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Introduction to functional programming syntax of Mathematica
Recently, I was browsing the Wolfram Community forum, and I came across the following question:
What are the symbols @, #, / in Mathematica?
I remember that grasping the basics of functional programming took me quite a lot of mental effort (well worth it, I think!) so here is my attempt at a guide to the process.
In Mathematica, there are only two things you can work with: the Symbol and the Atom. There is only one way to combine these things: you can provide them as arguments to each other. We denote “\(x\)
with arguments \(y\) and \(z\)” by “x[y,z]”.
What is an Atom? As the name suggests, it is something indivisible, like the number 2 or the string “Hello!”. So that the language isn’t too complicated to implement, we mean “indivisible without any
further work” - so the number 15 is “divisible” (in the sense that it’s 3x5), but not in our sense: it takes work to find the divisors of a number. Similarly, the string “Hello!” is “divisible” into
characters, but that again takes work.
A Symbol is something which we, as a programmer, tell Mathematica to give meaning to. We also tell it under what circumstances that Symbol has meaning. For instance, I might say to Mathematica, “In
future, when I ask you the Symbol $MachinePrecision, you will pretend I said instead the Atom 15.9546.” Something else I might say to Mathematica is, “In future, when I ask you for the Symbol Plus,
combined with the arguments 1 and 2, you will pretend I said instead the Atom 3.”
In Mathematica’s syntax, we write the above as:
$MachinePrecision = 15.9546;
Plus[1, 2] = 3;
(The semicolons prevent Mathematica from printing the value we gave. Without the semicolons, it would print out 15.9546 and 3. In fact, the semicolons are a shorthand for the Symbol
CompoundExpression, but that’s not important.)
Furthermore, we can ask Mathematica, “In future, when I ask you for Plus combined with zero and any other argument x, return that argument x”. In Mathematica’s syntax, that is:
Plus[0, Pattern[x, Blank[]] ] := x
More compactly:
Plus[0, x_] := x
Now, we have had to be careful. Mathematica needs a way of distinguishing the Symbol x from the “free argument” x. We want the “free argument” - that is, we want to be able to supply any argument we
like, and just temporarily call it x. We do that using the Pattern symbol, better recognised as x_ . I won’t go into how Pattern works in terms of the Symbol/Atom idea, but just recognise that x_
matches things, rather than being a thing.
Now, we’ll assume that there is already a “plus one” method - that Mathematica already knows how to do Plus[1, x_]. Let’s also assume that it knows what Plus[-1, x_] is (not hard to do, in principle,
once we know Plus[1, x_]). Then we can define Plus over the positive integers:
Plus[x_, y_] := Plus[Plus[-1, x], Plus[1, y]]
And so forth. This is how we build up functions out of Symbols and Atoms.
Now, there is a shorthand for f[x]. We can instead write f@x. This means exactly the same thing.
A really important Symbol is List. List[x, y, z] (or, in shorthand, {x, y, z}) represents a collection of things. There’s nothing “special” about List - it’s interpreted in exactly the same way as
everything else - but it’s a convenient, conventional way to lump several things together. (It would all have worked in exactly the same way if the creators of the language had decided that Yerpik
would be the symbol that represented a generic collection; even Plus could be used this way, if we made sure to tell Mathematica that “Plus” should not be evaluated in the usual way. You could even
use the number 2 as the list-indicating symbol, or even use it as Plus usually is used, leading to expressions like 2[5,6] == 11.) We can define functions like Length[list_], so Length[{1, 2, 3}] is
just 3.
Since everything is essentially function application (“apply a symbol to an expression”), we might explore ways to apply several functions at once, or to apply a function to several different parts
of an expression. It turns out that a really useful thing to do is to be able to apply a function to all the inside bits of a List. We call this “mapping”:
Map[f, {a, b, c}] == {f[a], f[b], f[c]}
More generally, Map[f, s[a1, a2, … ]] == s[f[a1], f[a2], …], but we use List instead of s for convenience. There is a shorthand, reminiscent of the f@x notation: we use f /@ {a, b, c} to denote
It’s all very well to want to map a function across the arguments to a symbol (let’s call that symbol, which has those arguments, the Head of an expression, so Head[f[x,y]] is just f), but what about
if we want to apply the function to the Head symbol? Actually, this turns out to be quite rare (the function is Operate[p, f[x,y]] to give (p[f])[x,y] ), but it’s much more common to want to replace
the Head completely. For instance, we might want to supply a List as arguments to a function, as follows:
f[x_, y_] := x + y^2
How would we get f to act on the List {5, 6}? We can’t just say f[{5, 6}] because f requires two inputs, not the one that is List[5, 6]. Mathematica’s syntax is that instead of f@{5,6}, we use f@@{5,
6}. This is shorthand for Apply[f, {5,6}], and it returns f[5, 6], which is 41.
More generally, f@@g[x, y] == f[x, y]. (Note, however, that Mathematica evaluates things as much as possible before doing these transformations, so f@@Plus[5,6] doesn’t give you f[5,6] but f@@11, an
expression which makes no sense. Mathematica’s convention is that Atoms don’t really have a Head, so replacing the Head with f does nothing; hence f@@11 will return 11.)
Particularly in conjunction with Map, it can be useful to Apply a function not to an expression, but to the arguments of the expression. That is, given a List {{1, 2}, {3, 4}}, which is {List[1, 2],
List[3, 4]}, we might want to output {f[1, 2], f[3, 4]}. We do this with the shorthand f@@@{{1, 2}, {3, 4}}, which is really Apply[f, {{1, 2}, {3, 4}}, 2]. This situation might arise if we wanted to
“transpose” two strings “ab” and “cd” to return “ac” and “bd” (imagine writing the strings out in a table, and reading the answer down the columns instead of across the rows). We could use
StringJoin@@@Transpose@Map[Characters, {"ab", "cd"}]. Indeed, what does this expression do? The first thing that will actually change when it is evaluated is Map[Characters, {"ab", "cd"}]. This will
return {{"a", "b"}, {"c", "d"}}. Then Transpose sees that new list, and flips things round to {{"a", "c"}, {"b", "d"}}, which is {List["a", "c"], List["b", "d"]}. Then StringJoin is asked not to hit
the outer List, or even to hit the inner Lists, but to replace the List head on the inner Lists: the expression becomes {StringJoin["a", "c"], StringJoin["b", "d"]}, or {"ac", "bd"}.
Now, it’s all very well to have functions that work like this. But what if we wanted to take the second character of a string? There’s a function for that - StringTake - but it needs arguments. We
could define a new function takeSecondChars[str_] := StringTake[str, {2}], but that’s unwieldy if we only want this function once - and what about if we wanted the third character instead, the next
There is a really useful way to define functions without names. Unsurprisingly, they look like:
Function[{x, y, …}, …]
So in the above example, we’d have Function[{str}, StringTake[str, {2}]]. And then to map it across a list would look like:
Function[{str}, StringTake[str, {2}]] /@ {"str1", "str2", "str3"}
We can also apply it to a string: Function[{str}, StringTake[str, {2}]]["string"], or Function[{str}, StringTake[str, {2}]]@"string".
There’s a really compact shorthand. Instead of Function[{args}, body] we use (body)&. We don’t even bother naming the arguments; we use the Slot[i] function to get the ith argument. Slot[i] is more
neatly written as #i, while just the # symbol is interpreted as #1.
Hence our function becomes StringTake[#, {2}]&, and its mapping looks like:
StringTake[#, {2}]& /@ {"str1", "str2", "str3"}
It takes some getting used to, but after a while it becomes extremely natural. In my most recent coursework project, there are almost no programs I wrote which don’t use this syntax, even though the
coursework is aimed at the language Matlab which is almost the antithesis of this idea of “symbols with arguments”. Once you become able to see problems in this way - mapping small functions over
expressions, and so forth - you start seeing it everywhere. The idea is about sixty years old - it’s the principle of Lisp - and it’s ridiculously powerful. Since functions are just expressions, you
can use them to alter themselves. For instance, memoisation is trivial:
fibonacci[n_] := (fibonacci[n] = fibonacci[n-1] + fibonacci[n-2])
fibonacci[1] = 1;
That is, “Whenever I ask you for fibonacci[n], you will set the value of fibonacci[n] to be the sum of the two previous values.” Note that this is “set the value of fibonacci[n] to be”, not “return”
- this is a permanent change (well, as permanent as the Mathematica session), and it means that the value of fibonacci[36] is instantly available forever after once you’ve calculated it once.
You can also get some crazy things with Slot notation, because #0 (which is Slot[0]) represents the function itself. Off the top of my head, an example is:
(Boole[# < 10] #0[# + 1] + #) &[1]
This generates the tenth triangle number. (The function Boole[arg] returns 1 if arg is True, and 0 otherwise.) This is because the function evaluates to exactly its input unless that input is less
than 10; in that case, the function evaluates to (its input, plus “this function evaluated at input+1”). Recursively expanded, it is f[x_] := If[x < 10, f[x+1]+x, x], evaluated at the input 1. It
gets quite mind-bending quite quickly, and I don’t think I have ever used #0 in earnest. Another example I came up with quickly was:
If[Cos[#] == #, #, #0[Cos[#]]] &[1.]
This finds a fixed point of the function Cos, starting at the initial input 1. (It has to be a numerical input, otherwise Mathematica will just keep going forever with better and better symbolic
expressions for this fixed point, like Cos[Cos[Cos[1]]]. It rightly recognises that, for instance, Cos[Cos[Cos[1]]] is not equal to Cos[Cos[Cos[Cos[1]]]], so it never stops.)
The last really useful piece of shorthand I can think of at the moment is // which is another way to apply functions.
Instead of f@x, we can use x//f . This has the benefit of making it a bit clearer what is actually contentful, and what is just afterthoughts, because the functions which are evaluated last actually
appear at the end:
CharacterRange["a","z"] // StringJoin
Of course, the usual function notation can be used:
1 // (Boole[# < 10] #0[#+1] + # &)
Phew, that was a whistlestop tour in rather more words than I had hoped - turns out there are far more Mathematica concepts that I’ve internalised than I had thought, all of which are really quite
fundamental and indispensable. I understand much better why people say Mathematica has a steep learning curve, and why it is derided as a “write-only language” - that final example is ridiculous! | {"url":"https://www.patrickstevens.co.uk/posts/2014-01-24-introduction-to-functional-programming-syntax-of-mathematica/","timestamp":"2024-11-05T13:57:24Z","content_type":"text/html","content_length":"21799","record_id":"<urn:uuid:05f56413-e9ba-410d-b802-490cb079ff86>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00587.warc.gz"} |
Turbulent Flow in Channels: Channel Flow, Separation, and Reattachment in context of Turbulent Flow
27 Aug 2024
Turbulent Flow in Channels: Understanding Channel Flow, Separation, and Reattachment
Turbulent flow is a fundamental concept in fluid dynamics that plays a crucial role in various engineering applications, including pipe flow, channel flow, and boundary layer flows. In this article,
we will delve into the world of turbulent flow in channels, exploring the concepts of channel flow, separation, and reattachment.
Channel Flow
Channel flow refers to the flow of fluids through a conduit or channel with a fixed cross-sectional area. The flow is characterized by a velocity profile that varies across the width of the channel.
In laminar flow, the velocity profile is parabolic, with the highest velocities near the centerline and decreasing velocities towards the walls.
In turbulent flow, the velocity profile is more complex, featuring irregular fluctuations in both space and time. The Reynolds number (Re) is a key parameter that determines whether the flow is
laminar or turbulent:
Re = ρUL/μ
where ρ is the fluid density, U is the average velocity, L is the characteristic length, and μ is the dynamic viscosity.
For Re > 2,000-4,000, the flow becomes turbulent, characterized by eddies and swirls that enhance mixing and heat transfer. In channels, the Reynolds number can be estimated using the following
Re = (ρUH)/μ
where H is the channel height.
As the fluid flows through the channel, it encounters regions of high and low velocities. At certain points, the flow separates from the wall, creating areas of recirculation or stagnation. This
phenomenon is known as separation.
Separation occurs when the velocity gradient near the wall becomes negative, indicating a region of decelerating flow. The boundary layer thickness (δ) can be estimated using the following formula:
δ = 5ν/U
where ν is the kinematic viscosity and U is the average velocity.
As the fluid flows downstream, it reattaches to the wall, creating a new boundary layer. This process is known as reattachment. The reattachment point marks the end of the separated region and the
beginning of a new laminar or turbulent flow regime.
The reattachment point can be estimated using the following formula:
x = 0.5H/Re
where x is the distance from the separation point to the reattachment point, H is the channel height, and Re is the Reynolds number.
Consequences of Turbulent Flow in Channels
Turbulent flow in channels has significant consequences for various engineering applications, including:
1. Pressure Drop: Turbulent flow increases pressure drop due to increased frictional losses.
2. Heat Transfer: Turbulent flow enhances heat transfer by increasing mixing and convective transport.
3. Mass Transport: Turbulent flow affects mass transport by altering the velocity profile and creating areas of recirculation.
Turbulent flow in channels is a complex phenomenon that plays a crucial role in various engineering applications. Understanding channel flow, separation, and reattachment is essential for designing
efficient systems and predicting fluid behavior. By applying the formulas presented in this article, engineers can better model and simulate turbulent flows in channels, ultimately leading to
improved system performance and reduced costs.
1. White, F. M. (2010). Fluid Mechanics. McGraw-Hill.
2. Schlichting, H. (1979). Boundary Layer Theory. McGraw-Hill.
3. Anderson, J. D. (2005). Computational Fluid Mechanics. McGraw-Hill.
Additional Resources
1. NASA Glenn Research Center: Turbulent Flow in Channels
2. University of Michigan: Turbulent Flow in Channels
3. American Institute of Aeronautics and Astronautics (AIAA): Turbulent Flow in Channels
Related articles for ‘Turbulent Flow’ :
• Turbulence in Compressible Flows: Shock-Turbulence Interactions, Mach Number Effects, and Prandtl-Meyer Flow in context of Turbulent Flow
• Laminar-Turbulent Transition: The Role of Boundary Layers and Wall Distances in context of Turbulent Flow
• Turbulent Flow in Boundary Layers: Bluff Body Effects, Wake Formation, and Vortex Dynamics in context of Turbulent Flow
• Theory of Turbulence: Navier-Stokes Equations, Reynolds Stress Models, and Closure Assumptions in context of Turbulent Flow
• Boundary Layer Turbulence: Skin Friction, Momentum Transport, and Heat Transfer in context of Turbulent Flow
• Turbulence in Rotating Flows: Centrifugal Effects, Taylor Columns, and Ekman Layers in context of Turbulent Flow
• Reading: Turbulent Flow in Channels: Channel Flow, Separation, and Reattachment in context of Turbulent Flow
• Experimental Techniques for Turbulent Flow Measurements: PIV, LDA, and Hot-Wire Anemometry in context of Turbulent Flow
• Turbulent Flow Characteristics: Reynolds Number, Eddy Viscosity, and Kolmogorov Length Scale in context of Turbulent Flow
Calculators for ‘Turbulent Flow’ | {"url":"https://blog.truegeometry.com/tutorials/education/c60260a7a5212ea2c86d0cf7081f06e8/JSON_TO_ARTCL_Turbulent_Flow_in_Channels_Channel_Flow_Separation_and_Reattach.html","timestamp":"2024-11-08T11:21:45Z","content_type":"text/html","content_length":"18954","record_id":"<urn:uuid:f89becb0-4ca2-4fbe-8f50-3859cebcbe5f>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00536.warc.gz"} |
if a man reads 2 chapters a day and there are 6 pages per chapter in a 798 page book how long will it take him to finish the book
HESI A2
HESI A2 Math Practice Test 2023
1. If a man reads 2 chapters a day and there are 6 pages per chapter in a 798-page book, how long will it take him to finish the book?
• A. 133 days
• B. 399 days
• C. 75 days
• D. 66.5 days
Correct answer: D
Rationale: To find the total number of chapters, divide the total number of pages by the number of pages per chapter: 798 ÷ 6 = 133 chapters. If he reads 2 chapters a day, it will take him 133 ÷ 2 =
66.5 days to finish the book. Therefore, the correct answer is 66.5 days. Choice A is incorrect because it miscalculates the number of days required. Choices B and C are incorrect as they do not
consider the correct calculation based on the given information.
2. Which word is used incorrectly in the following sentence? For whom was that email intended?
• A. For
• B. whom
• C. that
• D. intended
Correct answer: B
Rationale: The word 'whom' should be used instead of 'who' in the sentence to make it grammatically correct. 'Who' is used as the subject of a sentence, while 'whom' is used as the object. Therefore,
the correct form of the sentence should be: 'For whom was that email intended?' Choices A, C, and D are used correctly in the sentence and do not need any changes.
3. You need 4/5 cups of water for a recipe. You accidentally put 1/3 cups into the mixing bowl with the dry ingredients. How much more water in cups do you need to add?
• A. 7/15 cups
• B. 2/3 cups
• C. 1/3 cups
• D. 1/15 cups
Correct answer: A
Rationale: To find how much more water is needed, subtract 1/3 cup from 4/5 cup. First, find a common denominator: The least common denominator between 5 and 3 is 15. Convert the fractions: 4/5 = 12/
15, 1/3 = 5/15. Now, subtract: 12/15 - 5/15 = 7/15. Therefore, you need to add 7/15 cups of water. Choice B (2/3 cups) is incorrect because it does not represent the correct amount of additional
water needed. Choice C (1/3 cups) is incorrect because this is the amount of water that was accidentally added. Choice D (1/15 cups) is incorrect as it does not reflect the correct calculation of the
additional water required.
4. A store is offering a 25% discount on all items. If an item costs $120, what is the discounted price?
• A. $90
• B. $80
• C. $75
• D. $95
Correct answer: A
Rationale: To calculate the discounted price after a 25% discount on $120, you first find the discount amount by multiplying $120 by 0.25, which equals $30. Subtracting the discount amount from the
original price gives the discounted price: $120 - $30 = $90. Therefore, the correct answer is $90. Choice B, $80, is incorrect as it does not consider the 25% discount. Choice C, $75, is incorrect as
it is lower than the correct calculation. Choice D, $95, is incorrect as it does not reflect the reduction from the discount.
5. What is a nit?
• A. abscess
• B. parasite
• C. bandage
• D. infection
Correct answer: B
Rationale: A nit is a kind of parasite, specifically the egg of a louse. Nits are typically found attached to hair close to the scalp and are commonly associated with head lice infestations. Choice
A, 'abscess,' is incorrect as an abscess is a collection of pus caused by an infection. Choice C, 'bandage,' is incorrect as it is a material used for covering wounds. Choice D, 'infection,' is
incorrect as it refers to the invasion and multiplication of microorganisms in body tissues.
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[Solved] Find the area of the triangle having the | SolutionInn
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Verified Expert Solution
Find the area of the triangle having the given measurements. Round to the nearest square unit. 13) C=100, a 3 yards, b = 8
Find the area of the triangle having the given measurements. Round to the nearest square unit. 13) C=100, a 3 yards, b = 8 yards Use Heron's formula to find the area of the triangle. Round to the
nearest square unit. 14) a = 10 meters, b = 14 meters, c = 6 meters Solve the equation on the interval [0, 2). 15) cos 2x = 2 16) sin x-2 sin x cos x = 0
There are 3 Steps involved in it
Step: 1
1 Find the area of the triangle with given measurements Solution Given Angle C 100 C Side a 3 yards Side b 8 yards We can use the formula for the area ...
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Multiplication Chart 50 50 PrintableMultiplication | Multiplication Chart Printable
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Multiplication Chart 50 50 PrintableMultiplication – A Multiplication Chart is a handy tool for children to find out exactly how to multiply, separate, as well as find the smallest number. There are
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A multiplication chart can be utilized to assist children discover their multiplication facts. Multiplication charts been available in numerous types, from complete web page times tables to single
page ones. While specific tables work for providing pieces of information, a complete page chart makes it less complicated to evaluate realities that have actually already been grasped.
The multiplication chart will normally include a left column and a leading row. When you desire to locate the product of two numbers, select the first number from the left column and also the second
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A multiplication chart is a layout that reveals how to increase two numbers. You pick the initial number in the left column, relocate it down the column, and also after that choose the 2nd number
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Multiplication charts are valuable for many reasons, consisting of assisting kids learn exactly how to split and also streamline fractions. Multiplication charts can additionally be handy as workdesk
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Multiplication charts are additionally useful for assisting trainees memorize their times tables. They help them find out the numbers by reducing the variety of steps required to complete each
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Kullback-Leibler KL divergence between two normal rvs
The previous post looked at the best approximation to a normal density by normal density with a different mean. Dan Piponi suggested in the comments that it would be good to look at the
Kullback-Leibler (KL) divergence.
The previous post looked at the difference from between two densities from an analytic perspective, solving the problem that an analyst would find natural. This post takes an information theoretic
perspective. Just is p-norms are natural in analysis, KL divergence is natural in information theory.
The Kullback-Leibler divergence between two random variables X and Y is defined as
There are many ways to interpret KL(X || Y), such as the average surprise in seeing Y when you expected X.
Unlike the p-norm distance, the KL divergence between two normal random variables can be computed in closed form.
Let X be a normal random variable with mean μ[X] and variance σ²[X] and Y a normal random variable with mean μ[Y] and variance σ²[Y]. Then
If μ[X] = 0 and σ[X] = 1, then for fixed μ[Y] the value of σ²[Y] that minimizes KL(X || Y) is
KL divergence is not symmetric, hence we say divergence rather than distance. More on that here. If we want to solve the opposite problem, minimizing KL(X || Y), the optimal value of σ²[Y] is simply | {"url":"https://www.johndcook.com/blog/2023/11/05/kl-divergence-normal/","timestamp":"2024-11-13T13:04:00Z","content_type":"text/html","content_length":"49672","record_id":"<urn:uuid:37010e49-568d-478d-84ec-cb5c06aa4613>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00275.warc.gz"} |
October 10, 2016 (first version May 20, 2015)
Temporal Planning by Satisfiability modulo Theories
Planning as satisfiability (see Planning by SAT) is a powerful approach to domain-independent planning (in artificial intelligence) first proposed by Henry Kautz and Bart Selman in their SATPLAN
system in the 1990s.
The work of Kautz and Selman limited to the simplest case of planning, with asynchoronous discrete systems expressible in terms of states consisting of Boolean state variables.
Later extensions to the work include
• numeric state variables: Wolfman and Weld, The LPSAT engine & its application to resource planning, IJCAI 1999.
• continuous time model with continuous change: Shin and Davis, Processes and continuous change in a SAT-based planner, Artificial Intelligence Journal, 166(1), 2005.
Both of these extensions include integer and real-valued state variables, which usually cannot be effectively handled in the basic SAT framework. What is needed is what is now known as Satisfiability
modulo Theories (SMT), which extends the language of propositional logic with non-Boolean theories such as linear real arithmetics.
Modeling languages
• languages based on timed automata and their extensions (such as hybrid automata)
These languages are favored in the Computer Aided Verification community, due to the focus on model-checking, in which system models are verified with respect to complex specifications expressed
in temporal logics, and both the models and the specifications can be translated to timed or hybrid automata.
• temporal extensions of PDDL
These are modeling languages that extend the original PDDL 1.0 specification (McDermott et al., 1998). The languages' main limitations are their limited data types (Booleans, reals), awkward
syntax and semantics (with no formal semantics in existence), and a non-standard concurrency model that is largely incompatible with efficient constraint-based search and reasoning methods.
• other modeling languages developed by the AI planning community, but none enjoy wide use.
• NDL This is a rich modeling language supporting a wide range of data-types, and in comparison to PDDL, higher abstraction level with explicit resources, and more mainstream model of concurrent
actions. My papers in IJCAI and AAAI have demonstrated that natural representations of most standard PDDL benchmarks in SMT, CP, etc. have a significant performance penalty in comparison to NDL
models. This is the reason why we exclusively have focused on NDL, and do not recommend using PDDL, even for research purposes.
Efficient reductions to SMT
Although reductions of temporal and hybrid systems planning to SMT have been known for a long time (see work of Shin and Davis, 2005), SMT has not been viewed as a competitive approach. Making SMT
competitive is based on the following observations.
• Encodings have to be compact, preferably linear size.
• Number of real-valued variables has to be minimized, and references to real-valued variables have to be made simple.
1. Discretization (Rintanen, 2015) sometimes allows very compact encodings with no real-valued variables, solvable with efficient SAT solvers.
2. Real-valued variables refer to absolute time and time relative to the start of an action or an event. The number of linear inequalities referring to these variables has to be minimized.
• Same trade-offs as in classical planning (Rintanen, 2004; Rintanen et al. 2006) hold w.r.t. the number of steps in plans and the difficulty of finding plans.
The encodings of temporal planning in SMT (and all other constraint-based formalisms) follows the ideas first presented by Kautz and Selman (see Planning by SAT for details), with several extensions
to handle the far more complex model with real-valued timelines and concurrency of actions.
• Values of all state variables are represented at each step. A step corresponds to one time point in the real or rational-valued timeline.
• For every action and every step there is a Boolean state variable indicating if the action is taken at that step.
• Unlike in classical (asynchronous) planning, the absolute time of each step needs to be represented, either implicitly in terms of the difference in time between a step and its predecessor, or as
the absolute time of the step.
• There are constraints preventing a co-occurrence of two actions that use the same resources. This is similar, but different from the constraints for parallel plans in SAT-encodings of classical
planning: parallelism in the classical planning case is simply a form of partial-order reduction (possibility to reorder the actions in a plan without affecting the plans outcome; see (Rintanen
et al. 2006) for details), whereas the concurrency constraints for temporal planning express the possibilities of having different actions to overlap on the real-valued timeline. Resources are
the main mechanism to limit concurrency at the level of modeling languages.
• The effects of a temporal action can take place at the same step, the next step, or a step arbitrarily far in the future. How this delay between an action and its effects is represented is one of
the key issues in constraint-based representations of temporal planning.
Benchmark sets
We have translated standard PDDL 2.1 benchmarks into NDL. The benchmark set is a straightforward translation from the PDDL 2.1 version of the benchmarks expressed in terms of multi-valued state
variables. With this modeling, it is not necessary to automatically recognize exactly-one (or at-most-one) invariants (although in some cases there are still some invariants adding which into the SMT
encoding has a performance advantage.)
• translators from NDL into SMT (available later)
Jussi Rintanen. Temporal planning with clock-based SMT encodings. In Proceedings of the International Joint Conference on Artificial Intelligence (IJCAI), AAAI Press, 743-749, 2017. (slides)
Jussi Rintanen. Schematic invariants by reduction to ground invariants. In Proceedings of the AAAI Conference on Artificial Intelligence, AAAI Press, 2017. (© 2017 American Association for Artificial
Intelligence. AAAI)
Jussi Rintanen. Models of actions concurrency in temporal planning. In Proceedings of the International Joint Conference on Artificial Intelligence (IJCAI), AAAI Press, pages 1659-1665, 2015.
Jussi Rintanen. Discretization of temporal models with application to planning with SMT. In Proceedings of the AAAI Conference on Artificial Intelligence, AAAI Press, pages 3349-3355, 2015. (© 2015
American Association for Artificial Intelligence. AAAI)
Jussi Rintanen. Impact of modeling languages on the theory and practice in planning research. In Proceedings of the AAAI Conference on Artificial Intelligence, AAAI Press, pages 4052-4056, 2015. (©
2015 American Association for Artificial Intelligence. AAAI) (slides)
Jussi Rintanen. Constraint-based algorithm for computing temporal invariants. In Proceedings of the European Conference on Logic in Artificial Intelligence, Lecture Notes in Computer Science 8761,
pages 665-673, Springer-Verlag, 2014.
Jussi Rintanen. Planning as satisfiability: heuristics, Artificial Intelligence Journal, 193, 45-83, December 2012.
J. Rintanen, Evaluation strategies for planning as satisfiability, in R. Lopez de Mantaras and Lorenza Saitta, eds., ECAI 2004. Proceedings of the 16th European Conference on Artificial Intelligence,
pages 682-687, IOS Press, 2004. [additional material on slides of ECAI'04 talk, 8 on 1]
J. Rintanen, K. Heljanko and I. Niemelä, Planning as satisfiability: parallel plans and algorithms for plan search, Artificial Intelligence, 170(12-13), pages 1031-1080, 2006.
1. Ji-Ae Shin and Ernest Davis, Processes and continuous change in a SAT-based planner, Artificial Intelligence, 166(1), pages 194-253, 2005. (Introduces SAT-based encodings for planning with timed/
hybrid systems as modelled in PDDL 2.1.)
2. S. A. Wolfman and D. S. Weld, The LPSAT engine & its application to resource planning, In Dean, T., ed., Proceedings of the 16th International Joint Conference on Artificial Intelligence,
310-315, Morgan Kaufmann Publishers, 1999. | {"url":"https://users.aalto.fi/~rintanj1/PlanningBySMT.html","timestamp":"2024-11-03T03:40:46Z","content_type":"text/html","content_length":"11137","record_id":"<urn:uuid:2262bcea-6ac5-425b-93b8-224aa5305674>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00594.warc.gz"} |
Characterization (mathematics)
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In mathematics, the statement that "Property P characterizes object X" means that not only does X have property P, but that X is the only thing that has property P. In other words, P is a defining
property of X. Similarly a set of properties P is said to characterize X when these properties distinguish X from all other objects.
It is also common to find statements such as "Property Q characterises Y up to isomorphism". The first type of statement says in different words that the extension of P is a singleton set. The second
says that the extension of Q is a single equivalence class (for isomorphism, in the given example — depending on how up to is being used, some other equivalence relation might be involved).
A reference on mathematical terminology notes characteristic is from Greek kharax, "a pointed stake". "From Greek kharax came kharakhter, an instrument used to mark or engrave an object. Once an
object was marked, it became distinctive, so the character of something came to mean its distinctive nature. The Late Greek suffix -istikos converted the nown character into the adjective
characteristic, which, in addition to maintaining its adjectival meaning, later became a noun as well."^[1]
Just as in chemistry, the characteristic property of a material will serve to identify a sample, or in the study of materials, structures and properties determine characterization (materials science)
, so in mathematics there is a continual effort to express properties that will distinguish a desired feature in a theory or system. Characterization is not unique to mathematics, but since the
science is abstract, much of the activity can be described as "characterization". For instance, in Mathematical Reviews, as of 2018, more than 24,000 articles contain the word in the article title,
and 93,600 somewhere in the review.
In an arbitrary context of objects and features, characterizations have been expressed via the heterogeneous relation aRb meaning that object a has feature b. For example, b may mean abstract or
concrete. The objects can be considered the extensions of the world, while the features are expression of the intensions. A continuing program of characterization of various objects leads to their
• A parallelogram is a quadrilateral with opposite sides parallel. One of its characterizations is that the diagonals bisect each other. This means that the diagonals in all parallelograms bisect
each other, and conversely, that any quadrilateral where the diagonals bisect each other must be a parallelogram. The latter statement is only true if inclusive definitions of quadrilaterals are
used (so that, for example, rectangles count as parallelograms), which is the dominant way of defining objects in mathematics nowadays.
• "Among probability distributions on the interval from 0 to ∞ on the real line, memorylessness characterizes the exponential distributions." This statement means that the exponential distributions
are the only such probability distributions that are memoryless. (See also Characterization of probability distributions.)
• "According to Bohr–Mollerup theorem, among all functions f such that f(1) = 1 and x f(x) = f(x + 1) for x > 0, log-convexity characterizes the gamma function." This means that among all such
functions, the gamma function is the only one that is log-convex. (A function f is log-convex iff log(f) is a convex function. The base of the logarithm does not matter as long as it is more than
1, but conventionally mathematicians take "log" with no subscript to mean the natural logarithm, whose base is e.)
• The circle is characterized as a manifold by being one-dimensional, compact and connected; here the characterization, as a smooth manifold, is up to diffeomorphism.
See also[edit] | {"url":"https://static.hlt.bme.hu/semantics/external/pages/v%C3%A9ges_%C3%A1llapot%C3%BA_transzducereket_(FST)/en.wikipedia.org/wiki/Characterization_(mathematics).html","timestamp":"2024-11-13T06:28:40Z","content_type":"text/html","content_length":"38347","record_id":"<urn:uuid:ab923bc6-8a14-496a-87b2-d5e9bf973d10>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00247.warc.gz"} |
Fiona buys different amounts of gas at $2.25 per gallon she has a graph that shows the amount she should pay as a function of the amount of gas she buys. what constraints are there on the domain of the function?
Answer:See belowStep-by-step explanation:Fiona has a function that shows the amount of money she has to pay ( f(.) ) as a function of the gas the buys (g). So, the function can be written as: f(g).
As each litter costs $2.25 we can say that:f(g) = 2.25gLets try to find put which constraints the function f(g) has in its domain, it is, what are the restrictions for the values of g. Is evident
that g cannot be negative, as one cannot buy negative liters of gasoline. We can buy any positive amount of gasoline, doesn't matter if it is an integer number or not. it is possible to buy 5 liters
and also to buy 5.45454545545454 liters. A zero amount of liters is also possible, having zero cost. Thus, as any positive or zero value is positive, but negative values are not, we can restrict the
domain to every non-negative real number:Domain(f) = [0, infinite ) orDomain(f) = Real non negative | {"url":"https://thibaultlanxade.com/general/fiona-buys-different-amounts-of-gas-at-2-25-per-gallon-she-has-a-graph-that-shows-the-amount-she-should-pay-as-a-function-of-the-amount-of-gas-she-buys-what-constraints-are-there-on-the-domain-of-the-function","timestamp":"2024-11-04T13:56:27Z","content_type":"text/html","content_length":"31318","record_id":"<urn:uuid:565baee0-c3b4-4ebe-b136-86e4064684cf>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00514.warc.gz"} |
Elegant handling of possibly-state-dependent parameters in physical simulation
For a couple years now, I have been using OrdinaryDiffEq for developing a handful of models for describing the process of pharmaceutical lyophilization. The more time I spend, the more I learn about
more aspects of Julia or packages that would have made my life easier if I had known about them sooner (e.g. Parameters.jl, ComponentArrays.jl). In that light, I am wondering if my current headache
already has a good solution.
These models have anywhere between 10 and 30 physical properties that need to be defined for simulation. These range from universal physical properties (e.g. universal gas constant) to things that
don’t often need tinkering (e.g. density of glass) to things that are different for every simulation (input temperature/pressure profiles). As I compare these models to experiment and refine them, I
periodically find that I need to add temperature-dependence (or pressure-dependence, or frequency-dependence) to a parameter that I was previously treating as constant.
Up til now, my approach has been to rewrite the code in such a way that, instead of taking a constant value, the ODE function takes a callable (or anonymous function) which it evaluates everywhere it
needs the property. This works, of course. But every time it happens I inevitably end up re-running the code several times finding all the places where I am now adding an anonymous function to a
number, and I feel like it clutters the input script, e.g. turning
p = 10.0u"Pa"
p = t->10.0u"Pa"
which, in addition to being a little uglier, also makes error messages not as nice and confuses the non-Julia-users when I need to hand off my code to someone else.
I have also experimented with creating a custom type with pretty printing, with a single-argument constructor that essentially acts as a nicer-looking version of the anonymous function.
julia> p = RampedVariable(10.0u"Pa")
julia> p(50u"s")
julia> p(Inf)
I implemented that type as below, for any who are curious. But I find that this method still ends up being more typing, and means I need to document more interfaces myself, etc.
My question is: is there a more elegant solution to this problem than, say, implementing every single parameter every time as a callable which may not actually depend on any of its inputs?
For reference, sometimes I am thinking of time-dependent external controls, weakly temperature-dependent parameters like thermal conductivity, or things like the dielectric loss coefficient which
varies with temperature and electric field frequency.
struct RampedVariable
function RampedVariable(initial)
RampedVariable([initial], [], [], [0])
function RampedVariable(setpts, ramprate)
if length(ramprate) == 0 || length(setpts) == 1
@error "If no ramp necessary, construct RampedVariable with only one argument." ramprate
if length(ramprate) >= 2 || length(setpts) > 2
@error "For multiple ramps, need at least one hold time. Construct RampedVariable with three arguments." ramprate
if length(setpts) != 2
@error "Number of set points should be 1 more than ramps, since initial is included"
timestops = fill(0.0*setpts[1]/ramprate[1], 2)
timestops[2] = timestops[1] + (setpts[2]-setpts[1])/ramprate
RampedVariable(setpts, [ramprate], [], timestops)
function RampedVariable(setpts, ramprates, holds)
if (length(ramprates) != length(holds) + 1 ) || (length(ramprates)==0)
@error "Number of ramps should be zero or number of holds + 1"
if length(setpts) != length(ramprates) + 1
@error "Number of set points should be 1 more than ramps, since initial is included"
timestops = fill(0.0*setpts[1]/ramprates[1], length(ramprates) + length(holds) + 1)
(ramp, rest) = Iterators.peel(ramprates)
timestops[2] = timestops[1] + (setpts[2]-setpts[1])/ramp
for (i, ramp) in enumerate(rest)
timestops[2i+1] = timestops[2i] + holds[i]
timestops[2i+2] = timestops[2i+1] + (setpts[i+2]-setpts[i+1])/ramp
RampedVariable(setpts, ramprates, holds, timestops)
function (rv::RampedVariable)(t)
if length(rv.timestops) == 1
return rv.setpts[1]
im = findlast(rv.timestops .<= t)
if im == length(rv.timestops)
return rv.setpts[end]
elseif iseven(im)
return rv.setpts[im÷2+1]
ip = im+1
return (rv.setpts[ip÷2+1] - rv.setpts[ip÷2])/(rv.timestops[ip] - rv.timestops[im])*(t - rv.timestops[im]) + rv.setpts[ip÷2]
The way I would approach this would be to use a modeling tool like ModelingToolkit, and treat what you call parameters as system inputs. When you want a constant parameter, you connect a constant
source, and when you want something else, you change the input signal source.
