6 vertices and 10 edges. If one starts at a vertex in our graph, then moves along previously unused edges and returns to the original vertex, one travels a circuit. In the graph above examples of circuits are: u, v, s, u; and v (via edge 1), w, v (via edge 2). Note that we will consider two circuits the same if they are listed in reverse cyclic order and that there are many ways of writing down the same circuit that look a bit different. Thus, s, w, x, t, s and w, x, t, s, w and w, s, t, x, w are all different notations for the same circuit. Graphs have many applications: designing routes for street sweepers, showing relationships between workers and jobs they are qualified for, or coding the moves in a game. However, interest in graphs from the perspective of those who study matroids is not these applications per se, but certain underlying structural features that all graphs share. As a simple example of a structural theorem about graphs, consider the question: How few edges can a connected graph with n vertices have? Some experimentation should convince you that the answer is (n-1) edges. The concept of a matroid constructed from a graph (known as graphic matroids) starts with the set of edges of the graph together with the subsets of edges which form the circuits of the graphs (including circuits which are loops and formed from multiple edges). Now that you are familiar with the concept of a circuit we can use a slightly simpler notation for them, which works without ambiguity. What we do is list the edges that make up the circuit. Loops show up with a single edge, and multiple edges define circuits with 2 edges. The edges in each circuit will be set off by parentheses rather than set brackets. Here is a list of the circuits for the graph in Figure 1: (10), (3,4,5), (1,2), (2,5,6), (1,5,6), (3,4,6,2), (6,9,8,7), (3,4,9,8,7,2), (2,5,9,8,7), (1,5,9,8,7), and (3,4,6,1). There are 11 circuits. We can try to abstract the properties of circuits in a graph in the same way that we abstracted the properties of being independent in a vector space. Suppose we have a collection of subsets C on a set E which obeys the rules: (Circuit 1) The null set is in C. (Circuit 2) If C[1] and C[2] are members of C where C[1 ] [2] then C[1] = C[2. ] A circuit in a graph can not be a proper subset of another circuit, so we are turning this property of circuits into a rule. (Circuit 3) Suppose that C[1] and C[2] are distinct subsets of C, and e is a member of C[1] [2] then there is a member of C[3] in C such that: C[3] [1] [2]) - e Circuit 3 captures the property of circuits which says that when two circuits have an edge e in common, one can create a new circuit from the edges of the two circuits which does not use edge e. Another approach to matroids is to say that given the set of edges E of a graph G, the set of circuits of G is the circuit set C of a matroid. This matroid is known as the cycle matroid of the graph. (Although one often sees the words circuit and cycle used interchangeably for the circuit concept in a graph, an advantage of using the word cycle for what I have called circuits in a graph is that one has the word circuit to use for the elements of the set C of a matroid.) Now it may not be easy to see the connections between the abstraction of independence for a vector space and that of circuits for a graph, but they can be seen by using some definitions. Since in the independent set situation the special subsets of E that are in I are called independent, it seems natural to call a subset of E that is not in I to be dependent. Those sets in I which are maximal, that is, if one adds any element of E to them they are no longer in I, are called bases. One can prove that in any matroid the bases all have the same number of elements, just as is true for the bases of a vector space. If maximal independent sets are of interest, what about the minimal dependent sets? There are the sets with the property that if any element is dropped from them, one gets an independent set. We will define these sets, which are minimally dependent, to be circuits! Note that once the independent sets of a matroid are specified, the circuits are automatically determined, and conversely. Now using Independence rules 1 - 3, one can prove that the set of circuits just defined obeys the rules Circuit 1 - 3, thereby justifying the seemingly strange choice of name. If one takes any spanning tree of a connected graph (e.g. a subgraph which is a tree and includes all the vertices) and adds one edge e of the graph that is not already in the tree, one gets a unique circuit. If one removes one edge of this circuit different from e, one gets a new spanning tree. Spanning trees all have the same number of vertices; this may remind you of the property that the bases of a vector space have. Furthermore, the way one can get from one spanning tree to another is similar to the way one can get from one independent set to another (Independence 2). The phenomena which one observes - of ways in which ideas that come up in vector spaces (independence and span) and ideas that come up in graph theory (circuits and spanning trees) have certain similarities - is what drives the abstraction built into the theory of matroids. In addition to the two axiomatic approaches looked at here, independent sets and circuits, there are other axiomatic approaches involving rank functions, closure, and bases, all of which lead to the same place, a theory of matroids.	{"url":"http://www.ams.org/publicoutreach/feature-column/fcarc-matroids3","timestamp":"2024-11-11T11:54:26Z","content_type":"text/html","content_length":"57893","record_id":"<urn:uuid:95d7262b-f656-4cd8-9de2-f3bac842e446>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00521.warc.gz"}
CVP Analysis Equation, Graph and Example - Curarte The first step in setting up the spreadsheet is to organize the data that will be used to create the cost volume profit graph. This includes the sales revenue, variable costs, and fixed costs. CVP analysis is only reliable if costs are fixed within a specified production level. All units produced are assumed to be sold, and all fixed costs must be stable in CVP analysis. The worried owner was relieved to discover that sales could drop over 35 percent from initial projections before the brewpub incurred an operating loss. The contribution margin per unit is the amount each unit sold contributes to (1) covering fixed costs and (2) increasing profit. We calculate it by subtracting variable costs per unit (V) from the selling price per unit (S). The contribution margin ratio with the unit variable cost increase is 40%. The additional $5 per unit in the variable cost lowers the contribution margin ratio 20%. Each of these three examples could be illustrated with a change in the opposite direction. 1. This is the point of production where sales revenue will cover the costs of production. 2. Thus if the sales price per unit increases from $250 to $275, the break-even point decreases from 500 units (calculated earlier) to 400 units, which is a decrease of 100 units. 3. CyclePath Company produces two different products that have the following price and cost characteristics. 4. This is best illustrated by comparing two companies with identical sales and profits but with different cost structures, as we do in Figure 3.6 “Operating Leverage Example”. Recilia Vera is vice president of sales at Snowboard Company, a manufacturer of one model of snowboard. The hardest part in these situations involves determining how these changes will affect sales patterns – will sales remain relatively similar, will they go up, or will they go down? Once sales estimates become somewhat reasonable, it then becomes just a matter of number crunching and optimizing the company’s profitability. While fixed costs remain constant at $33,050, total costs increase in proportion to units. Once sales and total costs intersect at the break-even point, all you see is profit. For accrual method businesses, depreciation and amortization count as fixed costs because they don’t change with the number of units your company sells. Variable and fixed cost concepts are useful for short-term decision making. The short-term period varies, depending on a company’s current production capacity and the time required to change capacity. When you plug all the known variables into the target sales volume formula, you learn that Sleepy Baby needs to sell about 692 pajama sets to reach $50,000 in profit. Mastering Formulas In Excel: What Is The Formula For Standard Deviation Break Even analysis only identifies the sales volume required to break even. It is a subset of CVP analysis focused on finding the point where total revenue equals total costs, resulting in zero profit or loss. It helps determine the minimum sales volume needed to cover costs. Cost–volume–profit (CVP), in managerial economics, is a form of cost accounting. It is a simplified model, useful for elementary instruction and for short-run decisions. The variable cost is the cost to make the sandwich (this would be the bread, mustard, and pickles). This cost is known as “variable because it “varies” with the number of sandwiches you make. The contribution margin ratio with the selling price increase is 67%. The additional $5 per unit in unit selling price adds 7% to the contribution margin ratio. Contribution margin remains at 60% regardless of the sales volume. As grant writing fees sales increase, variable costs increase proportionately. CVP analysis is conducted to determine a revenue level required to achieve a specified profit. The revenue may be expressed in number of units sold or in dollar amounts. A cost volume profit analysis example Similarly, the break-even point in dollars is the amount of sales the company must generate to cover all production costs (variable and fixed costs). The cost volume profit graph in Excel allows you to visualize how changes in sales volume impact the overall profitability of the business. By analyzing the slope of the profit line, you can determine the extent to which an increase or decrease in sales volume will affect profits. This insight is valuable for making informed decisions about pricing strategies, production levels, and cost management. The break-even point is a crucial milestone for any business, as it represents the volume of sales at which total costs are equal to total revenue. On the graph, the break-even point is where the total revenue line intersects with the total cost line. Through research, you discover that you can sell each sandwich for $5. For each of the independent situations in requirements 2 through 4, assume that total sales remains at 2,000 units. Assume that each scenario that follows is independent of the others. Unless stated otherwise, the variables are the same as in the base case. Carefully review Figure 3.5 “Sensitivity Analysis for Snowboard Company”. The column labeled Scenario 1 shows that increasing the price by 10 percent will increase profit 87.5 percent ($17,500). In Video Production’s income statement, the $ 48,000 contribution margin covers the $ 40,000 fixed costs and leaves $ 8,000 in net income. The equation above demonstrates 100 percent of income ($100) https://simple-accounting.org/ minus $60 from variable costs equals $40 contribution margin. The equation below demonstrates revenues doubling to $200 and deducting fixed costs of $120, that results in $80 contribution margin. Cost-Volume-Profit (CVP) Analysis (With Formula and Example) After all, we will have debt of well over $1 million, and I don’t want anyone coming after my personal assets if the business doesn’t have the money to pay! ” Although all three owners felt the financial model was reasonably accurate, they decided to find the break-even point and the resulting margin of safety. Thus if the sales price per unit increases from $250 to $275, the break-even point decreases from 500 units (calculated earlier) to 400 units, which is a decrease of 100 units. Since they’re non-cash expenses that don’t affect your business’s cash profits, you might choose to leave depreciation and amortization off your CVP calculation. CVP comprises a collection of formulas that shed light on the relationship among product costs, sales volume, selling prices, and profits. A careful and accurate cost-volume-profit (CVP) analysis requires knowledge of costs and their fixed or variable behavior as volume changes. Businesses use this formula to determine how changes in sales volume, fixed costs, and variable costs can affect a business’s profitability. Several equations for price, cost, and other variables are used to run a CVP analysis, and these equations are then plotted out on an economic graph. The light green line represents the total sales of the company. This line assumes that as more units are produced more units are sold. The point where the total costs line crosses the total sales line represents the breakeven point. This point is where revenues from sales equal the total expenses. The total cost line is the sum total of fixed cost ($3,000) and variable cost of $15 per unit, plotted for various quantities of units to be sold. Subtracting variable costs from both costs and sales yields the simplified diagram and equation for profit and loss. These are costs that remain constant (in total) over some relevant range of output.	{"url":"https://www.curarte.com.ar/cvp-analysis-equation-graph-and-example/","timestamp":"2024-11-07T03:11:36Z","content_type":"text/html","content_length":"63152","record_id":"<urn:uuid:d4872b5b-f0d6-435e-b957-7ecaa188cac7>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00004.warc.gz"}
Estimating Single-Table Expression Cardinalities \| Teradata Vantage - Using Join Index Statistics to Estimate Single-Table Expression Cardinalities - Analytics Database - Teradata Vantage This topic describes how the Optimizer can use statistics collected on complex expressions specified in the select list of a single-table join index to more accurately estimate the cardinalities of complex expressions specified on base table columns in a query predicate. The use of the term join index anywhere in this topic refers only to single-table join indexes or an equivalent hash index. You should consider hash indexes to be equivalent to non-sparse single-table join indexes for this topic. Anything written about a non-sparse single-table join index applies equally to an equivalent hash index, substituting the term column list for select list. Although you cannot collect statistics on complex base table expressions, creating a single-table join index that specifies the same expression in its select list transforms the expression into a simple join index column, and you can then collect statistics on that column. The Optimizer can use those statistics to estimate the single-table cardinality of evaluations of the expression when it is specified in a predicate of a query on the base table. This capability extends the application of join indexes in query optimization to functions beyond simple query rewrite to include using their statistics for single-table expression cardinality estimation when statistics have not been, or cannot be, collected on complex expressions coded in predicate expressions that refer to their base table columns. To provide the enhanced single-table expression cardinality estimation capability, the following criteria must be met: • The join index from which the statistics are collected must specify the relevant complex expression in its select list. • You must collect statistics on the join index column set that specifies the relevant complex expression. Definitions of Simple and Complex Expressions You can collect statistics on a simple join index column created from a complex expression that is specified in its select list. The Optimizer can use statistics collected on the join index column to estimate more accurately the selectivity of complex expressions specified in a query predicate that specifies the matching base table expression. A simple expression is one with only a simple column reference on the left hand side of the predicate, while a complex expression is one that specifies something other than a simple expression on its left hand side. For example, the following predicate specifies a simple column reference: The following predicate specifies a complex expression because the term on its left hand side is not a simple column reference: Not all complex expressions can benefit from this optimization. The Optimizer can only use statistics collected on a join index column derived from a complex expression if that expression can be mapped completely to a simple join index expression. The following predicate, for example, cannot be mapped completely to a simple join index expression, so it cannot take advantage of this This expression cannot be mapped to a simple join index expression on which statistics can be collected. Matching Predicates and Mapping Expressions There are several specific cases where join index statistics can provide more accurate cardinality estimates than are otherwise available for base table predicates written using complex date • The case where an EXTRACT expression specified in a query predicate can be matched with a join index predicate. • The case where an EXTRACT DATE expression specified in a query predicate condition can be mapped to an expression specified in the select list of a join index. The Optimizer uses expression mapping when it detects an identical query expression or a matching query expression subset within a non-matching predicate. When this occurs, the Optimizer maps the predicate to the identical column of the join index, which enables it to use the statistics collected on the join index column to estimate the cardinality of the expression result. Using Predicate Matching to Estimate Single-Table Cardinalities for Complex Expressions This topic examines the case where an EXTRACT/DATE expression specified in a query predicate condition can be matched with an expression specified in the select list of a join index. If you have collected statistics on the matching join index column, the Optimizer can then use those statistics to estimate the cardinality of the predicate result. For example, consider the following join index created on base table t100_a: CREATE JOIN INDEX ji_a3 AS SELECT i1, c1 FROM t100_a WHERE EXTRACT(YEAR FROM d2)>=1969; You then collect statistics on the default nonunique primary index for ji_a3, ji_a3.i1. COLLECT STATISTICS ON ji_a3 INDEX (i1); Join index ji_a3 is designed to support the following queries on base table t100_a: Query 1 SELECT * FROM t100_a WHERE EXTRACT(YEAR FROM d2)>=1969; Query 2 SELECT * FROM t100_a WHERE EXTRACT(YEAR FROM d2)>=1969 AND i1>2; Because ji_a3 specifies a predicate (EXTRACT(YEAR FROM d2)>=1969) that is identical to the predicates specified in both queries 1 and 2, and query 2 specifies a predicate condition (i1>2) on a column you have collected statistics on for ji_a3, both queries can use the statistics collected on ji.i1 to enhance the ability of the Optimizer to estimate the cardinalities of their predicate The true cardinality for this example is 100 rows. FROM t100a WHERE EXTRACT(YEAR FROM d2)>=1970; The following shows a portion of the EXPLAIN output: 3) We do an all-AMPs RETRIEVE step from df2.t100_a by way of an all-rows scan with a condition of ("(EXTRACT(YEAR FROM (df2.t100_a.D2 )))= 1970") into Spool 1 (all_amps), which is built locally on the AMPs. The size of Spool 1 is estimated with high confidence to be 100 rows (49,800 bytes). The estimated time for this step is 0.03 seconds. The Optimizer estimates the cardinality of Spool 1 (highlighted in boldface type), which is derived from t_100a, to be 100 rows, an exact match with the true cardinality. Using Expression Mapping to Estimate Single-Table Cardinalities for Complex Expressions This topic examines the case where an EXTRACT/DATE expression specified in a query predicate condition can be mapped to an expression specified in the select list of a join index. Mapping converts the EXTRACT expression to an expression written on a DATE column. When the Optimizer makes a cardinality estimate by mapping from the base table to a join index, the normalization between the base table DATE column that contains the desired year value and the EXTRACT function in the join index definition applies, where join index statistics can provide more accurate cardinality estimates for single-table predicates written using complex expressions specified on the base table than are otherwise available. The following example demonstrates the case where the query specifies a predicate on DATE column d2, and join index ji_a4 specifies an EXTRACT function on the same DATE column as a simple join index For this case, the Optimizer attempts to convert the predicate DATE condition to its equivalent EXTRACT form before mapping to the join index column. The Optimizer first transforms the query predicate condition d2>=1969-01-01’ to the equivalent expression EXTRACT(YEAR FROM d2)>=‘1969’ and then further transforms that expression to ji.yr>=1969. Because ji.yr>=1969 is a simple expression on a join index column, the Optimizer can use the statistics collected on column ji.yr to estimate the cardinality of the expression result. Consider the following join index: CREATE JOIN INDEX ji_a4 AS SELECT i1, EXTRACT(YEAR FROM d2) AS yr, c1 FROM t100_a WHERE i1<30; You then collect the following statistics on the index: • Column i1 is the default nonunique primary index for ji_a4. COLLECT STATISTICS ON ji_a4 INDEX(i1); • The expression alias yr represents the value of the function EXTRACT(YEAR FROM d2). COLLECT STATISTICS ON ji_a4 COLUMN(yr); The true cardinality for this example is 30 rows. The following query specifies a DATE expression in its predicate that can be mapped to the EXTRACT function specified in the select list of join index is a matching subset of the select list expression EXTRACT(YEAR FROM d2) AS yr within a non-matching predicate: EXPLAIN SELECT i1 FROM t100a WHERE i1<30 AND d2>='1969-01-01'; The following shows a portion of the EXPLAIN output: 3) We do an all-AMPs RETRIEVE step from df2.JI_A4 by way of an all-rows scan with a condition of ("df2.JI_A4.yr >= 1969") into Spool 1 (all_amps), which is built locally on the AMPs. The size of Spool 1 is estimated with high confidence to be 30 rows (960 bytes). The estimated time for this step is 0.03 The Optimizer estimates the cardinality of Spool 1 (highlighted in boldface type), which is derived from t_100a, to be 30 rows, an exact match with the true cardinality. Join index ji_a4 is defined to alias the complex expression EXTRACT(YEAR FROM DATE) as yr=2010. This definition enables the Optimizer to map the complex predicate expression EXTRACT(YEAR FROM DATE)= 2010 to the simple expression yr=2010, allowing statistics collected on yr to be used to estimate the single-table cardinality of a predicate expression written using base table column references. But what if you were to code a query predicate expression such as EXTRACT(YEAR FROM DATE)/2 = 1005? The Optimizer can map this expression to the somewhat simpler expression ji_a4.yr/2 = 1005, but this mapping does not enable the use of join index statistics to estimate the single-table cardinality of the expression. The reason this mapping cannot facilitate more accurate cardinality estimation is that while the mapped expression has a less complicated appearance, it remains a complex expression. Because of the complexity of the expression, the Optimizer cannot use statistics collected on yr to estimate the single-table cardinality of the predicate expression EXTRACT(YEAR FROM DATE)/2=1005 whether it can be mapped to ji_a4.yr/2=1005 or not. The next example is a slightly more complicated example of expression mapping. This case uses statistics from matching complex expressions specified in the select list of a join index to an expression specified in a query predicate. This example applies to both sparse and non-sparse join indexes, though the join index used for this example is sparse. Consider the following SELECT request: SELECT * FROM perfcurrnew WHERE BEGIN(vt) <= CURRENT_DATE AND END(vt) > CURRENT_DATE AND END(tt) IS UNTIL_CHANGED; Vantage does not support collecting statistics from perfcurrnew on the complex expressions specified in this query predicate that can be used to estimate the cardinality of the result set. But suppose you create the following join index on perfcurrnew. The select list of this join index specifies expressions that are components of the query predicate written against the base table perfcurrnew in the example SELECT request. CREATE JOIN INDEX ji, NO FALLBACK, CHECKSUM = DEFAULT AS SELECT i, j, BEGIN(vt)(AS bvt), END(vt)(AS evt), END(tt)(AS ett) FROM perfcurrnew WHERE (evt>DATE) AND END(tt) IS UNTIL_CHANGED PRIMARY INDEX (i); After creating join index , you can collect the following statistics to support the SELECT request on • Column i is the nonunique primary index for ji. COLLECT STATISTICS ON ji INDEX(i); • Column vt is defined in base table perfcurrnew with a Period type. The expressions aliased as bvt and evt represent the BEGIN and END bound functions, respectively, of the Period column perfcurrnew.vt, and you can collect statistics on them as they are defined in the join index as ji.bvt and ji.evt. COLLECT STATISTICS ON ji COLUMN(bvt); COLLECT STATISTICS ON ji COLUMN(evt); • Column tt is also defined in perfcurrnew with a Period type, and you can collect statistics on the END bound function of the Period column ji.tt, aliased as ett. COLLECT STATISTICS ON ji COLUMN(ett); The true cardinality of the base table perfcurrnew for this example is 691 rows. The SELECT request in the following example specifies predicates that use the CURRENT_DATE function and the IS UNTIL_CHANGED predicate variable. IS UNTIL_CHANGED represents a “forever” or “until changed” date value for the END Period bound function specified for the Period column perfcurrnew.tt. Join index ji is defined using a subset of the predicates specified in the SELECT request, so the cardinality estimate for that predicate represents the number of rows selected by the predicates specified on END(vt) and END(tt). The Optimizer maps the predicate BEGIN(vt)<=CURRENT_DATE-2000 to the expression ji.bvt<=CURRENT_DATE-2000 and uses the statistics collected on ji.bvt. The combined cardinality estimate made from these statistics represents an evaluation of the cardinality of the result returned by all three of the predicates specified in the original SELECT In this case, the predicates in the query are the same as the predicates specified in the select list of , and the cardinality of is the same as the cardinality of the result set of the query, so statistics collected on are not required to make the cardinality estimate. FROM perfcurrnew WHERE BEGIN(vt) <= CURRENT_DATE - 2000 AND END(vt) > CURRENT_DATE AND END(tt) IS UNTIL_CHANGED; The following shows a portion of the EXPLAIN output: 3) We do an all-AMPs RETRIEVE step from a single partition of df2.perfcurrnew with a condition of ( "((BEGIN(df2.perfcurrnew.vt ))<= DATE '2004-11-24') AND (((END(df2.perfcurrnew.tt ))= TIMESTAMP '9999-12-31 23:59:59.999999+00:00') AND ((END(df2.perfcurrnew.vt ))> DATE '2010-05-17'))") into Spool 1 (all_amps), which is built locally on the AMPs. The size of Spool 1 is estimated with high confidence to be 723 rows (161,952 bytes). The estimated time for this step is 0.07 seconds. The estimated cardinality of Spool 1 (highlighted in boldface type), which is derived from perfcurrnew, is 723 rows, which errs by only 32 rows.	{"url":"https://docs.teradata.com/r/Enterprise_IntelliFlex_VMware/SQL-Request-and-Transaction-Processing/Query-Rewrite-Statistics-and-Optimization/Using-Join-Index-Statistics-to-Estimate-Single-Table-Expression-Cardinalities","timestamp":"2024-11-10T08:57:46Z","content_type":"text/html","content_length":"103276","record_id":"<urn:uuid:f0d04a8b-b87b-4f60-805d-0d8648ae750e>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00695.warc.gz"}
How to Generate Random Numbers In Julia? To generate random numbers in Julia, you can use the built-in rand() function. This function returns a random floating-point number between 0 and 1. For example, running the rand() function would look like this: If you want to generate random integer values within a specific range, you can multiply the result of rand() by the range and round it to the nearest integer. Here's an example: 1 random_integer = round(rand() * range) You can also generate an array of random numbers by specifying the dimensions. The following code generates a 3x3 matrix of random numbers: 1 random_matrix = rand(3, 3) If you need to generate random numbers from a different distribution, Julia provides various distributions in the Distributions package. You can use functions like rand or randn from this package to generate random numbers with specific distributions, such as normal distribution or uniform distribution. To use the randn function, you'll need to install the Distributions package first using the following command in the Julia REPL: 1 import Pkg 2 Pkg.add("Distributions") Once you have installed it, you can use randn() to generate random numbers from a standard normal distribution: 1 using Distributions 2 random_number = randn() These are some of the basic ways to generate random numbers in Julia. Keep in mind that random number generation may vary depending on the version of Julia you are using. What is the purpose of generating random numbers when testing algorithms? The purpose of generating random numbers when testing algorithms is to evaluate the performance, reliability, and efficiency of the algorithm under different random scenarios. It helps to quantify and measure the algorithm's behavior in a variety of scenarios that may not be predictable in advance. By generating random numbers as inputs, it allows for a more comprehensive testing process, covering a wide range of possibilities and edge cases that the algorithm may encounter in real-world usage. Furthermore, random number generation helps in examining the algorithm's robustness and fault tolerance, as it provides a diverse set of inputs that may challenge its assumptions or expose any weaknesses. How to generate random complex numbers in Julia? To generate random complex numbers in Julia, you can use the rand or randn functions in combination with the Complex constructor. Here are a few examples: 1. To generate a single random complex number with real and imaginary parts sampled from a uniform distribution between 0 and 1: 1 z = Complex(rand(), rand()) 1. To generate an array of random complex numbers with real and imaginary parts sampled from a uniform distribution between -1 and 1: 1 n = 5 2 z_array = [Complex(rand(-1:1), rand(-1:1)) for _ in 1:n] 1. To generate an array of random complex numbers with real and imaginary parts sampled from a standard normal distribution: 1 n = 5 2 z_array = [Complex(randn(), randn()) for _ in 1:n] Note that randn generates random numbers from a normal distribution with mean 0 and standard deviation 1, while rand generates random numbers from a uniform distribution between 0 and 1. You can adjust the range of the random numbers by passing custom arguments to rand or randn. How to generate random numbers from an exponential distribution in Julia? To generate random numbers from an exponential distribution in Julia, you can use the rand function from the Distributions package. Here's an example code: 1 # Import the Distributions package 2 using Distributions 4 # Set the lambda parameter for the exponential distribution 5 lambda = 2.0 7 # Create an Exponential distribution object 8 dist = Exponential(lambda) 10 # Generate a random number from the exponential distribution 11 random_number = rand(dist) 13 println(random_number) In this example, we set the parameter lambda for the exponential distribution to 2.0. Then, we create an Exponential distribution object dist using the specified lambda. Finally, we use the rand function to generate a random number from the distribution and store it in random_number. How to generate random numbers in parallel using multiple threads in Julia? To generate random numbers in parallel using multiple threads in Julia, you can make use of the Threads module along with the @threads macro and the Random module. Here's an example: 1 using Base.Threads, Random 3 # Set the number of threads to use 4 # On Julia 1.6 and above, you can use Threads.nthreads() 5 # Otherwise, set it to the desired number of threads 6 nthreads = Threads.nthreads() 8 # Function to generate random numbers concurrently 9 function generate_random_numbers(start_idx::Int, end_idx::Int) 10 @threads for i = start_idx:end_idx 11 # Set the seed for each thread 12 # This is important to maintain reproducibility 13 Random.seed!(i) 15 # Generate a random number 16 rand() 18 # Do something with the random number 19 # ... 20 end 21 end 23 # Divide the range of random numbers across threads 24 start_indices = range(1, nthreads, length = nthreads) 25 end_indices = vcat(start_indices[2:end]-1, [nthreads]) 27 # Spawn multiple tasks to generate random numbers concurrently 28 task_ids = Threads.@spawnat for i = 1:nthreads 29 generate_random_numbers(start_indices[i], end_indices[i]) 30 end 32 # Wait for all tasks to complete 33 fetch.(task_ids) In this example, we use the generate_random_numbers function to generate random numbers within a given range. The function is then invoked in parallel by multiple threads using the @threads macro. The range of random numbers is divided across the threads using the start_indices and end_indices vectors. By setting the seed for each thread separately (Random.seed!(i)), we ensure reproducibility of random number generation across different runs. Finally, we spawn multiple tasks to generate random numbers concurrently using Threads.@spawnat, and then wait for all tasks to complete using fetch.(task_ids). Make sure to adjust the code according to your specific use case and requirements. How to generate random numbers following a specific probability distribution in Julia? In Julia, you can use various functions and packages to generate random numbers following a specific probability distribution. Here are a few methods: 1. Using the Distributions package: This package provides several probability distributions and related functions. First, make sure the package is installed by running ] add Distributions in the Julia terminal. Then, you can generate random numbers from a specific distribution using the rand() function. For example, to generate 10 random numbers from a standard normal distribution, you can use the following code: 1 using Distributions 2 dist = Normal() 3 rand(dist, 10) 1. Using the Random package: This package provides a variety of random number generators. First, ensure that the package is installed by running ] add Random in the Julia terminal. To generate random numbers following a specific probability distribution, you can create a custom sampler using the Random.Sampler() function. For example, to generate 10 random numbers from a gamma distribution, you can use the following code: 1 using Random, RandomNumbers 2 dist = Gamma(2, 1) 3 sampler = Random.Sampler(dist) 4 rand(sampler, 10) 1. Using the StatsBase package: This package provides additional functionality for statistical computations. First, ensure that the package is installed by running ] add StatsBase in the Julia terminal. You can generate random numbers following a specific probability distribution using the rand() function. For example, to generate 10 random numbers from a Poisson distribution with a mean of 5, you can use the following code: 1 using StatsBase 2 rand(Poisson(5), 10) These are just a few examples of how to generate random numbers following a specific probability distribution in Julia. Depending on your requirements, you may find other packages or functions more What is the significance of random number generation in statistical simulations? Random number generation is an essential component of statistical simulations due to its significance in producing data that exhibits randomness and variability, mimicking real-world scenarios. Here are some key points highlighting the significance of random number generation in statistical simulations: 1. Replicating real-world uncertainty: Randomness is a fundamental aspect of many real-world phenomena, such as financial markets, weather patterns, or population growth. By generating random numbers, statistical simulations can effectively model and replicate this inherent uncertainty, allowing researchers to study and make predictions about complex systems. 2. Simulating data samples: Statistical simulations often involve creating multiple samples from a population to determine patterns, estimate parameters, or perform hypothesis testing. Random number generation enables the creation of these samples by randomly selecting individuals or observations from the population, resulting in representative and unbiased data sets. 3. Testing statistical methods: Random number generation plays a crucial role in assessing the performance and accuracy of statistical methods or models. Simulating random data using known statistical properties allows researchers to verify if their methods correctly identify patterns, estimate parameters, or make predictions, ensuring the validity and reliability of statistical 4. Sensitivity analysis: In statistical simulations, researchers often need to evaluate how sensitive their results are to various input parameters or assumptions. Random number generation enables the generation of multiple scenarios by randomly sampling from the parameter space, allowing sensitivity analysis to gauge the impact of different factors on the results of the simulation. 5. Monte Carlo simulations: Monte Carlo simulations are widely used in statistical modeling, optimization, and decision-making problems. These simulations involve repeated random sampling to estimate complex mathematical or statistical quantities and analyze their distributions. Random number generation is a fundamental requirement for these simulations, enabling the generation of random inputs that drive the modeling process. 6. Stochastic processes: Many statistical simulations model processes that evolve randomly over time or space, known as stochastic processes. Random number generation is essential to simulate the randomness inherent in these processes, allowing researchers to study and predict their behavior. In summary, random number generation is critical in statistical simulations to replicate uncertainty, create representative data samples, test statistical methods, conduct sensitivity analysis, drive Monte Carlo simulations, and model stochastic processes. It enables researchers and analysts to explore complex systems, make predictions, and draw meaningful conclusions from simulated data.	{"url":"https://topminisite.com/blog/how-to-generate-random-numbers-in-julia","timestamp":"2024-11-07T05:44:07Z","content_type":"text/html","content_length":"398710","record_id":"<urn:uuid:67f67ada-7a56-4306-a7f8-e7a27ebc9310>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00487.warc.gz"}
Calculus can be understood using simple analogies Calculus can be understood using simple analogies that relate to everyday experiences. Here are some analogies to help explain calculus concepts: 1. Derivatives – Speedometer on a Car: □ Imagine you’re driving a car, and the speedometer tells you how fast you’re going at any moment. The speedometer measures the rate of change of your distance traveled with respect to time. If you’re accelerating, the speedometer shows a higher speed, indicating a positive rate of change. If you’re slowing down, the speedometer shows a lower speed, indicating a negative rate of change. In calculus, derivatives tell us how fast something is changing at any given moment, just like the speedometer tells us how fast we’re driving. 2. Integrals – Adding Up Small Pieces: □ Think about stacking up coins of different values to make a total amount of money. Each coin represents a small piece of something, and when you add them all up, you get the total value. In calculus, integrals work similarly. They add up infinitely many small pieces (represented by tiny intervals on a graph) to find the total area under a curve. It’s like adding up all the small changes to find the overall effect. 3. Limits – Getting Closer and Closer: □ Imagine shooting arrows at a target and trying to hit the bullseye. At first, your shots might be far from the center, but as you practice, you get closer and closer to the bullseye. In calculus, limits represent this idea of getting closer and closer to a particular value or point. It’s like zooming in on a point on a graph and seeing where it’s headed. 4. Rate of Change – Growing Plants: □ Consider a plant growing in a garden. Initially, it may grow slowly, but as time goes by, it starts to grow faster and faster. The rate at which the plant grows changes over time. Calculus helps us understand how the rate of change of something, such as the height of the plant, varies with respect to time. It’s like observing how fast the plant is growing at different stages of its growth. 5. Optimization – Finding the Best Option: □ Picture yourself planning a picnic and trying to decide which route to take to avoid traffic and arrive at the park in the shortest time possible. Calculus helps us optimize decisions by finding maximum or minimum values. It’s like figuring out the best option among many possibilities, whether it’s maximizing profit, minimizing cost, or optimizing performance. By using these simple analogies, calculus concepts become more intuitive and relatable, making it easier to grasp the fundamental ideas behind calculus and its applications in various fields. No responses yet	{"url":"https://conceptslab.in/calculus-can-be-understood-using-simple-analogies/","timestamp":"2024-11-07T07:31:30Z","content_type":"text/html","content_length":"184763","record_id":"<urn:uuid:10d5cd95-b521-4035-998a-70f6a2d14ec0>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00740.warc.gz"}
A circle has a center at (7 ,6 ) and passes through (2 ,1 ). What is the length of an arc covering pi/8 radians on the circle? \| Socratic A circle has a center at #(7 ,6 )# and passes through #(2 ,1 )#. What is the length of an arc covering #pi/8# radians on the circle? 1 Answer $\frac{5 \sqrt{2} \pi}{8}$ We know the center and a point on the circle. The distance between the two points is the radius (draw a picture and convince yourself this). We know that the distance between two points $\left({x}_{1} , {y}_{1}\right)$ and $\left({x}_{2} , {y}_{2}\right)$ on the Euclidean plane is given by $d = \sqrt{{\left({x}_{1} - {x}_{2}\right)}^{2} + {\left({y}_{1} - {y}_{2}\right)}^{2}}$. Thus, radius $r = \sqrt{{\left(7 - 2\right)}^{2} + {\left(6 - 1\right)}^{2}} = \sqrt{{5}^{2} + {5}^{2}} = \sqrt{2 \cdot {5}^{2}} = 5 \sqrt{2}$. By definition, a radian is the angle subtended by an arc of length equal to the radius. Thus, an arc which subtends an angle of $\theta$ has a length of $\theta r$. In this case, $\theta = \frac{\pi} {8}$, so arc length = $\frac{5 \sqrt{2} \pi}{8}$. Hint to remember Note that the circumference subtends and angle of $2 \pi$ at the center, and has a length of $2 \pi r$. By unitary method, an arc which subtends an angle of $\theta$ has a length of $\theta r$. Impact of this question 3062 views around the world	{"url":"https://socratic.org/questions/a-circle-has-a-center-at-7-6-and-passes-through-2-1-what-is-the-length-of-an-arc#217624","timestamp":"2024-11-09T10:28:41Z","content_type":"text/html","content_length":"35643","record_id":"<urn:uuid:a21b43a4-a71c-4e1a-bfea-d272d4f14bb4>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00841.warc.gz"}
u-boot-imx/arch/nios2/include/asm/bitops/atomic.h - nest-learning-thermostat/5.7.1/u-boot - Git at Google #ifndef _ASM_GENERIC_BITOPS_ATOMIC_H_ #define _ASM_GENERIC_BITOPS_ATOMIC_H_ #include <asm/types.h> #include <asm/system.h> #ifdef CONFIG_SMP #include <asm/spinlock.h> #include <asm/cache.h> /* we use L1_CACHE_BYTES / / Use an array of spinlocks for our atomic_ts. * Hash function to index into a different SPINLOCK. * Since "a" is usually an address, use one spinlock per cacheline. # define ATOMIC_HASH_SIZE 4 # define ATOMIC_HASH(a) (&(__atomic_hash[ (((unsigned long) a)/L1_CACHE_BYTES) & (ATOMIC_HASH_SIZE-1) ])) extern raw_spinlock_t __atomic_hash[ATOMIC_HASH_SIZE] __lock_aligned; /* Can't use raw_spin_lock_irq because of #include problems, so * this is the substitute / #define _atomic_spin_lock_irqsave(l,f) do { \ raw_spinlock_t s = ATOMIC_HASH(l); \ local_irq_save(f); \ __raw_spin_lock(s); \ } while(0) #define _atomic_spin_unlock_irqrestore(l,f) do { \ raw_spinlock_t s = ATOMIC_HASH(l); \ __raw_spin_unlock(s); \ local_irq_restore(f); \ } while(0) # define _atomic_spin_lock_irqsave(l,f) do { local_irq_save(f); } while (0) # define _atomic_spin_unlock_irqrestore(l,f) do { local_irq_restore(f); } while (0) NMI events can occur at any time, including when interrupts have been * disabled by _irqsave(). So you can get NMI events occurring while a _bit function is holding a spin lock. If the NMI handler also wants to do bit manipulation (and they do) then you can get a deadlock * between the original caller of _bit() and the NMI handler. by Keith Owens * set_bit - Atomically set a bit in memory * @nr: the bit to set * @addr: the address to start counting from * This function is atomic and may not be reordered. See __set_bit() * if you do not require the atomic guarantees. * Note: there are no guarantees that this function will not be reordered * on non x86 architectures, so if you are writing portable code, * make sure not to rely on its reordering guarantees. * Note that @nr may be almost arbitrarily large; this function is not * restricted to acting on a single-word quantity. static inline void set_bit(int nr, volatile unsigned long addr) unsigned long mask = BIT_MASK(nr); unsigned long p = ((unsigned long )addr) + BIT_WORD(nr); unsigned long flags; _atomic_spin_lock_irqsave(p, flags); p \|= mask; _atomic_spin_unlock_irqrestore(p, flags); * clear_bit - Clears a bit in memory * @nr: Bit to clear * @addr: Address to start counting from * clear_bit() is atomic and may not be reordered. However, it does * not contain a memory barrier, so if it is used for locking purposes, * you should call smp_mb__before_clear_bit() and/or smp_mb__after_clear_bit() * in order to ensure changes are visible on other processors. static inline void clear_bit(int nr, volatile unsigned long addr) unsigned long mask = BIT_MASK(nr); unsigned long p = ((unsigned long )addr) + BIT_WORD(nr); unsigned long flags; _atomic_spin_lock_irqsave(p, flags); p &= ~mask; _atomic_spin_unlock_irqrestore(p, flags); * change_bit - Toggle a bit in memory * @nr: Bit to change * @addr: Address to start counting from * change_bit() is atomic and may not be reordered. It may be * reordered on other architectures than x86. * Note that @nr may be almost arbitrarily large; this function is not * restricted to acting on a single-word quantity. static inline void change_bit(int nr, volatile unsigned long addr) unsigned long mask = BIT_MASK(nr); unsigned long p = ((unsigned long )addr) + BIT_WORD(nr); unsigned long flags; _atomic_spin_lock_irqsave(p, flags); p ^= mask; _atomic_spin_unlock_irqrestore(p, flags); * test_and_set_bit - Set a bit and return its old value * @nr: Bit to set * @addr: Address to count from * This operation is atomic and cannot be reordered. * It may be reordered on other architectures than x86. * It also implies a memory barrier. static inline int test_and_set_bit(int nr, volatile unsigned long addr) unsigned long mask = BIT_MASK(nr); unsigned long p = ((unsigned long )addr) + BIT_WORD(nr); unsigned long old; unsigned long flags; _atomic_spin_lock_irqsave(p, flags); old = p; p = old \| mask; _atomic_spin_unlock_irqrestore(p, flags); return (old & mask) != 0; test_and_clear_bit - Clear a bit and return its old value * @nr: Bit to clear * @addr: Address to count from * This operation is atomic and cannot be reordered. * It can be reorderdered on other architectures other than x86. * It also implies a memory barrier. static inline int test_and_clear_bit(int nr, volatile unsigned long addr) unsigned long mask = BIT_MASK(nr); unsigned long p = ((unsigned long )addr) + BIT_WORD(nr); unsigned long old; unsigned long flags; _atomic_spin_lock_irqsave(p, flags); old = p; p = old & ~mask; _atomic_spin_unlock_irqrestore(p, flags); return (old & mask) != 0; test_and_change_bit - Change a bit and return its old value * @nr: Bit to change * @addr: Address to count from * This operation is atomic and cannot be reordered. * It also implies a memory barrier. static inline int test_and_change_bit(int nr, volatile unsigned long addr) unsigned long mask = BIT_MASK(nr); unsigned long p = ((unsigned long )addr) + BIT_WORD(nr); unsigned long old; unsigned long flags; _atomic_spin_lock_irqsave(p, flags); old = p; p = old ^ mask; _atomic_spin_unlock_irqrestore(p, flags); return (old & mask) != 0; #endif / _ASM_GENERIC_BITOPS_ATOMIC_H */	{"url":"https://nest-open-source.googlesource.com/nest-learning-thermostat/5.7.1/u-boot/+/refs/heads/master/u-boot-imx/arch/nios2/include/asm/bitops/atomic.h?autodive=0%2F%2F","timestamp":"2024-11-15T05:18:01Z","content_type":"text/html","content_length":"64276","record_id":"<urn:uuid:d6a82a71-624d-467e-9f1b-1ad69a48167d>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00106.warc.gz"}
A Contracting Dynamical System Perspective toward Interval Markov Decision Processes (Conference Paper) \| NSF PAGES We propose a Bayesian decision making framework for control of Markov Decision Processes (MDPs) with unknown dynamics and large, possibly continuous, state, action, and parameter spaces in data-poor environments. Most of the existing adaptive controllers for MDPs with unknown dynamics are based on the reinforcement learning framework and rely on large data sets acquired by sustained direct interaction with the system or via a simulator. This is not feasible in many applications, due to ethical, economic, and physical constraints. The proposed framework addresses the data poverty issue by decomposing the problem into an offline planning stage that does not rely on sustained direct interaction with the system or simulator and an online execution stage. In the offline process, parallel Gaussian process temporal difference (GPTD) learning techniques are employed for near-optimal Bayesian approximation of the expected discounted reward over a sample drawn from the prior distribution of unknown parameters. In the online stage, the action with the maximum expected return with respect to the posterior distribution of the parameters is selected. This is achieved by an approximation of the posterior distribution using a Markov Chain Monte Carlo (MCMC) algorithm, followed by constructing multiple Gaussian processes over the parameter space for efficient prediction of the means of the expected return at the MCMC sample. The effectiveness of the proposed framework is demonstrated using a simple dynamical system model with continuous state and action spaces, as well as a more complex model for a metastatic melanoma gene regulatory network observed through noisy synthetic gene expression data. more » « less	{"url":"https://par.nsf.gov/biblio/10480527-contracting-dynamical-system-perspective-toward-interval-markov-decision-processes","timestamp":"2024-11-05T06:52:30Z","content_type":"text/html","content_length":"243218","record_id":"<urn:uuid:f7fa567b-914a-487b-bb1c-e7588698f31a>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00423.warc.gz"}
Bug Free Programming - Wikibooks, open books for an open world Please share your thoughts about whether to keep this book at Wikibooks Maintain this notice so this book can receive a fair review and people have time to address reasonable concerns through concrete improvements. Remove or replace this notice after concludes and a community decision is made. Please also consider notifying the primary contributors on their discussion page with {{subst:Rfd warning\|Bug Free Programming\|~~~~}} Note: Take the article below with a grain of salt. Writing bug-free code is impossible. The best developers focus on making code that is reliable and maintainable, but anyone who says a piece of code is bug-free has not done their research. Research has shown that there is a factor of 20 between the productivity of the best and the worst programmers. That is, top programmers work 20 times faster than the worst ones, and around 5 to 10 times faster than average programmers. This means that a very good developer can do in a day what others may need a month for. There are a couple of ways to increase productivity, but the most important factor is accuracy. Accuracy strongly correlates with development speed. The best developers distinguish themselves by writing bug free code right away. If you are able to write code that does not contain any mistakes, you don't have to waste your time hunting bugs. Ever wondered why that guy at the next table who works so leisurely gets things done a lot faster than you? That's because he does the right thing at the first try. It is of course impossible to prevent bugs entirely, but: Working fast means working accurately rather than working hard. So how to write bug free code? You'd think that most bugs are introduced by thoughtlessness, but in fact, most bugs are introduced by the programmer not entirely understanding his own code. If you get a NullPointerException in Java, that's usually not because you forgot to think about null references, but because you didn't really understand the role of the variable that was null. The trick to bug free programming is a rock solid understanding of the code you write. To write bug free code, you first have to learn how to prove that a piece of code works. A lot of developers study these techniques at the university, but they never really learn to apply them in real-life coding. Other developers (especially those without a degree) never learn formal methods in the first place, so it's a good point to start. It may look too theoretical at first sight, but this is the key to bug free programming. Let's start out with a very simple program. The goal here is that variable x of type int contain the value 10 when the program terminates. We will use C++ for demonstration throughout this book. int x; x = 10; This code is obviously bug free. To prove it, let's add conditions enclosed by braces before and after each line of code. Conditions are logical statements that are true at the point of the condition in execution time no matter how execution arrives there. int x; x = 10; {x == 10} The first two conditions are true, because we don't have any information at those points (note that C++ doesn't preinitialize variables). The only thing we know is true there is true itself. However, after the line x = 10 is executed, we know that the value of x is 10, because this is the definition of assignment. We can state this in more general terms: {true} x = a {x == a} It means that whatever was true before, after an assignment of a to x, x==a will hold true, i. e. the value of x will be a. We will treat it as an axiom of constant assignment. The following program also assigns the value 10 to x, but it does it differently. int x, y; y = 10; x = y; To prove that it's correct, annotate it with conditions again: int x, y; y = 10; {y == 10} x = y; {x == 10} After the assignment to y, we can state that its value is 10. Well, if y is 10, and we assign it to x, x will also be 10. This leads to the full axiom of assignment: {y == a} x = y {x == a} Similar rules apply to when there is an expression