These inputs can then be propagated through the model, so that you only define the source at a single point, and if you change it it propagates through the entire model.
Take this pendulum on a cart as an example
It’s modeled in ModelingToolkit, the main system model Cartpole includes the component
motor = TranslationalModelica.Force(use_support = true)
which is an input component. I then simulate the system with a sinusoidal input by instantiating the cartpole and an input source, Blocks.Sine and connect the two, connecting the sine to the force
@components begin
world = World()
cartpole = Cartpole()
input = Blocks.Cosine(frequency=1, amplitude=1)
@equations begin
connect(input.output, :u, cartpole.motor.f)
Later on, I instead connect an LQG controller to the same input instead of the sine and simulate that.
This example is rather large and elaborate, the documentation of MTK includes much simpler examples. When you use MTK, MTK serves as a modeling layer, and OrdinaryDiffEq takes care of the actual
integration when you are done with your model. | {"url":"https://discourse.julialang.org/t/elegant-handling-of-possibly-state-dependent-parameters-in-physical-simulation/121640","timestamp":"2024-11-04T01:18:48Z","content_type":"text/html","content_length":"28202","record_id":"<urn:uuid:79ad32ed-cc39-467c-a775-3b55d55a7706>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00856.warc.gz"} |
Selecting the number of clusters with silhouette analysis on KMeans clustering
Go to the end to download the full example code or to run this example in your browser via JupyterLite or Binder
Selecting the number of clusters with silhouette analysis on KMeans clustering¶
Silhouette analysis can be used to study the separation distance between the resulting clusters. The silhouette plot displays a measure of how close each point in one cluster is to points in the
neighboring clusters and thus provides a way to assess parameters like number of clusters visually. This measure has a range of [-1, 1].
Silhouette coefficients (as these values are referred to as) near +1 indicate that the sample is far away from the neighboring clusters. A value of 0 indicates that the sample is on or very close to
the decision boundary between two neighboring clusters and negative values indicate that those samples might have been assigned to the wrong cluster.
In this example the silhouette analysis is used to choose an optimal value for n_clusters. The silhouette plot shows that the n_clusters value of 3, 5 and 6 are a bad pick for the given data due to
the presence of clusters with below average silhouette scores and also due to wide fluctuations in the size of the silhouette plots. Silhouette analysis is more ambivalent in deciding between 2 and
Also from the thickness of the silhouette plot the cluster size can be visualized. The silhouette plot for cluster 0 when n_clusters is equal to 2, is bigger in size owing to the grouping of the 3
sub clusters into one big cluster. However when the n_clusters is equal to 4, all the plots are more or less of similar thickness and hence are of similar sizes as can be also verified from the
labelled scatter plot on the right.
For n_clusters = 2 The average silhouette_score is : 0.7049787496083262
For n_clusters = 3 The average silhouette_score is : 0.5882004012129721
For n_clusters = 4 The average silhouette_score is : 0.6505186632729437
For n_clusters = 5 The average silhouette_score is : 0.561464362648773
For n_clusters = 6 The average silhouette_score is : 0.4857596147013469
import matplotlib.cm as cm
import matplotlib.pyplot as plt
import numpy as np
from sklearn.cluster import KMeans
from sklearn.datasets import make_blobs
from sklearn.metrics import silhouette_samples, silhouette_score
# Generating the sample data from make_blobs
# This particular setting has one distinct cluster and 3 clusters placed close
# together.
X, y = make_blobs(
center_box=(-10.0, 10.0),
) # For reproducibility
range_n_clusters = [2, 3, 4, 5, 6]
for n_clusters in range_n_clusters:
# Create a subplot with 1 row and 2 columns
fig, (ax1, ax2) = plt.subplots(1, 2)
fig.set_size_inches(18, 7)
# The 1st subplot is the silhouette plot
# The silhouette coefficient can range from -1, 1 but in this example all
# lie within [-0.1, 1]
ax1.set_xlim([-0.1, 1])
# The (n_clusters+1)*10 is for inserting blank space between silhouette
# plots of individual clusters, to demarcate them clearly.
ax1.set_ylim([0, len(X) + (n_clusters + 1) * 10])
# Initialize the clusterer with n_clusters value and a random generator
# seed of 10 for reproducibility.
clusterer = KMeans(n_clusters=n_clusters, n_init="auto", random_state=10)
cluster_labels = clusterer.fit_predict(X)
# The silhouette_score gives the average value for all the samples.
# This gives a perspective into the density and separation of the formed
# clusters
silhouette_avg = silhouette_score(X, cluster_labels)
"For n_clusters =",
"The average silhouette_score is :",
# Compute the silhouette scores for each sample
sample_silhouette_values = silhouette_samples(X, cluster_labels)
y_lower = 10
for i in range(n_clusters):
# Aggregate the silhouette scores for samples belonging to
# cluster i, and sort them
ith_cluster_silhouette_values = sample_silhouette_values[cluster_labels == i]
size_cluster_i = ith_cluster_silhouette_values.shape[0]
y_upper = y_lower + size_cluster_i
color = cm.nipy_spectral(float(i) / n_clusters)
np.arange(y_lower, y_upper),
# Label the silhouette plots with their cluster numbers at the middle
ax1.text(-0.05, y_lower + 0.5 * size_cluster_i, str(i))
# Compute the new y_lower for next plot
y_lower = y_upper + 10 # 10 for the 0 samples
ax1.set_title("The silhouette plot for the various clusters.")
ax1.set_xlabel("The silhouette coefficient values")
ax1.set_ylabel("Cluster label")
# The vertical line for average silhouette score of all the values
ax1.axvline(x=silhouette_avg, color="red", linestyle="--")
ax1.set_yticks([]) # Clear the yaxis labels / ticks
ax1.set_xticks([-0.1, 0, 0.2, 0.4, 0.6, 0.8, 1])
# 2nd Plot showing the actual clusters formed
colors = cm.nipy_spectral(cluster_labels.astype(float) / n_clusters)
X[:, 0], X[:, 1], marker=".", s=30, lw=0, alpha=0.7, c=colors, edgecolor="k"
# Labeling the clusters
centers = clusterer.cluster_centers_
# Draw white circles at cluster centers
centers[:, 0],
centers[:, 1],
for i, c in enumerate(centers):
ax2.scatter(c[0], c[1], marker="$%d$" % i, alpha=1, s=50, edgecolor="k")
ax2.set_title("The visualization of the clustered data.")
ax2.set_xlabel("Feature space for the 1st feature")
ax2.set_ylabel("Feature space for the 2nd feature")
"Silhouette analysis for KMeans clustering on sample data with n_clusters = %d"
% n_clusters,
Total running time of the script: (0 minutes 1.269 seconds) | {"url":"https://scikit-learn.org/1.3/auto_examples/cluster/plot_kmeans_silhouette_analysis.html","timestamp":"2024-11-06T01:31:34Z","content_type":"text/html","content_length":"40377","record_id":"<urn:uuid:7a566a54-b36d-4032-b8e4-d1f94e1a74bc>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00421.warc.gz"} |
Maths Archives - Le Rayon Vert
John Baez has an excellent post on the journal Chaos, Solitons and Fractals. In particular he is interested in the curious fact that one of the editors has had an amazing 322 papers published in the
journal. On closer examination Baez suggests that a number of these papers are essentially numerology hiding behind a bit of genuine maths and physics. Anyone with an interest in maths &/or physics
should follow the link to read the details, but it is also of more general interest for anyone who has followed the controversy over the big scientific publishers, particularly the much criticised
Baez writes
Now, I get crud like this in my email every day. I delete it without comment. What makes this case different is that El Naschie gets to publish these papers in a superficially respectable
journal that he actually edits.
The fact that Elsevier would let Naschie edit this journal and publish large numbers of papers like this in it shows that their system for monitoring the quality of their journals is broken.
The fact that this journal costs $4520 per year would be hilarious, except that libraries are actually buying it — at a reduced rate, bundled in with other Elsevier journals, but still!
It is worth following the long comments thread at the n-Category Café as various readers find unusual things in Chaos, Solitons and Fractals, including two near identical papers, a number of
sockpuppets come to El Naschie’s defence, and readers puzzle over the details of his background and current affiliation. No doubt there is more to this story which is still to emerge.
See also Backreaction, The Quantum Pontiff and Ars Technica.
My Wordle
This is fun. I put my PhD thesis into it.
Maths Degrees at Half Price
The Labor party propose to halve the cost of maths and science degrees. An excellent suggestion, though in my opinion it doesn’t really get to the heart of the current problems with the mathematical
sciences in Australia as outlined in this recent study (see also commentary on this at Larvatus Prodeo), the real problem is the massive funding cuts to universities by the Howard government which
have hit less industry-oriented faculties like maths and pure sciences (and also others such as the Arts) particularly hard as universities depend more on outside funding. Furthermore there is a
change towards more vocational courses (and less demanding ones) as universities compete for the student dollar, leaving the fundamental disciplines struggling.
So what has Howard done for the mathematical sciences lately? Well, he asked Australia’s first Fields medalist Terence Tao what country he was from. Of course if Howard knew more about the state of
the mathematical sciences he’d actually ask “what country did you go to?” (link to PDF).
UPDATE: The education minister disagrees with Labor’s plan, basically because it won’t fix the problems … the problems that her Government created that is. The universities on the other hand are
rather keen on the plan. Of course what the minister says is not in total disagreement with what I said, I also don’t think it will fix everything, but I do think that something has to be done and
this would achieve some good. | {"url":"https://www.frogworth.com/stuart/blog/?cat=13","timestamp":"2024-11-05T19:37:35Z","content_type":"text/html","content_length":"47534","record_id":"<urn:uuid:c645dffd-ec28-4756-8981-892f4f103754>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00070.warc.gz"} |
car loan calculator monthly payment
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2. For the figures above, the loan payment formula would look like: 0.06 divided by 12 = 0.005. 0.005 x $20,000 = $100. In this example, you’d pay $100 in interest in the first month. As you ...
3. How to calculate car loan interest payments There are several ways to calculate your monthly auto loan interest payment. You can use an online loan payment calculator or work directly with a
4. Here’s how to calculate the interest on an amortized loan: Divide your interest rate by the number of payments you’ll make that year. If you have a 6 percent interest rate and you make monthly
5. Amortization calculator. An amortization calculator is used to determine the periodic payment amount due on a loan (typically a mortgage), based on the amortization process. The amortization
repayment model factors varying amounts of both interest and principal into every installment, though the total amount of each payment is the same.
6. Annual percentage rate. Parts of total cost and effective APR for a 12-month, 5% monthly interest, $100 loan paid off in equally sized monthly payments. The term annual percentage rate of charge
(APR), [1][2] corresponding sometimes to a nominal APR and sometimes to an effective APR (EAPR), [3] is the interest rate for a whole year (annualized ...
7. The fixed monthly payment for a fixed rate mortgage is the amount paid by the borrower every month that ensures that the loan is paid off in full with interest at the end of its term. The monthly
payment formula is based on the annuity formula. The monthly payment c depends upon: r - the monthly interest rate. Since the quoted yearly percentage ...
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Calculus I Tutoring Services | Costa Comprehensive Tutoring
Calculus is the study of change. The subject primarily involves looking at functions and limits. In your Calculus I class, you will examine limits, continuity, derivatives, functions, integrals, and
differential equations. Your Calculus I class will create a foundation for your higher level math classes, such as Calculus II and III. | {"url":"https://costatutoring.com/calculus-1-tutoring/","timestamp":"2024-11-13T22:27:24Z","content_type":"text/html","content_length":"151446","record_id":"<urn:uuid:dd3adfd1-4cde-44cb-a5df-33537734937f>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00504.warc.gz"} |
Fracturing Search
Author: Benjamin Qi
A simple solution to "Robotic Cow Herd" that generalizes.
• some familiarity with "Robotic Cow Herd" analysis
General Outline
Suppose that you have a rooted tree where each vertex $i$ has a value $v_i$. Also, if $i$ is not the root then $i$ has a parent $p_i$ satisfying $v_{p_i} \le v_i$. Given that each vertex has at most
$D$ children, find the $K$ smallest values in the tree.
Approach 1: Use a priority queue initially containing only the root. At each step, extract the vertex with smallest value from the priority queue and insert all of its children into the queue. Since
we insert $\mathcal{O}(KD)$ vertices in the priority queue, this runs in $\mathcal{O}(KD\log (KD))$ time. You can think of this as Dijkstra on a tree.
Approach 2: Suppose that we know that the $K$-th smallest value is an integer in the range $[0,A]$. Then for any $x\in [0,A]$ we can check whether there are less than $K$ values in the tree less than
or equal to $x$ in $\mathcal{O}(KD)$ time with a simple DFS that breaks once you find $K$ values. This approach runs in $\mathcal{O}(KD\log A)$ time.
We'll focus on the first approach.
Optional: A Faster Solution
There are ways to do this in $\mathcal{O}(K)$ time for a binary tree if you don't need to return the values in sorted order (see here).
Suppose that you want to find the $K$ objects with the smallest values in some (potentially very large) search space.
• First, we need to impose a tree structure satisfying the properties mentioned above. Say that $b$ lies in the subtree of $a$ if $a$ lies above (or is equal to) $b$ in the tree.
• Let the "root" be the object of smallest value. Every object must lie in the subtree of the root.
• The children of the root should partition the entire search space (aside from the root) into a bounded number of disjoint subspaces.
• Of course, each child should also have the smallest value in its subtree.
Essentially, we start with the entire search space and then we fracture it into subspaces based on the children of the root. Then we can finish with either of the two approaches.
$K$-th Smallest Spanning Tree (USACO Camp 2018)
Let's look at an example.
Given a graph with $N\le 50$ vertices and at most $\binom{N}{2}$ edges, find the $K$-th ($K\le 10^4$) smallest spanning tree.
For this problem, the objects are spanning trees. The root is the minimum spanning tree (which can be calculated with Kruskal's algorithm), and contains all objects in its subtree.
The idea is to designate a small number of children of the root, each of which should be formed by modifying the root slightly. If we can somehow ensure that each object has at most $N$ "children"
then we only need to consider $\mathcal{O}(NK)$ spanning trees in order to find the $K$-th smallest.
The first step is to consider the easier problem of finding the second MST. To do this, we can choose to exclude one edge of the MST and then find the smallest possible replacement for it. Let the
edges in the MST be labeled $1\ldots N-1$. Then one idea is to let the $i$-th child subspace of the root to consist of all spanning trees not including edge $i$ of the minimum spanning tree for each
$i\in [1,N-1]$.
Unfortunately, this doesn't work because the child subspaces overlap. We can instead let the $i$-th child subspace contain all spanning trees that
• include the first $i-1$ edges of the MST
• do not include the $i$-th edge of the MST
for each $i\in [1,N-1]$. Every spanning tree other than the root is contained within exactly one of these child subspaces, which is what we want. After sorting the edges in increasing order of weight
once, we can compute the MST within each child subspace in $\mathcal{O}(M\alpha (N))$ time with DSU.
Overall, the runtime is $\mathcal{O}(NMK\alpha(N))$ for storing the information about each spanning tree and $\mathcal{O}(NK\log (NK))$ for maintaing the priority queue of objects so that we can
extract the minimum. Note that with the second approach mentioned in the first section the running time would instead be $\mathcal{O}(NMK\alpha(N)\log ans)$, which may be too slow.
#include <bits/stdc++.h>
using namespace std;
typedef bitset<1225> B;
typedef vector<int> vi;
struct DSU { // for Kruskal's
vi e;
void init(int n) { e = vi(n, -1); }
int get(int x) { return e[x] < 0 ? x : e[x] = get(e[x]); }
bool sameSet(int a, int b) { return get(a) == get(b); }
Robotic Cow Herd
Focus Problem – try your best to solve this problem before continuing!
As with the analysis, for each location you should
• sort the controllers of that location by cost
• add the controller of minimum cost for each location to the cost of the cheapest robot
• subtract that minimum cost from every controller at that location (so now the minimum cost controller for each location is just zero)
Importantly, we should then sort the locations by their respective second-minimum controller costs.
Approach 1
Binary search on the cost $c$ of the $K$-th robot. If we can compute the costs of all robots with cost at most $c$ or say that there are more than $K$ in $\mathcal{O}(K)$ time, then we can solve this
problem in $\mathcal{O}(N\log N+K\log \max(c))$ time (similar to "Approach 2" above). This is the approach that the first analysis solution takes, although it includes an extra $\log N$ factor due to
upper_bound. I have removed this in my solution below.
#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
typedef vector<int> vi;
typedef pair<ll, ll> pl;
#define f first
#define s second
Approach 2
There's also an $\mathcal{O}(N\log N+K\log K)$ time solution with a priority queue that constructs the robots in increasing order of cost. As before, we want each robot to have a bounded number of
"child" robots. However, if you look at my DFS function above, it seems that every robot can have up to $N$ children! Nevertheless, the DFS takes $\mathcal{O}(K)$ rather than $\mathcal{O}(KN)$ time
due to the break statement, which works since we sorted by second-cheapest robot.
In fact, we can modify the DFS function so that every robot has at most three rather than $N$ children.
void dfs(int pos, ll cur, int id) {
if (cur > mx || num == K) return;
res += cur;
if (id + 1 < v[pos].size()) dfs(pos, cur + v[pos][id + 1] - v[pos][id], id + 1);
if (pos + 1 < v.size()) {
if (id == 1) dfs(pos + 1, cur - v[pos][1] + v[pos + 1][1], 1);
if (id) dfs(pos + 1, cur + v[pos + 1][1], 1);
Now I'll describe how the priority queue solution works:
First start with the robot of minimum cost. The robot with second-minimum cost can be formed by just choosing the second-minimum controller for the first location. After this, we have a few options:
• We can choose the third-minimum controller for the first location.
• We can discard the second-minimum controller for the first location and select the second-minimum controller for the second location (and never again change the controller selected for the first
• We can keep the second-minimum controller for the first location and select the second-minimum controller for the second location (and never again change the controller selected for the first
None of these options can result in a robot of lower cost. In general, suppose that we have a robot and are currently selecting the $j$-th cheapest controller for the $i$-th location. Then the
transitions are as follows:
• Select the $j+1$-th cheapest controller for the $i$-th location instead.
• If $j=2$, select the $1$-st cheapest controller for the $i$-th location instead and also select the $2$-nd cheapest controller for the $i+1$-st.
• Keep the $j$-th cheapest controller for the $i$-th location and also select the $2$-nd cheapest controller for the $i+1$-st.
Since there exists exactly one way to get from the cheapest robot to every possible robot, we can use a priority queue.
#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
typedef pair<int, int> pi;
typedef vector<int> vi;
typedef pair<ll, pi> T;
#define f first
#define s second
Other Problems
Status Source Problem Name Difficulty Tags
Baltic OI 2019 - Olympiads Normal
CCO Shopping Plans Very Hard
YS K-th Shortest Walk Insane
Module Progress:
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Tips for Working with Calculated Fields in Tableau
To help you become more efficient with creating and editing calculated fields in Tableau, this article lists several tips for working in the calculation editor.
Note: The GIFs in this topic show an older version of the UI. The Data pane no longer calls out Dimensions and Measures.
Drag and drop fields into the calculation editor
When creating fields in the calculation editor, you can drag existing fields from the Data pane into the editor at any time.
Drag and drop formulas from the calculation editor to the Data pane
When typing a calculation in the calculation editor, you can highlight all or part of the formula and drag it to the Data pane to create a new calculated field. You can then rename the field by
typing a name. For more information, see Ad-Hoc Calculations(Link opens in a new window).
Use the functions reference in the calculation editor
When typing a calculation in the calculation editor, you can use the functions reference to browse all the functions available in Tableau.
To open the functions reference:
In the calculation editor, click the triangle icon on the right-side of the editor.
To add a function from the reference to a formula:
In the function reference, double-click a function.
Take advantage of auto-complete for formulas
As you type a formula in the calculation editor, Tableau suggests options to complete items in your formula. Tableau suggests functions, fields in your data source, parameters, sets and bins that
begin with or contain the string you type. The list of suggestions update as you type.
To add an item from auto-complete to a formula:
Press Enter on your keyboard to select the highlighted suggestion. You can use the up and down arrows on your keyboard to move between items in the auto-complete list.
Drag table calculations into the calculation editor to edit them
When you create a table calculation, you can drag it into the calculation editor to review or make changes to the formula.
To edit a table calculation in the calculation editor:
1. In the Analysis menu, select Create Calculated Field...
2. From the worksheet, drag the table calculation into the calculation editor.
3. When finished, click OK.
Resize text in the calculation editor
You can adjust the size of the text in the calculation editor as you create or edit calculations.
To increase text size in the calculation editor:
Press the CTRL and + keys on your keyboard (Command + on a Mac)
To decrease text size in the calculation editor:
Press the CTRL and - keys on your keyboard (Command - on a Mac).
Note: Text size persists until you close the editor. The next time you open the editor, text is at the default size.
See which sheets are using a calculated field
As you edit a calculated field, you can click Sheets Affected to see which other sheets are using the field. These sheets will also be updated when you commit your changes.
Format numbers and dates
Tip: Sometimes a calculation isn't needed, just some formatting.
There are times that a number or date value will be correct but it doesn't appear how you'd like. For example, a date displaying as 2027/02/02 instead of 2Feb27. This doesn't need a date calculation
to fix. Instead, format the date to the visual presentation you desire.
Similarly, the results of the ROUND() function can sometimes display oddly due to details of the data source. To control how the results appear, set the number format to specify the number of decimal
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Difference between Set and Subset
| Difference between Set and Subset
Difference between Set and Subset
June 14, 2023
The key difference between a set and a subset is that a set is a collection of distinct elements, while a subset is a subset of a larger set and contains elements that are also present in the larger
Set Vs Subset
Set Subset
A collection of distinct elements. A subset is a subset of a larger set.
Can contain any number of elements. Contains elements that are also present in the larger set.
Examples: {1, 2, 3}, {a, b, c, d}, {red, blue, green}. Examples: {1, 2} is a subset of {1, 2, 3}, {red, green} is a subset of {red, blue, green}.
Difference between Set and Subset
• A collection of distinct elements.
• Can contain any number of elements.
• Examples: {1, 2, 3}, {a, b, c, d}, {red, blue, green}.
• A subset is a subset of a larger set.
• Contains elements that are also present in the larger set.
• Can have fewer or the same number of elements as the larger set.
• Examples: {1, 2} is a subset of {1, 2, 3}, {red, green} is a subset of {red, blue, green}. | {"url":"https://eduinput.com/difference-between-set-and-subset/","timestamp":"2024-11-11T04:46:36Z","content_type":"text/html","content_length":"149183","record_id":"<urn:uuid:f3dcd7df-4e79-4cd5-b298-958b465cb0f7>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00111.warc.gz"} |
How Mortgage Loans Work
Excluding property taxes and insurance, a traditional fixed-rate mortgage payment consist of two parts: (1) interest on the loan and (2) payment towards the principal, or unpaid balance of the loan.
Many people are surprised to learn, however, that the amount you pay towards interest and principal varies dramatically over time. This is because mortgage loans work in such a way that the early
payments are primarily in interest, and the later payments are primarily towards the principal.
In the beginning... you pay interest
To help calculate monthly payments for loans based on different interest rates, lenders long ago developed what are known as "amortization tables." These tables also make it fairly easy to calculate
how much money of each payment is interest, and how much goes towards the principal balance.
For example, let's calculate the principle and interest for the very first monthly payment of a 30-year, $100,000 mortgage loan at 7.5 percent interest. According to the amortization tables, the
monthly payment on this loan is fixed at $699.21.
The first step is to calculate the annual interest by multiplying $100,000 x .075 (7.5 %). This equals $7,500, which we then divide by 12 (for the number of months in a year), which equals $625.
If you subtract $625 from the monthly payment of $699.21, we see that:
• $625 of the first payment is interest
• $74.21 of the first payment goes towards the principal
Next, if we subtract $74.21 (the first principal payment) from the $100,000 of the loan, we come up with a new unpaid principal balance of $99,925.79. To determine the next month's principal and
interest payments, we just repeat the steps already described.
Thus, we now multiply the new principal balance (99,925.79) times the interest rate (7.5%) to get an annual interest payment of $7,494.43. Divided by 12, this equals $624.54. So during the second
month's payment:
• $624.54 is interest
• $74.67 goes towards the principal.
Note: In Canada, payments are compounded semi-annually instead of monthly.
As you can see from the above example, even though you pay a lot of interest up front, you're also slowly paying down the overall debt. This is known as building equity. Thus, even if you sell a
house before the loan is paid in full, you only have to pay off the unpaid principal balance--the difference between the sales price and the unpaid principle is your equity.
In order to build equity faster--as well as save money on interest payments--some homeowners choose loans with faster repayment schedules (such as a 15-year loan).
Time versus savings
To help illustrate how this works, consider our previous example of a $100,000 loan at 7.5 percent interest. The monthly payment is around $700, which over 30 years adds up to $252,000. In other
words, over the life of the loan you would pay $152,000 just in interest.
With the aggressive repayment schedule of a 15-year loan, however, the monthly payment jumps to $927-for a total of $166,860 over the life of the loan. Obviously, the monthly payments are more than
they would be for a 30-year mortgage, but over the life of the loan you would save more than $85,000 in interest.
Bear in mind that shorter term loans are not the right answer for everyone, so make sure to ask your lender or real estate agent about what loan makes the best sense for your individual situation. | {"url":"https://kigarrealty.com/how-a-mortgage-works.php","timestamp":"2024-11-07T00:03:02Z","content_type":"application/xhtml+xml","content_length":"21864","record_id":"<urn:uuid:fed2dc3b-8557-4e8f-9f5b-de1e15c22ce8>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00771.warc.gz"} |
Coalition of cubic graphs of order at most 10
title = "Coalition of cubic graphs of order at most 10",
abstract = "The coalition in a graph G consists of two disjoint sets of vertices V1 and V2, neither of which is a dominating set but whose union V1 ∪ V2, is a dominating set. A coalition partition in
a graph G is a vertex partition π={V1, V2, ..., Vk} such that every set Vi∈Π is not a dominating set but forms a coalition with another set Vj∈Π which is not a dominating set. The coalition number C
(G) equals the maximum κ of a coalition partition of G. In this paper, we compute the coalition numbers of all cubic graphs of order at most 10.",
keywords = "coalition, cubic graphs, Petersen graph",
author = "Saeid Alikhani and Хамидреза Голмохаммади and Константинова, {Елена Валентиновна}",
note = "The research by Hamidreza Golmohammadi and Elena V. Konstantinova was supported by the Russian Science Foundation under grant no. 23-21-00459.",
year = "2024",
month = sep,
doi = "10.22049/CCO.2023.28328.1507",
language = "English",
volume = "9",
pages = "437--450",
journal = "Communications in Combinatorics and Optimization",
issn = "2538-2128",
publisher = "Azarbaijan Shahid Madani University",
number = "3",
TY - JOUR
T1 - Coalition of cubic graphs of order at most 10
AU - Alikhani, Saeid
AU - Голмохаммади, Хамидреза
AU - Константинова, Елена Валентиновна
N1 - The research by Hamidreza Golmohammadi and Elena V. Konstantinova was supported by the Russian Science Foundation under grant no. 23-21-00459.
PY - 2024/9
Y1 - 2024/9
N2 - The coalition in a graph G consists of two disjoint sets of vertices V1 and V2, neither of which is a dominating set but whose union V1 ∪ V2, is a dominating set. A coalition partition in a
graph G is a vertex partition π={V1, V2, ..., Vk} such that every set Vi∈Π is not a dominating set but forms a coalition with another set Vj∈Π which is not a dominating set. The coalition number C(G)
equals the maximum κ of a coalition partition of G. In this paper, we compute the coalition numbers of all cubic graphs of order at most 10.
AB - The coalition in a graph G consists of two disjoint sets of vertices V1 and V2, neither of which is a dominating set but whose union V1 ∪ V2, is a dominating set. A coalition partition in a
graph G is a vertex partition π={V1, V2, ..., Vk} such that every set Vi∈Π is not a dominating set but forms a coalition with another set Vj∈Π which is not a dominating set. The coalition number C(G)
equals the maximum κ of a coalition partition of G. In this paper, we compute the coalition numbers of all cubic graphs of order at most 10.
KW - coalition
KW - cubic graphs
KW - Petersen graph
UR - https://www.scopus.com/record/display.uri?eid=2-s2.0-85195019633&origin=inward&txGid=8f13530b439bf419eeb17b0680f34a38
UR - https://www.mendeley.com/catalogue/5878cf63-5c82-3bf6-96e4-7a56f824795c/
U2 - 10.22049/CCO.2023.28328.1507
DO - 10.22049/CCO.2023.28328.1507
M3 - Article
VL - 9
SP - 437
EP - 450
JO - Communications in Combinatorics and Optimization
JF - Communications in Combinatorics and Optimization
SN - 2538-2128
IS - 3
ER - | {"url":"https://pure.nsu.ru/portal/en/publications/coalition-of-cubic-graphs-of-order-at-most-10(88752511-8f88-4dbc-8994-0ce26ddef464)/export.html","timestamp":"2024-11-10T19:34:11Z","content_type":"application/xhtml+xml","content_length":"18489","record_id":"<urn:uuid:c5239735-9ccb-41f7-b4c1-a8c94aa1f91d>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00889.warc.gz"} |
Python Challenge Answers - HackMD
# Python Challenge Answers Challenges for 10-26-18 - python answers ``` # cumulative sum x=[3,10,4,12,55] cs=[0]*5 for i in range(0,len(x)): cs[i]=sum(x[0:(i+1)]) print(cs) x=[3,10,4,12,55] cs=list()
for i in range(0,len(x)): cs.append(sum(x[0:(i+1)])) print(cs) #how long to 5 5's x=[5,3,2,5,5,1,2,5,3,5,1,5,1] count=0 i=0 while count < 5: if x[i]==5: count+=1 i+=1 print(i) # answers for Lecture
11 extra practice import pandas as pd # 0 - calculate sum of female and male wages in wages.csv wages=pandas.read_csv("wages.csv",header=0,sep=",") femaleSum=0 maleSum=0 for i in range(0,len
(wages),1): if wages.gender[i]=="female": femaleSum=femaleSum+wages.wage[i] else: maleSum=maleSum+wages.wage[i] femaleSum maleSum sum(wages.gender=="female") sum(wages.gender=="male") wages.tail() #
find runs # load file findRuns=pd.read_csv("findRuns.txt",header=None,sep="\t") # create a variable out that is currently undefined out=pd.DataFrame(columns=['startIndex','runLength']) # I will use
this variable cur to hold onto the previous number in the vector; # this is analagous to using findRuns[i-1] cur=findRuns.iloc[0,0] #cur=findRuns[i-1] # this is a counter that I use to keep track of
how long a run of repeated values is; # if there are not repeated values than this count equals 1 count=1 # loop through each entry of our vector (except the 1st one, which we set to cur above) for i
in range(1,50,1): # test if the ith value in the vector findRuns equals the previous (stored in cur) if findRuns.iloc[i,0]==cur: # test whether count is 1 (we aren't in the middle of a run) or >1 (in
the middle of a run) if count==1: # if the ith value in the vector equals the previous (stored in cur) and count is 1, we # are at the beginning of a run and we want to store this value (we
temporarily store it in 'start') start=(i-1) # we add one to count because the run continued based on the ith value of findRuns being equal to # the previous (stored in cur) count=count+1 # if the
ith value in findRuns is not the same as the previous (stored in cur) we either are not in a run # or we are ending a run else: # if count is greater than 1 it means we were in a run and must be
exiting one if count>1: # add a row to 'out' that will hold the starting positions in the first column and the length # of runs in the second column; this appends rows to out after finding and
counting each run out.loc[len(out)]=[start,count] # reset count to 1 because we just exited a run count=1 # remember cur holds the previous element in findRuns, so we need to update this after each
time # we go through the for loop cur=findRuns.iloc[i,0] cur out ``` | {"url":"https://hackmd.io/@NFpEogXySTuWExLvDQQHig/S1oyAag3Q?type=view","timestamp":"2024-11-09T06:26:27Z","content_type":"text/html","content_length":"32954","record_id":"<urn:uuid:d8230bd1-a4f5-4a6f-9dc5-7cc5616b0d5e>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00756.warc.gz"} |
Fill series by hour | Microsoft Community Hub
I think there should be a simple solution to this, but it is escaping me. I'm creating a food diary, and want to make one column show the date, and the next column show every hour of that day. Ea...
• If i enter "12:00 AM" in cell C2 i can use the "fill handle" to drag the cell down. | {"url":"https://techcommunity.microsoft.com/discussions/excelgeneral/fill-series-by-hour/3688617/replies/3688668","timestamp":"2024-11-10T02:59:19Z","content_type":"text/html","content_length":"214458","record_id":"<urn:uuid:2c6846c9-131c-47f9-8946-a539014efbe9>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00338.warc.gz"} |
Heap | CodingCargo
This page provides links to solutions that use the Deque approach.
A heap is a specialized tree-based data structure that satisfies the heap property. It's commonly implemented as a binary tree where each node's value is less than or equal to (in a min-heap) or
greater than or equal to (in a max-heap) the values of its children. In a max-heap, the maximum value is at the root node, and in a min-heap, the minimum value is at the root.
How to Spot These Problems
You can identify heap problems if the problem requires you to:
• Efficiently retrieve the minimum or maximum element from a collection.
• Solve optimization problems, such as finding the k-th largest or smallest element in an array.
Leetcode Problem Set | {"url":"https://www.codingcargo.com/data-structures-and-algorithms/heap","timestamp":"2024-11-09T07:14:40Z","content_type":"text/html","content_length":"17564","record_id":"<urn:uuid:0b0f0a4d-aae4-48a1-9945-37696938cd87>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00566.warc.gz"} |
Bytes, bits, megabytes, teras ... learn to convert between units - MeTimeTech
In a digitized society in which information is so important, one of the basic principles is to know some of the main concepts that we move about. For this reason, you have probably heard many times
about megabytes, gigabytes, teras and other units of measure, and you may want to know what they are and how to convert from one to the other so as not to confuse them. In addition, we must bear in
mind that these terms can not only refer to storage units, but also to internet speed.
What is bit and byte?
The bit It is the basic unit of information in computing. It consists of a binary language digit that expresses a value of 0 or 1. It is used to represent the contraposition between two values that
are open and closed, off and on, false and true). This is the minimum unit of information in computing. Although one bit can represent two specific values, infinite combinations can be made. Another
relevant measure is the byte, not to be confused with the first. The latter can be quantized as 8 bits.
In computing and telecommunications, since the structure of computers is based on binary numbers, bytes are counted in powers of two. Hence, a kilobyte equals 1024 bytes and not 1000, since 1024 is
the result of 2 raised to 10.
We are going to show the equivalences between one and the other units of measurement based on these basic information units.
Units of measurement and equivalences
To give us a general idea of what we have just mentioned, a gig is capable of storing up to a total of a thousand data more than a mega. Specifically, a giga is equivalent to 1,024 megabytes.
Starting from the bytes as minimum unit, the different units of measurement are expressed as follows:
• One Kilobyte is 1024 bytes
• One Megabyte is 1024 KB
• One Gigabyte is 1024 MB
• One Terabyte is 1024 GB
• One Petabyte is 1024 TB
• One Exabyte is 1024 PB
• One Zettabyte is 1024 EB
• One Yottabyte is 1024 ZB
These units of measurement are often used when buying a new computer, tablet, mobile device, pendrive, hard drive or device in general. Obviously, we are not going to look for the same storage on a
smartphone than on a computer. Therefore, we must know these units of measurement and look for a device that suits our specific needs, depending on what we use the device for and if we are going to
store more or less things.
Unit Symbol Equivalence
1 kilobyte kB 1024 bytes
1 megabyte MB 1024 KB
1 gigabyte GB 1024 MB
1 terabyte TB 1024 GB
1 petabyte PB 1024 TB
1 exabyte EB 1024 PB
1 zettabyte ZB 1024 EB
1 yottabyte YB 1024 ZB
To measure the contracted internet connection speed, we use a measure based on the bits transmitted per second, although it should be noted that in fiber optics and ADSL more use is made of megabits
per second (Mbps) and gigabits per second (Gbps).