at the right hand side of the assignment. Another important axiom is that assignment doesn't change any variables that are not assigned to. {y == a} x = b {y == a} This is still very trivial and is only needed to build the rest around it. You've probably realized by now that there are two conditions to consider about each assignment: one that holds true before it, and one that holds true after it. The former is called precondition, the latter is called postcondition. We will prove the correctness of a program by chaining them: the postcondition of a statement is the precondition of the next statement. if {P} S1 {R} and {R} S2 {Q} then {P} S1; S2 {Q} Here S1; S2 means two statements executed after each other, i. e. in a sequence. There are two other types of control structures: conditionals and loops. We'll learn to reason about them later. One final axiom we need to start proper correctness proving is the notion that: if P => R and {R} S {T} and T => Q then {P} S {Q} A => B means A implies B in the logical sense. If this is all Greek to you, don't panic. We'll cover enough examples for these thigs to start making sense. Now let's do our first formal correctness proof. Prove that the following program assigns 10 to x. int x, y, z; z = 1; y = 2 * z; x = y * 5; Annotate it: int x, y, z; z = 1; {z == 1} y = 2 * z; {y == 2} x = y * 5; {x == 10} We have the annotations, but we have to prove that they indeed hold true at the point where they are written. {true} int x, y, z {true} This is trivial because variable definitions do nothing from a correctness point of view. Now we have to prove that: {true} z = 1 {z == 1} This is because of the axiom of assignment. Since it is an axiom, we don't have to further prove it. We can now use this as the precondition of the next statement: {z == 1} y = 2 * z {y == 2} and again: {y == 2} x = y * 5 {x == 10} Why does this prove the original statement? If we name the statements and the conditions like this: {Q1: true} S1: int x, y, z; {Q2: true} S2: z = 1; {Q3: z == 1} S3: y = 2 * z; {Q4: y == 2} S4: x = y * 5; {Q5: x == 10} and use the axiom of sequence three times, we get: {Q1} S1 {Q2} and {Q2} S2 {Q3} => {Q1} S1; S2 {Q3} {Q1} S1; S2 {Q3} and {Q3} S3 {Q4} => {Q1} S1; S2; S3 {Q4} {Q1} S1; S2; S3 {Q4} and {Q4} S5 {Q5} => {Q1} S1; S2; S3; S4 {Q5} We essentially proved that the program as a whole goes from {true} (whatever) to {x == 10} (x having a value of 10). This is what we had to prove to make 100%, absolutely sure that the code is bug free. Note that now there is no doubt that this code is correct (unless we did a mistake in the proof, of course). What made this possible is the fact that we had a very clear understanding about what was going on in the code at every possible point of execution. Annotating the code with conditions made our expectations explicit, and having to prove them in a formal way made us think about why these expectations were true. Nobody has the time to do formal correctness proofs in real life situations, but knowing how to do it helps a lot in creating bug free code. The trick is to be explicit about expectations. This prevents you from forgetting about null references, among other things. Let's move on to conditionals. int x, y; if (x % 2 != 0) y = x + 1; y = x; We want to prove here that y is even at the end. The axiom of conditional is as follows: given {P && R} S1 {Q} and {P && !R} S2 {Q} => {P} if R S1; else S2 {Q} To prove that a conditional is correct, you have to prove two things. First, you have to prove that when the condition is true, the "then" clause gives correct results. Second, you have to prove that when the condition is false, the "else" clause gives correct results. Annotating the former example: int x, y; if (x % 2 != 0) y = x + 1; y = x; {y % 2 == 0} As for the variable definitions: {true} int x, y {true} Check. We didn't explicitly assign a value to x and yet we will prove the correctness of a program that uses its value. C++ doesn't preinitialize local variables. The value of the variable is some memory garbage. We have a conditional here, so we have to check for the then and the else clause separately. {Q1: true && x % 2 != 0} y = x + 1 {R1: true && y == x + 1 && x % 2 != 0} true && y == x + 1 && x % 2 != 0 => (y - 1) % 2 != 0 => y % 2 == 0 Now if you use the axiom of relaxation: {Q1} y = x + 1 {R1} and R1 => y % 2 == 0 {Q1} y = x + 1 {y % 2 == 0} Now let's go for the else clasue {Q2: true && !(x % 2 != 0)} y = x {R1: true && y == x && !(x % 2 != 0)} true && y == x && !(x % 2 != 0) => y == x && x % 2 == 0 => y % 2 == 0 Again, using the axiom of relaxation: {Q2} y = x {R2} and R2 => y % 2 == 0 {Q2} y = x {y % 2 == 0} Now combine the results with the axiom of conditionals: {true && R} y = x + 1 {y % 2 == 0} and {true && !R} y = x {y % 2 == 0} {true} if R y = x + 1; else y = x {y % 2 == 0} I know you don't believe that it's of actual real life use, but keep reading. We will eventually drop the maths and only use the core notions of the proofs to produce bug free code. Let's write our first program with some actual value. Say we have an array of int 's, and we want to put the sum of its elements into a variable called sum. We will learn how to reason about loops in this example, and this will be the first time you'll see how it helps you write correct code right away. We will start with a pre made program, but later we will investigate how to turn the techniques used in correctness proofs to design the code. int[] array; int length; // Fill the array. {Q: array has been created and has at least length elements and length >= 0} int i, sum; i = 0; sum = 0; while (i != length) sum = sum + array[i]; i = i + 1; {R: sum = array[0] + array[1] + ... + array[length-1]} Precondition Q is stated to makes sure we don't try to read unallocated memory. We don't care how the array is allocated or filled with numbers for this example. Loops are the most complicated control structures of the three, and the most prone to mistakes. The difficulty is that a loop is a series of conditionals in disguise, and you don't even know how many conditionals it is, because it depends on the condition which is evaluated at runtime. If you try to reason about loops by visualizing how they execute, you are in trouble, particularly if you have more complicated loops than the previous one. Visualization is an important step in getting a rough idea about how to code something, but if you don't want to spend your evening before the deadline debugging corner cases you didn't visualize, you'd better reason about your code statically. That does not necessarily mean mathematically. We will eventually drop the math completely. The trick is to convert the dynamic nature of loops into a static condition. If we could prove a condition that is true every time before the condition is evaluated, we could reason about the loop without relying on visualization. This condition is called the invariant of the loop. {Q: array has been created and has at least length elements and length >= 0} int i, sum; i = 0; {Q && i == 0} sum = 0; {Q && i == 0 && sum == 0} while (i != length) sum = sum + array[i]; i = i + 1; {R: sum = array[0] + array[1] + ... + array[length-1]} We'll only care about the loop in this example because the rest is trivial. The invariant is that sum contains the sum of the elements from index 1 to index i-1 every time before the condition is evaluated, including when execution first arrives at the loop. {I: sum == array[0] + ... + array[i-1]} When execution hits the loop: {Q && i == 0 && sum == 0} holds. Does it imply I? Of course: Q && i == 0 && sum == 0 => sum = 'sum of elements from 0 to -1' This is true, because there are no elements between index 0 and index -1, and the sum of 0 numbers is 0. That is, the sum of all the elements of an empty array is 0. If the array is not empty, the body of the loop will execute, and the condition will be evaluated again. At this point I is true (we don't consider loop conditions with side effects, more on that later). The condition of the loop is also true, because otherwise, we wouldn't have entered the loop body. What we have to prove is: {I && i != length} sum = sum + array[i]; i = i + 1; That is, the invariant is preserved across iterations. What's the condition to use between the statements? sum has been updated, but i hasn't. Try: {Q1: sum == array[0] + ... + array[i-1] && i != length} sum = sum + array[i]; {Q2: sum == array[0] + ... + array[i-1] + array[i] && i != length} i = i + 1; Can we prove that {Q1} sum = sum + array[i] {Q2} ? We have a slight problem here, because we can't prove that array[i] is a valid expression. i must be smaller than length and at least 0 for it to be valid. Additionally, we need array to be an allocated array with at least length elements (let's denote this condition with T to save space). Therefore, we have to modify our invariant to include this information. {I: T && 0 <= i && i <= length && sum == array[0] + ... + array[i-1]} Note this important step in understanding the code we write. We didn't make all our expectations explicit. This is how bugs sneak in. With this new invariant we have to start over. Is it true when execution hits the loop? T && length >= 0 && i == 0 && sum == 0 => T && 0 <= i && i <= length && sum = 'sum of elements from 0 to -1' This is true. For example, we know that i <= length because i == 0 and length >= 0. Is it preserved across iterations? {Q1: T && 0 <= i && i <= length && sum == array[0] + ... + array[i-1] && i != length} sum = sum + array[i]; i = i + 1; Find the right condition after updating sum: {Q1: T && 0 <= i && i <= length && sum == array[0] + ... + array[i-1] && i != length} sum = sum + array[i]; {Q2: T && 0 <= i && i <= length && sum == array[0] + ... + array[i-1] + array[i] && i != length} i = i + 1; We know that array[i] is valid, because of the constraints on i included in the condition. What about {Q2} i = i + 1 {I} {Q2: T && 0 <= i && i <= length && sum == array[0] + ... + array[i-1] + array[i] && i != length} i = i + 1; {Q3: T && 0 <= i-1 && i-1 <= length && sum == array[0] + ... + array[i-1] && i-1 != length} We changed i to i-1 because it's the original value of i (before the statement was executed). Q3 => I T && 0 <= i-1 && i-1 <= length && sum == array[0] + ... + array[i-1] && i-1 != length => T && 0 <= i && i <= length && sum == array[0] + ... + array[i-1] For example 0 <= i-1 implies 1 <= i which implies 0 <= i. i-1 <= length and i-1 != length together imply that i-1 < length which in turn implies that i <= length. When the loop terminates, its condition is false. Additionally, the invariant is true. This is because it's true every time before the condition is evaluated. If we can prove that I && !(i != length) => R we prove that R is true when the loop terminates. It's trivial: T && 0 <= i && i <= length && sum == array[0] + ... + array[i-1] && i == length => sum = array[0] + array[1] + ... + array[length-1] We have proven that the program gives correct results if it terminates. However, note that we haven't proven that it actually terminates. This is called partial correctness. Full correctness means to also prove that the program terminates. We will come back to it later. Make sure you understand the above. I only gave a couple of examples, because this is not a book on formal correctness proving. If you feel you need to dive deeper into it, look up some book on Hoare's method. The rest of the book will not make much sense if you don't understand it. So how does this help you work faster? You don't have the time to work out the formal proof of your code all the time, but knowing how to do it will help you even when you aren't using it. A good understanding of preconditions, postconditions, and invariants will help you write much better code. Now we will cover plenty of examples that show how.	{"url":"https://en.wikibooks.org/wiki/Bug_Free_Programming","timestamp":"2024-11-09T14:31:43Z","content_type":"text/html","content_length":"70747","record_id":"<urn:uuid:69b00473-08ff-42a4-adf4-b42481dfea02>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00556.warc.gz"}
Engineering Reference — EnergyPlus 9.4 Absorption Chiller[LINK] The input object Chiller:Absorption provides a model for absorption chillers that is an empirical model of a standard absorption refrigeration cycle. The condenser and evaporator are similar to that of a standard chiller, which are both water-to-water heat exchangers. The assembly of a generator and absorber provides the compression operation. Low-pressure vapor from the evaporator is absorbed by the liquid solution in the absorber. A pump receives low-pressure liquid from the absorber, elevates the pressure of the liquid, and delivers the liquid to the generator. In the generator, heat from a high temperature source (hot water or steam) drives off the vapor that has been absorbed by the solution. The liquid solution returns to the absorber through a throttling valve whose purpose is to provide a pressure drop to maintain the pressure difference between the generator and absorber. The heat supplied to the absorber can be waste heat from a diesel jacket, or the exhaust heat from diesel, gas, and steam turbines. For more information on absorption chillers, see the Input/Output Reference Document (Object: Chiller:Absorption). The part-load ratio of the absoprtion chiller’s evaporator is simply the actual cooling effect produced by the chiller divided by the maximum cooling effect available. PLR is the part-load ratio of chiller evaporator ˙Qevap is the chiller evaporator load [W] ˙Qevap,rated is the rated chiller evaporator capacity [W]. This absorption chiller model is based on a polynomial fit of absorber performance data. The Generator Heat Input Part Load Ratio Curve is a quadratic equation that determines the ratio of the generator heat input to the demand on the chiller’s evaporator (Qevap). The Pump Electric Use Part Load Ratio Curve is a quadratic equation that determines the ratio of the actual absorber pumping power to the nominal pumping power. Thus, the coefficient sets establish the ratio of heat power in-to-cooling effect produced as a function of part load ratio. The ratio of heat-power-in to cooling-effect-produced is the inverse of the coefficient of performance. If the operating part-load ratio is greater than the minimum part-load ratio, the chiller will run the entire time step and cycling will not occur (i.e. CyclingFrac = 1). If the operating part-load ratio is less than the minimum part-load ratio, the chiller will be on for a fraction of the time step equal to CyclingFrac. Steam (or hot water) and pump electrical energy use are also calculated using the chiller part-load cycling fraction. CyclingFrac is the chiller part-load cycling fraction PLRmin is the chiller minimum part-load ratio ˙Qgenerator is the generator input power (W) ˙Qpump is the absorbtion chiller pumping power (W). The evaporator water mass flow rate is calculated based on the Chiller Flow Mode as follows. Constant Flow Chillers: Variable Flow Chillers: ˙mevap is the chiller evaporator water mass flow rate (kg/s) ˙mevap,max is the chiller design evaporator water mass flow rate (kg/s) ΔTevap is the chiller evaporator water temperature difference (∘C) Tevap,in is the chiller evaporator inlet water temperature (∘C) Tevap,SP is the chiller evaporator outlet water setpoint temperature (∘C) Cp is the = specific heat of water entering evaporator (J/kg-∘C). The evaporator outlet water temperature is then calculated based on the cooling effect produced and the evaporator entering water temperature. Tevap,out is the chiller evaporator outlet water temperature (∘C) Tevap,in is the chiller evaporator inlet water temperature (∘C) Cp,evap is the specific heat of chiller evaporator inlet water (J/kg-∘C) ˙mevap is the chiller evaporator water mass flow rate (kg/s). The condenser heat transfer and condenser leaving water temperature are also calculated. ˙Qcond is the chiller condenser heat transfer rate (W) Tcond,out is the chiller condenser outlet water temperature (∘C) Tcond,in is the chiller condenser inlet water temperature (∘C) Cp,cond is the specific heat of chiller condenser inlet water (J/kg-∘C) ˙mcond is the chiller condenser water mass flow rate (kg/s). The absorption chiller can model the impact of steam or hot water entering the generator, although the connection of the steam (hot water) nodes to a plant is not actually required. The calculations specific to the generator depend on the type of fluid used and are described here in further detail. Steam Loop Calculations[LINK] When a steam loop is used and the inlet and outlet node names are specified (i.e. the nodes are connected to a steam loop), the generator outlet node steam mass flow rate and temperature are calculated based on the generator input power, latent heat of steam, the specific heat of water, and the amount of subcooling in the steam generator. The model assumes dry saturated steam enters the absorption chiller’s generator and exits the generator as a subcooled liquid. The temperature leaving the generator is calculated based on the user entered amount of liquid subcooling in the generator. The effect of subcooling of the liquid (condensate) in the pipe returning to the boiler is not modeled. ˙msteam is the chiller steam mass flow rate (kg/s) hfg is the latent heat of steam (J/kg) cp,water is the specific heat of saturated water in the generator (J/kg-∘C) ΔTsc is the amount of subcooling in steam generator (∘C) Tgenerator,out is the generator steam outlet node temperature (∘C) Tgenerator,in is the generator steam inlet node temperature (∘C). Hot Water Loop Calculations[LINK] When a hot water loop is used and the inlet and outlet node names are specified (i.e. the nodes are connected to a hot water loop), the generator outlet node temperature is calculated based on the generator input power, mass flow rate of water, and the specific heat of water entering the hot water generator. The calculations are based on the Chiller Flow Mode as follows. Constant Flow Chillers: Variable Flow Chillers: ˙mgenerator is the generator hot water mass flow rate (kg/s) ˙mgenerator,max is the generator design hot water mass flow rate (kg/s) ΔTgenerator is the generator design hot water temperature difference (∘C). Indirect Absorption Chiller[LINK] The Chiller:Absorption:Indirect object is an enhanced version of the absorption chiller model found in the Building Loads and System Thermodynamics (BLAST) program. This enhanced model is nearly identical to the existing absorption chiller model (Ref. Chiller:Absorption) with the exceptions that: 1) the enhanced indirect