How to calculate
Starting from the bytes as minimum unit, and taking into account that we are talking in decimal base, a gigabyte is the storage unit (GB), equivalent to 1,000,000,000 bytes. It is not to be confused
with the gibibyte (GiB), which is equal to 1024 MiB. The gigabyte (GB) equals 1000 MB. The latter is the one used for the storage capacity or size of the files. A unit larger than this is the
Terabyte, increasingly present in more devices, equivalent to 1,000 GB.
Taking into account these differentiations, we are going to see how we can easily go from one unit to another by making their equivalence and taking into account what we are based on to make the
Megabytes to gigabytes
In many cases, you will find that the smallest unit of measure used (which does not exist) is the megabyte, so it is good to know how many megabytes are in a gigabyte and even how many are in a
In order to make the equivalence of megabytes to gigabytes, we find two ways of expressing ourselves, which are the decimal base (which starts from a base 10) and the binary base (which starts from a
base 2).
Name Decimal Name Binary unit
Kilobyte (kB) 10^3 Kibibyte (KiB) 2^10
Megabyte (MB) 10^6 Mebibyte (MiB) 2^twenty
Gigabyte (GB) 10^9 Gibibyte (GiB) 2^30
Terabyte (TB) 10^12 Tebibyte (TiB) 2^40
In this way, we find the following equivalences depending on the base from which we start and the unit of measurement, although in this case we will give priority to the decimal base:
• 1000 megabytes (MB) = 1 gigabyte (GB)
• 1024 mebibyte (MiB) = 1 gibibyte (GiB)
• 953,674 mebibyte (MiB) = 1 gigabyte (GB)
Gigabytes to Terabytes
To make the move from gigabytes to terabytes, increasingly used in different devices such as the most modern hard drives, we are going to continue using the two ways of expressing ourselves, which
are the binary base and the decimal base.
• 1000 gigabyte (GB) = 1 terabyte (TB)
• 1024 gibibyte (GiB) = 1 tebibyte (TiB)
• 931,323 gibibyte (GiB) = 1 terabyte (TB)
In most manufacturers today the decimal base is used as the unit of measurement, although computers are using the binary base.
Although we have already said it before, it is good to emphasize that starting from the decimal system 1 GB is equivalent to 1000 MB and 1 TB is equivalent to 1000 GB. So 1GB = 1000MB and 1TB =
Terabytes Gigabytes
1 TB 1000 GB
2 TB 2000 GB
3 TB 3000 GB
Computers use the binary system, so it is very important that when purchasing a new device you really know the amount of storage they are offering you to know what you are going to have and you can
also compare it with the real capacity of your team.
1 KB (kilobyte) 0.976563 KiB (kibityte)
1 MB (megabyte) 0.953674 MiB (mebibyte)
1 GB (gigabyte) 0.931323 GiB (gigibyte)
1 TB (terabite) 0.909495 TiB (tebibyte)
Gigabits per second to megabits per second
Now that we have made the equivalence in units of measure in the case of storage devices or file capacity, it is necessary to calculate the measure of transmission of the information, which is the
one used in internet speeds.
The gigabits per second (Gbps) are the measure used to describe the bandwidth of an internet connection such as fiber optic or the less and less used ADSL.
We must know that 1 Gbps is equivalent to 1000 Mbps. When we talk about a 1 Gbps internet connection, this is much higher than others such as 100 Mbps or 300 Mbps that are being offered by different
companies. Specifically, with respect to 100 Mbps the speed is 10 times higher, which means that it is capable of downloading or sending a file, depending on the case, in much less time.
In the internet connections Megabit per second is often used as a measure, so 1000 megabits per second equals 1 gigabit per second. 1000 gigabits per second would equal 1 terabit per second in your
case. So it would be 1000 Mbps = 1 Gbps. For reference, we must know that 10 Gbps is 1.25 GB / s. In the same way, one gigabit per second is equivalent to 125 MB / s, since it must be divided by 8.
Gigabit per second Megabyte per second
1 Gbps 125 MB / s
2 Gbps 250 MB / s
3 Gbps 375 MB / s
4 Gbps 500 MB / s
5 Gbps 625 MB / s
6 Gbps 750 MB / s
7 Gbps 875 MB / s
8 Gbps 1000 MB / s
10 Gbps 1250 MB / s
The gigabits per second (We have already said previously that in the case of the internet we will start from the base of the bits and not the bytes) they are equivalent to 1,000,000,000 bits. A
gigabit (Gb) is equivalent to 1/8 of a gigabyte, since it must be remembered that a byte is 8 bits.
The post Bytes, bits, megs, teras … learn to convert between units appeared first on ADSLZone. | {"url":"https://metimetech.com/bytes-bits-megabytes-teras-learn-to-convert-between-units/","timestamp":"2024-11-05T18:42:02Z","content_type":"text/html","content_length":"88233","record_id":"<urn:uuid:22b22ef0-4617-45ef-9f62-5a984afc0754>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00226.warc.gz"} |
Logistic growth in the human world
Today I presented my research for upper secondary school teachers as part of a day on ‘Mathematics in the human world’ (‘Den mänskliga världens matematik’). This is probably one of the most important
messages to get out from applied mathematics. Mathematics isn’t just for physicists, engineers and computer scientists. It is needed everywhere in society, not least in understanding society itself.
I am impressed that the Swedish Royal Society (KVA), who organized the day, had the foresight to choose this theme. I hope we gave some inspiration to the 50 or so teachers who gave up there time to
came along to listen.
Given the vast subject of ‘humanity and mathematics’, it was interesting that three of the four scientific talks (my own, Tom Britton’s and Kimmo Eriksson’s) all included a detailed description of
the logistic growth equation. In modeling three completely different contexts---disease spread, adoption of mobile phone usage and applause after an academic talk---this innocuous little equation
plays a central role.
The logistic equation can be derived from simple assumptions about social behavior. Assume you have heard a rumour and you tell 3 randomly chosen people about it during 24 hours. If there are N
people and X of them have heard the rumour, then the probability that each of these random people have not already heard the rumour is 1-X/N. On average, you will ‘infect’ 3(1-X/N) people with the
rumour. Now, if all the X people who have heard the rumour behave in the same way as you do then the average hourly increase in people knowing the rumour will be
dX = (3/24)X(1-X/N)
which is the logistic growth equation. The growth dX is small when either X is small (no-one has heard the rumour) or X is nearly equal to N (everyone has heard the rumour). The rumour spreads
fastest when X=N/2 and half the population know about the rumour. This leads to the sigmoidal growth curve shown here.
We published a paper earlier in the year looking at clapping in a small audience. First, we showed that both the onset and cessation of applause followed a logistic growth curve. Then we tried to
address a problem that Kimmo raised during his talk today: “lots of mechanisms produce logistic growth curves, how do we know that it is ‘social contagion’ in any particular case?” We looked at
clapping events, and compared the fit of a social contagion model with various models where individuals chose a randomly distributed number of claps to do irrespective of the behavior of others.
Social contagion models were the most important factor in predicting the individual clap patterns seen in the data.
This paper got a lot of media attention, mainly because of our prediction that long applauses can occur not just because a presentation is good, but also because of a failure to co-ordinate stopping.
Richard Mann gave a well-balanced interview on BBC about this. Richard also did a fun analysis, again using the logistic curve, of the media contacts he received after publication.
Of course, neither my presentation today nor the others at the humanity mathematics day were limited to logistic growth. I talked about Schelling’s model of segregation and collective motion, Kimmo
about popularity of names and cultural evolution, and Tom about disease networks. Klas Markström, who helped plan the day together with Ingvar Isfeldt at KVA, described the mathematics and paradoxes
of voting and democracy. There is a diverse mathematics to humanity. | {"url":"https://collective-behavior.blogspot.com/2013/11/logistic-growth-in-human-world.html","timestamp":"2024-11-03T17:10:30Z","content_type":"text/html","content_length":"72822","record_id":"<urn:uuid:84bbe525-d597-44c8-acaf-d25b2ba51844>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00059.warc.gz"} |
Problem Set 4 | ECON 306 — Microeconomic Analysis
Please submit on Blackboard Assignments by the end of the day Friday April 1.
Please read the instructions for completing homeworks.
Concepts and Critical Thinking
Please answer the following questions briefly (1-3 sentences). Use examples as necessary. Be sure to label graphs fully, if appropriate.
Question 1
What is the difference between accounting profit and economic profit? Is it possible for a firm to be profitable in an accounting sense but not an economic sense? Is it possible for a firm to be
profitable in an economic sense but not an accounting sense?
Question 2
In a competitive industry, with identical firms (e.g. all firms have the same costs), why are profits normal (zero) in the long run?
Question 3
In a competitive industry, even among firms with significant cost differences (e.g. there are economic rents), why do profits tend to return to normal (0) in the long run?
Question 4
For each industry below, indicate (i) whether (and why) it is likely a constant cost industry, increasing cost industry, or decreasing cost industry and (ii) what would happen to the long run
equilibrium price and output in the industry as a result of an increase in industry demand.
1. Software distribution
2. Pencil manufacturing
3. Gold mining
Question 5
For each of the following pairs of goods, which do you expect to be more elastically supplied, and why?
1. Toothpicks vs. Scotch whisky
2. Construction workers in Frederick, MD vs. construction workers in the State of Maryland
3. Supply of breakfast cereal vs. supply of food
4. Original Van Gogh paintings vs. reproductions of Van Gogh paintings
5. Gasolone tomorrow vs. gasoline over the next 10 years
Show all work for calculations. You may lose points, even if correct, for missing work. Be sure to label graphs fully, if appropriate.
Question 6
Mike’s Bikes produces racing bicycles. Consider the following graph, which illustrates the short run average total cost curves corresponding to three possible plant sizes Mike could produce with: a
small plant, a medium plant, and a large plant.
Part A
If Mike wanted to produce 125 bikes, what size plant should be used, and why? What about 150 bikes?
Part B
If Mike wanted to produce 250 bikes, what size plant should be used, and why? What about 275 bikes?
Part C
Draw the long run average cost curve on the graph provided (or sketch one yourself).
Part D
Suppose Mike’s long run total cost function can be roughly expressed as:
\[LRC(q)=\frac{1}{64}q^3-6.25q^2+725q\] with a long run marginal cost function of
Find the quantity of bikes where long run average cost is minimized. Plot this point on your graph. At what range of production does Mike experience economies of scale? At what range of production
does Mike experience diseconomies of scale?
Question 7
Daniel’s Midland Archers has the following cost structure for producing archery bows:
\[\begin{align*} C(q)&=2q^2+3q+50 \\ MC(q)&=4q+3\\ \end{align*}\]
Part A
Write an equation for fixed costs, \(f\).
Part B
Write an equation for variable costs, \(VC(q)\).
Part C
Write an equation for average fixed costs, \(AFC(q)\).
Part D
Write an equation for average variable costs, \(AVC(q)\).
Part E
Write an equation for average (total) costs, \(AC(q)\).
Part F
At what price does Daniel’s Midland Archers break even?
Part G
Below what price would Daniel’s Midland Archers shut down in the short-run?
Part H
Write an equation for the firm’s short-run supply curve, and sketch a rough graph.
Part I
What differences would there be between Daniel’s Midland Archers supply curve in the short run versus the long run?
Part J
In the long run, with many identical sellers of archery bows, what would we expect to be the equilibrium price in the market?
Question 8
The supply of movie tickets in a small town is given by:
Part A
Write the inverse supply function.
Part B
Calculate the price elasticity of supply at a price of $6 per ticket.
Part C
Calculate the price elasticity of supply at a price of $8 per ticket. | {"url":"https://micros22.classes.ryansafner.com/assignments/04-problem-set/","timestamp":"2024-11-06T23:31:14Z","content_type":"text/html","content_length":"22528","record_id":"<urn:uuid:180bed54-a60b-43b2-8750-b48b15270ff5>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00477.warc.gz"} |
Multiplication Of Fractions Area Model Worksheets
Mathematics, particularly multiplication, creates the keystone of various academic self-controls and real-world applications. Yet, for numerous learners, mastering multiplication can posture a
difficulty. To resolve this hurdle, teachers and moms and dads have accepted a powerful tool: Multiplication Of Fractions Area Model Worksheets.
Introduction to Multiplication Of Fractions Area Model Worksheets
Multiplication Of Fractions Area Model Worksheets
Multiplication Of Fractions Area Model Worksheets -
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Teaching With A Mountain View Multiplying Fractions
Teaching With A Mountain View Multiplying Fractions
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Our multiplying fractions using visual models worksheets with answer keys help children instantly key into exercises on finding the product of two fractions using number lines arrays
Printable PDF Area Model Worksheets with Answers Area models are a powerful tool for deepening students understanding of multiplication and algebraic concepts Cazoom Maths
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Multiplying Fractions Area Model Worksheet
Multiplying Mixed Fractions
Check more of Multiplication Of Fractions Area Model Worksheets below
Multiplying Fractions Worksheets Multiplying Fractions Fractions
Dividing Fractions By Area Model YouTube
50 Multiplying Fractions Area Model Worksheet Chessmuseum Template
Multiplying Fractions Using The Area visual Model Nate Mack Fifth Grade
How To Model Multiplication Of Fractions Area Model YouTube
Print Area Model Multiplication Worksheets Easy
These worksheets explain how to describe the area of a shape by using a multiplication sentence Your students will use these worksheets to multiply fractions as represented by shaded sections of
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6 Free Multiplying Fractions With Area Models
https://youvegotthismath.com › multiplyi…
These multiplying fractions with area models worksheets will help to visualize and understand the multiplication of fractions 3rd and 4th Grade Students will learn multiplication of fractions with
area model methods
These worksheets explain how to describe the area of a shape by using a multiplication sentence Your students will use these worksheets to multiply fractions as represented by shaded sections of
shapes in order to determine
These multiplying fractions with area models worksheets will help to visualize and understand the multiplication of fractions 3rd and 4th Grade Students will learn multiplication of fractions with
area model methods
Multiplying Fractions Using The Area visual Model Nate Mack Fifth Grade
Dividing Fractions By Area Model YouTube
How To Model Multiplication Of Fractions Area Model YouTube
50 Multiplying Fractions Area Model Worksheet Chessmuseum Template
Multiplying Fractions Area Model Worksheet
Multiplying Fractions Area Model Worksheet
The Worksheet For Adding And Subming Fraction Numbers
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Introduction to Using Properties of Rectangles, Triangles, and Trapezoids
What you’ll learn to do: Use properties of rectangles, triangles, and trapezoids to solve problems
This summer, you decide to grow veggies to add to your garden. However, you first need to build raised garden beds in your backyard. These garden beds are in different shapes and sizes, including
rectangles, triangles, and trapezoids. To build these garden beds, you will need to understand measurement, in linear, square, and cubic metrics. You will also learn about the properties of
rectangles, triangles, and trapezoids to make the best garden beds possible.
Before you get started, take this readiness quiz.
readiness quiz
If you missed this problem, review this video.
Simplify: [latex]\frac{1}{2}\left(6h\right)[/latex]
Solution: [latex]3h[/latex]
If you missed this problem, review the following video.
In this section, we’ll continue working with geometry applications. We will add some more properties of triangles, and we’ll learn about the properties of rectangles and trapezoids. | {"url":"https://courses.lumenlearning.com/wm-developmentalemporium/chapter/use-properties-of-rectangles-triangles-and-trapezoids/","timestamp":"2024-11-07T23:28:39Z","content_type":"text/html","content_length":"51122","record_id":"<urn:uuid:6dfbe31b-86cc-4875-8734-970d0ba45e46>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00078.warc.gz"} |
Quadrilaterals and Polygons | Geometry | Achievable CLT
Quadrilateral questions (questions about four-sided figures) on the CLT typically focus on rectangles (and squares, which are a type of rectangle), but they sometimes verge into other shapes like
parallelograms and trapezoids. We will also cover questions about polygons with more than four sides, since they are fairly rare and are solved by dividing them into smaller shapes like triangles and
quadrilaterals. You’ll see several word problems in the examples below, as the CLT frequently asks you to think about quadrilaterals in a real-life situation.
Approach Question
A park contains a rectangular swath of grass surrounded by a path uniformly feet in width. If the outer rim of the path has dimensions feet by feet, what is the area of the path, in square feet?
A. 400
B. 625
C. 1,000
D. 1,200
This problem is an excellent example of why drawing diagrams on your scratch paper is so important; it’s very difficult to visualize the situation correctly without a diagram. Give it a try! Your
diagram should look something like this:
Note that the question describes an x setup on the outside of the path, not the inside (along the grass). This would be the first place where we could go wrong. Once we have those dimensions and the
5-foot-wide path clearly shown, we can approach the problem in one of two ways:
1. We can divide the path into four rectangles, as shown by the dotted lines in the diagram. The rectangles on the top and bottom of the bath have dimensions of x ; the rectangles on the sides,
which included the entire 50-foot edge, have dimensions of x . We can find those areas by multiplying length times width in each case and adding the results together: . The answer is D.
2. A somewhat simpler approach involves recognizing that many CLT problems involve finding an area by subtracting a smaller area from a larger area outside it. (This is often true in problems where
a region of the diagram is shaded.) But be careful: to get the dimensions of the smaller rectangle, we have to subtract , not , from each dimension because the path is on both sides of the
figure. Choice lurks if you make this mistake, but done correctly, the larger rectangle’s area is and the smaller rectangle’s area is . .
In addition to definitions, and since it’s helpful to see all the kinds of quadrilaterals and how they relate to each other, we are including the diagram below. We challenge you to recopy this for
yourself and practice it until you can do it by memory!
Topics for Cross-Reference
The main alternative to a simple quadrilateral question is one that pairs a quadrilateral with another figure–usually a triangle or a circle. When a question has multiple figures, typically the first
thing to identify is what those shapes have in common. For example, when a square is inscribed in a circle, the diagonal of the square is the same as the diameter of the circle.
Flashcard Fodder
The most important formulas for quadrilaterals are for rectangles and squares:
• Area of a rectangle = x (this is in the CLT reference list)
• Perimeter of a rectangle = +
• Area of a square = , where is the side length
• Perimeter of a square =
We’ll also repeat here the formulas regarding angles in a polygon.
• Total degrees in the angles of a polygon: , where n is the number of sides
• Degrees in each angle of a regular polygon:
Sample Questions
Difficulty 1
A student is planning a remodel of her bedroom including the purchase of a new desk. She wants the surface of the desk to take up no more than of the room’s square footage. If the area of the
room is square feet and the desks available are all feet wide, how long, at most, can the student’s desk be? (Ignore the height of the desk in making your calculation.)
A. 3 feet
B. 4 feet
C. 5 feet
D. 6 feet
The answer is B. Sometimes CLT geometry problems will include a percentage or a fraction when considering comparisons in area. Fortunately, the total square footage is an even here so the maximum
surface area of the desk is a clean square feet ( of ). If the desks are feet wide, then we need to divide by , because ( = x ). The desk can be up to feet long.
Difficulty 2
Rectangle is similar to rectangle . If the area of is square units and has a width of , which of the following must be true?
A. Both rectangles are squares.
B. Rectangle MNOP has a length of 12.
C. Side AB is perpendicular to side BC.
D. Rectangle ABCD has a length of 15.
The answer is C. This problem is tricky in that can be tempting to think we know more than we actually do. We have the width of one rectangle but we don’t have a corresponding width for the other
rectangle; we have an area of one triangle but no corresponding area of the other. This means that we should stay away from answers like choices and ; there’s no way to find any of the unknown
widths, lengths, or areas. Choice would be an eternal truism if it read the other way; all squares are rectangles, but since not all rectangles are squares, we don’t know if choice A is true.
The right answer is something that, surprisingly, we would have known all along without any information except that the two polygons are both rectangles. Since rectangles have all 90° angles, any two
of their adjacent sides must be perpendicular. Side must be adjacent (connecting) to because they both contain , which marks a vertex of the rectangle.
Difficulty 3
The area of a rectangle is given by . If one of the sides measures , find the perimeter of the rectangle.
The answer is A. As with many other problems, we should draw the rectangle here, labeling one of the sides . (It doesn’t matter which side we choose; although it is traditional to make a rectangle’s
length longer than its width, standardized tests don’t typically make this distinction. Also, since we’re just trying to come up with , the perimeter of the rectangle, it won’t matter which side we
call and which side we call .)
We can use the rectangle area to understand that times something must equal . What is that something? Do you see the ratio between the terms of both binomials? This suggests that we might want to
factor out something from ; the greatest common factor of that binomial is . If we factor out we get, you guessed it, . So we now know the two dimensions: by . Adding them together and doubling the
result (don’t forget to double for the perimeter! If you forget, you’ll get answer ) gives us .
Difficulty 4
A student is thinking about parallelograms and comes up with the following conjecture:
The diagonals of a parallelogram are always perpendicular.
Which of the following is a counterexample that disproves the above statement?
A. A parallelogram in which all sides are units long.
B. A parallelogram whose diagonals create four right triangles.
C. A parallelogram with interior angles that are all .
D. A parallelogram with two sides of length units and two sides of length units.
The answer is D. You may remember, if you really excelled in geometry in school, that there’s a certain kind of parallelogram whose diagonals are perpendicular. (If you remember that, we’re
impressed!) Let’s assume you don’t remember that property exactly and use some diagramming to support your thinking. First, picture a long, skinny parallelogram like the one in the figure below. (Is
it us, or are we basically drawing the state of Tennessee?) Next to it, draw a parallelogram that is more squat, so that it appears its sides might be somewhere close to congruent (hint hint).
Now draw the two diagonals of both parallelograms. You may notice that the diagonals in the long, skinny parallelogram don’t seem to be anywhere close to perpendicular, but the diagonals of the other
parallelogram could be. This moves us closer to the truth about perpendicular diagonals, that among parallelograms, only the rhombus (all sides congruent) has perpendicular diagonals.
With this in mind, we can eliminate answer choice A because it agrees with the student’s statement rather than violating it. Reading choice carefully shows us that it’s saying something identical to
the student’s conjecture; perpendicular lines, by definition, create right angles. Choice is trickier because it brings up , but we’re looking for angles at the intersection of the diagonals, not at
the vertices. What choice is telling us is that we have a rectangle. But the question is, is that rectangle a square? A non-square rectangle would be a nice counterexample here, but since we can’t
rule out a square, and a square is a type of rhombus, choice leaves us uncertain. Choice is clearer because it resembles our first parallelogram: the long, skinny one. Understanding now that the
student is talking only about parallelograms that are also rhombuses, we can safely choose answer as the counterexample.
Difficulty 5
A stop sign (octagonal in shape) has two identical horizontal, parallel scratches in it that stretch from one diagonal to another and divide the octagon into a central rectangle and two isosceles
trapezoids above and below the rectangle. If each side of the stop sign has a length of inches, what is the combined surface area of the two trapezoids created by the parallel scratches?
The answer is D. This is another figure we should draw so that we can see clearly what’s happening. The question asks us to focus just on the trapezoids, so we can ignore the rectangle unless that
helps us with the trapezoid. It looks like the only thing we know about each trapezoid is all the slant heights are 6 inches long because they are identical to the sides of the octagon. We would do
well to divide each trapezoid into a central rectangle with triangles on the end, like so:
These triangles appear to be isosceles right triangles (), and indeed they are. The proof of that fact has to do with the fact that octagons have interior angles, which in this case divide into and .
But even if you are uncertain of that, it makes sense on standardized tests to assume that shapes appearing in familiar ways are in fact familiar (for example, if it looks like a right angle, it
probably is!). Since the hypotenuse of these little triangles is , we work backward to the legs using the relationship and discover that the legs are in length. So the area of each of the triangles
is . also becomes a side of the rectangle, while the other side is because it mirrors the side of the octagon. So each of the two rectangles has an area of . Multiplying that area by two because
there are two rectangles (one inside each trapezoid) and multiplying the triangle area by to account for all the triangles, then adding it all together, we come to .
That result doesn’t match any of the answer choices. What went wrong? Nothing, as it turns out; we have to remember that, according to the rules of algebra, a radical (in this case a square root) in
the denominator means the fraction can be further simplified. So we must transform the second term in our answer. We rationalize the denominator by multiplying both top and bottom by . This results
in in the denominator, which divides evenly into the numerator of . The result is , answer choice . | {"url":"https://app.achievable.me/study/clt/learn/geometry-quadrilaterals-polygons","timestamp":"2024-11-12T05:25:44Z","content_type":"text/html","content_length":"299376","record_id":"<urn:uuid:2dd9124b-cfe7-43f6-be8e-4248abf361d3>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00582.warc.gz"} |
Aspect Ratio -> 1 and copy-and-paste of figures
• To: mathgroup at smc.vnet.net
• Subject: [mg72278] Aspect Ratio -> 1 and copy-and-paste of figures
• From: Virgil Stokes <vs at it.uu.se>
• Date: Mon, 18 Dec 2006 06:55:35 -0500 (EST)
I am using vers. 5.2.0.0 on a Win2k platform and after some careful testing have
found that the option AspectRatio in Graphics is not working properly on my system.
When I execute the following two commands (the first comes directly from Help on
AspectRatio) the figure plotted on the screen (and on my printer's output) is
not a circle.
For example on my home PC monitor, the displayed circles have a width
(horizontal) of 9.05cm and a height (vertical) of 9.6cm which means that the
"circle" is not plotted as a circle; but an ellipse (aspect ratio measured > 1).
Note, I have been very careful in making these measurements directly on two
different monitors and neither gives a perfect circle! I have also generated
perfect circles in Corel Draw (vers. 13) and what is shown on my monitors and
outputted to my printers are indeed perfect circles. Interestingly, if I copy
and paste these circles into Corel Draw, they have an aspect ratio of 1.
However, when I copy and paste some of my other figures (created with
AspectRatio->1, in Mathematica using Plot or ParametricPlot), edges of these are
often lost and the aspect ratio is incorrect. I have been unable to identify any
consistent pattern in the missing part of the image, except that it is often the
lower right parts of the image. For example, if I plot some function in the
region 0 <= x <= 6, 0 <= y <= 6, with an AspectRatio->1, and
PlotRange->{{0,6},{0,6}} then again the height of the figure produced on my
monitor is greater than its width. If I copy and paste this figure into Corel
Draw, then the x-axis is terminated at x=5.4 (not at x=6 as in the figure shown
in Mathematica) and the lower half of the tic labels along the x-axis are lost.
The following code shows both these problems on my system:
vars = {x[t], y[t]};
f = x[t] (p - q*y[t]);
g = y[t] (-P + Q*x[t]);
eqns = {x'[t] == f, y'[t] == g}
p = 3.0; q = 2;
P = 2.5; Q = 1;
inits = {x[0] == 2.5, y[0] == 0.5};
sol = vars /. NDSolve[Join[eqns, inits], vars, {t, 0, 10}][[1]];
p0 = ParametricPlot[sol, {t, 0, 2.5}, PlotRange -> {{0, 6}, {0,
6}}, AspectRatio -> Automatic, AxesLabel -> {"prey", "predator"}];
These two problems, loss of parts of a figure in a copy-and-paste, and incorrect
display of aspect ratio may be related. I will be glad to provide a Mathematica
notebook which contains further examples of both these problems.
--V. Stokes | {"url":"http://forums.wolfram.com/mathgroup/archive/2006/Dec/msg00413.html","timestamp":"2024-11-09T01:35:32Z","content_type":"text/html","content_length":"32520","record_id":"<urn:uuid:a653299b-4f7e-4e81-914b-7e61149dd8c1>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00564.warc.gz"} |
Reflection on the 2018 AP Chemistry Free Response Questions
Kaleb Underwood | Sun, 05/13/2018 - 20:32
Now that the 2018 administration of the AP Chemistry Exam is in the books, all of us AP Chemistry teachers now have an opportunity to reflect on the year as we turn our attention toward preparing for
the fall.
One part of this process is the review of the released Free Response Questions from this year’s exam. Every year the College Board releases the FRQs from the operational exam (Form O, the version of
the exam most students in the United States take) forty-eight hours following the conclusion of the administration. The released FRQs since the redesign of the exam in 2014 can be found HERE and
questions from 1999-2013 can be found HERE.
The official scoring guidelines will be posted in late summer after the reading of the exam, followed by the extremely valuable Chief Reader Report, scoring statistics, and sample student responses.
Until then, teachers around the world post their draft answers on the National AP Chemistry Teachers Facebook Page or the AP Chemistry Teacher Community and reflect on the exam and what we can learn
from it.
I strongly suggest that all AP teachers take the time to fully answer every released FRQ in order to gain a deeper understanding and “feel” of the test. Additionally, read the Chief Reader Report
when it is released, read what others post online, and engage in the conversation around how the exam is scored. At BCCE this summer, the AP Chief Reader Dr. Paul Bonvallet will give a presentation
in which he reviews the exam. It is always a very informative talk.
My draft answers to the 2018 FRQs, as well as my prediction of how points will be awarded are attached to this post as a PDF. I will now reflect on each of the questions, the exam overall, and
highlight things that I think teachers, especially new teachers, can take away from this test.
Question 1
(a)-(c) - The question is very straightforward at the beginning, and I enjoyed the unique way the limiting reactant question was presented. I appreciated the use of more than two reactants and that
the limiting reactant could be determined without doing much (or really showing any!) mathematics, if a student had a strong understanding of proportional reasoning and immediately recognized the
equimolar solutions.
(d) - Reading graphs and glassware to proper levels of precision is an important skill and has shown up repeatedly since the redesign. Only one point on every exam will be awarded for proper
significant figures and it will always be a lab scenario. Note that it may or may not be specified for the student to report the value with the proper number of sig figs. My bet is that this was the
sig fig question, though others have suggested part (b) given that the volume of the solution provided was so precise (100.00 mL). See 2010 #3 for a very similar question.
(e) - Part (i) is very straightforward, and I liked that they gave a value for the heat capacity of the solution that differed from that of water. I may modify some of my practices to do this,
instead of always assuming it to be 4.184 J/g·°C.
Part (ii) is likely going to stir the pot AP teacher community due to the use of the “per mole of reaction” concept. For those who are not familiar, “per mole of reaction” basically means “every time
the reaction runs as written”. We will have to wait to see until after the reading which calculated value(s) earned credit. To learn more about this concept, read this article written by James
Spencer , the co-chair of the AP Exam Redesign Commission.
(f) - Would a student get credit for simply citing that thermodynamic quantities are intensive properties? Likely not based on conversations I’ve had with previous AP readers. The student would have
to explain in their answer that intensive properties do not vary with amount or state their reasoning in terms of “per mole.”
(g) - Remember, just because Question 4 from the Legacy Exam is gone, that does not mean that students do not have to write equations. In fact, this year they had to write three! This was very
straightforward, as sodium was the only spectator ion.
Question 2
(a) - I really liked this question. It was a unique way of testing the qualitative aspects of stoichiometry and a reminder of the importance of particle diagrams both for exam success, but also for
strong stoichiometric understanding. Have your students draw a ton of particle diagrams, especially in first-year chemistry, when balancing equations and doing stoichiometry.^[1] I wonder how many
students did not realize this was a limiting reactant problem…
(b)-(c) - These questions deal with equilibrium and are very straightforward. For b (ii) I wonder how many students got stuck when they hit the quadratic and didn’t know what to do. I hope not many,
as the question does not ask for a specific value to be calculated. This is a good reminder that quantitative proficiency does not equate to conceptual understanding and that the AP will never
require students to solve a quadratic equation to get an answer to any problem.
(d) - This was a very straightforward bonding question. Have your students practice filling in skeletal structures like this, they pop up frequently for more complex molecules. Two different
structures are possible, and would both earn points since students were not asked to consider formal charge.
(e)-(f) - A very simple, straightforward weak acid /strong base titration with predictable questions.
Question 3
(a) - Students will likely forget that the 4s electrons will be removed from an iron atom before the 3d electrons. Some teachers have expressed concern over Aufbau exceptions, but I see no problem
here as the Course and Exam Description (CED) does not require knowledge of any exceptions on the exam.
(b)-(c) - Basic atomic structure and Coulomb’s Law questions. I do wonder how specific or vague the wording will need to be for (b) to earn the point.
(d)-(e) - A simple redox titration. I am surprised at the numerous uses of molarity in the FRQs this year. Maybe I am wrong that it is abnormal, but at this point on the exam I was thinking, “wow,
more solution stoichiometry!”
(f) - An absurd question, if you know what you’re doing, that got a lot of hilarious posts on social media. Make sure student know proper equipment vocabulary!
(g)-(i) - The wording on part (i) will cause students points here I am afraid and perhaps it could have been phrased a bit more clearly. But with an FRQ that has already been criticized by students
and teachers for its length (9 parts, tying the longest FRQs of the redesigned exam with 2014 #1 and 2015 #1). This is likely only to be a single point.
Question 4
(a) - Again students were asked to explain how a non-polar substance can have a higher boiling point than a polar substance. This is a favorite question, it seems, as the test writers try to exploit
the common misconception that comes from simply memorizing that LDFs are “weak”.
(b) - A pleasant surprise that must be worth two points. As I say to my students, “Give me the points!”
Question 5
(a) - A good use of particle diagrams to test understanding of weak acids. I would rather them have had to choose which was the most accurate representation. The fact that they were told Figure 1 was
better was a huge advantage!
(b) - Students will assume “-x” is negligible even though they calculated it by determining [H^+]. This will cost many a point. The writers seem to be testing students understanding of when to ignore
“-x” in various ways in recent years.
(c) - This will be a bloodbath, just as it was in 2016 #6. Use Q v. K, it is your friend
Both (b) and (c) harken back to 2016 #6 where the modal score was a 0/4. See what was said about this question on what is now called the Chief Reader Report. This is a great question to foster good
class discussion.
Question 6
(a) - As far as I know, this is the first time students have been asked to explain the purpose of the salt bridge, as opposed to drawing the direction of ion flow, so it will be interesting to see
what phrasing earns credit. Many probably still think electrons flow through it…
(b) - Very simple, get those points! Though the algebraic sign of E^o will likely get many students.
Question 7
(a) - Simple PES identification. It would have been nice to see a follow up question that had more depth, but that means that there was an easy point available for rate constant units in part (b).
(b)-(c) - Straightforward first-order kinetics and half-life question. Teachers and students have been grumbling that it is unfair because nuclear chemistry is not part of the course. However, I do
not share their frustration. The CED clearly mentions radioactive decay as an example of first-order kinetics in Essential Knowledge Statement 4A.3(e) under Learning Objective 4.3 and even if it did
not, students should clearly recognize that if the half-life is citable, then it must be constant and therefore the process is first-order, no matter what that process is.
I would hope this question scores very high.
Overall Impressions
This was a very fair free response section that was well balanced between conceptual and quantitative understanding. The last few years we have seen a consistency develop in the style of the exam and
evidence that it is very faithful to the CED. One of the goals of the redesign was to create an exam that was faithful to a set of standards, not just to previous iterations of exams.^[2] The more
exams we see, the more I think that the Test Development Committee is succeeding in this aspect.
What are your thoughts of this years FRQs? I am excited to hear how the reading goes in a few weeks at to debrief at BCCE in South Bend, Indiana with everyone!
^[1] See the following articles and resources about visualizing chemistry using particle diagrams and using the table method (BCA) to solve stoichiometry problems.
Bridle, Chad A. and Ellen J. Yezierski. Evidence for the Effectiveness of Inquiry-Based, Particulate-Level Instruction on Conceptions of the Particulate Nature of Matter. Journal of Chemical
Education, 2012;89;(2), 192-198. DOI: 10.1021/ed100735u.
Dukerich, Larry. “Conceptual Chemistry.” https://www.chemedx.org/blog/conceptual-chemistry.
Hemling, Melissa. “Using Visual BCA Tables to Teach Limiting Reactants”, https://www.chemedx.org/blog/using-visual-bca-tables-teach-limiting-reactants.
Posthuma-Adams, Erica. “Simple Activities to Implement Particle-Level Diagrams", https://www.chemedx.org/blog/simple-activities-integrate-particle-level-diagrams.
Prilliman, Stephan J. “Integrating Particulate Representations in to AP Chemstry and Introductory Chemsitry Courses.” Journal of Chemical Education 2014 91 (9), 1291-1298. DOI: 10.1021/ed5000197.
Underwood, Kaleb. “A Visual and Intuitive Approach to Stoichiometry.” https://teachchemistry.org/professional-development/webinars/a-visual-and-intuitive-approach-to-stoichiometry.
^[2] David Yaron. Reflections on the Curriculum Framework Underpinning the Redesigned Advanced Placement Chemistry Course. Journal of Chemical Education201491;(9), 1276-1279. DOI: 10.1021/ed500103e
Join the conversation.
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Comments 2
These are my scoring guidelines.
Here are some of my thoughts about the answers and the point distributions. I think 3(g) will likely represent the sig fig question. I think most students will report the temperature change in 1(d)
to one decimal place without thinking about sig figs at all. The CB can make the sig fig question a little more subtle than that. They have already asked students in other parts of the FRQs to
convert from grams to moles. Therefore 3(g) is not really about grams to moles, but rather how should one report the answer to the proper number of sig figs. The moles of Fe2O3 must contain exactly 4
sig figs in order to earn credit because the mass of Fe2O3 was reported to 4 sig figs. Student must get their sig figs straight from the data table, but many will ignore this. For 1(e)(ii), I
think that an answer of 300 kJ/mol should earn zero points, because the student made two separate mistakes: wrong magnitude and wrong sign. An answer of –300 kJ/mol should earn 1 point (for the
correct sign but the wrong magnitude). I like giving them 1 point for having the negative sign, since this was mentioned in the stem of the question. We anticipate that we're going to see lots of
answers of "300" and not very many answers of "1200" In my opinion, the second point, magnitude of ΔHrxn, should be harder to earn. For question 2, I think part (b)(i) should be worth 1 point. You
either use the equation to calculate K correctly or you don't. In part (e)(i), you have to get the moles of KOH from determining the volume of KOH on the titration curve and doing the M x V
calculation. Then you have to recognize the 1-to-1 mole ratio between KOH and HNO2 and do the molarity calculation for the acid. That's a lot of manipulations for only 1 point. Therefore I think 2
points for 2(e)(i) seems reasonable. I'll be curious to see what represents the "minimum" acceptable answer in 5(c). Can a student simply state that diuluting a weak acid increases the %
ionization? Do they have to mention Q vs. K? Can they say "shift toward the side with more particles"? There were 2 points allocated in 2016 #6(b) for the dilution question, and it was interesting to
see what sort of responses did (or didn't) earn the justification point for that one. I think the most popularly missed portion of #7 is going to be the units in 7(b). And yet, are we going to
award the point for an answer of "0.70" in 4(b), even if they don't include the units of atm? Hmm. I wish they had asked students to include units in 4(b), considering that they could get an answer
in either atm or torr. Well, this post is long enough, so I guess I'll stop there. Thanks for sharing your reflections.
Thanks for your always insightful comments, Michael!
The more I think about it, the more agree with you on the sig fig point. The math works out too nicely for the graph to be it. I also agree with your point redistribution in question 2, and in fact
you identified where I was torn in where to award points.
I was surprised to see the unit point appear in #7, but I can't imagine any other way the points could be distributed. The reason I was surprised is that for 2017 #2(e)(ii) it was all or nothing. But
I guess every exam will be scored different depending on the content of the exam, even on similar (or almost identical) questions. | {"url":"https://chemedx.org/blog/reflection-2018-ap-chemistry-free-response-questions","timestamp":"2024-11-13T08:46:09Z","content_type":"text/html","content_length":"62373","record_id":"<urn:uuid:69585e02-3555-43bf-a7d8-3a597451a5f3>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00753.warc.gz"} |
Electronic excitations of the charged nitrogen-vacancy center in diamond obtained using time-independent variational density functional calculations
Aleksei V. Ivanov, Yorick L. A. Schmerwitz, Gianluca Levi, Hannes Jónsson
SciPost Phys. 15, 009 (2023) · published 13 July 2023
• doi: 10.21468/SciPostPhys.15.1.009
Elucidation of the mechanism for optical spin initialization of point defects in solids in the context of quantum applications requires an accurate description of the excited electronic states
involved. While variational density functional calculations have been successful in describing the ground state of a great variety of systems, doubts have been expressed in the literature regarding
the ability of such calculations to describe electronic excitations of point defects. A direct orbital optimization method is used here to perform time-independent, variational density functional
calculations of a prototypical defect, the negatively charged nitrogen-vacancy center in diamond. The calculations include up to 511 atoms subject to periodic boundary conditions and the excited
state calculations require similar computational effort as ground state calculations. Contrary to some previous reports, the use of local and semilocal density functionals gives the correct ordering
of the low-lying triplet and singlet states, namely ${}^{3}A_2 < {}^{1}E < {}^{1}A_1 < {}^{3}E$. Furthermore, the more advanced meta generalized gradient approximation functionals give results that
are in remarkably good agreement with high-level, many-body calculations as well as available experimental estimates, even for the excited singlet state which is often referred to as having
multireference character. The lowering of the energy in the triplet excited state as the atom coordinates are optimized in accordance with analytical forces is also close to the experimental estimate
and the resulting zero-phonon line triplet excitation energy is underestimated by only 0.15 eV. The approach used here is found to be a promising tool for studying electronic excitations of point
defects in, for example, systems relevant for quantum technologies.
This publication has been updated
Authors / Affiliations: mappings to Contributors and Organizations
See all Organizations.
Funders for the research work leading to this publication | {"url":"https://www.scipost.org/SciPostPhys.15.1.009","timestamp":"2024-11-10T18:08:14Z","content_type":"text/html","content_length":"39741","record_id":"<urn:uuid:54e1f0ce-c539-43c2-bc0f-aa0e873f4ba3>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00864.warc.gz"} |
Support Productive Struggle in Learning Mathematics – Video in the Middle
Contact us to learn more about Video in the Middle.
This class discussion highights how two students, James and Danielle, interpret x differently. Another student, Matt, uses money (quarters) to explain the distinction between the two interpretations.
As students discuss a variation of the Growing Dots task in which the starting point is shifted, one student shares that when the first dot is at time 0, the rule is 4x – 3. Another student, Angel,
asks, “Where is the –3 in the picture?”
This video focuses on the Polygons task and the interactions between Cindy (the teacher) and Stuart (a student) and his work. Cindy poses questions to Stuart to help understand his approach.
Pascal, Tammy, and Adam to go to the board and show their equation, n = (s + 2)4 – 4, explaining that each of the pool’s 4 sides is s + 2, giving you (s + 2)4. They add that since they counted each
of the 4 corners twice, they needed to subtract them out (– 4).
Maryann introduced the Cubes in a Line task by showing her students two cubes and asking the question, “If I put two cubes together, how many faces are there showing?” We drop in as several students
explain how they arrived at their totals. | {"url":"https://videointhemiddle.org/math-teaching-practice/support-productive-struggle-in-learning-mathematics/","timestamp":"2024-11-11T21:03:17Z","content_type":"text/html","content_length":"52460","record_id":"<urn:uuid:991159ee-aa0b-4dd6-b88a-48a585ff55e7>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00224.warc.gz"} |
Code System Display Definition
average terminology.hl7.org/ Average The mean of N measurements over the stated period.
maximum terminology.hl7.org/ Maximum The maximum value of N measurements over the stated period.
minimum terminology.hl7.org/ Minimum The minimum value of N measurements over the stated period.
count terminology.hl7.org/ Count The [number] of valid measurements over the stated period that contributed to the other statistical outputs.
http:// The total [number] of valid measurements over the stated period, including observations that were ignored because they did not contain valid result
total-count terminology.hl7.org/ Total values.
CodeSystem/ Count
median terminology.hl7.org/ Median The median of N measurements over the stated period.
std-dev terminology.hl7.org/ Standard
CodeSystem/ Deviation
sum terminology.hl7.org/ Sum The sum of N measurements over the stated period.
variance terminology.hl7.org/ Variance The variance of N measurements over the stated period.
20-percent terminology.hl7.org/ 20th The 20th Percentile of N measurements over the stated period.
CodeSystem/ Percentile
80-percent terminology.hl7.org/ 80th The 80th Percentile of N measurements over the stated period.
CodeSystem/ Percentile
4-lower terminology.hl7.org/ Lower The lower Quartile Boundary of N measurements over the stated period.
CodeSystem/ Quartile
4-upper terminology.hl7.org/ Upper The upper Quartile Boundary of N measurements over the stated period.
CodeSystem/ Quartile
http:// The difference between the upper and lower Quartiles is called the Interquartile range. (IQR = Q3-Q1) Quartile deviation or Semi-interquartile range
4-dev terminology.hl7.org/ Quartile is one-half the difference between the first and the third quartiles.
CodeSystem/ Deviation
http:// The lowest of four values that divide the N measurements into a frequency distribution of five classes with each containing one fifth of the total
5-1 terminology.hl7.org/ 1st population.
CodeSystem/ Quintile
http:// The second of four values that divide the N measurements into a frequency distribution of five classes with each containing one fifth of the total
5-2 terminology.hl7.org/ 2nd population.
CodeSystem/ Quintile
http:// The third of four values that divide the N measurements into a frequency distribution of five classes with each containing one fifth of the total
5-3 terminology.hl7.org/ 3rd population.
CodeSystem/ Quintile
http:// The fourth of four values that divide the N measurements into a frequency distribution of five classes with each containing one fifth of the total
5-4 terminology.hl7.org/ 4th population.
CodeSystem/ Quintile
http:// Skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be
skew terminology.hl7.org/ Skew positive or negative, or even undefined. Source: Wikipedia.
kurtosis terminology.hl7.org/ Kurtosis Kurtosis is a measure of the "tailedness" of the probability distribution of a real-valued random variable. Source: Wikipedia.
http:// Linear regression is an approach for modeling two-dimensional sample points with one independent variable and one dependent variable (conventionally,
regression terminology.hl7.org/ Regression the x and y coordinates in a Cartesian coordinate system) and finds a linear function (a non-vertical straight line) that, as accurately as possible,
CodeSystem/ predicts the dependent variable values as a function of the independent variables. Source: Wikipedia This Statistic code will return both a gradient
observation-statistics and an intercept value. | {"url":"https://terminology.hl7.org/5.0.0/ValueSet-observation-statistics.html","timestamp":"2024-11-03T09:21:38Z","content_type":"application/xhtml+xml","content_length":"24990","record_id":"<urn:uuid:431ae98d-0e69-418f-9df4-977dbe7c8e57>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00828.warc.gz"} |
Archives contributions by jacober
1. Users
2. jacober
This page only shows contributions to our file archives; you can find more information about jacober on their user profile page.
Authored files
As of 16 hours, 11 minutes ago, jacober had authored 4 files. The following statistics were current then, but may have changed in the intervening time.
With 4 files, jacober is at rank 26 among all users for number of files authored. These files have been downloaded 512 times, placing jacober at rank 409 for total downloads of their files.
Title Description Downloads Average
Orbit CE This program is designed to help students and enthusiasts understand orbital mechanics and the influence of mass and gravity on orbiting objects. Formulas are 249 —
provided for all calculations and constants programmed for the Earth, Moon, Mars, Jupiter and Kerbin. You can also input…
Delta-V CE This small program uses the rocket equation to calculate the delta-V of a vehicle using exhaust velocity as an input variable. It is excellent for students and 63 —
enthusiasts to understand better how a change in mass over time impacts the change in velocity of a rocket. The equatio…
Hull Speed This simple program calculates the theoretical hull speed of a sailing vessel measured in feet or meters. You decide your unit of measurement and input the LWL, and 87 —
CE the calculator will give you the theoretical hull speed in knots. This is a great app to have if you are lookin…
Stopping This handy program calculates the safe stopping distance (which aids in determining a safe following distance) of a vehicle travelling at any km/h speed. You input 117 —
Distance CE the reaction time in seconds, the vehicle's speed, the roadway's grade (+/-), and the friction coefficient. From th…
jacober has authored 5 reviews, for these files: | {"url":"https://dev.cemetech.net/downloads/users/jacober","timestamp":"2024-11-14T20:12:48Z","content_type":"text/html","content_length":"18974","record_id":"<urn:uuid:f60874ee-4d13-4963-a3e8-0af039315cd2>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00874.warc.gz"} |
: I guess
wanted this page, so here goes... :^)
: Oops, sorry, what I meant was
Shuffle a list
: Ah, well, at least I got to write another un/sorting story page.This page discusses shuffling algorithms and implementations thereof.
Version 1: sorting random reals edit
Here's a proc, adapted from
proc shuffle {data} {
set rand_data [list]
foreach elem $data {
lappend rand_data [list [expr {rand()}] $elem]
set result [list]
foreach elem [lsort -real -index 0 $rand_data] {
lappend result [lindex $elem 1]
return $result
It works, but it's seems suboptimal.
Version 2: sorting random integers edit
For starters, the real sort bothers me. I imagine sorting by integer would be more efficient, but then again producing random integers is (currently) slower than producing random reals. Change rand()
for int(0x7fffffff * rand()) and -real for -integer and it winds up taking longer. See the benchmark below. (By the way, how am I supposed to portably get the maximum and minimum integer?)Next,
having up to three copies of the list (one of which is augmented) appears to be a waste of RAM.
Version 3: in-place swaps edit
So would it be better to do an in-place shuffle? For each element of $data it could swap with another, randomly-selected element of $data. I'll take a shot at it:
proc shuffle {data} {
set length [llength $data]
for {set idx_1 0} {$idx_1 < $length} {incr idx_1} {
set idx_2 [expr {int($length * rand())}]
set temp [lindex $data $idx_1]
lset data $idx_1 [lindex $data $idx_2]
lset data $idx_2 $temp
return $data
Big improvement.
Benchmarking 1, 2, and 3 edit
Here's my magical test data:
set data [lrepeat 100000 x y z z y]
If you're trying this at home from your interactive tclsh, you might want to type "; puts -nonewline {}" after the end of the above line or else its printed result will scroll for a very long
time.Now I
the shuffles:
: : 400 MHz : 1500 MHz :
: 1, random reals : 21.847024 s : 9.541117 s :
: 2, random integers : 24.649960 s : 9.857447 s :
: 3, in-place swaps : 7.650484 s : 2.328508 s :
Wow, I think I will go with #3! Hold on while I update
... :^)
Version 4: in-place swaps, better random behavior edit
: Per
Lars H
's suggestion, I improved the code again. As Lars explains in
, the old code doesn't have a perfectly uniform distribution because, for any given list, the size of the set of permutations from which it chooses is not a multiple of the size of the set of all
possible permutations. This is similar to doing "random() % 3" in C and getting 0's more often than 2's.
proc shuffle {data} {
set length [llength $data]
for {} {$length > 1} {incr length -1} {
set idx_1 [expr {$length - 1}]
set idx_2 [expr {int($length * rand())}]
set temp [lindex $data $idx_1]
lset data $idx_1 [lindex $data $idx_2]
lset data $idx_2 $temp
return $data
That ought to do it! However, it's (theoretically) a little bit slower due to modifying more variables in the inner loop. I also made a less-readable version that
's idx_1 rather than recalculating it with
, for comparison's sake. Here are the numbers, using the same benchmark.
: : 400 MHz : 1500 MHz :
: 4 , more random : 8.428660 s : 2.286190 s :
: 4a, less readable : 7.824165 s : 2.163990 s :
For reasonably-sized lists (under 500,000 elements, I guess), the speed difference between #3, #4, and #4a is negligible. So I choose #4, which has a more uniform distribution and doesn't have grody
sources due to silly optimizations.I wonder why the 400 MHz results for #4 and #4a are slower than for #3 yet the 1500 MHz results are the other way around...? I guess this proc is random in more
than one way.
Here's another algorithm. Basically you randomly select an element from the input list, remove it from the input list, and place it at the end of the output list. The reason I thought of it is, for
it's not necessary to maintain an output list; simply
the elements as they come.But this algorithm involves a lot of copying---
is nice for in-place modifies, but there's no in-place element removal. So it's liable to be slower than #4. | {"url":"http://oldwiki.tcl-lang.org/13818","timestamp":"2024-11-13T14:42:54Z","content_type":"text/html","content_length":"14407","record_id":"<urn:uuid:47e19cac-9608-4921-86c7-ba363ee5e916>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00802.warc.gz"} |
AIPS HELP file
AIPS HELP file for IMFIT in 31DEC24
As of Sun Nov 3 21:40:18 2024
IMFIT: Task to fit gaussian models to an image
INNAME Input image name (name)
INCLASS Input image name (class)
INSEQ 0.0 9999.0 Input image name (seq. #)
INDISK 0.0 9.0 Input image disk drive #
BLC 0.0 4096.0 Bottom Left corner of fit
TRC 0.0 4096.0 Top Right corner of fit
OUTNAME Output image name (name)
OUTCLASS Output image name (class)
OUTSEQ -1.0 9999.0 Output image name (seq. #)
OUTDISK 0.0 9.0 Output image disk drive #
NGAUSS 0.0 4.0 Number of components
CTYPE 0.0 5.0 Model types; one for each
2=Zero level; 3=Zero+slope
5=see HELP IMFIT
Guess of model parameters
GMAX Peak of component (JY)
0-> Use maximum value
GPOS -50.0 16434.0 (X,Y) position (pixels)
0-> Use position of max
GWIDTH -180.0 180.0 (BMAJ, BMIN, PA) of comp.
0->Use clean beam
@ Fit model parameters
FMAX @ Peak of component (JY)
@ 0-> Use maximum value
FPOS @ -50.0 16434.0 (X,Y) position (pixels)
@ 0-> Use position of max
FWIDTH @ -180.0 180.0 (BMAJ, BMIN, PA) of comp.
@ (pixels,pixels,deg)
@ 0->Use clean beam
FSHIFT @ RA/Dec shift to put comp 1
@ on nearest pixel
DOMAX $ Solve for GMAX? >0 -> yes
$ returns the error
DOPOS $ Solve for GPOS? >0 -> yes
$ returns the error
DOWIDTH $ Solve for GWIDTH? >0-> yes
$ returns the error
BWSMEAR 0.0 0.1 Bandwidth smearing corr.
RADIUS Radius for finding RMS
NITER 0.0 4000.0 Maximum # of iteration
Solve for model parameters?
DOPRINT -4.0 132.0 <0 -> Print maps and
solutions on line printer
>0 -> print on terminal
=0 -> maps not printed
NDIG Number digits in printed
maps (3 - 6, else fit)
Disk file to save fit info
DOOUTPUT -1.0 2.0 >0 -> 1 Catalog residual map
OFFSET -1.0 1.0 Cutoff level. 0-> None
DOMODEL -1.0 1.0 > 0 => put solutions in a CC
file with input image
OUTVERS -1.0 MF table version number
-1 => none, 0 => new
STVERS -1.0 STars output file version
-1 => none, 0 => new
PBPARM Primary beam parameters:
(1) level to believe - <= 0
means do not apply a primary
beam (2) > 0 use (3)-(7)
EFACTOR 0.0 20.0 Scale width sigmas in the
deconvolution tests
Type: Task
Use: IMFIT is a task to fit a portion of an image with up to four (4)
gaussian components. It can also fit for a baseline term of up to
second order. The program estimates the error in the fits using
the image rms and theory. The rms for the latter is determined
from the image header keyword ACTNOISE (if present and positive)
or from fitting the histogram or from a robust determination of
the rms using the full (or a circular portion) image plane.
Note that pixels which are exactly zero are not used in this
fit, allowing blanked pixels to be REMAGed to zero if you wish.
The answers are returned in the parameters FMAX, FWIDTH and the
uncertainties are returned in DOMAX, DOPOS, and DOWIDTH.
INNAME......First image name (name). Standard defaults.
INCLASS.....First image name (class). Standard defaults.
INSEQ.......First image name (seq. #). 0 => highest.
INDISK......Disk drive # for the first image. 0 => any.
BLC.........Bottom left corner of area of image to fit.
TRC.........Top right corner of area of image to fit.
Maximum area is 40000 pixels (200x200)
OUTNAME.....Residual map name. Standard defaults.
OUTCLASS....Residual map class. Standard defaults.
OUTSEQ......Residual map seq. #. 0 => highest unique.
OUTDISK.....Residual map disk no. 0 => highest with room.
NGAUSS......The number of components to use in the fitting.
0->1. Maximum number is four.
CTYPE.......Each component type.
0=>1. Two-dimensional elliptical Gaussian
2=Solve for zero level
3=Solve for zero level and slope
4=Solve for zero, slope and curvature
5=Set the six baseline parameters as desired.
See EXPLAIN for use of GMAX,GPOS and GWIDTH
GMAX........The peak value guess for each component.
0=>Use value with largest absolute value in the
BLC,TRC window
GPOS........The position (X,Y) guess for components. The values are
in pixels in the order (X1,Y1,X2,Y2,X3,Y3,X4,Y4).
0=>Use pixel location with largest absolute value.
GWIDTH......The major axis, minor axis and position of major axis
guess for components. The values are pixels with
degrees for position angle and the order is
(MJ1,MN1,PA1,MJ2,MN2,PA2,...etc) 0->Use clean beam, if
available; otherwise it will be taken as a circular
gaussian of diameter 2.
********* The following are output adverbs:
FMAX........The peak value fit for each component is returned.
FPOS........The position (X,Y) fit for components is returned. The
values are in pixels in the order
FWIDTH......The major axis, minor axis and position of major axis
fit for components are returned. The values are pixels
with degrees for position angle and the order is
FSHIFT......The values that one could put in RASHIFT and DECSHIFT
to move the first component to the nearest integer
pixel. These values include any previous shifts and are
corrected for rotation.
********* The following are input/output adverbs:
DOMAX.......Flags for GMAX: if > 0 fit this parameter, else hold
fixed. Returned value is uncertainty in the fit
DOPOS.......Flags for GPOS: if > 0 fit this parameter, else hold
fixed. Returned value is uncertainty in the fit
DOWIDTH.....Flags for GWIDTH: if > 0 fit this parameter, else hold
fixed. Returned value is uncertainty in the fit
********* The following are input adverbs:
BWSMEAR.....If > 0, the Clean beam will be smeared by a Gaussian in
the radial direction of FWHM proportional to the radius
from the pointing position times BWSMEAR. Set it roughly
to the channel bandwidth divided by the center frequency.
The Clean beam is used as an initial estimate of the
source size and is used in the deconvolution attempt to
find the true size from the fit size. The peak
intensity printed will be corrected for this effect.
RADIUS......If = 0, the rms used for error estimates is taken from
the image header (keyword ACTNOISE) or found by fitting
the full image plane. If RADIUS > 0, then the rms is
found by fitting only those pixels within RADIUS of the
center of the BLC-TRC box. If RADIUS < 0, then
abs(RADIUS) is used as the rms. Pixels which are
exactly zero are not used in the fitting for rms. A
robust method is used and, if it fails within RADIUS,
the method will be applied to the full image plane.
NITER.......The maximum number of iterations to use in the
fitting. 0-> NGAUSS * 200.
DOPRINT.....<0 -> Plot map, model and residual map and list fit
information on the line printer
When FITOUT is not blank, DOPRINT=-2 suppresses
the page-feed character on page headers and
DOPRINT=-3 suppresses page headers and most other
header information.
DOPRINT=-4 produces output to FITOUT with one line
per source component in a special CfA format.
=0 -> List fit info in message file only
>0 -> Plot map, model, and residual map and list fit
information on the terminal
NDIG........Number of digits in printed maps. If NDIG = 3-6 and is
less than or equal the number that will fit, then NDIG
is used.
FITOUT......Disk file name in which to save the line printer output.
' ' => use scratch and print immediately for interactive
jobs - batch jobs use FITOUT = 'PRTFIL:BATCHjjj.nnn'
(jjj = job #, nnn = user #). When FITOUT is not blank,
multiple outputs are concatenated and the file is not
actually printed.
DOOUTPUT....>0 -> Catalog residual map with fitted components and
write them in a CC file attached to the output image. If
DOOUTPUT > 1.5, the components written to the CC file are
not deconvolved from the beam.
OFFSET......0-> Include all points in fitting area. Otherwise
disregard all points less than OFFSET*MAX, where MAX is
largest value in fitting window. If MAX is less than
zero, disregard all point greater than OFFSET*MAX.
DOMODEL.....If true (> 0), put the deconvolved solutions in a new CC
file attached to the input image.
OUTVERS.....The results are written into an MF (Model Fit) table file
with version OUTVERS unless OUTVERS is set < 0. If
OUTVERS points at a pre-existing table, the results are
appended to the file. OUTVERS = 0 always means to make a
new MF table.
STVERS......The results may also be written to a STars table
attached to the image. -1 => do not do this, 0 => write
a new one, > 0 => add to the specified version.
PBPARM......Primary beam parameters:
(1) Lowest beam value to believe: 0 -> do not do the
primary beam correction. This correction is done
to the printed parameters only. The beam value used
is max (PBPARM(1), that computed from (2)-(7)).
(2) > 0 => Use beam parameters from PBPARM(3)-PBPARM(7)
Otherwise use default parameters for the VLA (or
ATCA where appropriate)
(3-7)..For all wavelengths, the beam is described by the
1.0 + X*PBPARM(3)/(10**3) + X*X*PBPARM(4)/(10**7) +
X*X*X*PBPARM(5)/(10**10) + X*X*X*X*PBPARM(6)/(10**13)
where X is (distance from the pointing position in arc
minutes times the frequency in GHz)**2.
See EXPLAIN for details and defaults
EFACTOR.....The range of possible deconvolved widths is found by
trying the deconvolution with Major, Minor, and PA each
-EFACTOR*sigma, 0, +EFACTOR*sigma from the fit value.
The highest value found in the 27 tests and the lowest
value found are reported as well as the values found at
the nominal (fit) values. 0 -> 1.3.
IMFIT: Task to fit Gaussian models to an image.
DOCUMENTOR: E.B.Fomalont NRAO/VLA
IMFIT fits up to four Gaussian-shaped components to a selected
part of an image. One of the components can be a baseline function
with a zero level, slope and curvature term. IMFIT is most commonly
used to derive the position, peak and integrated intensity and angular
size of a source which is not too extended. An initial guess for the
parameters, some of which are picked as defaults, must be supplied
before running the task. Solution and error estimates are generated
and the residual image after the fit can be printed on the
line-printer. An arbitrary selection of parameters may be held
constant in the solution.
The fitting algorithm is based on the subroutine LMDER in the
Argonne National Laboratory Subroutine Package and the algorithm uses
a linearized least-square solution to obtain the parameters.
Occasionally, the solution will converge on an obviously unacceptable
fit. If this occurs when fitting one component to the source, try a
better first guess. When fitting several components to a blobby
source, the fitted parameters may be absurd. Careful selection of
fixed parameters will then be necessary.
COMMENTS ABOUT SOME PARAMETERS
BLC, TRC:
The fitting area should be chosen as small as possible; and
several disconnected components should be fit separately. The
fitting area is limited to an area of 10000 pixels.
The number of components to fit. The maximum number is four and
0-> 1.
The component types, placed in a scalar array of length 4.
0->1 Elliptical Gaussian component.
2 Zero level.
3 Zero level and slope.
4 Zero level, slope and curvature.
5 Insert baseline parameters as follows:
GMAX = zero level
GPOS = slope (intensity per pixel),
Orientation (deg N thru E)
GMAX = Curvature (intensity per pixel**2),
Ellipticity of curvature (-1 to 1),
Orientation (deg N thru E)
DOMAX, DOPOS and DOMAX are used to hold parameters fixed
for Types 1 and 5 only.
The initial guess of the model intensity may be supplied. The
units of GMAX are the same as those in the map. The default of 0 will
place the most extreme value in the fitting area (negative or
positive) in GMAX for the first component. Any subsequent components
with 0 default will be given the value of 0.1 times the extreme value.
The initial guess of the model position. The location must be
given by a pair of pixel coordinates. The default of 0 will insert
the location of the extreme value for all components. Note that GPOS
has the meaning of slope and orientation for a baseline component.
The defaults for the component widths are generally reasonable;
either the clean beam size of a circular beam of two pixels FWHP.
Because of poor convergence properties of the algorithm for circular
Gaussian models, the task will introduce a slight ellipticity before
beginning the fitting. This is not done if either axes is held
The number of iterations, NITER, is defaulted to 200*NGAUSS if it
is set to 0. If you are somewhat unsure if your model is reasonable
or is converging to an acceptable solution, especially for fits to
complicated sources with several Gaussians, set NITER=50 and check how
the convergence is going. The task has several other termination
conditions. If the solution is unchanging to a level of about 0.1 percent,
it will terminate. It some of the fitting parameters are obviously
ridiculous, it will also terminate.
In fitting complicated sources, it is common to hold some of the
component diameters fixed in order to obtain reasonable convergence.
Set DOPRINT = -1 in most cases. This produces an automatic hard
copy of the solutions and a digital map of the input image, the first
guess and the residuals. These maps are most useful for determining
the validity of the fit.
Catalog the residual map after the fit has been subtracted. If
all parameters are held fixed, no fitting is done and a residual map
is generated. Only the fitted area is cataloged. A CC file is also
written with this output image listing the components. If DOUTPUT >
1.5, the components written are not deconvolved from the beam. If 0 <
DOOUTPUT <= 1.5, the components are deconvolved if possible.
This adverb permits the exclusion of low valued points when doing
the fit. If the extremum value (MAX) in the fitted area is positive,
then all points less than OFFSET*MAX are ignored in the fit. If the
extremum value in the fitted area is negative, then all points greater
than OFFSET*MAX are ignored in the fit. If OFFSET = 0, then all
points are used.
IMFIT corrects an image for the primary beam attenuation of
the antennas. The function used to model the primary beam for normal
VLA frequencies
F(x) = 1.0
+ parm(3) * 10E-3 * x
+ parm(4) * 10E-7 * x*x
+ parm(5) * 10E-10 * x*x*x
+ parm(6) * 10E-13 * x*x*x*x
+ parm(7) * 10E-16 * x*x*x*x*x
where x is proportional to the square of the distance from the
pointing position in units of [arcmin * freq (GHz)]**2, and F(x)
is the multiplicative factor to divide into the image intensity at the
distance parameter x. For other antennas, the user may read
in appropraite constants in PBPARM(3) through PBPARM(7). The
flag, PBPARM(2) must be set to a positive number to invoke this
option and PBPARM(3) must not be zero.
This correction scales with frequency and has a cutoff
beyond which the map values are set to an undefined pixel value GIVEN
in PBPARM(1). At the VLA frequencies the default cutoff is
1.485 GHz 29.8 arcmin
4.885 GHz 9.13 arcmin
15 GHz 2.95 arcmin
22.5 GHz 1.97 arcmin
and occurs at a primary beam sensitivity of 2.3 percent of the value at
the beam center. Corrections factors < 1 are forced to be 1.
The estimated error of the algorithm is about 0.02 in (1/F(x))
and thus leads to very large errors for x>1500, or at areas
outside of the primary response of 20 percent. The cutoff level
may be specified with DPARM(1).
Default values of PBPARM for the VLA are given by Perley's fits:
0.0738 GHz -0.897 2.71 -0.242
0.3275 -0.935 3.23 -0.378
1.465 -1.343 6.579 -1.186
4.885 -1.372 6.940 -1.309
8.435 -1.306 6.253 -1.100
14.965 -1.305 6.155 -1.030
22.485 -1.417 7.332 -1.352
43.315 -1.321 6.185 -0.983
For the ATCA, these are by default:
1.5 GHz -1.049 4.238 -0.8473 0.09073 -5.004E-3
2.35 -0.9942 3.932 -0.7772 0.08239 -4.429E-3
5.5 -1.075 4.651 -1.035 0.12274 -6.125E-3
8.6 -0.9778 3.875 -0.8068 0.09414 -5.841E-3
20.5 -0.9579 3.228 -0.3807 0.0 0.0
For the Karl G Jansky VLA ("EVLA"), the defaults are frequency
dependent. If the observing frequency is between two tabulated
frequencies, then the beam is computed for each of the tabulated
frequencies and then interpolated to the observing frequency. The
values used are far too numerous to give here, see EVLA Memo 195,
"Jansky Very Large Array Primary Beam Characteristics" by Rick Perley,
revision dated June 2016. Obtain it from
COMMENTS ON THE USE OF IMFIT
For most simple cases the defaults in IMFIT adequately provide
starting values. Some examples are as follows: (Always insert the
appropriate input map and always set BLC and TRC to the smallest area
needed for the fit. The verb TVWINDOW can be used to set the window.)
1. Fit to one Gaussian component.
Nothing to specify except flags.
2. Fit to one Gaussian and zero level
Note that the zero level information is associated
with the second 'gaussian' with CTYPE=2
Additional flags can be specified.
3. Fit to one Gaussian, zero-level and slope
Note that the zero level and slope information is
associated with the second 'gaussian' with
Additional flags can be specified.
4. Fit to two Gaussians and a zero level
Depending on the source complexity it may be
important to set some of the fitting flags
5. Fit one Gaussian with a zero level, slope and
curvature only in E/W direction
When attempting to obtain the flux density of a well-resolved
source, the task IMEAN, which integrates the map values in a specified
rectangle, is often more accurate than fitting the source with several
Gaussian components and summing the integrated flux densities.
The verb MAXFIT, a simple fitting of the peak of a component with a
second degree interpolation, is much faster than IMFIT and useful to
obtain the approximate peak and position of a component.
An estimate of the each error is determined from theory based on
the actual rms (R) of the image (neglecting signal portions) or the
rms given in header parameter ACTNOISE (if present and positive).
AIPS task IMEAN or verb ACTNOISE may be used to set this header
parameter. Theory gives expressions for the errors in two limiting
cases: point source (the beam area > 0.9*the fitted gaussian area) and
expanded source (the beam area < 0.1*the fitted gaussian area). The
formulae are taken from J. Condon paper 'Errors in Elliptical Gausian
Fit', AA, 1996. The intermediate case is handled by interpolation
between the two limit cases. The formulae now used are:
M = 1 (Clean beam area > 0.9 fit area)
M = sqrt (8 * ARbeam/ARimag) (Clean beam area < 0.1 fit area)
M = sqrt (0.8 + 0.25*(ARbeam/ARimag-0.1)) (else)
Delta(P) = M * R
Delta(W) = W * Delta(P) / P
Delta(PA) = sqrt(2) * (Smaj+Smin)/(Smaj^2+Smin^2) * Delta(P) / P
Delta(X) = sqrt[(Delta(Smaj)*sin(PA))^2 + (Delta(Smin)*cos(PA))^2]
/ sqrt (8 * ln (2.0))
Delta(Y) = sqrt[(Delta(Smaj)*cos(PA))^2 + (Delta(Smin)*sin(PA))^2]
/ sqrt (8 * ln (2.0))
Delta(F) = Delta(P) * ARimag/ARbeam * sqrt(1+2*ARbeam/ARimag)
where P is fit peak, W = Smaj or Smin are fit widths, X and Y are
positions, PA is position angle, ARimag = Smaj*Smin, ARbeam=Bmaj*Bmin.
When fitting to a clean map, IMFIT deconvolves the Clean beam from
the fitted component size. The nominal deconvolution is obtained by
deconvolving the fit from the Clean beam (corrected for bandwidth
smearing). A value of 0.0 means that the source is smaller than the
corrected Clean beam in some dimension. The minimum and maximum
values are obtained by deconvolving the source beam parameters with
all 27 combinations of EFACTOR * (-1, 0, 1) * uncertainties in the
major axis, minor axis, and position angle. The extremas in these
parameters over all 27 tries are listed. The default EFACTOR (1.3)
appears to work well with the considerations below.
An estimate is given concerning whether the source should be
viewed as resolved or unresolved. The task assumes that the
component is probably unresolved if:
(a) the deconvolution of the fit answers has the major axis 0
(b) the fit total flux minus the error in the total flux is
less than the peak AND the minimum deconvolved major axis
is 0.
The task assumes the conponent is probably resolved if
(c) the total fit flux minus the error in the total fit flux is
greater than peak flux AND the minimum deconvolved major axis
is greater than 0.
The task is undecided about resolution if
(d) the deconvolution of the fit answers has the major axis greater
than zero.
(e) the fit total flux minus the error in the total flux is
less than the peak BUT the minimum deconvolved major axis
is greater than 0.
(f) the total fit flux minus the error in the total fit flux is
greater than peak flux BUT the minimum deconvolved major axis
is 0.
Note: the total flux and its error are corrected for primary beam and
the peak is corrected for primary beam and bandwidth smearing (if such
corrections are requested) in the tests described above. If the
component is unresolved, the best estimate of its total flux is its
peak brightness. In cases where the task is uncertain, use caution in
deciding if the component is resolved. Noise seems preferentially to
make sources appear resolved when they are not. Note too that a
"resolved" source may be clearly unresolved along the minor axis. | {"url":"https://www.aips.nrao.edu/cgi-bin/ZXHLP2.PL?IMFIT","timestamp":"2024-11-04T04:40:18Z","content_type":"text/html","content_length":"33630","record_id":"<urn:uuid:320e97ca-b5ef-4a0d-92d3-f7c1e0e707ad>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00867.warc.gz"} |
Convert Pressure Units: from Inch of water to Ton force per square foot
Convert Pressure Units:
from Inch of water to Ton force per square foot
Inch of water to Ton force per square foot Conversion Table
In this table you can find unit conversions from Inch of water (inH₂O) to Ton force per square foot (tsf), both units of Pressure.
The Inch of water (inH₂O) values range from 1 to 10. The equivalent Ton force per square foot (tsf) values are expressed in scientific notation for convenience and consistent precision.
inH₂O tsf
1 2.601165e-3
2 5.202330e-3
3 7.803495e-3
4 1.040466e-2
5 1.300582e-2
6 1.560699e-2
7 1.820815e-2
8 2.080932e-2
9 2.341048e-2
10 2.601165e-2
Learn everything about units of measurement. Use our smart App to convert units in real-time with just a few keystrokes, both metric and US units. | {"url":"https://www.unitscenter.com/convert/pressure/inch-of-water/ton-force-per-square-foot","timestamp":"2024-11-08T17:15:41Z","content_type":"text/html","content_length":"61959","record_id":"<urn:uuid:69d3e5b9-6504-42e9-9f70-f433811f74f9>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00824.warc.gz"} |
Proof By Contradiction
It is often possible to prove a statement is false by assuming that it is true. If it is true, then it should be consistent with all the other mathematical theorems which have been proved to be true.