absorption chiller model provides more flexible performance curves and 2) chiller performance now includes the impact of varying evaporator, condenser, and generator temperatures. Since these absorption chiller models are nearly identical (i.e., the performance curves of the enhanced model can be manipulated to produce similar results to the previous model), it is quite probable that the Chiller:Absorption model will be deprecated in a future release of The indirect absorption chiller’s condenser and evaporator are similar to that of a standard chiller, which are both water-to-water heat exchangers. The assembly of a generator and absorber provides the compression operation. A schematic of a single-stage absorption chiller is shown in the figure below. Low-pressure vapor from the evaporator is absorbed by the liquid solution in the absorber. A pump receives low-pressure liquid from the absorber, elevates the pressure of the liquid, and delivers the liquid to the generator. In the generator, heat from a high temperature source (hot water or steam) drives off the vapor that has been absorbed by the solution. The liquid solution returns to the absorber through a throttling valve whose purpose is to provide a pressure drop to maintain the pressure difference between the generator and absorber. The heat supplied to the generator can be either hot water or steam, however, connection to an actual plant loop is not required. For more information on indirect absorption chillers, see the Input/Output Reference Document (Object: Chiller:Absorption:Indirect). The chiller cooling effect (capacity) will change with a change in condenser water temperature. Similarly, the chiller cooling effect will change as the temperature of the evaporator water changes. The chiller cooling effect will also change with a change in or generator inlet water temperature and only applies when Hot Water is used as the generator heat source. A quadratic or cubic equation is used to modify the rated chiller capacity as a function of both the condenser and generator inlet water temperatures and the evaporator outlet water temperature. If any or all of the capacity correction factor curves are not used, the correction factors are assumed to be 1. CAPFTgenerator=i+j(Tgenerator)+k(Tgenerator)2+l(Tgenerator)3 (Hot Water only) CAPFTevaporator is the capacity correction (function of evaporator temperature) factor CAPFTcondenser is the capacity correction (function of condenser temperature) factor CAPFTgenerator is the capacity correction (function of generator temperature) factor Tevaporator is the evaporator outet water temperature (∘C) Tcondenser is the condenser inlet water temperature (∘C) Tgenerator is the generator inlet water temperature (∘C) ˙Qevap,max is the maximum chiller capacity (W) ˙Qevap,rated is the rated chiller capacity (W). The part-load ratio of the indirect absoprtion chiller’s evaporator is simply the actual cooling effect required (load) divided by the maximum cooling effect available. PLR is the part-load ratio of chiller evaporator ˙Qevap is the chiller evaporator operating capacity (W). The generator’s heat input is also a function of several parameters. The primary input for determining the heat input requirements is the Generator Heat Input function of Part-Load Ratio Curve. The curve is a quadratic or cubic equation that determines the ratio of the generator heat input to the chiller’s maximum capacity (Qevap,max) and is solely a function of part-load ratio. Typical generator heat input ratios at full load (i.e., PLR = 1) are between 1 and 2. Two additional curves are available to modifiy the heat input requirement based on the generator inlet water temperature and the evaporator outlet water temperature. GeneratorHIR is the ratio of generator heat input to chiller operating capacity GenfCondT is the heat input modifier based on generator inlet water temperature GenfEvapT is the heat input modifier based on evaporator outlet water temperature. The Pump Electric Use function of Part-Load Ratio Curve is a quadratic or cubic equation that determines the ratio of the actual absorber pumping power to the nominal pumping power. If the chiller operating part-load ratio is greater than the minimum part-load ratio, the chiller will run the entire time step and cycling will not occur (i.e. CyclingFrac = 1). If the operating part-load ratio is less than the minimum part-load ratio, the chiller will be on for a fraction of the time step equal to CyclingFrac. Generator heat input and pump electrical energy use are also calculated using the chiller part-load cycling fraction. CyclingFrac is the chiller part-load cycling fraction PLRmin is the chiller minimum part-load ratio ˙Qgenerator is the generator heat input (W) ˙Qpump is the chiller pumping power (W). The evaporator water mass flow rate is calculated based on the Chiller Flow Mode as follows. Constant Flow Chillers: Variable Flow Chillers: ˙mevap is the chiller evaporator water mass flow rate (kg/s) ˙mevap,max is the chiller design evaporator water mass flow rate (kg/s) ΔTevap is the chiller evaporator water temperature difference (∘C) Tevap,in is the chiller evaporator inlet water temperature (∘C) Tevap,SP is the chiller evaporator outlet water setpoint temperature (∘C) Cp,evap is the specific heat of water entering evaporator (J/kg-∘C). The evaporator outlet water temperature is then calculated based on the cooling effect produced and the evaporator entering water temperature. Tevap,out is the chiller evaporator outlet water temperature (∘C) Tevap,in is the chiller evaporator inlet water temperature (∘C) Cp,evap is the specific heat of chiller evaporator inlet water (J/kg-∘C) ˙mevap is the chiller evaporator water mass flow rate (kg/s). The condenser heat transfer and condenser leaving water temperature are also calculated. ˙Qcond = chiller condenser heat transfer rate (W) Tcond,out = chiller condenser outlet water temperature (∘C) Tcond,in = chiller condenser inlet water temperature (∘C) Cp,cond = specific heat of chiller condenser inlet water (J/kg-∘C) ˙mcond = chiller condenser water mass flow rate (kg/s). The absorption chiller can model the impact of steam or hot water entering the generator, although the connection of the steam (hot water) nodes to a plant is not actually required. The calculations specific to the generator depend on the type of fluid used and are described here in further detail. Steam Loop Calculations[LINK] When a steam loop is used and the inlet and outlet node names are specified (i.e. the nodes are connected to a steam loop), the generator outlet node steam mass flow rate and temperature are calculated based on the generator heat input, latent heat of steam, the specific heat of water, and the amount of subcooling in the steam generator. The model assumes dry saturated steam enters the generator and exits the generator as a subcooled liquid. The temperature leaving the generator is calculated based on the user entered amount of liquid subcooling in the generator. The effect of subcooling of the liquid (condensate) in the pipe returning to the boiler is also modeled using the user entered abount of steam condensate loop subcooling. ˙msteam is the chiller steam mass flow rate (kg/s) hfg is the latent heat of steam (J/kg) cp,water is the specific heat of water (J/kg-∘C) ΔTsc is the amount of subcooling in steam generator (∘C) ΔTsc,loop is the amount of condensate subcooling in steam loop (∘C) Tgenerator,out is the generator steam outlet node temperature (∘C) Tgenerator,in is the generator steam inlet node temperature (∘C). Hot Water Loop Calculations[LINK] When a hot water loop is used and the inlet and outlet node names are specified (i.e. the nodes are connected to a hot water loop), the generator outlet node temperature is calculated based on the generator heat input, mass flow rate of water, and the specific heat of water entering the hot water generator. The calculations are based on the Chiller Flow Mode as follows. Constant Flow Chillers: Variable Flow Chillers: ˙mgenerator is the generator hot water mass flow rate (kg/s) ˙mgenerator,max is the generator design hot water mass flow rate (kg/s) ΔTgenerator is the generator design hot water temperature difference (∘C). Combustion Turbine Chiller[LINK] The input object Chiller:CombustionTurbine provides a chiller model that is the empirical model from the Building Loads and System Thermodynamics (BLAST) program. Fitting catalog data to a third order polynomial equations generates the chiller performance curves. Three sets of coefficients are required to model the open centrifugal chiller as discussed in the section, titled, ‘Electric Chiller Based on Fluid Temperature Differences’. The gas turbine-driven chiller is an open centrifugal chiller driven directly by a gas turbine. The BLAST model of an open centrifugal chiller is modeled as standard vapor compression refrigeration cycle with a centrifugal compressor driven by a shaft power from an engine. The centrifugal compressor has the incoming fluid entering at the eye of a spinning impeller that throws the fluid by centrifugal force to the periphery of the impeller. After leaving the compressor, the refrigerant is condensed to liquid in a refrigerant to water condenser. The heat from the condenser is rejected to a cooling tower, evaporative condenser, or well water condenser depending on which one is selected by the user based on the physical parameters of the plant. The refrigerant pressure is then dropped through a throttling valve so that fluid can evaporate at a low pressure that provides cooling to the evaporator. The evaporator can chill water that is pumped to chilled water coils in the building. For more information, see the Input/Output Reference Document. This chiller is modeled like the electric chiller with the same numerical curve fits and then some additional curve fits to model the turbine drive. Shown below are the definitions of the curves that describe this model. The chiller’s temperature rise coefficient which is defined as the ratio of the required change in condenser water temperature to a given change in chilled water temperature, which maintains the capacity at the nominal value. This is calculated as the following ratio: TCEntrequired is the required entering condenser air or water temperature to maintain rated capacity TCEntrated is the rated entering condenser air or water temperature at rated capacity TELvrequired is the required leaving evaporator water outlet temperature to maintain rated capacity TELvrated is the rated leaving evaporator water outlet temperature at rated capacity. The Capacity Ratio Curve is a quadratic equation that determines the Ratio of Available Capacity to Nominal Capacity. The defining equation is: where the Delta Temperature is defined as: TempCondIn is the temperature entering the condenser (water or air temperature depending on condenser type) TempCondInDesign is the temperature of the design condenser inlet from user input above TempEvapOut is the temperature leaving the evaporator TempEvapOutDesign is the temperature of the design evaporator outlet from user input above TempRiseCoefficient is based on user input from above. The following three fields contain the coefficients for the quadratic equation. The Power Ratio Curve is a quadratic equation that determines the Ratio of Full Load to Power. The defining equation is: The Full Load Ratio Curve is a quadratic equation that determines the fraction of full load power. The defining equation is: The Fuel Input Curve is a polynomial equation that determines the Ratio of Fuel Input to Energy Output. The equation combines both the Fuel Input Curve Coefficients and the Temperature Based Fuel Input Curve Coefficients. The defining equation is: where FIC represents the Fuel Input Curve Coefficients, TBFIC represents the Temperature Based Fuel Input Curve Coefficients, Rload is the Ratio of Load to Combustion Turbine Engine Capacity, and ATa ir is the difference between the current ambient and design ambient temperatures. The Exhaust Flow Curve is a quadratic equation that determines the Ratio of Exhaust Gas Flow Rate to Engine Capacity. The defining equation is: where GTCapacity is the Combustion Turbine Engine Capacity, and ATair is the difference between the current ambient and design ambient temperatures. The Exhaust Gas Temperature Curve is a polynomial equation that determines the Exhaust Gas Temperature. The equation combines both the Exhaust Gas Temperature Curve Coefficients (Based on the Part Load Ratio) and the (Ambient) Temperature Based Exhaust Gas Temperature Curve Coefficients. The defining equation is: where C represents the Exhaust Gas Temperature Curve Coefficients, TBC are the Temperature Based Exhaust Gas Temperature Curve Coefficients, RLoad is the Ratio of Load to Combustion Turbine Engine Capacity, and ATair is the difference between the actual ambient and design ambient temperatures. The Recovery Lubricant Heat Curve is a quadratic equation that determines the recovery lube energy. The defining equation is: where PLoad is the engine load and RL is the Ratio of Load to Combustion Turbine Engine Capacity The UA is an equation that determines the overall heat transfer coefficient for the exhaust gasses with the stack. The heat transfer coefficient ultimately helps determine the exhaust stack temperature. The defining equation is: Chiller Basin Heater[LINK] This chiller’s basin heater (for evaporatively-cooled condenser type) operates in the same manner as the Engine driven chiller’s basin heater. The calculations for the chiller basin heater are described in detail at the end of the engine driven chiller description (Ref. Engine Driven Chiller). This model (object name ChillerHeater:Absorption:DirectFired) simulates the performance of a direct fired two-stage absorption chiller with optional heating capability. The model is based on the direct fired absorption chiller model (ABSORG-CHLR) in the DOE-2.1 building energy simulation program. The EnergyPlus model contains all of the features of the DOE-2.1 chiller model, plus some additional capabilities. This model simulates the thermal performance of the chiller and the fuel consumption of the burner(s). This model does not simulate the thermal performance or the power consumption of associated pumps or cooling towers. This auxiliary equipment must be modeled using other EnergyPlus models (e.g. Cooling Tower:Single Speed). Model Description[LINK] The chiller model uses user-supplied performance information at design conditions along with five performance curves (curve objects) for cooling capacity and efficiency to determine chiller operation at off-design conditions. Two additional performance curves for heating capacity and efficiency are used when the chiller is operating in a heating only mode or simultaneous cooling and heating mode. The following nomenclature is used in the cooling equations: AvailCoolCap is the available full-load cooling capacity at current conditions (W) CEIR is the user input “Electric Input to Cooling Output Ratio” CEIRfPLR is the electric input to cooling output factor, equal to 1 at full load, user input “Electric Input to Cooling Output Ratio Function of Part Load Ratio Curve Name” CEIRfT is the electric input to cooling output factor, equal to 1 at design conditions, user input “Electric Input to Cooling Output Ratio Function of Temperature Curve Name” CFIR is the user input “Fuel Input to Cooling Output Ratio” CFIRfPLR is the fuel input to cooling output factor, equal to 1 at full load, user input “Fuel Input to Cooling Output Ratio Function of Part Load Ratio Curve Name” CFIRfT is the fuel input to cooling output factor, equal to 1 at design conditions, user input “Fuel Input to Cooling Output Ratio Function of Temperature Curve Name” CondenserLoad is the condenser heat rejection load (W) CoolCapfT is the cooling capacity factor, equal to 1 at design conditions, user input “Cooling Capacity Function of Temperature Curve Name” CoolElectricPower is the cooling electricity input (W) CoolFuelInput is the cooling fuel input (W) CoolingLoad is the current cooling load on the chiller (W) CPLR is the cooling part-load ratio = CoolingLoad / AvailCoolCap HeatingLoad is the current heating load on the chiller heater (W) HFIR is the user input “Fuel Input to Heating Output Ratio” HPLR is the heating part-load ratio = HeatingLoad / AvailHeatCap MinPLR is the user input “Minimum Part Load Ratio” NomCoolCap is the user input “Nominal Cooling Capacity” (W) RunFrac is the fraction of time step which the chiller is running Tcond is the entering or leaving condenser fluid temperature (∘C). For a water-cooled condenser this will be the water temperature returning from the condenser loop (e.g., leaving the cooling tower) if the entering condenser fluid temperature option is used. For air- or evap-cooled condensers this will be the entering outdoor air dry-bulb or wet-bulb temperature, respectively, if the entering condenser fluid temperature option is used. Tcw,l is the leaving chilled water temperature (∘C). Five performance curves are used in the calculation of cooling capacity and efficiency: 1) Cooling Capacity Function of Temperature Curve 2) Fuel Input to Cooling Output Ratio Function of Temperature Curve 3) Fuel Input to Cooling Output Ratio Function of Part Load Ratio Curve 4) Electric Input to Cooling Output Ratio Function of Temperature Curve 5) Electric Input to Cooling Output Ratio Function of Part Load Ratio Curve The cooling capacity function of temperature (CoolCapfT) curve represents the fraction of the cooling capacity of the chiller as it varies by temperature. This a biquadratic curve with the input variables being the leaving chilled water temperature and either the entering or leaving condenser fluid temperature. The output of this curve is multiplied by the nominal cooling capacity to give the full-load cooling capacity at specific temperature operating conditions (i.e., at temperatures different from the design temperatures). The curve should have a value of 1.0 at the design temperatures and flow rates specified in the input data file by the user. The biquadratic curve should be valid for the range of water temperatures anticipated for the simulation. The available cooling capacity of the chiller is then computed as follows: The fuel input to cooling output ratio function of temperature (CFIRfT) curve represents the fraction of the fuel input to the chiller at full load as it varies by temperature. This a biquadratic curve with the input variables being the leaving chilled water temperature and either the entering or leaving