If we can show that it is not consistent with one or more of these other statements then we have shown that it is false.
Disproving a statement in this way is called 'proof by contradiction'. Proof by contradiction can be a tricky skill to learn.
Example: Prove that if
Suppose that
We can then write
Raising both sides to the power of
This means that amongs all the powers of 3, there is at least one power of 5, and amongst all the powers of 5 there is at least one power of 3. Both of these statements are false, so the statement, '
Example: Prove that
Suppose that
Raise 2 to the power of both sides to give
Both these statements are false, so the statement ' | {"url":"https://astarmathsandphysics.com/ib-maths-notes/proofs/1079-proof-by-contradiction.html","timestamp":"2024-11-12T06:54:38Z","content_type":"text/html","content_length":"33152","record_id":"<urn:uuid:2b427bf2-7466-41ad-b223-a687a4b5f81d>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00445.warc.gz"} |
derive critical speed of ball mill
WEBA section cutthrough of ball mills. A ball mill is a type of grinder filled with grinding balls, used to grind or blend materials for use in mineral dressing ... Critical speed can be
understood as that speed after which the steel balls that are responsible for the grinding of particles start rotating along the direction of the cylindrical ...
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WEBMar 1, 2003 · the mass of mill load, kg. N. speed of mill rotation in rpm or percentage of critical speed. P. mill power in watts. P i. power drawn by layer i. R, R i. distance from ball
center to mill center, i, layer of balls. r b. radius of ball, m. R d. radius of Davis circle, m. R m. radius of the 2D mill, m. T. torque, Nm. x cog
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WEBformula calculates the critical speed of a ball mill. Critical Speed Calculation Of Ball Mill – Raymond Grinding Mill. .. CEMENT MILL FORMULAS MILL CRITICAL VELOCITY = 76 / (D)^1/2 MILL.. Ball
Mill 1. n = C (AB ..
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WEBA Ball Mill Critical Speed (actually ball, rod, AG or SAG) is the speed at which the centrifugal forces equal gravitational forces at the mill shell's inside surface and no balls will fall
from its position onto the shell. The imagery below helps explain what goes on inside a mill as speed varies. Use our online formula [.]
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WEBtwo types of mill where the capacity of the pot was 60 L and 300 L. The mill was filled with 15 mm nylon coated iron balls, and the rotational speed N of the mill was varied in a range from
40%– using the critical rotational speed Nc defined by (Eq. 11) as a reference. The critical rotational speed Nc is the limiting speed where the
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WEBJul 26, 2023 · The critical speed of a ball mill can be derived using the formula nc = 1/2π√(g/R), where nc is the critical speed, g is the gravitational acceleration, and R is the radius of
the mill. Understanding the critical speed is important for the design and operation of ball mills, as it determines the optimal rotational speed at which the mill ...
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WEBNov 08, 2012 ·nbsp Ball mills are normally operated at around 75% of critical speed, so a mill with diameter 5 metres . Ball mill is the key equipment for . hartl pc1055j . Ball mill is the
key equipment for . hartl pc1055j .
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WEBMill Critical Speed Formula Derivation Grinding . The formula to calculate critical speed is given below N c = 42305 /sqt(Dd) N c = critical speed of the mill D = mill diameter specified in
meters d = diameter of the ball In practice Ball Mills are driven at a speed of 5090% of the critical speed, the factor being influenced by economic .
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WEBMar 23, 2022 · where g is m/s 2, R radius of the cylinder (m), r radius of the ball (m), and n c critical speed (rps). The operating speed of the ball mill is kept at 65–80% of the critical
speed. The lower values are kept for the wet grinding in viscous solution, while a higher value is kept for dry grinding. Burr Mill or Plate Mill
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WEBThe Formula derivation ends up as follow: Critical Speed is: Nc =766 (D05) where: Nc is the critical speed,in revolutions per minute, D is the mill effective The formula to calculate critical
speed is given below N c = 42305 /sqt(Dd) N c = critical speed of the mill D = mill diameter specified in meters d = diameter of the ball In Mill ...
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WEBJan 1, 2022 · The filling levels M* was taken as 30%, 40% and 50% of full mill and the mill speed N* was selected as, and of the critical speed. The critical speed is the speed at which a mill
drum rotates such that balls are stick to the drum, which is given by 2 g / D − d where D and d are the mill diameter and particle diameter in meters ...
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WEBderive expression of critical speed of ball mill. WebT05:05:40+00:00 Ball Mill Critical Speed Mineral Processing Metallurgy. 19/06/2015 A Ball Mill Critical Speed (actually ball, rod, AG or
SAG) is the speed at which the centrifugal forces equal gravitational forces at the mill shell's inside surface and no balls will fall from its ...
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WEBDerive E Pression For Critical Speed Of Ball Mill derive expression for critical speed of ball mill lesotho A Ball Mill Critical Speed (actually ball, rod, AG or SAG) is the speed at which the
centrifugal forces equal gravitational forces at the mill shell's inside surface and no balls will fall from its position onto the shell.
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WEBIn solid mechanics, in the field of rotordynamics, the critical speed is the theoretical angular velocity that excites the natural frequency of a rotating object, such as a shaft, propeller,
leadscrew, or gear. As the speed of rotation approaches the object's natural frequency, the object begins to resonate, which dramatically increases system ...
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WEBBall Mill Critical Speed 911 Metallurgist. The Formula derivation ends up as follow: Critical Speed is: Nc =766 (D05) where: Nc is the critical speed,in revolutions per minute, D is the mill
effective Contribute to changjiangsx/ development by creating an account on / derivation of critical speed in a ball mill pdfmd at main
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WEBderive an equation to find critical speed of ball mill in ... The Formula derivation ends up as follow: Critical Speed is: N c =(D ) where: N c is the critical speed,in revolutions per minute,
D is the mill effective inside diameter, in feet mill critical speed formula derivation grinding
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WEBIn the ball mill both shearing and impact forces are utilized in the size reduction. The unit consists of a horizontal, slow speedrotating cylinder containing a charge of steel balls or flint
stones. ... For efficient milling the critical speed should not be exceeded. This is defined as the speed at which a small sphere inside the mill just ...
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WEBsbm critical speed of ball mill wikipediaBall Mill Critical Speed 911 Metallurgist 17 Mar Ball Mill Critical Speed (actually ball,rod,AG or SAG) is the speed at which the centrifugal forces
equal gravitational forces at the mill shell#39;snbsp;. Ball mill Wikipedia Critical speed can be understood as that speed after which the steel ...
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WEBThe critical speed of a ball mill in rpm whose diameter is 12 inches with grinding balls diameter of 1⁄2 in is approximately _____ rpm. Your solution's ready to go! Our expert help has broken
down your problem into an easytolearn solution you can count on.
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WEBThe point where the mill becomes a centrifuge is called the "Critical Speed", and ball mills usually operate at 65% to 75% of the critical speed. Ball mills are generally used to grind
material 1/4 inch and finer, down to the particle size of 20 to 75 microns.
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WEBHow to derive ball mill critical speed mill critical speed mineral processing ball mill critical speed actually ball, rod, the formula derivation ends up as follow critical speed is n c where
n c is the critical speed,in revolutions per minute, d is . Read On
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WEBOct 19, 2015 · Power draw is related directly to mill length, and, empirically to the diameter to the power Theoretically, this exponent should be (Bond, 1961). Power draw is directly related
to mill speed (in, or, fraction of critical speed) over the normal operating range.
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WEBGiven that R =400 mm and r =, what is the critical speed? a) b) c) d) View Answer. Answer: b ... Ball Mill ; Mechanical Operations Questions and Answers – Medium Peripheral Speed Mill ; Food
Engineering Questions and Answers – Unit Operations – Size Reduction and Seperation2 ...
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WEBQuestion: BALL MILL Objective: To determine the (a) Critical speed (b) Actual speed (c) Optimum speed (d) Reduction ratio (e) Constants for i. Rittinger's Law ii. Kick's Law iii. Bond's Law.
Equipment and Materials required: Ball mill, sieves, weight balance, brick. Theory: The ball mill is used for fine grinding of soft materials.
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WEBPrompt : Caesar is a famous mining equipment manufacturer wellknown both at home and abroad, major in producing stone crushing equipment, mineral separation equipment, limestone grinding
equipment, etc. Derivation Of Critical Velocity Of Ball Mill. Derivation of critical speed of grinding mill Wiki Answers. derivation of critical speed of ball ...
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WEBJun 19, 2015 · The approximate horsepower HP of a mill can be calculated from the following equation: HP = (W) (C) (Sin a) (2π) (N)/ 33000. where: W = weight of charge. C = distance of centre
of gravity or charge from centre of mill in feet. a = dynamic angle of repose of the charge. N = mill speed in RPM. HP = A x B x C x L. Where.
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WEBBall mills are often operated at higher speeds than rod mills, so that the larger balls aract and impact on the ore particles. The work input to a mill increases in proportion to the speed,
and ball mills are run at as high a speed as is possible without centrifuging. Normally this is 70–80% of the critical speed.
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WEBshow the critical speed of ball mill derivationball mill equation zacarafarm. derivation for the critical speed of ball mill show the critical speed of ball mill formula derivation Type or
paste a DOI name into the text box Click Go Your ball mill equipment equations principle Ball mill is vital equipment in industries viz mineral dressing ore processing fertilizers .
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WEBFigure shows a laboratory planetary mill. Large diameter ball mills. For processing larger batches of powder, however, there has been a trend recently to use conventional horizontal ball mills
with larger. diameters ( to m) to achieve high energy by rotating it just below the critical speeds ωc (up to ωc ).
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WEBderive an equation to find critical speed of ball mill in ... critical speed grinding mill equation . Critical speed of ball mill formula derivation. 3 apr 2018 semiautogenous grinding sag
mill and a ball bonds equation to industrial mills, which differ from the standard, for each mill there is a critical speed that creates centrifuging figure 37c .
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WEBderive expression of critical speed of ball mill – Crusher . Costea **, Silaghi Helga Maria**, Rohde L. Ulrich (the critical speed) Expression (1) illustrates the elements from the mill Ball
Mill,Proceedings of 2010 .
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WEBSep 21, 2021 · The critical speed is the angular velocity that excites the natural frequency of the rotating objects like rotors, shafts.etc., resulting in severe vibration of the shaft in the
transverse direction. Critical speed is also known as the whirling speed of the shaft. Let us derive the governing equation of the critical speed.
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WEBPrompt : Caesar is a famous mining equipment manufacturer wellknown both at home and abroad, major in producing stone crushing equipment, mineral separation equipment, limestone grinding
equipment, etc.
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WEBderive expression of critical speed of ball mill crusher and ... Ball Mill Operating Speed Mechanical Operations . The critical speed of ball mill is given by, where R = radius of ball mill; r
= radius of ball. For R = 1000 mm ...
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WEBContribute to luoruoping/id development by creating an account on GitHub.
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Explore projects · GitLab
• Method for identification of the five-parameter model: second-order process with zero plus dead time. Model parameters can be analytically calculated from the process’ closed- or open-loop time
response or the arbitrary process
• A repository to hold code for DMPs in various programming languages.
• MATLAB's implementation of the PhD thesis entitled Parametric and Nonparametric PI/PID Controller Tuning Methods for Integrating Processes.
• Matlab/Simulink simulation model of a PWR reactor | {"url":"https://repo.ijs.si/explore?archived=true&language=66&sort=stars_desc","timestamp":"2024-11-02T15:45:44Z","content_type":"text/html","content_length":"59903","record_id":"<urn:uuid:185ad35c-bbb0-4dad-928c-3e3841260585>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00859.warc.gz"} |
Number Theory
Number theory is an exciting area of mathematics, with many practical applications. Some day, you may need to use factors to help you build a yard with a specific area, or use multiples to solve a
scheduling problem at work. Many topics from elementary number theory are covered on Math Goodies. You have been selected to explore these topics, and to apply your knowledge through critical
The Task
You will explore resources on Math Goodies to answer specific questions on number theory. The key to learning in this WebQuest is to become engaged in finding information, and to learn from what you
have found. These tasks will require a computer, access to the Internet, paper and pencil. Familiarity with Math Goodies is helpful, but not necessary.
The Process
Process Part I: Number Theory Unit and Worksheets
1. Browse our instructional unit on Elementary Math (Number Theory), which covers factors, GCF, Multiples, LCM, prime and composite numbers, divisibility tests and exponents.
2. Complete all five exercises at the end of each core lesson in this unit.
3. Complete our Worksheet on The Sieve of Eratosthenes
Elementary Number Theory Questions
Answer each question, in order, and write down your answer on paper in your own words. You may work alone or in groups.
1. What is elementary number theory?
2. Give an example of how can multiples help you with gardening? (You may provide a specific example you found on our site.)
3. What is a factor?
4. What is 1,729,463 raised to the zero power?
5. Is 2^5 the same as 2 x 5? Explain why or why not using full sentences.
6. What is the Fibonacci sequence? (Hint: It can be found on several pages.)
7. Write the Fibonacci sequence on paper.
8. What is the rule for this sequence? Write your answer using your own words.
9. Is the number 31 prime or composite? Explain your answer.
10. Is the number 747 prime or composite? Explain your answer.
11. How did you arrive at your answer to questions 9 and question 10? What method did you use for each?
12. Why do we use divisibility tests?
13. Which test(s) will determine if the number 876 is prime or composite? Explain your answer.
14. What is the Sieve of Eratosthenes?
15. Find all prime numbers less than 100.
16. What is the smallest prime number?
17. Which prime number is even?
18. What is an emirp?
19. List all emirps between 1 and 100.
1. Our instructional unit on Elementary Math.
2. Our worksheet on The Sieve of Eratosthenes.
1. Switch your answers with a classmate or group member.
2. Review each other’s answers.
3. Exchange ideas: Discuss what you have learned.
4. Go through the answer key to our worksheet on The Sieve of Eratosthenes.
5. Assess your ability to find information in this WebQuest.
6. Assess your ability to learn from the information you found.
7. Which tasks were easy for you?
8. Which tasks did you struggle with?
Congratulations! You learned about topics in number theory and extended your knowledge through critical thinking. You did this by embarking on a quest for information, and by learning from the
information you found.
Number theory is a branch of mathematics that focuses on the study of whole numbers, or integers, and the relationships between them. It explores patterns, properties, and behaviors of numbers, often
in their purest form without considering their application to real-world problems. Number theory delves into various topics, including prime numbers, divisibility, congruences, and arithmetic
One of the central concepts in number theory is prime numbers. These are numbers greater than 1 that can only be divided by 1 and themselves without leaving a remainder. Understanding prime numbers
is crucial in cryptography, a field that deals with secure communication systems.
Number theory also deals with divisibility rules and the relationships between numbers when they are divided. For instance, it explores concepts like greatest common divisors and least common | {"url":"https://mathgoodies.com/webquests/number_theory/","timestamp":"2024-11-05T15:24:33Z","content_type":"text/html","content_length":"39088","record_id":"<urn:uuid:9516f3b7-01c0-4393-984b-c6be9e1fb871>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00496.warc.gz"} |
LibGuides: Statistics Resources: Point Biserial
The Point-Biserial Correlation is a special case of the Pearson Correlation and is used when you want to measure the relationship between a continuous variable and a dichotomous variable, or one that
has two values (i.e. male/female, yes/no, true/false).
1. No outliers (continuous variable) - assessed through a visual examination of the scatterplot
2. Approximately normally distributed (continuous variable)
3. Homogeneity of variance of the continuous variable between both groups of the dichotomous variable - assessed through Levene's Test
Running Point-Biserial Correlation in SPSS
1. Analyze > Correlate > Bivariate
2. Move variables of interest to the "Variables" box.
3. Ensure "Pearson" is the only option selected for the test.
4. You may use the "Options" button to select descriptive statistics you wish to include as well.
5. Click "OK" to run the test.
Interpreting the Output
As with the Pearson and Spearman results, SPSS will generate the results in a matrix. You can ignore any boxes that show a "1" as the correlation value as these are simply the variable correlated
with itself. These values will form a diagonal across the matrix that can be used to help you focus on the correct values. You only need to explore the correlation values on half of the matrix. APA
Style uses the bottom half.
With the release of SPSS 27, users now have the option to only produce the lower half of the table, which is in line with APA Style and makes it easier to identify the correct correlation values.
Reporting Results
When reporting the results of the correlation analysis, APA Style has very specific requirements on what information should be included. Below is the key information required for reporting the Point
Biserial Correlation results. You want to replace the red text with the appropriate values from your output.
r[pb](degrees of freedom) = the r[pb] statistic, p = p-value
A point-biserial correlation was run to determine the relationship between income and gender. There was a negative correlation between the variables, which was statistically significant (r[pb](38), p
- .023).
• When reporting the p-value, there are two ways to approach it. One is when the results are not significant. In that case, you want to report the p-value exactly: p = .24. The other is when the
results are significant. In this case, you can report the p-value as being less than the level of significance: p < .05.
• The r statistic should be reported to two decimal places without a 0 before the decimal point: .36
• Degrees of freedom for this test are N - 2, where "N" represents the number of people in the sample. N can be found in the correlation output. | {"url":"https://resources.nu.edu/statsresources/Pointbiserial","timestamp":"2024-11-07T13:39:02Z","content_type":"text/html","content_length":"48922","record_id":"<urn:uuid:70f3a221-5b9a-44c9-b2d2-cc99b00b9c4f>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00497.warc.gz"} |
What is 30 as a Fraction? - StuffSure
We all know that fractions can be a bit tricky, but don’t let that discourage you! In this blog post, we’ll show you how to tackle the question “What is 30 as a fraction?” We’ll walk you through each
step so you can confidently work with fractions.
30 as a Fraction
30 as a fraction is 30/1.
30 as a Decimal
30 as a decimal is 0.3. You can also writeThirty as a fraction. In this case, it would be written as 30/100.
30 as a Percent
30 as a percent is equal to 30/100. To convert 30 percent to a decimal, simply divide 30 by 100. The answer is 0.3.
30 as a Ratio
30 can be expressed as a ratio in a few different ways. The simplest way to write 30 as a ratio is to write it as 30:1. This means that there are 30 parts, and that each part is 1/30th of the whole.
Another way to express 30 as a ratio is to write it as 15:2. This means that there are 15 parts, and that each part is 2/30th of the whole.
You can also express 30 as a fraction by writing it as 3/10. This means that 3 out of 10 parts are equal to 30/100, or 1/3 of the whole. | {"url":"https://stuffsure.com/what-is-30-as-a-fraction/","timestamp":"2024-11-13T04:46:28Z","content_type":"text/html","content_length":"57991","record_id":"<urn:uuid:08b6c296-16f4-4e69-ba73-cca1022ddb45>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00877.warc.gz"} |
HSC Logic 2nd Paper Question with Solution 2022 যুক্তিবিদ্যা দ্বিতীয় পত্র উত্তর
HSC Question
HSC Logic 2nd Paper Question with Solution 2022 যুক্তিবিদ্যা দ্বিতীয় পত্র উত্তর
HSC 2022 Logic 2nd Paper MCQ Answer Dhaka Education Board
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Examination Name: Higher Secondary Certificate (HSC)
Subject Name: Logic 2nd Paper
Date: 17 November 2022
Time: 11.00 am to 1 pm
Question Type: MCQ and Written
এইচএসসি 2022 পরীক্ষার সকল প্রশ্ন সমাধান দেখতে চোখ রাখুন আমাদের ফেইজবুক পেজে– Click Here
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Last Word
Through today’s article we have published the solution of HSC 2022 logic second paper exam mcq question of all the education boards of the country. For any information related to the solution of the
question, please comment us. | {"url":"https://nextresultbd.com/hsc-logic-2nd-paper-mcq-solve/","timestamp":"2024-11-09T09:18:08Z","content_type":"text/html","content_length":"77123","record_id":"<urn:uuid:a4aaac94-3f44-4c6c-b9f3-ed3c40982ef2>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00734.warc.gz"} |
Daily Calendar Discussion
Number of Days in School:
• Create a number line that counts the number of days in school, adding a number for each day of class.
• As the chart expands, have students count by different numbers (e.g., count by 2s, 3s, etc.) and identify patterns.
□ Document emerging patterns by marking the numbers with different symbols.
• Visually represent the number of days by placing straws in three separate jars, each representing either the ones, tens, or hundreds place.
□ Add a straw to the ones jar each day. When there are 10 straws in the ones jar, remove them and explain that they will be replaced by one straw in the tens jar.
• Visually represent the number of days using magnetic square tiles. The goal is to create a bigger square/rectangle with the magnetic squares.
□ Have students create relevant multiplication or division equations. Begin a discussion on remainders if appropriate.
Monthly Calendars:
• Explore why each month starts on a different day: create small pre-cut months and attach the past month to the current month (printable calendars for every year can be downloaded here). Students
will notice that all months fit together like a puzzle!
• Invite students to create their own monthly calendars by providing each group of students a calendar outline (Appendix A).
□ Have students input the year, month, and dates. Special events can be included, such as
birthdays, field trips, etc.
• Hang these up with the current month so that students can see it better.
□ Append a monthly tooth chart (Appendix B) beside the calendars where students who have lost a tooth can add their name. At the end of the month, add up how many teeth were lost.
Days of the Week Wheel:
• Print a Days of the Week wheel (Appendix C) and glue the back to cardstock to strengthen.
• Keep track of how many times each day of the week appears in a month by placing a clothes pin on the appropriate sector each day.
• Have students make observations, such as on the fourteenth there will be two clothespins on each day because there are seven days in a week, and seven plus seven is 14.
Working with Today’s Date:
• Keep a tally of the number of days in the month by adding a mark every day. Use different colours to show repeating patterns.
□ Cluster in groups of five to promote practice counting by fives.
□ At the end of each month, tally how many days there were.
• Use multilink cubes to represent today’s date and select different colours to visually represent odd and even numbers, which grow and change every day (see Figure to the right).
□ Ask students how they know today’s date is an odd or even number.
Daily Weather:
• Chart the daily weather using the weather graph (Appendix D).
□ At the end of every month, chart the cumulative number of days with sunny, cloudy, rainy and snowy weather (Appendix E).
• Indicate today’s high and low temperatures on the paper thermometer (Appendix F).
• Record the daily high using a line plot, making sure to use titles, labels, and scales.
□ This is a good introduction to positive and negative numbers and increases interest in the weather forecasts.
• Provide a context and purpose for knowing the temperature (e.g., if the temperature is 0°C, do students need to wear their coat for recess?).
Accumulation of Data:
• At the end of each month, hang the data up so that each month’s data can be compared. | {"url":"https://wordpress.oise.utoronto.ca/robertson/portfolio-item/daily-calendar/","timestamp":"2024-11-06T15:00:39Z","content_type":"text/html","content_length":"122640","record_id":"<urn:uuid:fee7767e-d475-4328-80f5-bc9252d76b0e>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00829.warc.gz"} |
Editing a matrix' dimensions
09-04-2015, 05:34 PM
(This post was last modified: 09-04-2015 05:39 PM by pwarmuth.)
Post: #1
pwarmuth Posts: 35
Junior Member Joined: Sep 2015
Editing a matrix' dimensions
Is there a better way to add rows/columns in the matrix editor than tediously inputting them one at a time via MORE -> INSERT -> ROW/COLUMN ? A hotkey? Perhaps a way to directly define the matrix'
and/or vector's dimensions? On this same line of thinking, I know the comma key will traverse along a row withinin the matrix template (on the home or CAS screen), but is there a carriage return
hotkey hidden away somewhere?
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Seminar on Advances in Probabilistic Machine Learning
Abstract: State space approximations and pseudo point approximations can be combined in a principled manner to yield scalable approximate inference algorithms for sums of separable Gaussian
processes. In this talk, I will: 1. show how this combination can be performed for variational pseudo point approximations via a simple conditional independence result, 2. discuss how existing exact
inference algorithms for state space models can be re-purposed for approximate inference, 3. interpret existing related work in light of our work, and 4. briefly discuss some experimental results in
a spatio-temporal context. For more info, please see our recent UAI paper.
Abstract: Gaussian Processes are a tool of choice for modelling functions with uncertainties. However, the inference is only tractable analytically for the classical case of regression with Gaussian
noise since all other likelihoods are not conjugate with the Gaussian prior. In this talk, I will show how one can transform a large class of likelihoods into conditional conjugate distributions by
augmenting them with latent variables. These augmented models have the advantage that, while the posterior inference is still not fully analytic, the full conditionals are! Consequently, one can work
easily (and efficiently!) with algorithms like Gibbs sampling or Coordinate Ascent VI (CAVI) and outperform existing inference methods.
Abstract: Scientists and engineers are often interested in learning the number of subpopulations (or components) present in a data set. A common suggestion is to use a finite mixture model (FMM) with
a prior on the number of components. Past work has shown the resulting FMM component-count posterior is consistent; that is, the posterior concentrates on the true, generating number of components.
But consistency requires the assumption that the component likelihoods are perfectly specified, which is unrealistic in practice. In this paper, we add rigour to data-analysis folk wisdom by proving
that under even the slightest model misspecification, the FMM component-count posterior diverges: the posterior probability of any particular finite number of components converges to 0 in the limit
of infinite data. Contrary to intuition, posterior-density consistency is not sufficient to establish this result. We develop novel sufficient conditions that are more realistic and easily checkable
than those common in the asymptotics literature. We illustrate the practical consequences of our theory on simulated and real data.
Abstract: Solving decision-making problems in a variety of domains such as healthcare or operations research requires experimentation. By performing interventions, one can understand how a system
behaves when an action is taken and thus infer the cause-effect relationships of a phenomenon. Experiments are usually expensive, time-consuming, and may present ethical issues. Therefore,
researchers generally have to trade-off cost, time, and other practical considerations to decide which experiments to conduct in order to learn about a system. In this talk, I will present two
methodologies that, by linking causal inference, experimental design and Gaussian process (GP) modelling, allow to efficiently learn the causal effects in a graph and identify the optimal
intervention to perform. Firstly, I will show how to construct a multi-task causal GP model, the DAG-GP model, which captures the non-trivial correlation structure across different experimental
outputs. By sharing experimental information, the DAG-GP model accurately estimates the causal effects in a variety of experimental settings while enabling proper uncertainty quantification. I will
then demonstrate how this model, and more generally GP models, can be used within decision-making algorithm to choose experiments to perform. Particularly, I will introduce the Causal Bayesian
Optimization algorithm, and I will show how incorporating the knowledge of the causal graph in Bayesian Optimization improves the ability to reason about optimal decision making while decreasing the
optimization cost and avoiding suboptimal solutions.
Abstract: In this talk, I will discuss recent work on developing generative models for efficient sampling and inference by incorporating inductive biases in the form of equivariances. I will begin by
introducing the Equivariant Stein Variational Gradient Descent algorithm – an equivariant sampling method based on Stein’s identity for sampling from densities with symmetries. Equivariant SVGD
explicitly incorporates symmetry information in a density through equivariant kernels, which makes the resultant sampler efficient both in terms of sample complexity and the quality of generated
samples. Subsequently, I will demonstrate the use of Equivariant SVGD by defining equivariant energy-based models to model invariant densities that are learned using contrastive divergence. I will
then discuss the applications of these equivariant energy models for modelling joint densities in regression and classification tasks for image datasets, many-body particle systems and molecular
structure generation. Finally, if time permits, I will touch on methods for sampling using diffusion models and neural transport augmented Monte Carlo methods for more efficient sampling in discrete
spaces with applications in denoising, Bayesian posterior sampling, and training light-weight Bayesian quantised neural nets.
Abstract: Automated decision-making systems are increasingly being deployed in areas with personal and societal impacts: from personalized ads to medical diagnosis and criminal justice. This led to
growing interest and need for trustworthy AI and ML systems - that is, models that are robust, explainable, fair, and so on. It is important to note that these guarantees only hold with respect to a
certain model of the world, with inherent uncertainties. In this talk, I will present how probabilistic modelling and inference, by incorporating a distribution, offer a principled way to handle
different kinds of uncertainties when reasoning about decision-making system behaviours. For example, labels in training data may be biased; I will show that probabilistic circuits, a class of
tractable probabilistic models (TPMs), can be effective in enforcing and auditing fairness properties by explicitly modelling a latent unbiased label. Another common source of uncertainty is missing
values at prediction time, which also leads to fairness and robustness queries that account for this to be computationally hard inference tasks. I will also demonstrate how TPMs can again tractably
answer these complex queries.
Abstract: Probabilistic graphical models are flexible models for representing complex high-dimensional distributions and for incorporating domain knowledge in an intuitive and expressive way. For
making probabilistic inference, one often relies on recursive message passing methods. While these methods are efficient for restricted model classes (e.g., for trees), they only serve as
approximation methods for more complex models. In this talk, I will show how we can enhance the performance of message passing methods from two opposing angles: i.e., by simplifying the model itself
with the utilized inference method in mind and by modifying the inference method with the underlying model in mind. Therefore, I will show how we can advance our understanding of message passing
methods by considering them as a dynamic system and by applying tools from system theory. These insights will then suggest various improvements. Recently, we have also complemented message passing
methods by neural networks. I will discuss how such hybrid models benefit from the flexibility of neural networks in combination with the implicit underlying model assumptions.
Abstract: Progressively applying Gaussian noise transforms complex data distributions to approximately Gaussian. Reversing this dynamic defines a generative model. When the forward noising process is
given by a Stochastic Differential Equation (SDE), Song et al. (2021) demonstrate how the time inhomogeneous drift of the associated reverse-time SDE may be estimated using score-matching. A
limitation of this approach is that the forward-time SDE must be run for a sufficiently long time for the final distribution to be approximately Gaussian. In contrast, solving the Schrödinger Bridge
problem (SB), i.e. an entropy-regularized optimal transport problem on path spaces, yields diffusions which generate samples from the data distribution in finite time. We present Diffusion SB (DSB),
an original approximation of the Iterative Proportional Fitting (IPF) procedure to solve the SB problem, and provide theoretical analysis along with generative modeling experiments. The first DSB
iteration recovers the methodology proposed by Song et al. (2021), with the flexibility of using shorter time intervals, as subsequent DSB iterations reduce the discrepancy between the final-time
marginal of the forward (resp. backward) SDE with respect to the prior (resp. data) distribution. Beyond generative modeling, DSB offers a widely applicable computational optimal transport tool as
the continuous state-space analogue of the popular Sinkhorn algorithm (Cuturi, 2013). Joint work with Valentin De Bortoli, Jeremy Heng and Arnaud Doucet.
Abstract: Topographic generative models can be seen as a class of generative models where the latent variables have an underlying topographic (or spatial) organization which determines their
correlation structure. Such structure is widely observed in biological neural networks, however, its computational value is still debated and thus lacks adoption by the deep learning community at
large. In this talk, we will describe the statistical motivations behind early topographic generative models like Topographic ICA, and show how such priors can be integrated into modern deep neural
networks by introducing the Topographic Variational Autoencoder (TVAE). Further, we will show how topographic representations can be seen as generalized structured representations, and demonstrate
how topographic organization over space and time can be leveraged to induce the learning of equivariant sets of features we call capsules. Finally, we will show preliminary results comparing the
representations learned by deep TVAEs with FMRI recordings, demonstrating the emergence of localized specialized regions similar to the face area observed in primates.
Abstract: Despite their ubiquitousness in modern data-driven decision-making systems, neural networks are not very well understood. A symptom of this is that network hyperparameters are almost always
chosen via cross-validation, an expensive approach that scales poorly in the number of hyperparameters. Additionally, obtaining robust uncertainty estimates for neural network predictions remains an
open problem. The probabilistic framework holds the promise of providing both an objective for model selection and reliable uncertainty estimates. However, for the case of neural networks exact
probabilistic inference is intractable. This talk introduces the Linearised Laplace approximation for Bayesian deep learning. We examine the assumptions behind linearised Laplace, particularly in
conjunction with model selection. We show that these interact poorly with some now-standard features of deep learning—stochastic approximation methods and normalisation layers—and make
recommendations for how to better adapt this classic method to the modern setting. We provide Theoretical support of our recommendations and validate them empirically on MLPs, classic CNNs, residual
networks with and without normalisation layers, generative autoencoders and transformers. As a case study, we deep dive into Bayesian deep learning methods for tomographic reconstruction. Using
linearised Laplace, we construct a probabilistic Deep Image Prior over reconstructed images. Inference in this model allows us to choose U-Net architecture parameters without the need for
cross-validation and yields state of the art uncertainty calibration for tomographic reconstruction.
Abstract: Coupling methods have recently been used to compute unbiased estimates of Markov chains Monte Carlo and particle smoothing expectations. However, in most cases, sampling from couplings has
a random run time, the variance of which can be infinite. This behaviour, acceptable in distributed computing, is highly problematic in the parallel computing framework. We propose a limited variance
coupled rejection sampling method for sampling from couplings of arbitrary distributions. We show how we can modify the coupled rejection method to propose an ensemble of proposals so as to
asymptotically recover a maximal coupling while decreasing the total run time of the algorithm. We then discuss the important special case of coupling Gaussian distributions with different means and
covariances, and show how the rejection sampling method can be optimised in this case. We then apply the method to sampling from couplings of Gaussian tails, perform coupled Gibbs sampling, couple
parallel resampling algorithms in particle filtering, and couple manifold MALA.
Abstract: Well-calibrated predictive uncertainty of neural networks—essentially making them know when they don’t know—is paramount in safety-critical applications. However, deep neural networks are
overconfident in the region both far away and near the training data. In our works, we study Bayesian neural networks (BNNs) and their extensions to mitigate this issue. First, we show that being
Bayesian, even just at the last layer and in a post-hoc manner, helps mitigate overconfidence in deep ReLU classifiers. Then, we provide a cost-effective Gaussian-process extension to ReLU BNNs that
provides a guarantee that ReLU nets will never be overconfident in the region far from the data. Furthermore, we propose two ways of improving the calibration of general BNNs in the
out-of-distribution (OOD) regions near the data by (i) training the uncertainty of Laplace approximations and (ii) by leveraging OOD data during training. Finally, we provide a simple library,
laplace-torch, to facilitate modern arts of Laplace approximations in deep learning. This library gives users a way to turn a standard pre-trained deep net into a BNN in a cost-efficient manner.
Abstract: Normalizing flows have shown great success as general-purpose density estimators. However, many real-world applications require the use of domain-specific knowledge, which normalizing flows
cannot readily incorporate. We propose embedded-model flows (EMF), which alternate general-purpose transformations with structured layers that embed domain-specific inductive biases. These layers are
automatically constructed by converting user-specified differentiable probabilistic models into equivalent bijective transformations. We also introduce gated structured layers, which allow bypassing
the parts of the models that fail to capture the statistics of the data. We demonstrate that EMFs can be used to induce desirable properties such as multimodality, hierarchical coupling and
continuity. Furthermore, we show that EMFs enable a high-performance form of variational inference where the structure of the prior model is embedded in the variational architecture. In our
experiments, we show that this approach outperforms state-of-the-art methods in common structured inference problems.
Abstract: Bayesian joint parameter and state inference in non-linear state-space models is a difficult problem due to the often high-dimensional state sequence. Particle Gibbs (PG) is well-suited for
solving this type of inference problem but produces correlated samples. In this talk, I describe how the correlation can be reduced by marginalizing out one or more parameters from the state update
when conjugacy relations exist between the parameter prior and the complete data likelihood. Deriving the marginalized conjugacy relations is often time-consuming, but probabilistic programming can
be employed to automate the process. I also introduce a marginalized PG sampler for multiple time series described by a common state-space model structure, where subsets of the parameters are shared
between different models. The spread of mosquito-borne diseases, where some parameters are location-specific, and some are disease-specific, is one example. In theory, it is possible to update all
models concurrently, but sequential Monte Carlo becomes inefficient as the number of time series increases. Our suggested marginalized PG sampler instead updates one model at a time, conditioned on
the remaining datasets, and can be formulated in a modular fashion that greatly facilitates its implementation.
Abstract: Designing sequences of adaptive experiments to maximise the information gathered about an underlying process is a key challenge in science and engineering. Bayesian Experimental Design
(BED) is a powerful mathematical framework for tackling the optimal design problem. Despite the huge potential of obtaining information more quickly and efficiently, the widespread adoption of
adaptive BED has been severely limited by the costly computations required at each experiment iteration. In this talk, I’ll present a new method, called Deep Adaptive Design (DAD), that alleviates
this problem. DAD marks a critical change from previous BED methods in that it optimises a policy instead of individual designs during the experiment. The policy is parametrised by a neural network,
taking as inputs past data and returning the design to use at the next experiment iteration. Using a single pass through the network, DAD enables quick and adaptive design decisions in real time.
Abstract: Despite trends in modern medicine and epidemiological control, the risk for novel outbreaks and previously existing pathogens is currently greater than ever. Indeed, the current outbreak of
SARS-CoV-2 has exposed the need for precise, robust, and principled mathematical modelling of disease outbreaks that can perform well with noisy and potentially biased data. To tackle these
challenges, I will present a unifying view of modelling infectious diseases that contributes to the new understanding of the spread of the diseases and their epidemiological properties. The unified
framework allows flexible probabilistic models that are capable of fitting complex and noisy data from different sources. I will touch upon how the new unified framework, built using Stan (numpyro),
has helped us to characterize the initial spread of SARS-CoV-2 and quantify the altered epidemiological characteristics of various ‘variants of concerns’ (VOCs).
Abstract: In this talk, I will introduce the Transformed Gaussian Processes, a stochastic process specified by transforming samples from a Gaussian process using an invertible transformation (warping
function). These processes can be easily made non-stationary by parameterizing the warping function through an input-dependent transformation. I show how this is achieved with a Bayesian Neural
Network implemented with Monte Carlo dropout with the additional benefit of incorporating uncertainties, effectively regularizing the model. This new model can match the performance of a Deep
Gaussian Process at a fraction of its cost and also allow us to incorporate inductive biases in the function that we are trying to model (e.g. positive constraints), among other benefits. Training
and predictions can be scaled using a sparse variational inference algorithm. We also show how the basic idea of Transformed Gaussian Processes can be used to create a set of C dependent function
priors which can provide similar or better results than an SVGP in classification problems with a big number of classes but one order of magnitude faster.
Abstract: I will describe a kernel-based nonparametric test of relative goodness of fit, where the goal is to compare two models, both of which may have unobserved latent variables, such that the
marginal distributions of the observed variables are intractable. Given the premise that “all models are wrong,” the goal of the test is to determine whether one model significantly outperforms the
other in respect of a reference data sample. The test generalises earlier kernel Stein discrepancy (KSD) tests to the case of latent variable models, a much more general class than the fully observed
models treated previously. The new test, with a properly calibrated threshold, has a well-controlled type-I error. In the case of models with low-dimensional latent structure and high-dimensional
observations, our test significantly outperforms the relative maximum mean discrepancy test, which is based on samples from the models, and does not exploit the latent structure. I will illustrate
the test on probabilistic topic models of arXiv articles.
Abstract: Optimizing expensive-to-evaluate black-box functions of discrete (and potentially continuous) design parameters is a ubiquitous problem in science and engineering applications. Bayesian
optimization (BO) is a popular sample-efficient method that selects promising designs to evaluate by optimizing an acquisition function (AF) over some domain based on a probabilistic surrogate model.
However, maximizing the AF over mixed or high-cardinality discrete search spaces is challenging as we cannot use standard gradient-based methods or evaluate the AF at every point in the search space.
To address this issue, we propose using probabilistic reparameterization (PR). Instead of directly optimizing the AF over the search space containing discrete parameters, we instead maximize the
expectation of the AF over a probability distribution defined by continuous parameters. We prove that under suitable reparameterizations, the BO policy that maximizes the probabilistic objective is
the same as that which maximizes the AF, and therefore, PR enjoys the same regret bounds as the underlying AF. Moreover, our approach admits provably converges to a stationary point of the
probabilistic objective under gradient ascent using scalable, unbiased estimators of both the probabilistic objective and its gradient, and therefore, as the number of starting points and gradient
steps increases, our approach will recover of a maximizer of the AF (an often-neglected requisite for commonly used BO regret bounds). We validate our approach empirically and demonstrate
state-of-the-art optimization performance on many real-world applications. PR is complementary to (and benefits) recent work and naturally generalizes to settings with multiple objectives and
black-box constraints.
Abstract: A number of variational autoencoders (VAEs) have recently emerged with the aim of modelling multimodal data, e.g., to jointly model images and their corresponding captions. Still,
multimodal VAEs tend to focus solely on a subset of the modalities, e.g., by fitting the image while neglecting the caption. We refer to this limitation as modality collapse. In this presentation, I
argue that this effect is a consequence of modality-specific gradients conflicting during the training of multimodal VAEs. After this talk, you will be able to detect which parts of your model’s
computational graph can suffer from gradients conflict (which I call impartiality blocks), as well as how to leverage existing gradient-conflict solutions from multitask learning to mitigate modality
collapse. In order words, you will learn how to encourage impartial optimization across modalities. The framework I introduce is general, and we have successfully applied it to several multimodal VAE
models, losses, and datasets from the literature, and empirically showed that it significantly improves the reconstruction performance, conditional generation, and coherence of the latent space
across modalities.
Abstract: The performance of many algorithms in the fields of hard combinatorial problem solving, machine learning or AI in general depends on hyperparameter tuning. Automated methods have been
proposed to alleviate users from the tedious and error-prone task of manually searching for performance-optimized configurations. However, there is still a lot of untapped potential. Existing
solution approaches often neglect the non-stationarity of hyperparameters where different hyperparameter values are optimal at different stages of an algorithms run. Taking the non-stationarity into
account in the optimization procedure promises much better performances but also poses many new challenges. In this talk we will discuss existing solution approaches to classical hyperparameter
optimization and explore ways of tackling the non-stationarity of hyperparameters in particular by means of reinforcement learning.
Abstract: State-of-the-art machine learning models can be vulnerable to very small input perturbations that are adversarially constructed. Adversarial attacks are a popular framework for studying
these vulnerabilities. They consider worst-case input disturbances designed to maximize model error and got a lot of attention due to their impact on the performance of state-of-the-art models.
Adversarial training considers extending model training with these examples and is an effective approach to defend against such attacks. This talk will explore adversarial attacks and training in
linear regression. There is a strong reason for this focus, for linear regression, adversarial training can be formulated as a convex and quadratic problem. Moreover, many interesting phenomena that
can be observed in nonlinear models are still present. The setup is used to study the role of high dimensionality in robustness. And to reveal the connection between adversarial training,
parameter-shrinking methods and minimum-norm solutions.
Abstract: Score-based generative models (SGMs) are a powerful class of generative models that exhibit remarkable empirical performance. Score-based generative modelling consists of a noising stage,
whereby a diffusion is used to gradually add Gaussian noise to data, and a generative model, which entails a denoising process defined by approximating the time-reversal of the diffusion. Existing
SGMs assume that data is supported on a Euclidean space, i.e. a manifold with flat geometry. In many domains such as robotics, geoscience or protein modelling, data is often naturally described by
distributions living on Riemannian manifolds and current SGM techniques are not appropriate. We introduce here Riemannian Score-based Generative Models (RSGMs), a class of generative models extending
SGMs to Riemannian manifolds. We demonstrate our approach on a variety of manifolds, and in particular with earth and climate science spherical data.
Abstract: While Bayesian deep learning has been a popular field of research in recent years, most of the work has focused on improving inference methods for better performance and lower computational
costs. Conversely, the priors have often been ignored and merely chosen to be isotropic Gaussian, for mathematical and computational convenience. In this talk, I will review recent work that calls
this popular practice into question and highlights pathologies that can arise from prior misspecification in Bayesian neural networks. I will then present different methods that can aid the selection
of better priors and I will discuss the advantages of function-space priors over weight-space ones.
Abstract: Simulator-based models are models for which the likelihood is intractable but simulation of synthetic data is possible. They are often used to describe complex real-world phenomena, and as
such can often be misspecified in practice. In this talk, I will present a novel algorithm based on the posterior bootstrap and maximum mean discrepancy estimators. This leads to a
highly-parallelisable Bayesian inference algorithm with strong robustness properties. This is demonstrated through an in-depth theoretical study which includes generalisation bounds, frequentist
consistency and robustness of our posterior guarantees. The approach is then illustrated on a range of examples including a g-and-k distribution and a toggle-switch model.
Abstract: Causal structure learning aims to discover the causal directed acyclic graph (DAG) responsible for generating a given dataset. However, a point estimate can be flawed due to limited data as
well as non-identifiability of the underlying DAG. Bayesian approaches to structure learning instead seek to characterize a full posterior distribution over DAGs, but are typically very
computationally expensive in high dimensions. In this talk, I will present Tractable Uncertainty for STructure learning (TRUST), a framework for approximate posterior inference that relies on
probabilistic circuits, a type of tractable probabilistic model, as the representation of our posterior belief over causal DAGs. In comparison to Monte-Carlo posterior approximations, our
representation can capture a much richer space of DAGs, while also being able to tractably reason about the uncertainty; for example, inferring the most likely completion of a partial graph, or the
expected linear causal effect. Finally, I will show how our posterior representations can be learned by exploiting existing structure learning algorithms together with variational inference, leading
empirically to improvement in both the quality of inferred structures and posterior uncertainty.
Abstract: The numerical simulation of differential equations underpins many modelling decisions made in the natural sciences. Solving differential equations with probabilistic numerical algorithms
promises better uncertainty quantification than with non-probabilistic approaches, but until recently, probabilistic solvers have been inefficient, unstable, and generally impractical. In this talk,
I will explain the fundamentals of probabilistic numerical algorithms for the simulation of ordinary differential equations. Building on this, I will survey the stable and efficient implementation of
probabilistic numerical solvers and discuss generalisations of the algorithm to partial differential equations.
Abstract: Comparative decisions, such as picking between two cars or deciding between two hiking trails, require the users to visit multiple webpages and contrast the choices along relevant aspects.
Given the impressive capabilities of pre-trained large language models, we ask whether they can help automate such analysis. We refer to this task as extractive aspect-based contrastive summarization
which involves constructing a structured summary that compares the choices along relevant aspects. In this paper, we propose a novel method called STRUM for this task that can generalize across
domains without requiring any human-written summaries or fixed aspect list as supervision. Given a set of relevant input webpages, STRUM solves this problem using two pre-trained T5-based large
language models: first one fine-tuned for aspect and value extraction, and second one fine-tuned for natural language inference. We showcase the abilities of our method across different domains,
identify shortcomings, and discuss questions that we believe will be critical in this new line of research.
Abstract: In this talk, I present extensive evidence against the common belief that variational Bayesian learning is ineffective for large neural networks. First, I show that a recent deep learning
method called sharpness-aware minimization (SAM) solves an optimal convex relaxation of the variational Bayesian objective. Then, I demonstrate that a direct optimization of the variational objective
with an Improved Variational Online Newton method (IVON) can consistently match or outperforms Adam for training large networks such as GPT-2 and ResNets from scratch. IVON’s computational costs are
nearly identical to Adam but its predictive uncertainty is better. The talk concludes with several new use cases of variational learning where we improve fine-tuning and model merging in Large
Language Models, accurately predict generalization error, and faithfully estimate sensitivity to data.
Abstract: A natural way of estimating heteroscedastic label noise in regression is to model the observed (potentially noisy) target as a sample from a normal distribution, whose parameters can be
learned by minimizing the negative log-likelihood. This formulation has desirable loss attenuation properties, as it reduces the contribution of high-error examples. Intuitively, this behaviour can
improve robustness against label noise by reducing overfitting. We propose an extension of this simple and probabilistic approach to classification, that has the same desirable loss attenuation
properties. We evaluate the effectiveness of the method by measuring its robustness against label noise in classification. In follow-up work, we improve the method’s robustness by modelling and
estimating a shift (non-zero mean) in the Gaussian noise distribution, which we show makes it possible for the method to correct noisy labels.
Abstract: We lay out three avenues in which we think generative models are especially valuable for modeling biomolecules. 1) Hard prediction tasks can be better addressed with generative models that
can suggest and rank multiple solutions (e.g., docking). 2) The dynamics and conformations of biomolecules can be captured with generative models (e.g., protein conformational ensembles and MD
trajectories). 3) Designing new biomolecules can be accelerated, informed by samples or likelihoods from generative models (e.g., protein binder or regulatory DNA design).
Abstract: Modeling partial differential equations (PDEs) is of crucial importance in science and engineering. Some of the most common tasks include 1) forecasting, where the aim is to predict future
states based on an initial one, as well as 2) inverse problems, such as data assimilation (DA), with the goal of reconstructing an aspect of the PDE (i.e. coefficient, initial condition, full
trajectory, etc.) given some partial observations of the solution to the PDE. However, most previous numerical and machine learning approaches that target forecasting cannot be applied out-of-the-box
for data assimilation. Recently, diffusion models have emerged as a powerful tool for conditional generation, being able to flexibly incorporate observations without retraining. In this talk, I will
discuss our recent work in this domain, where we perform a comparative study of score-based diffusion models for forecasting and assimilation of sparse observations. In particular, we focus on
diffusion models that are either presented with the conditional information during training, or conditioned after unconditional training. We address the shortcomings of previous work and develop
methods that are able to successfully tackle the combination of forecasting and data assimilation, a task commonly encountered in real-world scenarios such as weather modeling.
Abstract: We introduce a software package, Pigeons.jl, that provides a way to leverage distributed computation to obtain samples from complicated probability distributions, such as multimodal
posteriors arising in Bayesian inference and high-dimensional distributions in statistical mechanics. Pigeons.jl provides simple APIs to perform such computations single-threaded, multi-threaded, and
/or distributed over thousands of MPI-communicating machines. Our software provides several Markov kernels, including our newly proposed algorithm, autoMALA. This MCMC algorithm, based on the
Metropolis-adjusted Langevin algorithm, automatically sets its step size at each iteration based on the local geometry of the target distribution. Our experiments demonstrate that autoMALA is
competitive with related state-of the-art MCMC methods, in terms of the number of log density evaluations per effective sample, and it outperforms state-of-the-art samplers on targets with varying
Abstract: Probabilistic circuits are prominent tractable probabilistic models which can provide exact answers to a wide range of probabilistic queries in a tractable way. Given their sparse nature
and the structural constraints enabling exact inference, it is challenging to induce these models in high-dimensional real-world domains such as time series and raw images. In this talk, I show how
we can leverage spectral modeling and the clear probabilistic semantics of probabilistic circuits to learn models able to provide efficient and reliable predictions in these challenging domains, and
how to make these particular models more robust to distribution shift and out-of-distribution data.
Abstract: TBH
Abstract: TBA
Abstract: TBA | {"url":"https://aaltoml.github.io/apml/","timestamp":"2024-11-07T11:53:12Z","content_type":"text/html","content_length":"94048","record_id":"<urn:uuid:a433829f-c7bc-4571-8bd3-4c980edc9de8>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00135.warc.gz"} |
JaehyeonSeo Archives - Discrete Mathematics Group
Jaehyeon Seo (서재현) gave a talk on the rainbow Turán problem for color-critical graphs at the Discrete Math Seminar
On January 18, 2022, Jaehyeon Seo (서재현) from KAIST gave a talk at the Discrete Math Seminar on the rainbow Turán problem for color-critical graphs. The title of his talk was “A rainbow Turán
problem for color-critical graphs“.
Jaehyeon Seo (서재현), A rainbow Turán problem for color-critical graphs
For given $k$ graphs $G_1,\dots, G_k$ over a common vertex set of size $n$, what conditions on $G_i$ ensures a ‘colorful’ copy of $H$, i.e. a copy of $H$ containing at most one edge from each $G_i$?
Keevash, Saks, Sudakov, and Verstraëte defined $\operatorname{ex}_k(n,H)$ to be the maximum total number of edges of the graphs $G_1,\dots, G_k$ on a common vertex set of size $n$ having no colorful
copy of $H$. They completely determined $\operatorname{ex}_k(n,K_r)$ for large $n$ by showing that, depending on the value of $k$, one of the two natural constructions is always the extremal
construction. Moreover, they conjectured the same holds for every color-critical graphs and proved it for $3$-color-critical graphs.
We prove their conjecture for $4$-color-critical graphs and for almost all $r$-color-critical graphs when $r > 4$. Moreover, we show that for every non-color-critical non-bipartite graphs, none of
the two natural constructions is extremal for certain values of $k$. This is a joint work with Debsoumya Chakraborti, Jaehoon Kim, Hyunwoo Lee, and Hong Liu. | {"url":"https://dimag.ibs.re.kr/tag/jaehyeonseo/","timestamp":"2024-11-11T22:43:44Z","content_type":"text/html","content_length":"138316","record_id":"<urn:uuid:488973b1-ca4f-424a-b5dd-4a45c83e1e05>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00281.warc.gz"} |
Lesson 6.4 | CodeSciLab
top of page
Statistics Refresher
Checkpoint I
Checkpoint II
There are multiple ways to visualize data other than a histogram, with which it can be difficult to see the median of the distribution. We can also visualize medians (and distributions) with another
type of graph called a box plot. Let's look at our human weight data in box plot form.
This code should yield a box plot titled WeightBoxPlot.png that looks like this:
Let's break this down. The red line in the middle, as you have likely guessed, is the median of the data. The upper blue line represents the 75th percentile of the data. The 75th percentile is the
value at which 75% of the data points lie below it. So in this data, the 75th percentile is approximately 135, meaning that 75% of the data points are less than 135. The bottom blue line is the 25th
percentile, which is the value at which 25% of the data points lie below it. So in this data, the 25th percentile is 120, meaning that 25% of the data points are less than 120.
The lines extending from either side of the box are called "whiskers" (box plots are also called box-and-whisker plots for this reason). They denote the range of the data; points outside this range
are considered outliers and are plotted individually (the crosses on this graph).
So far, all the data we've looked at, even the data with a huge outlier, have been mostly symmetrical (except for the outlier itself) -- the shape of the histogram is relatively similar on both sides
of the mean.
Sometimes data is very asymmetric, though. For example, let's look at the histogram generated with the code below.
This generates a histogram titled AsymmetricHistogram.png, which should look something like this:
Lets calculate the exact mean and median of the above distribution to check our intuition.
This data is what we refer to as "skewed" -- one side of the histogram is significantly heavier than the other (in this case, the right side). This particular data would be called "right-skewed." The
right skew is what causes the mean and median to differ from each other. Specifically, the middle value (the median) is close to 0, since clearly about half of the values are above it and half are
below it. But the heavy right tail (the extra high values) pulls the mean upward so it is quite a bit higher than the median.
This example shows why we need different measures of our data ("summary statistics") -- for some distributions, the mean and median will be very similar, but in cases like this, they give very
different information.
bottom of page | {"url":"https://www.codescilab.org/copy-of-lesson-6-3","timestamp":"2024-11-13T13:11:11Z","content_type":"text/html","content_length":"491562","record_id":"<urn:uuid:18a0535d-e57b-424b-9e89-c8057e928d02>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00126.warc.gz"} |
American Air Flight 2705 from New York to San Francisco has
seats for 340 passengers. An...
American Air Flight 2705 from New York to San Francisco has seats for 340 passengers. An...
American Air Flight 2705 from New York to San Francisco has seats for 340 passengers. An average of 5% if people with reservations don’t show up, so American Air overbooks by accepting 350
reservations for the 340 seats. We can analyze this system by using a binomial distribution with n = 350 and p = 0.95 (the probability that someone with a reservation does not show up). Find the
probability that when 350 reservations are accepted for a particular flight, there are more passengers than seats. That is, find the probability of at lest 341 people showing up with reservation,
assuming that 350 reservations were accepted.
Let X be the number of with reservations show up
X follows Binomial(n = 350, p = 0.95)
The probability that when 350 reservations are accepted for a particular flight, there are more passengers than seats would be
P(X ≥ 341) = 1 - P(X ≤ 340)
We can use Binomial probabilities using =BINOM.DIST(number_s, trials, probability_s, cumulative) excel function. For P(X ≤ 340), =BINOM.DIST(340, 350, 0.95, TRUE) (cumulative TRUE since it is ≤)
P(X ≥ 341) = 1 - 0.982120005 = 0.017879995 | {"url":"https://justaaa.com/statistics-and-probability/295931-american-air-flight-2705-from-new-york-to-san","timestamp":"2024-11-09T07:31:23Z","content_type":"text/html","content_length":"37458","record_id":"<urn:uuid:59b2db0f-8e34-4e65-ba75-2fbbc0fe40fd>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00193.warc.gz"} |
Solving The Equation: Exploring The Roots Of 5X – 12 = 0
Are you struggling with the equation 5x – 12 = 0? Well, fret not, because in this article, we are going to unravel its solution in a simple and conversational manner. Don’t worry if math isn’t your
strong suit; we’ve got you covered. Let’s dive right in and explore how to crack this equation and find the value of x that satisfies it. Trust me, it’s easier than you might think!
Understanding 5x – 12 = 0: Solving Linear Equations Made Simple
In the realm of mathematics, equations play a vital role in solving problems and uncovering unknown values. One such equation that frequently arises in algebra is 5x – 12 = 0. This linear equation
may appear complex at first glance, but fear not! In this article, we will break it down step by step, demystifying the process to help you solve similar equations with ease. So, let’s dive in and
explore the world of linear equations together!
What is a Linear Equation?
Before we unravel the mystery of 5x – 12 = 0, let’s quickly review what a linear equation is. A linear equation is an equation where the highest degree of the variable(s) involved is one. In other
words, linear equations represent straight lines when graphed on a coordinate plane. These equations can be solved by isolating the variable on one side of the equation.
The Structure of 5x – 12 = 0
Now, let’s focus on our specific equation, 5x – 12 = 0, and dissect its structure to better understand its components. This equation consists of:
• A coefficient (5) multiplying the variable (x)
• A constant term (-12) on the other side of the equation
• An equals sign, indicating that the left and right sides of the equation are balanced
• An outcome of zero (0) on the right side
Understanding the structure of an equation helps us recognize the necessary steps to solve it accurately.
Step-by-Step Solving of 5x – 12 = 0
Now that we have a solid foundation, let’s go through the step-by-step process of solving 5x – 12 = 0. Remember, the goal is to isolate the variable (x) and determine its value.
Step 1: Isolate the Variable
To begin, we need to move the constant term (-12) to the other side of the equation. By adding 12 to both sides, we maintain balance.
5x – 12 + 12 = 0 + 12
Simplifying, we get:
5x = 12
Step 2: Solve for x
Now that the variable (x) is no longer hindered by the constant term, we can focus on finding its value. To achieve this, we need to isolate x by dividing both sides of the equation by the
coefficient (5).
5x/5 = 12/5
Simplifying further:
x = 12/5
Step 3: Final Solution
After performing the necessary calculations, we find that x equals 12/5. This is our final solution to the equation 5x – 12 = 0. The x-value represents the point at which the equation’s graph
intersects the x-axis.
Real-World Applications
Now that we have mastered the process of solving the equation 5x – 12 = 0, let’s explore some real-world applications where this knowledge can be applied. Understanding linear equations allows us to
solve problems and find solutions in various fields, such as:
Finance and Budgeting
Linear equations can be used to analyze financial situations and create budgets. By understanding how different variables interact, we can determine optimal strategies for saving, investing, and
managing expenses. For example, if we have a fixed income and want to calculate the amount we can save each month, linear equations can help us find that value.
Physics and Engineering
Linear equations are fundamental in physics and engineering, where they help describe the relationships between different physical quantities. From calculating velocity and acceleration to analyzing
electrical circuits, linear equations play a crucial role in developing innovative technologies and solving engineering problems.
Graphing and Data Analysis
Graphing data is an essential skill in many fields, including science, economics, and social sciences. Linear equations allow us to represent data on a coordinate plane, enabling us to analyze
trends, make predictions, and draw meaningful conclusions from the information at hand.
Tips and Tricks for Solving Linear Equations
While solving the equation 5x – 12 = 0 was relatively straightforward, some equations may present more complex challenges. To enhance your problem-solving abilities, here are a few tips and tricks to
keep in mind:
1. Simplify before Solving
Before diving into complex equations, simplify them as much as possible. Combine like terms, eliminate any unnecessary parentheses, and reduce fractions to their simplest form. This will make the
subsequent steps more manageable.
2. Clear Fractions
If an equation contains fractions, multiply both sides by the least common denominator (LCD) to eliminate them. This simplifies the equation and prepares it for further manipulation.
3. Be Cautious with Negative Signs
Pay close attention to negative signs while manipulating equations. It’s easy to overlook a negative sign or make errors when distributing negatives during multiplication or division. Double-check
your work to ensure accurate results.
4. Simplify Radicals
If an equation involves square roots or other radicals, isolate and square both sides to eliminate the radical sign. Remember to consider both the positive and negative square root solutions in your
final answer.
5. Check Your Solution
Once you have found a solution, always verify it by substituting the value back into the original equation. This step helps identify any errors made during the solving process and ensures the
solution is valid.
In conclusion, the equation 5x – 12 = 0 is a linear equation with a straightforward solution. By following the step-by-step process of isolating the variable and solving for x, we can determine its
value. Understanding linear equations provides a solid foundation for problem-solving and analysis in various fields. Whether you’re exploring the realms of finance, physics, or data analysis, the
ability to solve linear equations is a valuable skill. So, embrace the world of equations, boost your problem-solving prowess, and unlock countless possibilities!
How to solve the equation: 4x²-5x-12=0?
Frequently Asked Questions
What is the solution to the equation 5x – 12 = 0?
The solution to the equation 5x – 12 = 0 is x = 12/5 or x = 2.4. To find the solution, we need to isolate the variable x by performing mathematical operations. In this case, we add 12 to both sides
of the equation to eliminate the constant term -12. This leaves us with 5x = 12. Then, we divide both sides of the equation by 5 to find the value of x.
How do I solve the equation 5x – 12 = 0?
To solve the equation 5x – 12 = 0, you follow these steps:
1. Add 12 to both sides of the equation: 5x = 12
2. Divide both sides of the equation by 5: x = 12/5
Therefore, the solution to the equation 5x – 12 = 0 is x = 12/5 or x = 2.4.
Can I get a decimal value as the solution?
Yes, you can obtain a decimal value as the solution to the equation 5x – 12 = 0. When you divide 12 by 5, you get 2.4, which is a decimal value. So, the solution to the equation can be expressed as x
= 12/5 or x = 2.4.
Final Thoughts
In conclusion, the equation 5x – 12 = 0 is a simple linear equation that can be solved to find the value of x. By adding 12 to both sides of the equation, we eliminate the constant term -12,
resulting in the simplified equation 5x = 12. To isolate x, we divide both sides by 5, giving us the solution x = 12/5. It is essential to understand the process of solving linear equations to
successfully solve problems involving unknown variables. By applying these steps, we can easily find the value of x in equations like 5x – 12 = 0. | {"url":"https://holdingsbusiness.com/5x-12-0/","timestamp":"2024-11-03T06:31:11Z","content_type":"text/html","content_length":"134768","record_id":"<urn:uuid:62a91039-641b-42a0-9b14-05adfb6cef55>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00413.warc.gz"} |
Hermann Weyl
Based on four lectures given in 1951, Herman Weyl's
is a slender work which focuses on the mathematics of symmetry, starting simply but building up to increasing abstraction, but which also explores more widely, touching on aspects of symmetry in
philosophy, art and architecture, physics, and biology. Weyl was a leading mathematician, physicist and philosopher of the first half of the twentieth century and he offers a magisterial perspective.
Symmetry begins with bilateral symmetry, looking at its uses in art and architecture and at left-right symmetry in biology, in phylogeny and ontogeny. The only mathematics introduced in the first
lecture, and that only in passing, is the notion of congruence.
The second lecture moves on to translational and rotational symmetries, looking at frieze and cyclic patterns in art and architecture, and in the natural world at flowers, echinoderms, snowflakes,
and so forth. The concepts of automorphisms and groups are introduced here and Weyl presents a classification of the finite groups of proper and improper rotations in space (though proof of the
completeness of this is left to an appendix).
Lecture three is devoted to "Ornamental Symmetry", or the symmetries of regular tilings of the plane. This introduces some basic linear algebra and the notion of a lattice, and works through
classification of the 17 plane symmetry groups. It doesn't attempt "an explicitly algebraic description" of those groups, but it is still fairly involved.
It does include a discussion of symmetry in physics, but the final lecture on "The General Mathematical Idea of Symmetry" mostly offers further mathematics. Almost as if to scare off most readers, it
opens with a paragraph on the classification of "the unimodularly inequivalent discontinuous groups of non-homogeneous linear transformations which contain the translations with integral coordinates
but no other translations". And it goes on to touch on Galois theory.
The limitations of Symmetry are fairly obvious. Its illustrations, while they adequately illustrate the connections being made to art, architecture and biology, are grainy black and white halftones
which hardly grab the attention. And Weyl is writing for, or speaking to, a relatively sophisticated audience. He introduces mathematical ideas rapidly, more as a refresher than a genuine
introduction, and his presentation sometimes seems a little awkward; some of his terminology is also a little dated. So this is really not recommended as a first introduction to group theory.
General readers might enjoy the first half of Symmetry, however, and wing their way through the second half, while anyone with some familiarity with group theory, and curious about a broad
perspective on its application to symmetry, should find the whole work engaging.
March 2016
External links:
- buy from Amazon.com or Amazon.co.uk
- share this review on Facebook or Twitter
Related reviews:
- books about art + art history
- books about popular mathematics
- books published by Princeton University Press | {"url":"https://dannyreviews.com/h/Symmetry.html","timestamp":"2024-11-07T21:34:03Z","content_type":"text/html","content_length":"7643","record_id":"<urn:uuid:0049dde5-5b12-45e9-92fc-bc71a025152d>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00565.warc.gz"} |
Yaiza Canzani, UNC-Chapel Hill - Analysis & PDE Seminar - Department of Mathematics
Yaiza Canzani, UNC-Chapel Hill – Analysis & PDE Seminar
April 10, 2019 @ 4:00 pm - 5:00 pm
Title: Understanding the growth of Laplace eigenfunctions.
Abstract: In this talk we will discuss a new approach to understanding eigenfunction concentration. We characterize the features that cause an eigenfunction to saturate the standard supremum bounds
in terms of the distribution of L^2 mass along geodesic tubes emanating from a point. We also show that the phenomena behind extreme supremum norm growth is identical to that underlying extreme
growth of eigenfunctions when averaged along submanifolds. Using the description of concentration, we obtain quantitative improvements on the known bounds in a wide variety of settings.
Related Events | {"url":"https://math.unc.edu/event/yaiza-canzani-unc-chapel-hill-analysis-pde-seminar-2/","timestamp":"2024-11-12T06:15:03Z","content_type":"text/html","content_length":"112249","record_id":"<urn:uuid:ad0eed90-b9a6-4d3b-a83f-ff38e3ef414e>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00081.warc.gz"} |
Developmental Math Emporium
Learning Outcomes
• Factor polynomials with negative or fractional exponents
• Factor by substitution
Expressions with fractional or negative exponents can be factored using the same factoring techniques as those with integer exponents. It is important to remember a couple of things first.
• When you multiply two exponentiated terms with the same base, you can add the exponents: [latex]x^{-1}\cdot{x^{-1}}=x^{-1+(-1)}=x^{-2}[/latex]
• When you add fractions, you need a common denominator: [latex]\frac{1}{2}+\frac{1}{3}=\frac{3}{3}\cdot\frac{1}{2}+\frac{2}{2}\cdot\frac{1}{3}=\frac{3}{6}+\frac{2}{6}=\frac{5}{6}[/latex]
• Polynomials have positive integer exponents – if it has a fractional or negative exponent it is an expression.
First, practice finding a GCF that is a negative exponent.
Factor [latex]12y^{-3}-2y^{-2}[/latex].
Show Solution
Try It
Now let us factor a trinomial that has negative exponents.
Factor [latex]x^{-2}+5x^{-1}+6[/latex].
Show Solution
In the next example, we will see a difference of squares with negative exponents. We can use the same shortcut as we have before, but be careful with the exponent.
Factor [latex]25x^{-4}-36[/latex].
Show Solution
In the following video, you will see more examples that are similar to the previous three written examples.
Fractional Exponents
Again, we will first practice finding a GCF that has a fractional exponent.
Factor [latex]x^{\frac{2}{3}}+3x^{\frac{1}{3}}[/latex].
Show Solution
In our next example, we will factor a perfect square trinomial that has fractional exponents.
Factor [latex]25x^{\frac{1}{2}}+70x^{\frac{1}{4}}+49[/latex].
Show Solution
Try It
In our next video, you will see more examples of how to factor expressions with fractional exponents.
Factor Using Substitution
We are going to move back to factoring polynomials; our exponents will be positive integers. Sometimes we encounter a polynomial that looks similar to something we know how to factor but is not quite
the same. Substitution is a useful tool that can be used to “mask” a term or expression to make algebraic operations easier.
You may recall that substitution can be used to solve systems of linear equations and to check whether a point is a solution to a system of linear equations.
For example, consider the following equation:
To determine whether [latex]x=5[/latex], and [latex]y=1[/latex] is a solution to the equation, we can substitute the values [latex]x=5[/latex] and [latex]y=1[/latex] into the equation.
[latex]8=8[/latex] True
We replaced the variables with numbers and then performed the algebraic operations specified. In the next example, we will see how we can use a similar technique to factor a fourth degree polynomial.
Factor [latex]x^4+3x^2+2[/latex].
Show Solution
Try It
In the following video, we show two more examples of how to use substitution to factor a fourth degree polynomial and an expression with fractional exponents.
Factor Completely
Sometimes you may encounter a polynomial that takes an extra step to factor. In our next example, we will first find the GCF of a trinomial, and after factoring it out, we will be able to factor
again so that we end up with a product of a monomial and two binomials.
Factor [latex]6m^2k-3mk-3k[/latex] completely.
Show Solution
In our last example, we show why it is important to factor out a GCF, if there is one, before you begin using the techniques shown in this module.
In this section, we used factoring with special cases and factoring by grouping to factor expressions with negative and fractional exponents. We also returned to factoring polynomials and used the
substitution method to factor a [latex]4th[/latex] degree polynomial. The last topic we covered was what it means to factor completely. | {"url":"https://courses.lumenlearning.com/wm-developmentalemporium/chapter/read-factor-using-substitution-2/","timestamp":"2024-11-06T20:45:40Z","content_type":"text/html","content_length":"61483","record_id":"<urn:uuid:5c4cfcca-d5c3-4101-bb67-a0f1da63fd51>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00352.warc.gz"} |
AffineMatrix Class Properties
In This Topic
For a list of all properties of this type, see
Public Properties
Name Description
Determinant Gets the determinant of the matrix.
IsIdentity Gets a value indicating whether this AffineMatrix class is an identity matrix.
IsInvertible Gets a value indicating whether the matrix is invertible.
M11 Gets or sets the value of the first row and first column of the matrix (scale X).
M12 Gets or sets the value of the first row and second column of the matrix.
M21 Gets or sets the value of the second row and first column of the matrix.
M22 Gets or sets the value of the second row and second column of the matrix (scale Y).
OffsetX Gets or sets the value of the third row and first column of the matrix (offset X).
OffsetY Gets or sets the value of the third row and second column of the matrix (offset Y).
Type Gets the type of the matrix.
See Also | {"url":"https://www.vintasoft.com/docs/vsimaging-dotnet/Vintasoft.Imaging/Vintasoft.Imaging.AffineMatrix_properties.html","timestamp":"2024-11-09T16:01:48Z","content_type":"application/xhtml+xml","content_length":"11228","record_id":"<urn:uuid:de09f916-91a2-4b71-9b12-aaddfee1e271>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00386.warc.gz"} |
\dfrac{{Z(Z - 1){e^2}}}{{4\
Hint: mass of proton, neutron and nuclei of ${}_7^{15}Nand{}_8^{15}O$ using mass energy equivalence proposed by einstein which is $E = m{c^2}$ find the binding energy of ${}_7^{15}Nand{}_8^{15}O$ by
calculating energy of each individual neutrons and protons and then subtracting it from the total energy of those nuclei once you know the binding energy find the binding energy using the given
expression in the question and then compare to get the radius of the spherical nucleus.
Complete step by step answer:
Let's calculate the radius of ${}_7^{15}N$. It has 7 protons and 8 neutrons. Now total mass of 7 protons and 8 neutrons will be,
${m_{total}} = 7{m_{proton}} + 8{m_{neutron}}$
$ \Rightarrow {m_{total}} = 7(1.007825) + 8(1.008665)\\
\Rightarrow {m_{total}} = 15.124095u$
But given mass of ${}_7^{15}N$is $15.000109u$
So difference in mass will be,
$\Delta m = {m_{total}} - m({}_7^{15}N)$
$ \Rightarrow \Delta m = 15.124095u - 15.000109u \\
\Rightarrow \Delta m = 0.123386u$
Now using mass energy equivalence we will find the binding energy. Therefore, binding energy,
$E = \Delta m{c^2}\\
\Rightarrow E= 0.123386 \times 931.5 MeV \\
\Rightarrow E= 114.93 MeV$
Now using the formula of binding energy given in the question. We have,
$E = \dfrac{3}{5}\dfrac{{Z(Z - 1){e^2}}}{{4\pi {\varepsilon _o}R}}\\
\Rightarrow E= \dfrac{3}{5}\dfrac{{7(7 - 1)}}{R}1.44MeV$
As \[\dfrac{{{e^2}}}{{4\pi {\varepsilon _o}}} = 1.44MeVfm.\]
$ \Rightarrow E = \dfrac{3}{5}\dfrac{{42}}{R}1.44MeV \\
\Rightarrow E = \dfrac{{36.288}}{R}MeV$
Comparing both these energies we have,
$R = \dfrac{E}{{36.288}}fm \\
\therefore R= \dfrac{{114.93}}{{36.288}}fm \approx 3.03fm$
Hence Option B is correct.