condenser fluid temperature. The output of this curve is multiplied by the nominal fuel input to cooling output ratio (CFIR) to give the full-load fuel input to cooling capacity ratio at specific temperature operating conditions (i.e., at temperatures different from the design temperatures). The curve should have a value of 1.0 at the design temperatures and flow rates specified in the input data file by the user. The biquadratic curve should be valid for the range of water temperatures anticipated for the simulation. The fuel input to cooling output ratio function of part load ratio (CFIRfPLR) curve represents the fraction of the fuel input to the chiller as the load on the chiller varies at a given set of operating temperatures. The curve is normalized so that at full load the value of the curve should be 1.0. The curve is usually linear or quadratic. The fraction of the time step during which the chiller heater is operating is computed as a function of the cooling and heating part-load ratios and the user-input minimum part-load ratio: The cooling fuel input to the chiller is then computed as follows: The electric input to cooling output ratio as function of temperature (CEIRfT) curve represents the fraction of electricity to the chiller at full load as it varies by temperature. This a biquadratic curve with the input variables being the leaving chilled water temperature and either the entering or leaving condenser fluid temperature. The electric input to cooling output ratio function of part load ratio (CEIRfPLR) curve represents the fraction of electricity to the chiller as the load on the chiller varies at a given set of operating temperatures. The curve is normalized so that at full load the value of the curve should be 1.0. The curve is usually linear or quadratic. The cooling electric input to the chiller is computed as follows: All five of these cooling performance curves are accessed through EnergyPlus’ built-in performance curve equation manager (objects Curve:Linear, Curve:Quadratic and Curve:Biquadratic). It is not imperative that the user utilize all coefficients in the performance curve equations if their performance equation has fewer terms (e.g., if the user’s CFIRfPLR performance curve is linear instead of quadratic, simply enter the values for a and b, and set coefficient c equal to zero). The condenser load is computed as follows: The following nomenclature is used in the heating equations: AvailHeatCap is the available full-load heating capacity at current conditions (W) CPLRh is the cooling part-load ratio for heating curve = CoolingLoad / NomCoolCap HeatCapfCPLR is the heating capacity factor as a function of cooling part load ratio, equal to 1 at zero cooling load, user input “Heating Capacity Function of Cooling Capacity Curve Name” HeatCoolCapRatio is the user input “Heating to Cooling Capacity Ratio” HeatElectricPower is the heating electricity input (W) HeatFuelInput is the heating fuel input (W) HeatingLoad is the current heating load on the chiller (W) HEIR is the user input “Electric Input to Heating Output Ratio” HFIR is the user input “Fuel Input to Heating Output Ratio” HFIRfHPLR is the fuel input to heating output factor, equal to 1 at full load, user input “Fuel Input to Heat Output Ratio During Heating Only Operation Curve Name” HPLR is the heating part-load ratio = HeatingLoad / AvailHeatCap MinPLR is the user input “Minimum Part Load Ratio” NomCoolCap is the user input “Nominal Cooling Capacity” (W) RunFrac is the fraction of time step which the chiller is running TotalElectricPower is the total electricity input (W) TotalFuelInput is the total fuel input (W). Cooling is the primary purpose of the Direct Fired Absorption Chiller so that function is satisfied first and if energy is available for providing heating that is provided next. The two performance curves for heating capacity and efficiency are: 1) Heating Capacity Function of Cooling Capacity Curve 2) Fuel-Input-to Heat Output Ratio Function The heating capacity function of cooling capacity curve (HeatCapfCool) determines how the heating capacity of the chiller varies with cooling capacity when the chiller is simultaneously heating and cooling. The curve is normalized so an input of 1.0 represents the nominal cooling capacity and an output of 1.0 represents the full heating capacity. An output of 1.0 should occur when the input is The available heating capacity is then computed as follows: The fuel input to heat output ratio curve (HFIRfHPLR) function is used to represent the fraction of fuel used as the heating load varies as a function of heating part load ratio. It is normalized so that a value of 1.0 is the full available heating capacity. The curve is usually linear or quadratic and will probably be similar to a boiler curve for most chillers. The fuel use rate when heating is computed as follows: The fraction of the time step during which the chiller is operating is computed as a function of the cooling and heating part-load ratios and the user-input minimum part-load ratio: The heating electric input to the chiller is computed as follows: If the chiller is delivering heating and cooling simultaneously, the parasitic electric load will be double-counted, so the following logic is applied: IF ( HeatElectricPower < = CoolElectricPower ) THEN HeatElectricPower = 0.0 HeatElectricPower = HeatElectricPower - CoolElectricPower The total fuel and electric power input to the chiller is computed as shown below: This model (object name ChillerHeater:Absorption:DoubleEffect) simulates the performance of an exhaust fired two-stage (double effect) absorption chiller with optional heating capability. The model is based on the direct fired absorption chiller model (ABSORG-CHLR) in the DOE-2.1 building energy simulation program. The EnergyPlus model contains all of the features of the DOE-2.1 chiller model, plus some additional capabilities. The model uses the exhaust gas output from Microturbine. This model simulates the thermal performance of the chiller and the thermal energy input to the chiller. This model does not simulate the thermal performance or the power consumption of associated pumps or cooling towers. This auxiliary equipment must be modeled using other EnergyPlus models (e.g. Cooling Tower:Single Speed). Model Description[LINK] The chiller model uses user-supplied performance information at design conditions along with five performance curves (curve objects) for cooling capacity and efficiency to determine chiller operation at off-design conditions. Two additional performance curves for heating capacity and efficiency are used when the chiller is operating in a heating only mode or simultaneous cooling and heating mode. The following nomenclature is used in the cooling equations: AvailCoolCap is the available full-load cooling capacity at current conditions (W) CEIR is the user input “Electric Input to Cooling Output Ratio” CEIRfPLR is the electric input to cooling output factor, equal to 1 at full load, user input “Electric Input to Cooling Output Ratio Function of Part Load Ratio Curve Name” CEIRfT is the electric input to cooling output factor, equal to 1 at design conditions, user input “Electric Input to Cooling Output Ratio Function of Temperature Curve Name” TeFIR is the user input “Thermal Energy Input to Cooling Output Ratio” TeFIRfPLR is the thermal energy input to cooling output factor, equal to 1 at full load, user input “Thermal Energy Input to Cooling Output Ratio Function of Part Load Ratio Curve Name” TeFIRfT is the thermal energy input to cooling output factor, equal to 1 at design conditions, user input “Thermal Energy Input to Cooling Output Ratio Function of Temperature Curve Name” CondenserLoad is the condenser heat rejection load (W) CoolCapfT is the cooling capacity factor, equal to 1 at design conditions, user input “Cooling Capacity Function of Temperature Curve Name” CoolElectricPower is the cooling electricity input (W) CoolThermalEnergyInput is the cooling thermal energy input (W) CoolingLoad is the current cooling load on the chiller (W) CPLR is the cooling part-load ratio = CoolingLoad / AvailCoolCap HeatingLoad is the current heating load on the chiller heater (W) HFIR is the user input “Thermal Energy Input to Heating Output Ratio” HPLR is the heating part-load ratio = HeatingLoad / AvailHeatCap ˙mExhAir is the exhaust air mass flow rate from microturbine (kg/s) MinPLR is the user input “Minimum Part Load Ratio” NomCoolCap is the user input “Nominal Cooling Capacity” (W) RunFrac is the fraction of time step which the chiller is running Ta,o is the exhaust air outlet temperature from microturbine entering the chiller (∘C) Tabs,gen,o is the temperature of exhaust leaving the chiller (the generator component of the absorption chiller) Tcond is the entering condenser fluid temperature (∘C). For a water-cooled condenser this will be the water temperature returning from the condenser loop (e.g., leaving the cooling tower). For air- or evap-cooled condensers this will be the entering outdoor air dry-bulb or wet-bulb temperature, respectively. Tcw,l is the leaving chilled water temperature (∘C) The selection of entering or leaving condense fluid temperature can be made through the optional field-Temperature Curve Input Variable. Five performance curves are used in the calculation of cooling capacity and efficiency: 1) Cooling Capacity Function of Temperature Curve 2) Thermal Energy Input to Cooling Output Ratio Function of Temperature Curve 3) Thermal Energy Input to Cooling Output Ratio Function of Part Load Ratio Curve 4) Electric Input to Cooling Output Ratio Function of Temperature Curve 5) Electric Input to Cooling Output Ratio Function of Part Load Ratio Curve The cooling capacity function of temperature (CoolCapfT) curve represents the fraction of the cooling capacity of the chiller as it varies with temperature. This a biquadratic curve with the input variables being the leaving chilled water temperature and the entering condenser fluid temperature. The output of this curve is multiplied by the nominal cooling capacity to give the full-load cooling capacity at specific temperature operating conditions (i.e., at temperatures different from the design temperatures). The curve should have a value of 1.0 at the design temperatures and flow rates specified in the input data file by the user. The biquadratic curve should be valid for the range of water temperatures anticipated for the simulation. The available cooling capacity of the chiller is then computed as follows: The thermal energy input to cooling output ratio function of temperature (TeFIRfT) curve represents the fraction of the thermal energy input to the chiller at full load as it varies with temperature. This a biquadratic curve with the input variables being the leaving chilled water temperature and the entering condenser fluid temperature. The output of this curve is multiplied by the nominal thermal energy input to cooling output ratio (TeFIR) to give the full-load thermal energy input to cooling capacity ratio at specific temperature operating conditions (i.e., at temperatures different from the design temperatures). The curve should have a value of 1.0 at the design temperatures and flow rates specified in the input data file by the user. The biquadratic curve should be valid for the range of water temperatures anticipated for the simulation. The thermal energy input to cooling output ratio function of part load ratio (TeFIRfPLR) curve represents the fraction of the thermal energy input to the chiller as the load on the chiller varies at a given set of operating temperatures. The curve is normalized so that at full load the value of the curve should be 1.0. The curve is usually linear or quadratic. The fraction of the time step during which the chiller heater is operating is computed as a function of the cooling and heating part-load ratios and the user-input minimum part-load ratio: The cooling thermal energy input to the chiller is then computed as follows: To make sure that the exhaust mass flow rate and temperature from microturbine are sufficient to drive the chiller, the heat recovery potential is compared with the cooling thermal energy input to the chiller (CoolThermalEergyInput). The heat recovery potential should be greater than the CoolThermalEnergyInput. Heat recovery potential is calculated as: Tabs,gen,o is the minimum temperature required for the proper operation of the double-effect chiller. It will be defaulted to 176∘C. The electric input to cooling output ratio as function of temperature (CEIRfT) curve represents the fraction of electricity to the chiller at full load as it varies with temperature. This a biquadratic curve with the input variables being the leaving chilled water temperature and either the entering or leaving condenser fluid temperature. The electric input to cooling output ratio function of part load ratio (CEIRfPLR) curve represents the fraction of electricity to the chiller as the load on the chiller varies at a given set of operating temperatures. The curve is normalized so that at full load the value of the curve should be 1.0. The curve is usually linear or quadratic. The cooling electric input to the chiller is computed as follows: All five of these cooling performance curves are accessed through EnergyPlus’ built-in performance curve equation manager (objects Curve:Linear, Curve:Quadratic and Curve:Biquadratic). It is not imperative that the user utilize all coefficients in the performance curve equations if their performance equation has fewer terms (e.g., if the user’s TeFIRfPLR performance curve is linear instead of quadratic, simply enter the values for a and b, and set coefficient c equal to zero). A set of curves derived from manufacturer’s data are also provided in the dataset (ExhaustFiredChiller.idf) which is provided with the standard EnergyPlus installation. The condenser load is computed as follows: The following nomenclature is used in the heating equations: AvailHeatCap is the available full-load heating capacity at current conditions (W) CPLRh is the cooling part-load ratio for heating curve = CoolingLoad / NomCoolCap HeatCapfCPLR is the heating capacity factor as a function of cooling part load ratio, equal to 1 at zero cooling load, user input “Heating Capacity Function of Cooling Capacity Curve Name” HeatCoolCapRatio is the user input “Heating to Cooling Capacity Ratio” HeatElectricPower is the heating electricity input (W) HeatThermalEnergyInput is the heating thermal energy input (W) HeatingLoad is the current heating load on the chiller (W) HEIR is the user input “Electric Input to Heating Output Ratio” HFIR is the user input “Thermal Energy Input to Heating Output Ratio” HFIRfHPLR is the thermal energy input to heating output factor, equal to 1 at full load, user input “Thermal Energy Input to Heat Output Ratio During Heating Only Operation Curve Name” HPLR is the heating part-load ratio = HeatingLoad / AvailHeatCap MinPLR is the user input “Minimum Part Load Ratio” NomCoolCap is the user input “Nominal Cooling Capacity” (W) RunFrac is the fraction of time step which the chiller is running TotalElectricPower is the total electricity input (W) TotalThermalEnergyInput is the total thermal energy input (W) Cooling is the primary purpose of the Exhaust Fired Absorption Chiller so that function is satisfied first and if energy is available for providing heating that is provided next. The two performance curves for heating capacity and efficiency are: 1) Heating Capacity Function of Cooling Capacity Curve 2) Thermal Energy Input to Heat Output Ratio Function The heating capacity function of cooling capacity curve (HeatCapfCPLR) determines how the heating capacity of the chiller varies with cooling capacity when the chiller is simultaneously heating and cooling. The curve is normalized so an input of 1.0 represents the nominal cooling capacity and an output of 1.0 represents the full heating capacity. An output of 1.0 should occur when the input is The available heating capacity is then computed as follows: The thermal energy input to heat output ratio curve (HFIRfHPLR) function is used to represent the fraction of thermal energy used as the heating load varies as a function of heating part load ratio. It is normalized so that a value of 1.0 is the full available heating capacity. The curve is usually linear or quadratic and will probably be similar to a boiler curve for most chillers. The thermal energy use rate when heating is computed as follows: The fraction of the time step during which the chiller is operating is computed as a function of the cooling and heating part-load ratios and the user-input minimum part-load ratio: The heating electric input to the chiller is computed as follows: If the chiller is delivering heating and cooling simultaneously, the parasitic electric load would be double-counted, so the following logic is applied: If HeatElectricPower is less than or equal to CoolElectricPower:	{"url":"https://bigladdersoftware.com/epx/docs/9-4/engineering-reference/chillers.html","timestamp":"2024-11-02T06:19:57Z","content_type":"text/html","content_length":"1048935","record_id":"<urn:uuid:83ec155f-5b44-4b40-9175-88b3f474835f>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00407.warc.gz"}