Note: This question looks scary but if you read the question carefully and apply the basics of modern physics specially mass energy equivalence it is lot easier also here you don’t have to lot of
calculations as values of some of the constant are given in the question.one more thing its asking radius of either of the nuclei you don’t have to calculate the both because total number of nucleons
in both the nuclei are same so radius of both the nuclei will almost be same. | {"url":"https://www.vedantu.com/question-answer/the-electrostatic-energy-of-z-protons-uniformly-class-12-physics-cbse-5fb4a473d1d70752ffa3a230","timestamp":"2024-11-14T17:31:44Z","content_type":"text/html","content_length":"186593","record_id":"<urn:uuid:d4b949c7-10c7-4fc7-a608-9eeba8c73787>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00734.warc.gz"} |
Introductory Chemistry
Learning Objectives
1. Learn what is meant by the term gas laws.
2. Learn and apply Boyle’s law.
3. Learn and apply Charles’s law.
When seventeenth-century scientists began studying the physical properties of gases, they noticed simple relationships between some of the measurable properties of gases. Take pressure (P) and volume
(V), for example. Scientists noted that for a given amount of a gas (usually expressed in units of moles [n]), if the temperature (T) of the gas is kept constant, pressure and volume are related: as
one increases, the other decreases. As one decreases, the other increases. We say that pressure and volume are inversely related.
There is more to it, however: pressure and volume of a given amount of gas at a constant temperature are numerically related. If you take the pressure value and multiply it by the volume value, the
product is a constant for a given amount of gas at a constant temperature:
P × V = constant at constant n and T
If either volume or pressure changes while the amount and temperature stay the same, then the other property must change so that the product of the two properties still equals that same constant.
That is, if the original conditions are labelled P[1] and V[1] and the new conditions are labelled P[2] and V[2], we have
P[1]V[1] = constant = P[2]V[2]
where the properties are assumed to be multiplied together. Leaving out the middle part, we have simply
P[1]V[1] = P[2]V[2] at constant n and T
This equation is an example of a gas law. A gas law is a simple mathematical formula that allows you to model, or predict, the behaviour of a gas. This particular gas law is called Boyle’s law, after
the English scientist Robert Boyle, who first announced it in 1662. Figure 6.1 Boyle’s Law shows two representations of what Boyle’s law describes.
Figure 6.1 Boyle’s Law
A piston having a certain pressure and volume (left piston) will have half the volume when its pressure is twice as much (right piston). One can also plot P versus V for a given amount of gas at a
certain temperature; such a plot will look like the graph on the right.
Boyle’s law is an example of a second type of mathematical problem we see in chemistry—one based on a mathematical formula. Tactics for working with mathematical formulas are different from tactics
for working with conversion factors. First, most of the questions you will have to answer using formulas are word-type questions, so the first step is to identify what quantities are known and assign
them to variables. Second, in most formulas, some mathematical rearrangements (i.e., algebra) must be performed to solve for an unknown variable. The rule is that to find the value of the unknown
variable, you must mathematically isolate the unknown variable by itself and in the numerator of one side of the equation. Finally, units must be consistent. For example, in Boyle’s law there are two
pressure variables, and they must have the same unit. There are also two volume variables; they also must have the same unit. In most cases, it won’t matter what the unit is, but the unit must be the
same on both sides of the equation.
Example 3
A sample of gas has an initial pressure of 2.44 atm and an initial volume of 4.01 L. Its pressure changes to 1.93 atm. What is the new volume if temperature and amount are kept constant?
First, determine what quantities we are given. We are given an initial pressure and an initial volume, so let these values be P[1] and V[1]:
P[1] = 2.44 atm and V[1] = 4.01 L
We are given another quantity, final pressure of 1.93 atm, but not a final volume. This final volume is the variable we will solve for.
P[2] = 1.93 atm and V[2] = ? L
Substituting these values into Boyle’s law, we get
(2.44 atm)(4.01 L) = (1.93 atm)V[2]
To solve for the unknown variable, we isolate it by dividing both sides of the equation by 1.93 atm—both the number and the unit:
Note that, on the left side of the equation, the unit atm is in the numerator and the denominator of the fraction. They cancel algebraically, just as a number would. On the right side, the unit atm
and the number 1.93 are in the numerator and the denominator, so the entire quantity cancels:
What we have left is
Now we simply multiply and divide the numbers together and combine the answer with the L unit, which is a unit of volume. Doing so, we get
V[2] = 5.07 L
Does this answer make sense? We know that pressure and volume are inversely related; as one decreases, the other increases. Pressure is decreasing (from 2.44 atm to 1.93 atm), so volume should be
increasing to compensate, and it is (from 4.01 L to 5.07 L). So the answer makes sense based on Boyle’s law.
Test Yourself
If P[1] = 334 torr, V[1] = 37.8 mL, and P[2] = 102 torr, what is V[2]?
124 mL
As mentioned, you can use any units for pressure or volume, but both pressures must be expressed in the same units, and both volumes must be expressed in the same units.
Example 4
A sample of gas has an initial pressure of 722 torr and an initial volume of 88.8 mL. Its volume changes to 0.663 L. What is the new pressure?
We can still use Boyle’s law to answer this, but now the two volume quantities have different units. It does not matter which unit we change, as long as we perform the conversion correctly. Let us
change the 0.663 L to milliliters:
Now that both volume quantities have the same units, we can substitute into Boyle’s law:
The mL units cancel, and we multiply and divide the numbers to get
P[2] = 96.7 torr
The volume is increasing, and the pressure is decreasing, which is as expected for Boyle’s law.
Test Yourself
If V[1] = 456 mL, P[1] = 308 torr, and P[2] = 1.55 atm, what is V[2]?
119 mL
There are other measurable characteristics of a gas. One of them is temperature (T). One could vary the temperature of a gas sample and note what effect it has on the other properties of the gas.
Early scientists did just this, discovering that if the amount of a gas and its pressure are kept constant, then changing the temperature changes the volume (V). As temperature increases, volume
increases; as temperature decreases, volume decreases. We say that these two characteristics are directly related.
A mathematical relationship between V and T should be possible except for one question: What temperature scale should we use? We know from Chapter 2 “Measurements” that science uses several possible
temperature scales. Experiments show that the volume of a gas is related to its absolute temperature in Kelvin, not its temperature in degrees Celsius. If the temperature of a gas is expressed in
kelvins, then experiments show that the ratio of volume to temperature is a constant:
We can modify this equation as we modified Boyle’s law: the initial conditions V[1] and T[1] have a certain value, and the value must be the same when the conditions of the gas are changed to some
new conditions V[2] and T[2], as long as pressure and the amount of the gas remain constant. Thus, we have another gas law:
This gas law is commonly referred to as Charles’s law, after the French scientist Jacques Charles, who performed experiments on gases in the 1780s. The tactics for using this mathematical formula are
similar to those for Boyle’s law. To determine an unknown quantity, use algebra to isolate the unknown variable by itself and in the numerator; the units of similar variables must be the same. But we
add one more tactic: all temperatures must be expressed in the absolute temperature scale (Kelvin). As a reminder, we review the conversion between the absolute temperature scale and the Celsius
temperature scale:
K = °C + 273
where K represents the temperature in kelvins, and °C represents the temperature in degrees Celsius.
Figure 6.2 “Charles’s Law” shows two representations of how Charles’s law works.
Figure 6.2 Charles’s Law
A piston having a certain volume and temperature (left piston) will have twice the volume when its temperature is twice as much (right piston). One can also plot V versus T for a given amount of gas
at a certain pressure; such a plot will look like the graph on the right.
Example 5
A sample of gas has an initial volume of 34.8 mL and an initial temperature of 315 K. What is the new volume if the temperature is increased to 559 K? Assume constant pressure and amount for the gas.
First, we assign the given values to their variables. The initial volume is V[1], so V[1] = 34.8 mL, and the initial temperature is T[1], so T[1] = 315 K. The temperature is increased to 559 K, so
the final temperature T[2] = 559 K. We note that the temperatures are already given in kelvins, so we do not need to convert the temperatures. Substituting into the expression for Charles’s law
We solve for V[2] by algebraically isolating the V[2] variable on one side of the equation. We do this by multiplying both sides of the equation by 559 K (number and unit). When we do this, the
temperature unit cancels on the left side, while the entire 559 K cancels on the right side:
The expression simplifies to
By multiplying and dividing the numbers, we see that the only remaining unit is mL, so our final answer is
V[2] = 61.8 mL
Does this answer make sense? We know that as temperature increases, volume increases. Here, the temperature is increasing from 315 K to 559 K, so the volume should also increase, which it does.
Test Yourself
If V[1] = 3.77 L and T[1] = 255 K, what is V[2] if T[2] = 123 K?
1.82 L
It is more mathematically complicated if a final temperature must be calculated because the T variable is in the denominator of Charles’s law. There are several mathematical ways to work this, but
perhaps the simplest way is to take the reciprocal of Charles’s law. That is, rather than write it as
write the equation as
It is still an equality and a correct form of Charles’s law, but now the temperature variable is in the numerator, and the algebra required to predict a final temperature is simpler.
Example 6
A sample of a gas has an initial volume of 34.8 L and an initial temperature of −67°C. What must be the temperature of the gas for its volume to be 25.0 L?
Here, we are looking for a final temperature, so we will use the reciprocal form of Charles’s law. However, the initial temperature is given in degrees Celsius, not kelvins. We must convert the
initial temperature to kelvins:
−67°C + 273 = 206 K
In using the gas law, we must use T[1] = 206 K as the temperature. Substituting into the reciprocal form of Charles’s law, we get
Bringing the 25.0 L quantity over to the other side of the equation, we get
The L units cancel, so our final answer is
T[2] = 148 K
This is also equal to −125°C. As temperature decreases, volume decreases, which it does in this example.
Test Yourself
If V[1] = 623 mL, T[1] = 255°C, and V[2] = 277 mL, what is T[2]?
235 K, or −38°C
Key Takeaways
• The behaviour of gases can be modelled with gas laws.
• Boyle’s law relates a gas’s pressure and volume at constant temperature and amount.
• Charles’s law relates a gas’s volume and temperature at constant pressure and amount.
• In gas laws, temperatures must always be expressed in kelvins.
1. Define gas law. What restrictions are there on the units that can be used for the physical properties?
2. What unit of temperature must be used for gas laws?
3. Boyle’s law relates the _____________ of a gas inversely with the ___________ of that gas.
4. Charles’s law relates the _____________ of a gas directly with the ___________ of that gas.
5. What properties must be held constant when applying Boyle’s law?
6. What properties must be held constant when applying Charles’s law?
7. A gas has an initial pressure of 1.445 atm and an initial volume of 1.009 L. What is its new pressure if volume is changed to 0.556 L? Assume temperature and amount are held constant.
8. A gas has an initial pressure of 633 torr and an initial volume of 87.3 mL. What is its new pressure if volume is changed to 45.0 mL? Assume temperature and amount are held constant.
9. A gas has an initial pressure of 4.33 atm and an initial volume of 5.88 L. What is its new volume if pressure is changed to 0.506 atm? Assume temperature and amount are held constant.
10. A gas has an initial pressure of 87.0 torr and an initial volume of 28.5 mL. What is its new volume if pressure is changed to 206 torr? Assume temperature and amount are held constant.
11. A gas has an initial volume of 638 mL and an initial pressure of 779 torr. What is its final volume in liters if its pressure is changed to 0.335 atm? Assume temperature and amount are held
12. A gas has an initial volume of 0.966 L and an initial pressure of 3.07 atm. What is its final pressure in torr if its volume is changed to 3,450 mL? Assume temperature and amount are held
13. A gas has an initial volume of 67.5 mL and an initial temperature of 315 K. What is its new volume if temperature is changed to 244 K? Assume pressure and amount are held constant.
14. A gas has an initial volume of 2.033 L and an initial temperature of 89.3 K. What is its volume if temperature is changed to 184 K? Assume pressure and amount are held constant.
15. A gas has an initial volume of 655 mL and an initial temperature of 295 K. What is its new temperature if volume is changed to 577 mL? Assume pressure and amount are held constant.
16. A gas has an initial volume of 14.98 L and an initial temperature of 238 K. What is its new temperature if volume is changed to 12.33 L? Assume pressure and amount are held constant.
17. A gas has an initial volume of 685 mL and an initial temperature of 29°C. What is its new temperature if volume is changed to 1.006 L? Assume pressure and amount are held constant.
18. A gas has an initial volume of 3.08 L and an initial temperature of −73°C. What is its new volume if its temperature is changed to 104°C? Assume pressure and amount are held constant.
A gas law is a simple mathematical formula that allows one to predict the physical properties of a gas. The units of changing properties (volume, pressure, etc.) must be the same.
pressure; volume
amount of gas and temperature
2.62 atm
50.3 L
1.95 L
52.3 mL
260 K
444 K, or 171°C | {"url":"https://courses.lumenlearning.com/suny-introductory-chemistry/chapter/gas-laws/","timestamp":"2024-11-11T14:07:49Z","content_type":"text/html","content_length":"77054","record_id":"<urn:uuid:d0e43fe9-2a73-4ae2-ad7f-9d957ddb3157>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00641.warc.gz"} |
Vakhtang Putkaradze (University of Alberta), Applied Mathematics Colloquium - Department of Mathematics
Vakhtang Putkaradze (University of Alberta), Applied Mathematics Colloquium
November 11, 2016 @ 4:00 pm - 5:00 pm
Tea at 3:30 in 330 Phillips Hall
Title: Exact geometric approach to the discretization of fluid-structure interactions and the dynamics of tubes conveying fluid
Abstract: Variational integrators for numerical simulations of Lagrangian systems have the advantage of conserving the momenta up to machine precision, independent of the time step. While the theory
of variational integrators for mechanical systems is well developed, there are obstacles in direct applications of these integrators to systems involving fluid-structure interactions. In this talk,
we derive a variational integrator for a particular type of fluid-structure interactions, namely, simulating the dynamics of a bendable tube conveying ideal fluid that can change its cross-section
(collapsible tube). We start by deriving a fully three-dimensional, geometrically exact theory for flexible tubes conveying fluid. Our approach is based on the symmetry-reduced, exact geometric
description for elastic rods, coupled with the fluid transport and subject to the volume conservation constraint for the fluid. Using these methods, we obtain the fully three dimensional equations of
motion. We then proceed to the linear stability analysis and show that our theory introduces important corrections to previously derived results, both in the consistency at all wavelength and in the
effects arising from the dynamical change of the cross-section. Based on this theory, we derive a variational discretization of the dynamics based on the appropriate discretization of the fluid’s
back-to-labels map, coupled with a variational discretization of elastic part of the Lagrangian. Time permitting, we shall also discuss some fully nonlinear solutions and the results of experiments.
Joint work with F. Gay-Balmaz (ENS and LMD, Paris). The work was partially supported by NSERC and the University of Alberta Centennial Fund. | {"url":"https://math.unc.edu/event/vakhtang-putkaradze-university-of-alberta-applied-mathematics-colloquium/","timestamp":"2024-11-02T11:46:42Z","content_type":"text/html","content_length":"113737","record_id":"<urn:uuid:374de3ae-db27-48cb-b335-ec7ae0535321>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00692.warc.gz"} |
Optimize a metric function in binary classification after
maximize_boot_metric {cutpointr} R Documentation
Optimize a metric function in binary classification after bootstrapping
Given a function for computing a metric in metric_func, these functions bootstrap the data boot_cut times and maximize or minimize the metric by selecting an optimal cutpoint. The returned optimal
cutpoint is the result of applying summary_func, e.g. the mean, to all optimal cutpoints that were determined in the bootstrap samples. The metric function should accept the following inputs:
• tp: vector of number of true positives
• fp: vector of number of false positives
• tn: vector of number of true negatives
• fn: vector of number of false negatives
metric_func = youden,
pos_class = NULL,
neg_class = NULL,
summary_func = mean,
boot_cut = 50,
inf_rm = TRUE,
metric_func = youden,
pos_class = NULL,
neg_class = NULL,
summary_func = mean,
boot_cut = 50,
inf_rm = TRUE,
data A data frame or tibble in which the columns that are given in x and class can be found.
x (character) The variable name to be used for classification, e.g. predictions or test values.
class (character) The variable name indicating class membership.
metric_func (function) A function that computes a single number metric to be maximized. See description.
pos_class The value of class that indicates the positive class.
neg_class The value of class that indicates the negative class.
direction (character) Use ">=" or "<=" to select whether an x value >= or <= the cutoff predicts the positive class.
summary_func (function) After obtaining the bootstrapped optimal cutpoints this function, e.g. mean or median, is applied to arrive at a single cutpoint.
boot_cut (numeric) Number of bootstrap repetitions over which the mean optimal cutpoint is calculated.
boot_stratify (logical) If the bootstrap is stratified, bootstrap samples are drawn in both classes and then combined, keeping the number of positives and negatives constant in every resample.
inf_rm (logical) whether to remove infinite cutpoints before calculating the summary.
All cutpoints will be passed to summary_func that lead to a metric value in the interval [m_max - tol_metric, m_max + tol_metric] where m_max is the maximum achievable metric value.
tol_metric This can be used to return multiple decent cutpoints and to avoid floating-point problems.
(logical) If TRUE (default FALSE) the returned optimal cutpoint will be the mean of the optimal cutpoint and the next highest observation (for direction = ">") or the next lowest
use_midpoints observation (for direction = "<") which avoids biasing the optimal cutpoint.
... To capture further arguments that are always passed to the method function by cutpointr. The cutpointr function passes data, x, class, metric_func, direction, pos_class and neg_class to
the method function.
The above inputs are arrived at by using all unique values in x, Inf, and -Inf as possible cutpoints for classifying the variable in class. The reported metric represents the usual in-sample
performance of the determined cutpoint.
A tibble with the column optimal_cutpoint
See Also
Other method functions: maximize_gam_metric(), maximize_loess_metric(), maximize_metric(), maximize_spline_metric(), oc_manual(), oc_mean(), oc_median(), oc_youden_kernel(), oc_youden_normal()
cutpointr(suicide, dsi, suicide, method = maximize_boot_metric,
metric = accuracy, boot_cut = 30)
cutpointr(suicide, dsi, suicide, method = minimize_boot_metric,
metric = abs_d_sens_spec, boot_cut = 30)
version 1.1.2 | {"url":"https://search.r-project.org/CRAN/refmans/cutpointr/html/maximize_boot_metric.html","timestamp":"2024-11-12T16:32:50Z","content_type":"text/html","content_length":"7322","record_id":"<urn:uuid:4e2d69db-d06b-42cf-9699-13b54d474233>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00603.warc.gz"} |
America decides..
As I have said , many a time.. Politics do not interest me and I have little understanding when it comes to them..
However I am really looking forward to the results when USA go to the polls to elect a (New) president..
I really believe that Trump will get in , by a whisker.. | {"url":"https://letschat.club/index.php?PHPSESSID=p5sinqfu188vkmpmnrjhe230po&topic=5131.0","timestamp":"2024-11-04T11:57:11Z","content_type":"text/html","content_length":"69406","record_id":"<urn:uuid:4b8fa555-f424-40d7-9892-8291ed456602>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00590.warc.gz"} |
114. Derivatives of complex functions - Avidemia
So far we have supposed that \(y = \phi(x)\) is a purely real function of \(x\). If \(y\) is a complex function \(\phi(x) + i\psi(x)\), then we define the derivative of \(y\) as being \(\phi'(x) + i\
psi'(x)\). The reader will have no difficulty in seeing that Theorems (1)–(5) above retain their validity when \(\phi(x)\) is complex. Theorems (6) and (7) have also analogues for complex functions,
but these depend upon the general notion of a ‘function of a complex variable’, a notion which we have encountered at present only in a few particular cases.
$\leftarrow$ 113. General rules for differentiation Main Page 115. The notation of the differential calculus $\rightarrow$ | {"url":"https://avidemia.com/pure-mathematics/derivatives-of-complex-functions/","timestamp":"2024-11-10T04:20:59Z","content_type":"text/html","content_length":"75458","record_id":"<urn:uuid:76dc4706-1709-4a84-a5b9-1c98cb21902f>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00020.warc.gz"} |
toilet paper
Jan 30, 2016 · The easiest way is to treat the problem by thinking about the length as the sum of the whole circumferences starting by radius r and ending with radius R.
What does 12/24 mean in toilet paper?
Does toilet paper have calcium?
What is the brown stuff on my toilet paper?
How is toilet paper counted?
Dec 20, 2021 · Finding the Equation for the Length of the Toilet Paper. This derivation will begin with the combined length (L) of the toilet paper sheets.
Apr 1, 2024 · Toilet paper math requires the use of addition, subtraction, multiplication, division, decimals, place value, money, fractions and ratios.
Jan 14, 2019 · Supposedly each roll is equal to 6 rolls of regular Charmin. On the outside of the pack it states there are 426 2-ply sheets per roll. Divided ...
Oct 14, 2021 · Multiply sheets per roll by number of rolls in a package, and divide that by the price of a package to get sheets per dollar. Higher is better.
Oct 30, 2024 · Using the side of the roll, we can use math to determine how much toilet paper is left! If we measure the diameter of the roll, we ...
Jun 14, 2012 · Method 1 – One way of calculating the length of toilet paper is to calculate the volume of the toilet paper when it is has been rolled out and ...
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PhET Teacher Activities: Hooke's Law
Detail Page
published by the PhET
written by Jessica Mullins
This two-hour activity for high school physics was created specifically to accompany the PhET simulation
Masses & Springs
. In the first lesson, students will use the simulation to explore how displacement of a spring is mathematically related to the load applied to it. In the next day's exploration,
learners analyze the energy of a mass oscillating on a spring by observing distribution and transfer of kinetic, elastic potential, and gravitational potential energy. Materials include
learning goals, explicit directions for use of the simulation, homework problems, and answer key.
The spring motion simulation (which is required to complete this activity) is available from PhET at:
Masses & Springs
This lesson is part of the PhET (Physics Education Technology Project), a large collection of free interactive science simulations.
Please note that this resource requires Flash.
Subjects Levels Resource Types
Classical Mechanics
- Applications of Newton's Laws
- Work and Energy - Instructional Material
= Conservation of Energy = Activity
Oscillations & Waves - High School = Interactive Simulation
- Oscillations = Problem/Problem Set
= Damped Oscillators - Assessment Material
= Hooke's Law
= Springs and Oscillators
Appropriate Courses Categories Ratings
- Conceptual Physics - Activity
- Algebra-based Physics - Assessment
- AP Physics - New teachers
Intended Users:
Access Rights:
Free access
© 2007 PHET; University of Colorado at Boulder
Additional information is available.
Hooke's Law, damping, elastic potential energy, friction, harmonic oscillation, simple harmonic motion, spring constant, spring energy, springs, springs
Record Cloner:
Metadata instance created September 25, 2012 by Caroline Hall
Record Updated:
August 14, 2016 by Lyle Barbato
Last Update
when Cataloged:
April 14, 2008
AAAS Benchmark Alignments (2008 Version)
4. The Physical Setting
4E. Energy Transformations
• 6-8: 4E/M2. Energy can be transferred from one system to another (or from a system to its environment) in different ways: 1) thermally, when a warmer object is in contact with a
cooler one; 2) mechanically, when two objects push or pull on each other over a distance; 3) electrically, when an electrical source such as a battery or generator is connected in a
complete circuit to an electrical device; or 4) by electromagnetic waves.
• 6-8: 4E/M4. Energy appears in different forms and can be transformed within a system. Motion energy is associated with the speed of an object. Thermal energy is associated with the
temperature of an object. Gravitational energy is associated with the height of an object above a reference point. Elastic energy is associated with the stretching or compressing of
an elastic object. Chemical energy is associated with the composition of a substance. Electrical energy is associated with an electric current in a circuit. Light energy is associated
with the frequency of electromagnetic waves.
• 9-12: 4E/H9. Many forms of energy can be considered to be either kinetic energy, which is the energy of motion, or potential energy, which depends on the separation between mutually
attracting or repelling objects.
4F. Motion
• 9-12: 4F/H1. The change in motion (direction or speed) of an object is proportional to the applied force and inversely proportional to the mass.
• 9-12: 4F/H4. Whenever one thing exerts a force on another, an equal amount of force is exerted back on it.
11. Common Themes
11B. Models
• 6-8: 11B/M4. Simulations are often useful in modeling events and processes.
• 9-12: 11B/H1a. A mathematical model uses rules and relationships to describe and predict objects and events in the real world.
12. Habits of Mind
12E. Critical-Response Skills
• 6-8: 12E/M5b. Notice and criticize the reasoning in arguments in which the claims are not consistent with the evidence given.
Common Core State Standards for Mathematics Alignments
Ratios and Proportional Relationships (6-7)
Analyze proportional relationships and use them to solve real-world and mathematical problems. (7)
• 7.RP.2.b Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
Functions (8)
Use functions to model relationships between quantities. (8)
• 8.F.5 Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Materials
Sketch a graph that exhibits the qualitative features of a function that has been described verbally.
High School — Functions (9-12) Materials
Interpreting Functions (9-12)
• F-IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features
given a verbal description of the relationship.^?
• F-IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
• F-IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
Linear, Quadratic, and Exponential Models^? (9-12)
• F-LE.5 Interpret the parameters in a linear or exponential function in terms of a context.
NSES Content Standards
Physical Science
• 5-8: Motion & Forces
• 9-12: Motions & Forces
• 9-12: Conservation of Energy & Increase in Disorder
This resource is part of a Physics Front Topical Unit.
Periodic and Simple Harmonic Motion
Unit Title:
Conservation of Energy and Forces on a Spring
This two-hour activity was created to accompany the PhET simulation Masses & Springs. In the first lesson, students use the simulation to explore how displacement of a spring is
mathematically related to the load applied to it. On Day 2, learners analyze the energy interactions by observing distribution and transfer of kinetic, elastic potential, and
gravitational potential energy. Includes explicit directions for use of the simulation, homework problems, and answer key. Note: Only registered users have access to the PhET
teacher-contributed materials, but registration is free and easy.
Link to Unit:
ComPADRE is beta testing Citation Styles!
<a href="https://www.compadre.org/precollege/items/detail.cfm?ID=12426">Mullins, Jessica. PhET Teacher Activities: Hooke's Law. Boulder: PhET, April 14, 2008.</a>
J. Mullins, (PhET, Boulder, 2007), WWW Document, (https://phet.colorado.edu/en/contributions/view/2939).
J. Mullins, PhET Teacher Activities: Hooke's Law (PhET, Boulder, 2007), <https://phet.colorado.edu/en/contributions/view/2939>.
Mullins, J. (2008, April 14). PhET Teacher Activities: Hooke's Law. Retrieved November 9, 2024, from PhET: https://phet.colorado.edu/en/contributions/view/2939
Mullins, Jessica. PhET Teacher Activities: Hooke's Law. Boulder: PhET, April 14, 2008. https://phet.colorado.edu/en/contributions/view/2939 (accessed 9 November 2024).
Mullins, Jessica. PhET Teacher Activities: Hooke's Law. Boulder: PhET, 2007. 14 Apr. 2008. 9 Nov. 2024 <https://phet.colorado.edu/en/contributions/view/2939>.
@misc{ Author = "Jessica Mullins", Title = {PhET Teacher Activities: Hooke's Law}, Publisher = {PhET}, Volume = {2024}, Number = {9 November 2024}, Month = {April 14, 2008}, Year = {2007}
%A Jessica Mullins %T PhET Teacher Activities: Hooke's Law %D April 14, 2008 %I PhET %C Boulder %U https://phet.colorado.edu/en/contributions/view/2939 %O application/pdf
%0 Electronic Source %A Mullins, Jessica %D April 14, 2008 %T PhET Teacher Activities: Hooke's Law %I PhET %V 2024 %N 9 November 2024 %8 April 14, 2008 %9 application/pdf %U https://
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PhET Teacher Activities: Hooke's Law:
Requires PhET Simulation: Masses & Springs
A link to the Flash simulation Masses & Springs, which must be running to complete this activity.
relation by Caroline Hall
Same topic as Illinois PER Interactive Examples: Mass on a Vertical Spring
An interactive tutorial that provides comprehensive scaffolding to solve a problem involving a mass hanging from a vertical spring. Promotes understanding of when to use Conservation of
Mechanical Energy method.
relation by Caroline Hall
See details...
Know of another related resource? Login to relate this resource to it. | {"url":"https://www.compadre.org/precollege/items/detail.cfm?ID=12426","timestamp":"2024-11-09T06:46:00Z","content_type":"application/xhtml+xml","content_length":"53750","record_id":"<urn:uuid:67ea2005-f6fc-4db8-9478-7d14669d8e44>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00730.warc.gz"} |
Week 1
Monday: Course introduction [slides]
Lab sections: Orientation to Jupyter notebooks [html]
Wednesday: Data science lifecycle [slides]
Week 2
• HW1, BRFSS case study, due Monday, April 24 [html]
Monday: Tidy data [slides]
Lab sections: Pandas [html]
Wednesday: Dataframe transformations [slides]
Week 3
• Mini project 1, due Monday, May 1 [html]
Monday: Sampling, bias, and missingness [slides]
Lab sections: Exploring sampling bias through simulation [html]
Wednesday: Voter fraud case study [slides] [activity html]
Week 4
• HW2, SEDA case study, due Monday, May 8 [html]
Monday: Statistical graphics [slides]
Lab sections: Data visualization [html]
Wednesday: Principles of figure design [slides]
Week 5
Monday: Exploratory analysis and density estimation [slides]
Lab sections: Smoothing [html]
Wednesday: Multivariate KDE, mixture models, and scatterplot smoothing [slides] [activity html]
Week 6
• HW3, Diatom paleoclimatology case study, due Monday, May 22 [html]
Monday: Covariance, correlation, and spectral decomposition [slides]
NO lab sections this week
Wednesday: Principal components [slides]
Week 7
• Mini project 2, due Tuesday, May 30 [html]
Monday: Modeling concepts; least squares [slides]
Lab sections: Principal components [html]
Wednesday: The simple linear regression model [slides]
Week 8
• HW4, Discrimination in disability benefit allocation, due Wednesday, June 7 [html]
Monday: Prediction [slides]
Lab sections Fitting regression models [html]
Wednesday: Multiple regression [slides]
Week 9
• Course project due Friday, June 16 [html]
No class or lab sections Monday
Wednesday: Classification [slides]
Week 10
No readings or new assignments
No class Monday
Lab sections: Logistic regression (submission is optional) [html]
Wendesday: Clustering [slides] | {"url":"https://pstat100.tdruiz.com/content","timestamp":"2024-11-06T01:17:06Z","content_type":"application/xhtml+xml","content_length":"24592","record_id":"<urn:uuid:6c9d257e-9bc7-4d0e-bf60-f4f5ac691989>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00430.warc.gz"} |
portfolio Model - Disposition Assumptions
Disposition Assumptions Section
Welcome to this video. In this video, I will cover the disposition section in the ARGUS Portfolio Model. In this section, I can make different exit assumptions for each property to account for
various disposition possibilities.
Let’s start with 49 Road. 49 road is relatively small among the five properties, so I plan to sell it in 13 months with an 8% cap rate and 1.5% disposition cost. This early disposition should boost
up the IRR of the deal. I can see the sale price is $3.8 million, and the Net Operating Income is $300,000. The disposition cost is 1.5% of the sale price. I can see a couple more metrics. The sale
price per square foot is $95. 49 Road is 40,000 square feet, accounting for 9% of the portfolio square foot-wise. I can see from the percentage square foot. 165 roads and EFG are the main properties
in this portfolio accounting for 73% of the portfolio in terms of square feet. Therefore, they will be sold at last at the end of the 10-year holding period. I will assume that 165 Road will be sold
at a 7.5% cap rate and 1% disposition cost. EFG will be sold at a 5.25% cap rate and 1% disposition cost. Going back to the Basic Model Information section, I can see the holding period is 10 years
Let’s continue with the disposition assumptions, I assume that Rue 104 will be sold 5 years after closing with a 6.25% cap rate and 1% disposition cost. At last, Rue 124 is assumed to be sold in
month 25 with a 6% cap rate, and 1.5% disposition cost to boost up the returns. I am done with disposition assumptions.
Let’s check out if the cash flow reflects the assumptions I just input. I can see a drop in gross potential revenue in year 2 because 49 Road is sold, and its cash flow is no longer included in the
forecast. Scrolling down. I can see that the sale proceeds are used to partially pay off the senior debt and mezzanine. The amount of debt gets paid off depending on the 49 road’s NOI as a percentage
of the Portfolio’s NOI. Going to the Combined Monthly Cash Flow Summary tab, scrolling to month 13 when 49 Road is sold, I can see that 49 road accounts for 6% of the portfolio’s NOI. Therefore, 6%
of the senior debt and mezzanine’s outstanding balance get paid off.
Going back to the Annual Cash Flow Summary tab, I can see the gross potential revenue and expenses continue to decrease in year 3 because Rue 124 is sold at the beginning of year 3. Just like the
disposition of 49 Road, the senior debt and mezzanine are partially paid off. I can also see a similar situation in year 5 when Rue 104 is sold. The only difference is that the refinance debt is paid
off partially because the refinance has replaced the senior debt and mezzanine in year 5. The disposition assumptions have been accurately reflected in the cash flow. I am done with the disposition
assumptions section in the ARGUS Portfolio Model.
Thanks for watching this video. I will see you at the next one. | {"url":"https://www.financialexcelmodeling.com/portfolio_model_6","timestamp":"2024-11-01T23:23:24Z","content_type":"text/html","content_length":"16119","record_id":"<urn:uuid:6ebd2890-0982-42fd-a5ce-0a021b6d6e75>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00583.warc.gz"} |
CPM Homework Help
The class advisor was helping students plan an end-of-the-year trip. The students were surveyed regarding their choices. The results are in the circle graph below.
a. What percentage of the students chose the water park?
Find the unknown percentage by subtracting the known percentages from $100\%$.
b. Which two results are very close?
Which two trips have the highest percentages?
c. Write a recommendation to the class advisor regarding what the next step would be?
What should the class advisor do since there are two trips in high demand? | {"url":"https://homework.cpm.org/category/CON_FOUND/textbook/mc2/chapter/6/lesson/6.2.2/problem/6-92","timestamp":"2024-11-04T01:55:13Z","content_type":"text/html","content_length":"37198","record_id":"<urn:uuid:e349d773-84f1-4839-ae25-c13b9c50a9db>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00889.warc.gz"} |
Laser-beam propagation in a long solenoid
An analysis of the propagation of a laser beam in a cylindrical magnetically confined plasma with parabolic density profile is presented. The normal modes which are self-trapped are given. It is
found that the largest mode that can be trapped by the plasma is given by (1/2)(R-squared/w-squared-1) where R is the radius of the plasma column and w is the fundamental mode width. It is found that
all the trapped modes in a finite plasma can easily propagate distances of the order of one kilometer. An exact solution for the amplitude of the electric field for an incident Gaussian beam has been
obtained. The solution exhibits alternate focusing and defocusing of the beam. The effect of this on the plasma heating is discussed.
Nuclear Fusion
Pub Date:
June 1975
□ Laser Beams;
□ Laser Heating;
□ Light Transmission;
□ Magnetic Control;
□ Plasma Heating;
□ Propagation Modes;
□ Solenoids;
□ Coherent Light;
□ Defocusing;
□ Focusing;
□ Plasma Control;
□ Plasma Cylinders;
□ Plasma-Electromagnetic Interaction;
□ Plasmaguides;
□ Lasers and Masers | {"url":"https://ui.adsabs.harvard.edu/abs/1975NucFu..15..371M/abstract","timestamp":"2024-11-09T17:41:39Z","content_type":"text/html","content_length":"36071","record_id":"<urn:uuid:7127c821-908c-493f-911b-7ee80eed8087>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00068.warc.gz"} |
General Topology—Part 7 (Locally Compact Hausdroff Space)
General Topology — Part 7 (Locally Compact Hausdroff Space)
Photo by Dan Dennis on Unsplash
In the Locally Compact Hausdroff Space, when a compact set is contained by an open set, a “bump” function over this open set will exist. To prove this theorem, we need the following definition and
lemma first :
X denotes the topological space
f:X→R denotes the real-valued continuous function on X
C(X) denotes space of f on X
supp denotes the Support of f where f ∈ C(x):
supp(f) = Cl({ x ∈ X| f(x)≠0 })
The Support of a function is defined as the closure of the set of points where the function is non-zero. The closure is required here since this can ensure that isolated points where the function is
non-zero are also included. Remember that the closure of a set includes all the limit point and the given set.