Why the 15c should not come back OK, I got your attention! The 15c is a wonderful machine. after using a 32s, do you really think it makes sense to have keycodes instead of actual function names? after using a 32sii, where you can enter many algebraic expressions into the solver, do you really want to have to program, in RPN, line by line, for every function solve or integrate problem? after trying an editable Formula or true algebraic machine, such as a sharp pc 1500, (or even a Casio fx 115ms plus, or even HP30s), do you really really see an advantage to having a limited stack, intermediate operation machine, with no ablility to recall and edit and re-use an equation or expression, without turning it inside out in RPN? (The 17bii gives you the same feeling only better---editable, long variables, fully algebraic solver, returns the values to the stack...) Afer the 33s, with its 32 kb (8 kb really when efficiency isconsidered) do you really think there is a reason to spend real money on a scientific calculator with less than 500 bytes? After RPL, where you can evaluate an algebraic, DUP it, operate on it with stack logic, save it, edit it, merge it with other algebraics, re-use it, etc, is there really any reason for RPN? Afer you have tried all of these machines, seen what is possible, enjoyed the absolutely beautiful design of the 15c, savored its remarkable power, acclimated to and internalized its keycodes...... Now, think logically, clearly, and efficiently for the future....why would you spend real money on a new device (without true collector value both nostalgic and monetary) with the limitations of the The future is not RPN. But a stack could (and should even) be a part of the future. Algebraic objects should be normal standard fare---and they should be editable. Memory should be plentiful. Variables and registers, too. I really like the 15c. I enjoy it. I use it regularly. I greatly appreciate Valentin's challenges, Karl's tips---- because I have one, and know it, I will use it. I like the user mode---I put what I use frequently under labels a through E and have direct access--its great! But I can see that it is antiquated--a bit obtuse if you don't already know it......there is no reason to really produce it anymore in real nuimbers....you can do so much better! Why should a person learn math, and then have to go learn a new way of writing math? Why shouldn't the rules of precedence, the notation, etc be the same as it is for maths? Why should a user have to be molded by the arbitrary limits of the machine's architecture, when it is perfectly possible with available technology, to develop an interface which is consistent with mathematical notation? (I will note that most "algebraic" half-breeds (not good formula machines) are even worse when it comes to distortion--and the "4 function" types are abominable--as are old Ti's where clearing or overtyping after an incoorect operator will give unexpected results....at least RPN is consistent.) Rather than bringing back the 15c, I think we should push, through openRPN, to develop a truly powerful machine, which is capable of working in the new paradigm. Best regards, 07-21-2004, 03:59 PM All good points, and I agree that the 15c unchanged and unimproved would suffer from the failings you mention, buuuut: Like the HP33s, toss in gobs of cheap RAM. Give it a dot matrix display and basically beef it up to an Hp42s in a 15c housing. Toss in a cheap USB interface and file transfer. Give it some units conversions. And most of all keep it at its present size. Price it just a bit below the Hp48Gii or same price as the Hp 17Bii+ (ie retail approx $99 US or even better, less). Now you have a real pocket calculator. I would buy a couple. Would it have shortcomings, yes. But they would be shortcomings due to choice and technology, not bean counting and gimickery to appease market share or to not rob from Hp's precious graphics calculator sales. Dot matrix LCD's aren't as nice and crisp to view as the segmented display, but gives you lots more options. Toss in an alpha toggle key and a two line display like the 42s with USB and a SDK to develop programs on the PC and you eliminate concern about the awkward keyboard. Many people complain about this on the 42s, but if there were a serial port, that would eliminate most complaints. Just me Ranting and Raving, Again. I want a pocket calculator, not a graphics monster. 07-21-2004, 04:40 PM That's exactly my point. What we like most about the 15C are its form factor, its highly usable keyboard and display, and most of all its conceptual integrity. But IT IS true that keycodes are somewhat outdated, and at the time when the 15C was issued, were my primary reason to go for a casio 602p instead - programmability is so much simpler with readable ASCII op-opcodes. Well, in the meantime I got two 15C (along with a 2nd 602p), and as a professional EE I also admit that programmability has become less important than built-in matrix and complex numbers cability, but anyway, my point is: An improved 15C with dot matrix display (with equal contrast of course) and somewhat more memory would be the perfect machine for any practicing engineer. Students wouldn't go for it, but nobody said we need one calc for everybody. BTW, I remember having seen a mockup of an improved 15C (with the golden faceplate taken from the 12c) somewhere on the internet, meant as an april fools joke - it actually came pretty close to what I imagine as my dream machine. Anybody remember that link? 07-21-2004, 05:01 PM There is no reason to assume that I am suggesting that the form factor should grow from the original 15c target. There is also no reason to assume that because I seem to favor RPL, that it should be In fact, I use my RPL 48Gx far less than my regular machines--because the functions are "buried". I like the RPL stack, the Algebraic power, but find the lack of convenient access to many functions a RPL can be applied to a 2-line, non-graphing scientific. RPL could easily be modified to allow for stack depth control, or ability to load a constant, or even some block of standard memory "registers." I find myself using the 30s a lot now, though---and that is my point----if you bother to step outside our traditional RPN world, you will discover that there is so much more possible, right now, with available technology, that will make you never want to be pure RPN ever again. Until you have tried it, given it a chance, you really cannot appreciate (I didn't!) the power of being able to put in algebraic expressions, edit them, find them again, re-use them etc. Why shouldn't a modern, relatively expensive machine enable this? By way of comparison, I immediately saw the value of the equation list in my 32sii, when it replaced my 11c. I started to write equations instead of programs, for many tasks. It was easier to see what the "program" was for, when it is an equation. This was in 1995. And there was no editing allowed, and it still made sense! Then I found the 48 series--and was frustrated by too much horsepower and many simple things buried. But Algebraics a big plus---and if only my head were more pliable, the programming style might have been good, too! Then, I came across the 17bii---and the equation solver there blows the 32sii away! When I saw that the 30s allowed editing of an algebraic expression, and saved the last so many expressions, I became even more fascinated by the possibilities. Note that to a Sharp or Casio user, my discoveries are ridiculously old---like, all this was possible 15 years ago! For a good current screen, take a look at the 30s (minus the glary plastic over-screen). You have two lines, dot-matrix upper, segmented lower, big characters, adjustable display, very good contrast, and provision for alphanumerics. Much nicer than the 33s---better by far than the 48. What I am trying to do is to stop "copping out" with "I would be just fine with such and so." While that might work for us old folks, the fact is that the next generation should not be saddled with new products which are not as good as possible. (This is what bothers me most about the 33s---like, why not make the equation list editable!!!?). 07-21-2004, 05:12 PM And that is why I think the 27s is the 2nd best pocket calculator ever made. I prefer RPN, but the solver and the amount of memory available to the 27s is still unmatched by any other scientific pocket calculator except the 42s. If it had an RPN mode, it would have been a great favorite among the RPN crowd due to its simple to use solver (the very same as the 17Bii that you mention). It is just so simple to use, and that is what makes it both easy and powerful. 07-21-2004, 07:42 PM I own the HP 48gx, 42s, 17bII, 32sII, 15c, 12c and I have to say the 15c is still my day to day favourite, actually purchased a backup for the day when my original stops working. As others have said before, I can't see the same model going back into production but something of similar form factor with extra memory and two line dot-matrix display would be good. 07-22-2004, 02:21 AM Bill -- Interesting "short essays"; I have a few points: I agree: Owing to their limitations, the 15C (and 11C) probably would be not very successful in today's marketplace. As a comparison, the 1966 Ford Mustang is a classic design that remains common and relatively popular even today. However, even if it met modern emissions and safety standards, it wouldn't sell well as a re-released car. It is antiquated -- more so than the 15C -- in many respects. When I saw that the 30s allowed editing of an algebraic expression, and saved the last so many expressions, I became even more fascinated by the possibilities. Note that to a Sharp or Casio user, my discoveries are ridiculously old---like, all this was possible 15 years ago! Ah, but the 27S and 17B with the solver of algebraic expressions are also 15 years old! For a good current screen, take a look at the 30s (minus the glary plastic over-screen). You have two lines, dot-matrix upper, segmented lower, big characters, adjustable display, very good contrast, and provision for alphanumerics. Much nicer than the 33s---better by far than the 48. No argument here with the displays of the 33S and the 48-series. I've toyed with the 30S (two collegues have them, one to replace his employer-issued 28S that finally broke), and I don't share your enthusiasm for it. I find the digits too narrow in proportion to height -- Voyagers, and all Pioneers without the two-line display are more legible. I also don't find its "pure algebraic" system (shared by Casio and TI) very much of a help: • To perform a calculation on a previous result, you do [shift][Ans]...[=]. Heck, it was easier with RPN and old "hybrid algebraic". • Proofreading a long, parenthesis-laden expression isn't simple, either -- you can't see the whole expression; braces and brackets can't be used to distinguish pairs. 07-22-2004, 04:56 AM Hi all, Just to add to these very good comments, one thing that strikes me is the very small amount of machines that can do both formula programming AND have control structures for looping and branching. To my knowledge, the only non-graphing machines that falls into this category were: 1) The fx-4000P from Casio, made in... 1985 ! 2) The 32SII from HP, only made in 1991. And yet, this machine does not have editing capabilities... 3) The 67 Galaxy, made in 1992 only. There were some descendants of these machines, e.g. the Casio fx-4500P or the HP-33S, but still, the offer was quite limited. It is a pity, as I feel that formula programming only is too limited, while keystroke programming only is a pain to deal with complex equations. Programmable calculators seem to be out of fashion... That makes me think that Casio is sometimes rated unfairly. Think that, back in 1981, the fx-602P was a very good alternative to the HP-41, cheaper and faster; than as early as 1985, Casio invented both the formula + structure programmable with the FX-4000P AND the graphing calculator with the FX-7000G. I was a high school studend in 1988 and Casio was at the time the vendor of choice for us (I had a FX-8000G which I still like today). Only latter in 1990 I discovered HP with the fabulous HP-48SX. The fx-4000P was a wonderful nice little machine, with equation editing and replay, 550 steps memory, 96 registers with indirect addressing, and was even smaller than the HP-15C... Those were the days. I do agree that formula programming is missed on some HP wonderful machines, such as the 41C or the 15C. The ACE ROM does some effort for the 41C with 'PROG', but no editing, no replay, no integration with solver... HP has yet to produce something like the 4000P, inegrating the best of Casio (equation editing) and HP (infinite number of alphanumeric labels, multi-variable solver...). That would be a hit ! Long live Casio ! :) 07-22-2004, 10:46 AM Hp did indeed come out with that calculator. My wife had a Casio fx 4000 and really liked it. She later switched over to the Hp 27s. It is actually better than the Casio for everything except fitting in your pocket (the 27s fits in your pocket, just not as nicely). 07-22-2004, 06:10 PM May I respectfully disagree - the HP27S is a wonderful machine which I respect a lot, but is -not- a true programmable... IF and SOLVE are nice functions, but do not allow everything that the keystrokes programmable can do... My point is, there are a lot of keystroke programmable calculators out there, and there are tons of formula programmable too. But there are very few calcs than can do both ! 07-23-2004, 04:18 AM I have both the FX-4000P and the FX-4500P as well as a TI-67 Galaxy. I like the 4000P. The 4500 has a cramped keyboard, even worse than a 32SII. Pretty much impossible to find anything quickly, athough it looks interesting. The TI has a nice form factor, but IMO the process to store and recall equations is cumbersome. Also, its display is hard to read with a decimal point too small to The disadvantage of all three calculators and the reason I don't use them regularly is the algebraic input. Typing [SQRT] Ans Exe just to calculate the root of what's in the display isn't very intuitive. As for programming, I think it depends much on what kind of program you like to write. Formula-like problems are easier to implement, although I think keystroke programming is easier to learn. However, more complex branching is easier with keystroke programming (however it's even easier with a real programming language like Basic). I like the Casios, my first programmable was a 602P, too. And the 4000P and 4500P are dirt cheap nowadays. I bought mine on ebay for 2 and 10 Euros, respectively. But the most usable calc I have is without doubt the 42S while the most beautiful and fascinating is the 41CX. 07-23-2004, 11:13 AM Hi Holger, I largly agree with your comments. I am fascinated with the 41CX and I love the 42S too, but I still think that both lack an equation mode a la 32SII, with editing capabilities. Nothing is perfect: The 32SII really shines on some points (Equations, fraction mode, ease of use, speed) but is a real move backwards in some others (cramp memory, primitive complex numbers, lack of labels and registers). My dream would be a 42S + equation mode with editing + proper alphanumeric keyboard + franction mode + USB i/o. That would be my dream calc... I also think I forgot some interesting SHARP non-basic models in the very small club of non-graphing formula + control structure calculators. So, I think the complete list of such calcs are: - Casio: fx-3900P, fx-4000P, fx-4500P, fx-4800P - HP: 32SII, 33S - TI: 67 Galaxy - Sharp: EL-5050 (very nice dual keyboard, I would love to own one; the EL-5150 is horizontal version, but is a large a BASIC computer), EL-512H & EL-512II (more limited), and EL-5120 (too modern for my taste). That's a total of 12 machines in the whole calculator history, out of which only 4 (fx-4000P, 32SII, 67 Galaxy and EL-5050) are really interesting IMHO. Something the market should re-explore ! 07-23-2004, 02:28 PM I am curious - posters to this forum discuss in detail calculators from HP and often mention calculators from TI, Sharp and Casio. Are there currently any European companies that produce calculators? 07-26-2004, 02:39 PM Hi Holger, You mention an EL-5050, 5150, and 5120. Do you know anything about the EL-5020? Especially, where to find a (or copy of) manual? Best regards, 07-22-2004, 12:05 PM I disagree. I don't have a 42S, but based on its features, it is the only calculator that might surpass the 15C as a pocket calcultor for this old engineer. I had a 32SII at work for a while, but never liked it as well. In the day of the PC, my use of a calculator is perfectly met by the limited programability of the 15C. Its size, form, quality, durability, efficiency, and RPN are for me simply what a calculator is all about. There are simply too many tools that are just plain better at most of the other aspects of data aquisition, analysis, number crunching, and presentation than a calculator. Certainly other folks have different needs than I, but if my 12C, 15C, 41CV, 41CX, (many 41 modules) and 71B all were stolen or destroyed, the only replacement I could justify on the basis of need would be the 15C. The future may not be RPN, but for the next 5 or 10 years, the best calc tool for my job is the 15C. Bill S. 07-22-2004, 11:32 PM I certainly believe OpenRPN will produce the great calculators of the future. When I set out to found the project my goal was to make the calculator that you take everywhere you go, that does everything you need of it to do. Beyond that it must be the most durable calculator ever (if I can pull it off I would love to design it to be run over by a truck!). Ultimately, I think everyone will be impressed as soon as the first prototypes start to appear. Thanks to all for your support.	{"url":"https://archived.hpcalc.org/museumforum/thread-60966-post-61099.html#pid61099","timestamp":"2024-11-04T01:49:28Z","content_type":"application/xhtml+xml","content_length":"80513","record_id":"<urn:uuid:f39a04c4-fd97-4140-a468-6b4575d6c709>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00721.warc.gz"}
Lumped Parameter Model 6.2 Lumped Parameter Models The control theory approach has been successfully applied to simulate karst aquifers using mathematical methods for solving sets of ordinary differential equations. For example, the control theory approach is used to evaluate the filling and mixing of water in a tank, or a network of tanks, and a karst basin can be conceptualized as a group of interconnected tanks. The tanks are not assigned spatial coordinates as is done for cells of distributed parameters models, rather flow between tanks is based only on connections assigned by the modeler. Coefficients (that is, lumped parameters) for each tank are developed by calibrating a system of equations describing the connections between tanks (Wanakule and Anaya, 1993; Hartmann et al., 2014). Data preparation for lumped parameter models is simpler than that required for distributed parameter models, and computational time is shorter. However, detailed representation of the aquifer is not possible using this approach. Groundwater withdrawals and recharge are summed together for each tank which represents a geographic area (for example, a spring basin). Withdrawals from the model simulate spring discharge and inputs to the model may represent distributed infiltration, sink holes or sinking streams. Lumped parameter models may be adequate for providing gross estimates of the effects of changes in pumpage and/or recharge rates on spring discharge, as well as, for estimating recharge given natural discharge and pumpage. Mathematical filters can be applied to some of the input data, for example, to divide the annual basin recharge into varying recharge for each month, in order to achieve a better fit between observed and simulated spring discharge (Wanakule and Anaya, 1993; Dreiss, 1982). Many of the geochemical mixing models, such as piston flow, binary mixing, or exponential mixing models applied to atmospheric environmental tracers are also used for estimation of groundwater age. These are forms of lumped parameter models (Böhlke and Denver, 1995; Cook and Herzog, 2000). Example Application of a Lumped Parameter Model The lumped parameter modeling method is demonstrated in a simulation of spring flow at Comal and San Marcos Springs, Texas. The model input and calibration data were based on annual estimates of recharge and pumpage in nine surface watershed basins and an index water level in each sub-basin of the San Antonio part of the Edwards Aquifer (Wanakule and Anaya, 1993). The lumped parameter model of Wanakule and Anaya (1993) was one of the earliest applications of such models to the Edwards Aquifer. Barret and Charbeneau (1997) applied the lumped parameter method to the Barton Springs part of the Edwards Aquifer, but it was simpler because there were fewer basins, so the earlier model is used as an example in this book. Conceptually, each of the nine watershed basins is treated as an interconnected tank. The lumped parameter mathematical relation was developed much like a statistical regression model where recharge and pumpage in each lumped parameter block (tank) were treated as input to a set of linked tanks that transport water to the major springs. The mathematical description of each tank was formulated with an ordinary differential equation. Figure 67 shows the Edwards Aquifer and catchment area as well as the conceptualized tanks. Surface runoff from the catchment area infiltrates the Edwards Aquifer across its outcrop. Wanakule and Anaya (1993) used the recharge and pumpage data as inputs to the system. They developed filters to break up the annual estimates of recharge into monthly estimates using stream gage data and annual recharge estimates. Additionally, the monthly pumpage by county was reapportioned to each sub-watershed (Wanakule and Anaya, 1993). These values were then matched to average monthly groundwater levels in each basin and to the historical spring discharge at Comal and San Marcos Springs by calibrating values of parameters related to storage and transmissivity for each tank. The filtering method for the disaggregation of recharge falls into category 1 (fitting models), which includes time series techniques. Wanakule and Anaya (1993) were successful in refining estimates of monthly recharge and pumpage for each basin and simulating the spring discharge at Comal and San Marcos Springs (Figure 68) as well as water levels in each basin. which are not shown here, but were presented by Wanakule and Anaya (1993). Schulman and others (1995) developed equations for stochastically generating recharge volumes for the watershed basins. The equations have the same statistical properties as the historical (since 1934) recharge data for each basin. The wellbeing of endangered species at the springs are a concern, and the courts have required minimum discharges to be maintained at both springs. Given these minimums, groundwater use is restricted at times during summer months even as the population continues to grow. Thus, water-resource planners recognize that water may need to be imported to maintain discharge at the springs. Because of its computational speed and simplicity, the calibrated lumped parameter model of Wanakule and Anaya (1993), combined with the generated recharge scenarios, were used to screen water-supply options for the Edwards Aquifer (Watkins and McKinney, 1999).	{"url":"https://books.gw-project.org/introduction-to-karst-aquifers/chapter/lumped-parameter-models/","timestamp":"2024-11-14T23:27:20Z","content_type":"text/html","content_length":"83768","record_id":"<urn:uuid:607f6785-f95e-4d77-b969-df2889eddc0a>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00216.warc.gz"}
To Keep Students in STEM fields, Let’s Weed Out the Weed-Out Math Classes All routes to STEM (science, technology, engineering and mathematics) degrees run through calculus classes. Each year, hundreds of thousands of college students take introductory calculus. But only a fraction ultimately complete a STEM degree, and research about why students abandon such degrees suggests that traditional calculus courses are one of the reasons. With scientific understanding and innovation increasingly central to solving 21st-century problems, this loss of talent is something society can ill afford. Math departments alone are unlikely to solve this dilemma. Several of the promising calculus reforms highlighted in our report Charting a New Course: Investigating Barriers on the Calculus Pathway to STEM, published with the California Education Learning Lab, were spearheaded by professors outside of math departments. It's time for STEM faculty to prioritize collaboration across disciplines to transform math classes from weed-out mechanisms to fertile terrain for cultivating a diverse generation of STEM researchers and professionals. This is not uncharted territory. In 2013, life sciences faculty at the University of California, Los Angeles, developed a two-course sequence that covers classic calculus topics such as the derivative and the integral, but emphasizes their application in a biological context. The professors used modeling of complex systems such as biological and physiological processes as a framework for teaching linear algebra and a starting point for teaching the basics of computer programming to support students’ use of systems of differential equations. Creating this course, Mathematics for Life Scientists, wasn’t easy. The life sciences faculty involved, none of whom had a joint appointment with the math department, said they resorted to designing the course themselves after math faculty rebuffed their overture. The math faculty feared creating a ”watered-down” course with no textbook (though after the course was developed, one math instructor taught some sections of the class). Besides math, the life sciences faculty said they experienced “significant pushback” from the chemistry and physics departments over concerns that the course wouldn’t adequately prepare students for required courses in those disciplines. But the UCLA course seems to be successful, and a textbook based on it now exists. According to recently published research led by UCLA education researchers, students in the new classes ended up with “significantly higher grades” in subsequent physics, chemistry and life sciences courses than students in the traditional calculus course, even when controlling for factors such as demographics, prior preparation and math grades. Students’ interest in the subject doubled, according to surveys. The UCLA example highlights longstanding concerns about the relevance of traditional calculus for biology students. Traditionally, college math departments have overseen general education math courses for students in other majors. A little over two decades ago, biology faculty convened by the Mathematics Association of America advocated that, for biology majors, “statistics, modeling, and graphical representation should take priority” over calculus. But change has been slow, until life sciences departments got involved. Math education researchers consider more relevant and engaging curriculum to be an important strategy for increasing persistence rates, particularly among students traditionally excluded from STEM fields, such as Black, Latinx and Indigenous students, as well as women. Engineering departments also worry about calculus sequences driving attrition. In Ohio, Wright State University’s solution also involved revising math offerings. But rather than changing the content of the calculus course, they focused on preparing students for calculus by emphasizing “engineering motivation for math.” In lieu of traditional calculus prerequisites such as precalculus or college algebra, the engineering faculty launched a contextualized math course in 2004. Emphasizing problem-based learning, the course covers topics students need in sophomore engineering classes, including linear equations, quadratic equations, 2-D vectors and complex numbers. A modest redesign of the engineering curriculum allows students to delay taking a traditional calculus sequence until later in their programs. The approach enhanced opportunities for students with weaker math backgrounds to succeed in engineering and doubled the average graduation rate of engineering students without reducing the average grade of graduates. Students from groups historically underrepresented in STEM (female, Black, and Latinx), experienced the greatest gains, with those in the new problem-based course completing engineering degrees at three times the rate of engineering majors who did not take it. At least 15 other universities are replicating the strategy. Increasingly leading math and science organizations are recognizing the importance of interdisciplinary collaboration; the MAA has a history of convening faculty from partner disciplines, and a National Academies’ 2013 publication called for reassessing math education in a cross-disciplinary context. The National Science Foundation, which funded both the UCLA and Wright State innovations, recognizes the value of cross-disciplinary or “convergence” research, which is driven by a compelling scientific or societal problem. Low persistence in STEM majors and lack of diversity in the STEM fields are themselves pressing societal problems. Yet, math departments without jointly-appointed professors seem to be less interested in evidence-based contributions from other disciplines to enhance the effectiveness of math instruction—or even aware of successes to date. The shift toward more practical applications of calculus is missing one key academic endorsement: publication in widely-read journals, if the success of the courses is examined academically at all. The UCLA research appears in a life sciences education journal, and presentations on Wright State’s innovation reach an American Society for Engineering Education audience. Faculty responsible for undergraduate math education are not likely to see these journals. Ten years ago, at the University of California Berkeley, Lior Pachter used his joint appointments in math and biology to create a new course with the help of his fellow mathematicians and biologists. Methods of Mathematics: Calculus, Statistics, and Combinatorics (Math 10A and 10B), expands on the standard calculus curriculum to reflect how data and technology have transformed the field of biology. Today, most Berkeley life sciences majors take the sequence, and the campus is offering 15 sections of 10B beginning this spring. (Pachter now teaches at the California Institute of Technology, and says Berkeley’s math department has yet to publish any research on the calculus sequence, despite having institutionalized it.) Math learning is fundamental to all STEM fields, but the opposite also appears to be true: the STEM fields may be central to making math learning effective for more students. Involving other STEM disciplines in redesigning math classes is a key way to ensure those classes offer engaging and inclusive on-ramps to STEM.	{"url":"http://huiyangkeji.com/index-442.html","timestamp":"2024-11-02T09:02:32Z","content_type":"text/html","content_length":"93189","record_id":"<urn:uuid:ea4fb22c-8db4-4321-8628-6907965f1695>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00413.warc.gz"}
Going Quantum Last weekend, I was able to participate at iQuHACK 2021, the interdisciplinary Quantum HACKathon. iQuHACK is an annual hackathon organized by MIT. Unlike last year, this year’s event was entirely online so it was open to international participants, many of which in different timezones. With two peers from Hasso-Plattner-Institut, we devoted our weekend to learning about the state of quantum computing and its possibilities. At iQuHACK, there were three divisions: • Quantum Annealing. This was the division we participated in. The hardware was provided by D-Wave and conveniently accessed with the Ocean SDK. • Gate-based. This division used universal quantum hardware consisting of gates that manipulate the qubits. The hardware was provided by IonQ. • Hybrid. This division sought to combine quantum annealing with the gate-based model. What is Quantum Computing in general? Quantum Computing tries to use phenomena that occur at very small scales (quantum scales) for computing. Unlike classical computers which use voltages to represent the boolean values 0 and 1, quantum computers use qubits that can be in states between 0 and 1. These states are called superpositions of 0 and 1. When you measure a qubit, their superposition determines the probability of them being measured as 0 or 1. The real power of qubits comes from a property called entanglement. This means that two or more qubits have interacted in a way that their states cannot be described independently anymore. Just like multiple voltages are combined to bytes, one can combine multiple qubits to describe a larger state space. Different states of a quantum system contain different amounts of energy. This is what is called the energy landscape of the system. Usually, a system will tend to change towards a local energy What is Quantum Annealing? Quantum Annealing is a special algorithm to implement an adiabatic quantum computer. The “adiabatic” comes from the adiabatic theorem of quantum computing . The theorem says that if a change on a quantum system is acting slow enough, the system will stay in the state of minimal energy. To put it concretely, if we put the system in a global energy minimum with a known energy landscape and slowly change that landscape, the system will end up in the global minimum of the second landscape. To solve a problem with quantum annealing, we need to design an energy function (Hamiltonian) which is minimal when the system state is a solution. In contrast, gate-based quantum computing uses quantum logic gates similar to classical electron logic gates to manipulate a quantum state. It then measures and records the qubits to obtain a solution. This process is of inherently discrete nature (the qubits go through one gate at a time) whereas a quantum annealer works in an analog way. These two approaches to quantum computing have shown to be polynomially equivalent, but they each have advantages and disadvantages: Quantum annealers are not universal, i.e. they are not able to perform Shor’s algorithm, for example. Still, they are able to solve certain types of problems very well, even with large amounts of variables. Gate-based quantum computers are universal but still have issues integrating more qubits and reducing noise. How do you formulate problems for D-Wave systems? General problem formulation The primary programming model used is that of a Binary Quadratic Model (BQM). The term binary means that a variable has two possible values. The values used determine th variant of the BQM used. This model comes in two variants: Ising (values -1 and 1) and Quadratic Unconstrained Binary Optimization (QUBO, values 0 and 1) which I talk about here. To create the energy function that will be minimized by the quantum annealer, a bias term is introduced for every variable. Also, couplings between two qubits can be introduced. If $x_1$ and $x_2$ are the binary variables, $a$ and $b$ the biases, $c$ is the coupling term and $d$ is a constant, the energy of the system would be represented as $$E = ax_1 + bx_2 + cx_1x_2 + d$$ A nice metaphor for the model: Imaging having to decide which members of a sports team should play in a certain match. There are stronger and weaker players (bias term). Also, there are combinations of two players that boost the team (positive coupling) or weaken the team (negative coupling). A table for the values of E in terms of $x_1$ and $x_2$ would look like this: $x_1$ $x_2$ $E$ 0 0 $d$ 0 1 $b+d$ 1 0 $a+d$ 1 1 $a+b+c+d$ If you have set values for $E(x_1, x_2)$, it is possible to determine your coefficients $a,b,c,d$ by solving the linear system of equations. That way, you can implement basic logic operations, e.g. AND, OR, XOR etc. From there, implementing a one-hot encoding with additional variables requires a bit of algebra but is no rocket science. When you have your coefficients, you can give them to a D-Wave Sampler which tries to find the energy minimum. Our Project For the hackathon, we accepted a D-Wave challenge. The challenge was about the puzzle Star Battle / Two Not Touch by Krazydad. You can find his intro to the puzzle here. First, we had to decide on how to encode possible solutions. Since every cell of the board could either contain a star or be empty, one binary variable for each cell was appropriate. Second, we had to think about the thing we are trying to minimize. For a problem like the Travelling salesman problem, this would be the distance travelled. Since we did not want to minimize anything but instead comply with all constraints, we moved on to the next step. Third, we needed to find all constraints and encode them into the QUBO. 1. Each block, row and column needs to contain the specified number of stars. This could be done with truth tables and algebra, but we used a D-Wave helper function provided with the Ocean SDK. 2. No blocks are allowed to be adjacent. For this constraint, the energy should be minimal if there is at most one star in two adjacent cells. It should be higher when there are two stars in two adjacent cells. We added constraint 1 for every row, block and column to our model and constraint 2 for every pair of adjacent stars. Our program was able to solve puzzles of easy and medium difficulty correctly 100% of the time and puzzles of hard difficulty 90% of the time. We were very satisfied with the results as we didn’t expect a system with 196 variables system to work as well. Creating a nice user interface Additional to just implementing a solver, we created an interactive user interface so everyone (with a D-Wave Leap API token) could play against the quantum computer. Here are some screenshots of our Our GitHub repository is available under here. It contains a detailed README as well as final presentation slides that explain further how our solver works. Overall, I had a great experience at iQuHACK 2021. I learned a lot about the possibilities and challenges of quantum computing.	{"url":"https://rgwohlbold.de/2021/going-quantum/","timestamp":"2024-11-11T16:56:56Z","content_type":"text/html","content_length":"11154","record_id":"<urn:uuid:6bc0dad3-9163-48da-84fd-1d0ce49c4775>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00827.warc.gz"}