K(X) denotes the set of f ∈ C(x) having Compact Support:
K(X) = { f ∈ C(x) | supp(f) is compact }
I denotes the indicator function:
Given a subset A ⊆ X, IA: X → {0, 1}
IA(x) = 1 if x ∈ A
IA(x) = 0 if x ∉ A
Lemma A ( from Part 4 )
Let X be a normal space. For any closed set A ⊆ X and an open set C ∈ τ where A ⊆ C, there is an open set o∈τ s.t. A ⊆ o ⊆ Cl(o) ⊆ C
LCH1. Let X be a Locally Compact Hausdorff space. For any compact set k⊂X and let U be open set in X where k ⊆ U, there exists f ∈ K(X), satisfies Ik≤ f ≤ IU and k ⊆ supp(f) ⊆ U
U is open in X
⇒ U is open in X*
[ X* is the one-point compactification extension of X, so τ ⊂ τ* and X* is Compact Hausdorff ]
k is compact in X
⇒ k is compact in X* [ By CP4 where X is subspace of X* ]
X* is Hausdorff and k is compact in X* [ Since X* is Compact Hausdorff ]
⇒ k is closed in X* [ By HD1 ]
X* is Compact Hausdorff ⇒ X* is Normal [ By CH1 ]
⇒ ∃ o∈τ* s.t. k ⊆ o ⊆ ClX*(o) ⊆ U [ By Lemma A ]
k is closed in X*, X*\o is closed in X* and X* is Normal
⇒ ∃ g: X* → [0, 1] is continuous where g(k)=1, g(X*\o)=0 [ By Urysohn’s lemma in Part 4 ]
Let f: X → [0, 1], f = g|X is continuous [ Since Restriction function is continuous ]
f(k)=g(k)=1, f(X\o)=g(X\o)=0 [ Since X\o ⊂ X*\o ]
If x ∈ k ⇒ f(x)=1, Ik(x)=1, IU(x)=1 ⇒ Ik = f = IU
If x ∈ X\o ⇒ f(x)=0, Ik(x)=0, IU(x)=0 or 1 ⇒ Ik ≤ f ≤ IU
If x ∈ o\k ⇒ 0<f(x)<1, Ik(x)=0, IU(x)=1 ⇒ Ik < f < IU
⇒ Ik ≤ f ≤ IU
supp(f) = ClX({x ∈ X| f(x)≠0}) = ClX({x ∈ X| x∈ k ∪ (o\k)}) = ClX(o) [ Since k ⊆ o]
ClX*(o) is closed in X* [ By CL2, closure is smallest closed set containing o ]
⇒ ClX*(o) is compact in X* [ By CP1 where X* is compact ]
⇒ ClX*(o) is compact in X [ By CP4, compactness is preserved ]
ClX*(o) = ClX(o) [ By CL12, ClX(o) = X∩CIX*(o) = CIX*(o) since CIX*(o)⊆U⊆X ]
⇒ ClX(o) is compact in X
⇒ k ⊆ supp(f) = ClX(o) = ClX*(o) ⊆ U
⇒ supp(f) is compact ⇒ f has compact support
If the support of a continuous function is compact and can be contained by a finite union of open sets, then we can decompose this function as the sum of bump functions with compact support which is
contained by one of the open sets.
LCH2. Let X be a Locally Compact Hausdorff space. Let f ∈ K(X) and let U1,U2, ... Un be open subsets of X s.t. supp(f) ⊆ ∪{ Ui | 1≤i≤n }. Then, there are continuous f1,..,fn ∈ K(X) s.t. f = f1+..+fn
and for each i, supp(fi) ⊆ Ui. Furthermore, if f is non-negative, then each fi can be chosen to be non-negative as well.
Let us prove the LCH2 statement is true by induction.
When n = 1, the statement is true.
When n = 2:
supp(f) ⊆ U1∪U2
⇒ There exists compact sets K1, K2 s.t. supp(f)=K1∪K2 where K1⊆U1, K2⊆U2 [ By HD5 where X is Hausdorff, f ∈ K(X) and supp(f) is compact ]
⇒ There exists g1,g2 ∈ K(X), satisfies IK1≤ g1 ≤ IU1, IK2≤ g2 ≤ IU2 and K1⊆supp(g1)⊆ U1, K2⊆supp(g2)⊆ U2 [ By LCH1 ]
Let f1:X→R and f2:X->R be the real-valued functions on X: f1(x)=g1(x)/(g1(x)+g2(x))*f(x) if x ∈ supp(g1)
f1(x)=0 otherwise
f2(x)=g2(x)/(g1(x)+g2(x))*f(x) if x ∈ supp(g2)
f2(x)=0 otherwise [ The set {f1,f2} is also called Partition of Unity ]
If x ∈ supp(g1), g1/g1+g2 is the elementary combination of continuous function
If x ∉ supp(g1), f1=0 is a constant function which is continuous [ see part 2 ]
f1 is continuous in both cases
f2 is continuous [ By similar argument ] and f1,f2 are non-negative when f is non-negative.
supp(f1) = Cl({ x ∈ X| f1(x)≠0 }) = Cl({x ∈ X| f≠0} ∩ {x ∈ X| g1≠0})
⇒ supp(f1) ⊆ Cl({x ∈ X| f≠0}) ∩ Cl({x ∈ X| g1≠0}) = supp(f)∩supp(g1) [ By CL10, note that g1,g2 ≥0, g1+g2 ≠0 when g1≠0 ]
⇒ supp(f1) ⊆ supp(g1) ⊆ U1 and supp(f1) is compact [ Since supp(f1) is closed and supp(g1) is compact and by CP1 ]
⇒ supp(f2) ⊆ supp(g2) ⊆ U2 and supp(f2) is compact [ By similar argument ]
⇒ f1, f2 ∈ K(X)
If x ∉ supp(f)
⇒ f = 0
⇒ if x∈supp(g1), f1=g1/(g1+g2)*0, if x∉supp(g1), f1=0 ⇒ f1=0 in both cases
f2 = 0 [ By similar argument ]
⇒ f1+f2 = f = 0
If x ∈ supp(f)
If x ∈ supp(g1)∩supp(g2)
⇒ f1+f2 = [g1/(g1+g2) + g2/(g1+g2)]*f= f
If x ∈ supp(g1) ∩ X\supp(g2)
⇒ f1+f2 = g1/(g1+0)*f + 0 = f
If x ∈ X\supp(g1) ∩ supp(g2)
⇒ f1+f2 = 0 + g2/(0+g2)*f = f
If x ∈ X\supp(g1) ∩ X\supp(g2)
⇒ f1+f2 = 0 + 0 = 0, x ∈ X\K1∩X\K2=X\(K1∪K2)=X\supp(f)
[ since K1⊆supp(g1) and K2⊆supp(g2) ]
⇒ f = 0 , f1+f2=0 ⇒ f1+f2=f
⇒ f = f1+f2 in all cases
Hence, the statement is true for n=2
Assume the statement is true for n=k.
When n = k+1, there are U1,U2, … Uk+1 open subsets of X s.t. supp(f) ⊆ ∪{Ui | 1≤i≤k+1 }.
Let U=∪{ Ui | 1≤i≤k }
⇒ U, Uk+1 are open sets and supp(f) ⊆ U∪Uk+1 [ By T2, Part 1 ]
⇒ There exist continuous h,fk+1 ∈ K(X) s.t. f = h + fk+1, supp(h) ⊆ U and supp(fk+1) ⊆ Uk+1 and h,fk+1 are non-negative when f is non-negative [Since statement is true when n=2 ]
h ∈ K(X) and U1,U2, … Uk are open subsets of X and supp(h) ⊆∪{ Ui |1≤i≤k}
⇒ There exist continuous f1,..,fk ∈ K(X) s.t. h = f1+..+fk and for each i where 1≤i≤k, supp(fi) ⊆ Ui and f1,..fk are non-negative as h is non-negative [ Since statement is true when n=k ]
⇒ There exist continuous f1,..,fk,fk+1 ∈ K(X) s.t. f = f1+..+fk+fk+1 and for each i where 1≤i≤k+1, supp(fi) ⊆ Ui and f1,..fk,fk+1 are non-negative when f is non-negative
⇒ The statement is also true for n=k+1
⇒ Hence, by induction, the statement is true for n=1,2,..
In this post, we have found 2 nice properties of the Locally Compact Hausdroff Space. 1) When a compact set in this space is contained by an open set, a “bump” function over this open set will exist.
Outside this open set, the function will become zero. 2) If the support of a continuous function is compact and can be contained by a finite union of open sets, then we can decompose this function as
the sum of bump functions with compact support.
In the next article, I will list all the theorems we have learned so far.
Your feedback is highly appreciated and will help me to continue my articles.
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From UTXOs to zk-SNARKs: How Zcash Protects Your Privacy
•15 min read
Zcash is one of the first privacy preserving cryptocurrencies using ZK and FHE
Introduction to Zero Knowledge
It is a proof that there exists or that we know something, plus a zero knowledge aspect, that is the person verifying the proof only gains one piece of information that the proof is valid or invalid.
Actors in a Zero Knowledge Proof System
Creator, Prover and Verifier
Proving Systems
A statement is a proposition we want to prove. It depends on:
• Instance variables, which are public.
• Witness variables, which are private.
Given the instance variables, we can find a short proof that we know witness variables that make the statement true (possibly without revealing any other information).
Imagine you have a locked treasure chest (the statement) that you claim to have the key to (the witness variables). The treasure chest is placed in a public square, where everyone can see it (the
instance variables). Now, you want to prove to everyone that you have the key to open the chest, but you don’t want to reveal the key itself or any other information.
In zero-knowledge terms, instead of showing the key to everyone or revealing the combination to the lock (which would expose your private information), you perform a magic trick. You take the chest
into a private room, open it using your key, and bring it back outside with the chest still locked but the treasure inside rearranged in a specific, agreed-upon way.
Introduction to ZK-Snarks
Currently zkSNARKS are the most common proof system being used, for example they form the basis for the privacy provided in ZCash.
The S here stands for succinctness, which means that verifying time scales poly-logarithmically in n (without demanding quasi-linear proving time) and the N means non-interactive, which means that
after a pre-processing phase (which may be non-transparent), the proof system cannot allow any further interaction. Notice that according to this definition a non-succinct trusted setup phase is
allowed, and, generally speaking, the system need not be transparent, but it must be noninteractive
A zk-SNARK consists of three algorithms \(C, P, V\) defined as follows:
The Creator takes a secret parameter \(\lambda\) and a program \(C\), and generates two publicly available keys:
1. \(Proving~Key~(pk)\)
2. \(Verification~Key~(vk)\)
These keys are public parameters that only need to be generated once for a given program \(C\). They are also known as the Common Reference String.
The prover Peggy takes a proving key \(pk\), a public input \(x\) and a private witness \(w\). Peggy generates a proof \(pr = P(pk, x, w)\) that claims that Peggy knows a witness \(w\) and that the
witness satisfies the program \(C\).
The verifier Victor computes \(V(vk, x, pr)\) which returns \(True\) if the proof is correct, and \(False\) otherwise. Thus this function returns true if Peggy knows a witness \(w\) satisfying .
Toxic Waste
Note the secret parameter \(\lambda\) in the setup, this parameter sometimes makes it tricky to use zk-SNARK in real-world applications. The reason for this is that anyone who knows this parameter
can generate fake proofs. Specifically, given any program C and public input x a person who knows lambda can generate a proof \(pr2\) such that \(V(vk, x, pr2)\) evaluates to \(True\) without
knowledge of the secret \(w\).
Trusted Setups
"SNARKs require something called “the public parameters” . The SNARK public parameters are numbers with a specific cryptographic structure that are known to all of the participants in the system.
They are baked into the protocol and the software from the beginning.
The obvious way to construct SNARK public parameters is just to have someone generate a public/private keypair, similar to an ECDSA keypair, (See ZCash explanation) and then destroy the private key.
The problem is that private key. Anybody who gets a copy of it can use it to counterfeit money. (However, it cannot violate any user’s privacy — the privacy of transactions is not at risk from this.)
ZCash used a secure multiparty computation in which multiple people each generate a “shard” of the public/private keypair, then they each destroy their shard of the toxic waste private key, and then
they all bring together their shards of the public key to to form the SNARK public parameters. If that process works — i.e. if at least one of the participants successfully destroys their private key
shard — then the toxic waste byproduct never comes into existence at all.
FHE - Fully Homomorphous Encryption
[TODO - light explainer]
Homomorphic additive properties:
Given an encrypting function \(F\), we can then perform addition like this:
$$F(x) + F(y) = F(x+y)$$
This property allows us to verify if \(F(x) + F(y)\) is indeed \(F(x+y)\)without having to know the value of \(x\) and \(y\)
Zcash is an implementation of the Decentralized Anonymous Payment scheme Zerocash, with security fixes and improvements to performance and functionality. It bridges the existing transparent payment
scheme used by Bitcoin with a shielded payment scheme secured by zero-knowledge succinct non-interactive arguments of knowledge (zk-SNARKs).
Value in Zcash is either transparent or shielded. Transfers of transparent value work essentially as in Bitcoin and have the same privacy properties.
Addresses which start with "t" behave similarly to Bitcoin, exposing addresses and balances on the blockchain and we refer to these as "transparent addresses". Addresses which start with "z" or "zc"
or "zs" include the privacy enhancements provided by zk-proofs and we refer to these as "shielded addresses". It is possible to send ZEC between these two address types.
"In order to ensure the toxic waste did not come into existence, our team designed multi-party computation (MPC) protocols which allowed multiple independent parties to collaboratively construct the
parameters. These protocols had the property that, in order to compromise the final parameters, all of the participants would have had to be compromised or dishonest. Through 2018, Zcash had created
two distinct sets of public parameters. The first ceremony happened in October 2016 just before the launch of Zcash Sprout. The second was generated in2018, anticipating the Sapling network upgrade
later that year."
Imagine Alice has control of Note1, via her secret key $ska$
Alice $(sk_a) -> Note1$
She could send the Note1 to Bob by giving him the secret key sk that controls Note1, but then she would still have control of it. A better idea is Alice signs a transaction to cancel Note1 and allow
Bob to create Note2 of the same value
Alice $(sk_a)$ cancel $Note1$
Bob $(sk_b)$ -> $Note2$
Then Bob has control of the new Note.
ZCash follows the UTXO model and develops it further,
1. to the note we add a random value r as an identifier
2. rather than storing these values directly, we store a hash of those values. In order to cancel the notes we introduce the notion of a nullifier
The Commitment / Nullifier
A commitment scheme is defined by algorithms Commit and Open Given a message m and randomness r, compute as the output a value c
A commitment scheme has 2 properties:
1. Binding: Given a commitment c, it is hard to compute a different pair of message and randomness whose commitment is c. This property guarantees that there is no ambiguity in the commitment
scheme, and thus after c is published it is hard to open it to a different value.
2. Hiding: It is hard to compute any information about m given c.
In ZCash Pedersen hashes are used to create the commitments using generator points on an elliptic curve Given a value v which we want to commit to, and some random number r, commitment c:
$$c = v ∗ Gv + r ∗ Gr$$
Where Gv and Gr are generator points on an elliptic curve.
Nullifiers are used to signal that a note has been spent. Each note can deterministically produce a unique nullifier.
The nullifier is designed to uniquely identify a note when it is spent, without revealing which note it is or to whom it belongs. To achieve this, Zcash uses the spending key and a note-specific
value (typically called ρ or rho).
PRF (Pseudo-Random Function): This function is keyed by the spending key and takes the note-specific value ρ as input. The output is the nullifier, which is unique to this specific note and the
associated spending key
When spending a note,
1. The nullifier set is checked to ascertain whether that note has already been spent.
2. If no nullifier exists for that note, the note can be spent
3. Once the note has been spent its nullifier is added to the nullifier set
To each note there is cryptographically associated a note commitment . Once the transaction creating a note has been mined, the note is associated with a fixed note position in a tree of note
Computing the nullifier requires the associated private spending key. It is infeasible to correlate the note commitment or note position with the corresponding nullifier without knowledge of at least
this key.
An unspent valid note, at a given point on the block chain, is one for which the note commitment has been publically revealed on the block chain prior to that point, but the nullifier has not.
Each transaction has a list of Spend and Output Descriptions
We also create zkps to prove existence of the note in the merkle tree, and ownership of the note
Spend Descriptions spend existing notes , the spender needs to show that
1. The note exists
2. The spender owns the note
3. The note has not been spent before, by computing a nullifier unique to that note and checking this against the nullifier set.
The Spend description commits to a value commitment of the note and all necessary public parameters needed to verify the proof. The proof is constructed on private parameters to validate note
ownership and spendability. Output Descriptions create new notes
• Only the sender’s outgoing view key and recipient’s incoming view key can decrypt the details
• Only the recipient can spend the new note
• Which note was used is not revealed
• Who the sender, recipient, or the amount is not revealed
• The nullifier is unique to each note, and is revealed when spent
The nullifiers are necessary to prevent double-spending: each note on the block chain only has one valid nullifier, and so attempting to spend a note twice would reveal the nullifier twice, which
would cause the second transaction to be rejected.
Key Components and Address Derivation
• Spending Key ($sk$): This is the root private key, randomly generated.
• Full Viewing Key ($vk$): Derived from $sk$, includes components such as $ak$ (authorizing key) and $nk$ (nullifier key).
• Incoming Viewing Key ($ivk$): Derived from $vk$, used to derive the diversified public key.
• Diversified Public Key ($pk_d$):
where G is the base point on the elliptic curve.
• Diversifier (d): A small value used to derive distinct addresses from the same root key.
• Shielded Address ($z-addr$):
Encryption by Sender
Step 1: Generate Ephemeral Key Pair
• The sender generates an ephemeral private key $esk$ (ephemeral secret key).
• The ephemeral public key $epk$ is then computed as:
Step 2: Compute the Shared Secret
• The shared secret $K_{shared}$ is computed using the sender's ephemeral secret key $esk$ and the recipient's diversified public key $pk_d$:
• Since $pk_d$ is $ivk \times G$, this can also be expressed as:
Step 3: Derive the Symmetric Encryption Key
• The shared secret $K_{\text{shared}}$ is passed through a key derivation function (KDF) to produce a symmetric encryption key $k_{\text{enc}}$:
$$k_{enc} = KDF(K_{shared})$$
Step 4: Encrypt the Note
• The note $m$ (which contains the value, ρ, and other relevant data) is encrypted using the symmetric key $k_{\text{enc}}$:
where $c$ is the cyphertext
Decryption by Receiver
Step 1: Compute the Shared Secret
The recipient computes the shared secret $K_{\text{shared}}$ using their private key associated with $pk_d$ (denoted as $sk$) and the sender's ephemeral public key $epk$:
Since $epk$ is $esk \times G$, this can be expressed as:
Step 2: Derive the Symmetric Decryption Key
• The recipient derives the symmetric decryption key $k_{\text{enc}}$ using the same key derivation function:
Step 3: Decrypt the Note
• The recipient uses $k_{enc}$ to decrypt the ciphertext c:
This begs, the question if only the recipient and sender can decrypt the note, then how is this transaction verified by the Blockchain validators? That takes us into the next section.
Alice wants to send 5 ZEC to Bob.
Step 1: Alice Selects the UTXO to Spend
• Alice selects a note she owns, say with value 5 ZEC and unique identifier $ρ_A$. This note has been committed to in the blockchain with a commitment $C_A$.
• This note is associated with a nullifier $N_A$, which is derived from Alice's spending key and $ρ_A$:
Step 2: Alice Creates the New Note for Bob
• Alice generates a new note for Bob with:
□ Value: 5 ZEC
□ Unique identifier: $ρ_B$ (randomly generated by Alice)
□ Recipient's public key: $pk_{d_B}$
• Alice generates the note commitment for this new note:
$$C_B = Commit(5, \rho_b, pk_{d_b})$$
where $Commit$ is the commitment function. (Pederson Hash is used for ZCash Commitment)
Step 3: Alice Encrypts the Note for Bob
• Same as Explain above using $Bob's$ zk-addr
Step 4: Alice Prepares the Transaction
• Inputs: The transaction includes:
□ The nullifier $N_A$ of the note Alice is spending.
□ The encrypted note $c_B$ for Bob.
□ The ephemeral public key $epk_A$.
• Outputs: The transaction also includes:
□ The commitment $C_B$ of the new note for Bob.
• zk-SNARK Proof ($pr$): Alice generates a zk-SNARK proof $pr$ that proves:
□ The sum of inputs equals the sum of outputs (i.e., 5 ZEC = 5 ZEC).
□ Alice has the right to spend the note associated with $N_A$.
□ The commitment $C_B$ is correctly formed.
Verification by the Validator
When the transaction is included in the blockchain, the validator performs the following checks:
Step 1: Verify the zk-SNARK Proof
• The validator runs the zk-SNARK verification function $V$ to verify the proof $pr$:
If $V$ returns $True$, the transaction passes the zk-SNARK verification.
Step 2: Check the Nullifier Set
• The validator checks whether $N_A$ is in the global nullifier set:
• If $N_A$ is already in the set, the transaction is invalid (double-spending attempt).
• If $N_A$ is not in the set, the validator adds $N_A$ to the nullifier set to mark the note as spent
Step 3: Check the Commitment
• The validator checks that the new note commitment $C_B$ is valid and correctly formed (as proven by the zk-SNARK proof)
• The proof $pr$ already guarantees this, so the validator relies on the zk-SNARK proof for this verification.
This process shows how Alice can create a transaction for Bob, and how the validator verifies the transaction, ensuring that all cryptographic conditions are met and that privacy is preserved.
Commitment Scheme and Homomorphic Properties
Zcash uses a Pedersen commitment scheme, which has homomorphic properties. A Pedersen commitment allows you to commit to a value (like a note value) while keeping it hidden. The key property of
Pedersen commitments is that they are additively homomorphic. This means that the commitments can be combined in a way that the result is a commitment to the sum of the committed values
Homomorphic Property
The homomorphic property of Pedersen commitments means that:
In other words, the commitment to the sum of two values is the same as the sum of the commitments to each value.
In a Zcash transaction, the goal is to prove that the sum of the inputs (the notes being spent) equals the sum of the outputs (the notes being created) without revealing the actual values of the
inputs and outputs.
1. Commitments to Individual Notes:
• Suppose you have two notes with values $v_1$ and $v_2$, and randomness $r_1$ and $r_2$, respectively.
• The Pedersen commitments to these notes are:
• Here, G and H are fixed generators on the elliptic curve used in the commitment scheme.
1. Adding the Commitments:
• You can add the commitments of the two notes:
• This sum is a new commitment $C_3$
The resulting commitment $C_3$ is a commitment to the sum of the values $v_1 + v_2$ using the combined randomness $r_1 + r_2$
1. Output Note Commitment:
• When you create an output note with a value $v_3$ (which should equal $v_1 + v_2$) and randomness $r_3$ (which should equal $r_1 + r_2$), its commitment $C_3$ would be:
• If $v_3 = v_1 + v_2$ and $r_3 = r_1 + r_2$, then the output commitment $C_3$ will exactly match the sum of the input commitments.
$$C_3 = C_1 + C_2$$
1. Adjusting for randomness in Zcash
The randomness value $r$ for each note (which is part of the Pedersen commitment) is typically generated randomly and independently for each new note. This raises an important question: how can the
sum of the input commitments match the output commitments if the r values are generated randomly?
$$r3 \neq r1 + r2$$
$$C3 \neq C1 + C2$$
This is because in practice $r_3$ is generated independently for $r_1$ and $r_2$.
The zk-SNARK proof plays a critical role here by ensuring that the commitments are correctly formed even though the randomness values are different. The proof guarantees that:
1. Value Conservation: The sum of the input values equals the sum of the output values, i.e., $v_1 + v_2 = v_3$.
2. Correctness of Commitments: The commitments are correctly formed, taking into account the new randomness for the output note.
To account for the difference in randomness, the transaction may include an adjustment factor. Here's how it works:
• Adjustment Factor:
□ The zk-SNARK proof ensures that an additional term (let's call it $R_{adj}$) is introduced to account for the difference in randomness:
• The commitments are then adjusted as part of the proof to ensure that:
• This ensures that even though r_3 is randomly chosen and independent, the overall commitment equation holds.
Zero-knowledge proofs allow one to prove knowledge of information without revealing the information itself, ensuring privacy in cryptographic systems. zk-SNARKs are a widely used zero-knowledge proof
system, particularly in Zcash, where they enable privacy-preserving transactions. Zcash combines the UTXO model with zk-SNARKs to create shielded transactions that hide the sender, recipient, and
transaction amount. Commitments and nullifiers are key cryptographic tools in Zcash, ensuring that notes (UTXOs) can be committed to securely and spent only once. Pedersen commitments, which have
homomorphic properties, allow the sum of commitments to be verified without revealing the underlying values. Despite the randomness in these commitments, zk-SNARKs ensure the integrity of the
transactions by accounting for differences in randomness. The "trusted setup" phase in zk-SNARKs is crucial for generating secure public parameters, with careful handling required to prevent misuse.
Zcash’s design effectively balances privacy with transaction integrity, using advanced cryptographic techniques. Homomorphic encryption in Zcash allows operations on encrypted data, maintaining
privacy while enabling verification. The combination of zk-SNARKs and homomorphic properties makes Zcash a powerful tool for secure, private digital transactions. | {"url":"https://www.spaghetti-coder.com/from-utxos-to-zk-snarks-how-zcash-protects-your-privacy","timestamp":"2024-11-08T00:03:01Z","content_type":"text/html","content_length":"153127","record_id":"<urn:uuid:f6ac4271-a549-4847-8b23-55252d76ad76>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00284.warc.gz"} |
Connective Real $K$-Theory of Finite Groupssearch
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Connective Real $K$-Theory of Finite Groups
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Connective Real $K$-Theory of Finite Groups
Hardcover ISBN: 978-0-8218-5189-0
Product Code: SURV/169
List Price: $129.00
MAA Member Price: $116.10
AMS Member Price: $103.20
eBook ISBN: 978-1-4704-1396-5
Product Code: SURV/169.E
List Price: $125.00
MAA Member Price: $112.50
AMS Member Price: $100.00
Hardcover ISBN: 978-0-8218-5189-0
eBook ISBN: 978-1-4704-1396-5
Product Code: SURV/169.B
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• Mathematical Surveys and Monographs
Volume: 169; 2010; 318 pp
MSC: Primary 19; 55; 13; Secondary 20; 53
This book is about equivariant real and complex topological \(K\)-theory for finite groups. Its main focus is on the study of real connective \(K\)-theory including \(ko^*(BG)\) as a ring and \
(ko_*(BG)\) as a module over it. In the course of their study the authors define equivariant versions of connective \(KO\)-theory and connective \(K\)-theory with reality, in the sense of Atiyah,
which give well-behaved, Noetherian, uncompleted versions of the theory. They prove local cohomology and completion theorems for these theories, giving a means of calculation as well as
establishing their formal credentials. In passing from the complex to the real theories in the connective case, the authors describe the known failure of descent and explain how the \(\eta\)
-Bockstein spectral sequence provides an effective substitute.
This formal framework allows the authors to give a systematic calculation scheme to quantify the expectation that \(ko^*(BG)\) should be a mixture of representation theory and group cohomology.
It is characteristic that this starts with \(ku^*(BG)\) and then uses the local cohomology theorem and the Bockstein spectral sequence to calculate \(ku_*(BG)\), \(ko^*(BG)\), and \(ko_*(BG)\).
To give the skeleton of the answer, the authors provide a theory of \(ko\)-characteristic classes for representations, with the Pontrjagin classes of quaternionic representations being the most
Building on the general results, and their previous calculations, the authors spend the bulk of the book giving a large number of detailed calculations for specific groups (cyclic, quaternion,
dihedral, \(A_4\), and elementary abelian 2-groups). The calculations illustrate the richness of the theory and suggest many further lines of investigation. They have been applied in the
verification of the Gromov-Lawson-Rosenberg conjecture for several new classes of finite groups.
Graduate students and research mathematicians interested in connective \(K\)-theory.
□ Chapters
□ 1. Introduction
□ 2. $K$-theory with reality
□ 3. Descent, twisting and periodicity
□ 4. The Bockstein spectral sequence
□ 5. Characteristic classes
□ 6. Examples for cohomology
□ 7. Examples for homology
□ 8. Dihedral groups
□ 9. The $ko$-cohomology of elementary abelian 2-groups
□ 10. The $ko$-homology of elementary abelian groups (BSS)
□ 11. The structure of $TO$
□ 12. The $ko$-homology of elementary abelian groups (LCSS)
□ 13. Ext charts
□ 14. Conventions
□ 15. Indices
□ The book is very carefully written, including many diagrams and tables, and also a thorough review of the authors' previous work on the complex case.
Donald M. Davis, Mathematical Reviews
• Permission – for use of book, eBook, or Journal content
• Book Details
• Table of Contents
• Additional Material
• Reviews
• Requests
Volume: 169; 2010; 318 pp
MSC: Primary 19; 55; 13; Secondary 20; 53
This book is about equivariant real and complex topological \(K\)-theory for finite groups. Its main focus is on the study of real connective \(K\)-theory including \(ko^*(BG)\) as a ring and \(ko_*
(BG)\) as a module over it. In the course of their study the authors define equivariant versions of connective \(KO\)-theory and connective \(K\)-theory with reality, in the sense of Atiyah, which
give well-behaved, Noetherian, uncompleted versions of the theory. They prove local cohomology and completion theorems for these theories, giving a means of calculation as well as establishing their
formal credentials. In passing from the complex to the real theories in the connective case, the authors describe the known failure of descent and explain how the \(\eta\)-Bockstein spectral sequence
provides an effective substitute.
This formal framework allows the authors to give a systematic calculation scheme to quantify the expectation that \(ko^*(BG)\) should be a mixture of representation theory and group cohomology. It is
characteristic that this starts with \(ku^*(BG)\) and then uses the local cohomology theorem and the Bockstein spectral sequence to calculate \(ku_*(BG)\), \(ko^*(BG)\), and \(ko_*(BG)\). To give the
skeleton of the answer, the authors provide a theory of \(ko\)-characteristic classes for representations, with the Pontrjagin classes of quaternionic representations being the most important.
Building on the general results, and their previous calculations, the authors spend the bulk of the book giving a large number of detailed calculations for specific groups (cyclic, quaternion,
dihedral, \(A_4\), and elementary abelian 2-groups). The calculations illustrate the richness of the theory and suggest many further lines of investigation. They have been applied in the verification
of the Gromov-Lawson-Rosenberg conjecture for several new classes of finite groups.
Graduate students and research mathematicians interested in connective \(K\)-theory.
• Chapters
• 1. Introduction
• 2. $K$-theory with reality
• 3. Descent, twisting and periodicity
• 4. The Bockstein spectral sequence
• 5. Characteristic classes
• 6. Examples for cohomology
• 7. Examples for homology
• 8. Dihedral groups
• 9. The $ko$-cohomology of elementary abelian 2-groups
• 10. The $ko$-homology of elementary abelian groups (BSS)
• 11. The structure of $TO$
• 12. The $ko$-homology of elementary abelian groups (LCSS)
• 13. Ext charts
• 14. Conventions
• 15. Indices
• The book is very carefully written, including many diagrams and tables, and also a thorough review of the authors' previous work on the complex case.
Donald M. Davis, Mathematical Reviews
Permission – for use of book, eBook, or Journal content
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Math Colloquia - Nonlocal generators of jump type Markov processes
Empirical observations have shown that for an adequate description of many random phenomena non-Gaussian processes are needed. The paths of these Markov processes necessarily have jumps. Their
generators are nonlocal operators which admit a representation as pseudo-differential operators with so-called negative definite symbols.
The talk gives an introduction to the relationship between jump processes and this non classical type of pseudo-differential operators. A particular focus will lie on different possibilities to
construct the process starting from a given symbol. | {"url":"http://www.math.snu.ac.kr/board/index.php?mid=colloquia&sort_index=speaker&order_type=asc&page=2&l=en&document_srl=758698","timestamp":"2024-11-11T07:17:48Z","content_type":"text/html","content_length":"43819","record_id":"<urn:uuid:8761bdc8-5bb8-4f1d-a186-008b056a6f50>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00031.warc.gz"} |
Connections between Abstract Algebra and High School Algebra: A Few Connections Worth Exploring
by Erin Baldinger, University of Minnesota; Shawn Broderick, Keene State College; Eileen Murray, Montclair State University; Nick Wasserman, Columbia University; and Diana White, Contributing Editor,
University of Colorado Denver.
Mathematicians often consider knowledge of how algebraic structure informs the nature of solving equations, simplifying expressions, and multiplying polynomials as crucial knowledge for a teacher to
possess, and thus expect that all high school teachers have taken an introductory course in abstract algebra as part of a bachelor’s degree. This is far from reality, however, as many high school
teachers do not have a degree in mathematics (or even mathematics education) and have pursued alternative pathways to meet content requirements of certification. Moreover, the mathematics education
community knows that more mathematics preparation does not necessarily improve instruction (Darling-Hammond, 2000; Monk, 1994). In fact, some research has shown that more mathematics preparation may
hinder a person’s ability to predict student difficulties with mathematics (Nathan & Petrosino, 2003; Nathan & Koedinger, 2000). Nevertheless, the requirements for traditional certification to teach
secondary mathematics across the country continue to include an undergraduate major in the subject, and many mathematicians and mathematics educators still regard such advanced mathematics knowledge
as potentially important for teachers.
Given this, it is important that, as a field, we investigate the nature of the present mathematics content courses offered (and required) of prospective secondary mathematics teachers to gain a
better understanding of which concepts and topics positively impact teachers’ instructional practice. That is, we need to explore links not just between abstract algebra and the content of secondary
mathematics, but also to the teaching of that content (e.g., see Wasserman, 2015). In November 2015, a group of mathematicians and mathematics educators met as a working group around this topic at
the annual meeting of the North American Chapter of the Psychology of Mathematics Education. We began to probe the impact understanding connections such as those described above might have on
teachers’ instructional choices. For example, how does understanding the group axioms shift teacher instruction around solving equations? How does understanding integral domains shift teacher
instruction around factoring? Through answering questions such as these, mathematicians and mathematics educators can better support teachers to connect advanced mathematical understanding to school
mathematics in meaningful ways that enhance the quality of instruction.
In the remainder of this blog post, we explain and discuss three frequently cited examples of connections between abstract algebra and high school mathematics.
Example 1: Solving equations
Solving equations and simplifying expressions is a technique used in multiple settings within mathematics. It uses the precise axioms of a group, but this is often not made transparent to students
What would you do to solve this “one-step” equation? Many students are taught to subtract 5 from both sides to isolate the variable x, and they might write something like this (crossing out the 5s
on the left hand side):
x + 5 = 12
-5 -5
x = 7
However, on closer inspection, a variety of algebraic properties come to bear that the above work suppresses. (See Wasserman [2014] for a more complete elaboration and discussion.) An expanded
version might look like this, with justifications for each step.
(x + 5) + -5 = (12) + -5 (Additive Equivalence)
x + (5 + -5) = 12 + -5 (Associativity of addition)
x + 0 = 12 + -5 (Additive Inverse)
x = 12 + -5 (Identity Element for addition)
x = 7 (Closure under addition)
Similarly, if attention is given to algebraic properties used to solve equations, the solution to an equation of the form 5x=12 might appear as follows:
⅕*(5x)= ⅕*12 (Equivalence)
(⅕*5)x = ⅕*12 (Associativity of multiplication)
1*x = ⅕*12 (Multiplicative Inverse)
x = ⅕*12 (Identity Element for multiplication)
x = 12/5 (Closure)
These solution techniques can be related to students’ learning of matrix algebra in a course on linear algebra. Specifically, students learn, under appropriate conditions, to solve matrix equations
of the form AX = B using these same steps.
In each case above, the last four steps being used – the ones “hidden” from view in the one-step cancellation process – are the precise axioms for a group. In the first case, we’re working on the
additive group of integers, in the second on the nonzero multiplicative group of rational numbers, and in the last under the group of n by n square matrices with nonzero determinant (i.e.,
invertible) under matrix multiplication. Thus, these are three a priori separate problems, all united by the same algebraic structure of a group – and that structure becomes evident in the algebraic
solution process. Wasserman and Stockton (2013) discuss one vignette for how such knowledge might be incorporated into secondary instruction.
Example 2: Simplifying expressions
As a related example, consider the following two samples of student work:
In each case, clearly a form of “cancellation” is being attempted. But what, technically, results in “cancellation”? And what remains after the cancellation is complete? Do sin and sin-1 make “1”?
Is the “x” still an exponent? While we recognize this “cancellation” as attending to both the inverse elements and the meaning of the identity element in the group of invertible functions, these
are subtle issues that are often not clear to students, and they are often taught in isolation, without the underlying structure being made apparent.
In using the above two examples to illustrate, we do not intend to imply that teachers should require students to make explicit each and every use of a mathematical property when they solve
equations. Rather, we aim to draw attention to the importance of recognizing the consistency going on across all of these examples of solving equations. Moreover, it is the collective power of
individual properties – as they form the group (or ring/field) axioms – that allow for algebraic solution approaches and also help reconcile the meaning of “cancellation” in these different contexts
as an interaction of both inverse and identity elements.
Example 3: Polynomials and Factoring
As another example of the connection between abstract algebra and secondary mathematics, we consider the problem of multiplying two polynomials. (See Baldinger [2013, 2014] for additional examples of
this type.) In high school, students learn that the degree of the product of two nonzero polynomials is the sum of the degrees of the factors. Yet this does not hold in all types of algebraic
settings. Consider, for example, the product of the following two polynomials when working modulo 7 versus modulo 8.
As mathematicians, we of course recognize that the the degree of the product of two polynomials is the sum of the degrees of the factors — when the coefficients are elements of an integral domain,
but that this relationship need not hold in other settings. Students, however, may be mystified when they first encounter an example like this in modular arithmetic, as their prior conceptions and
understandings are being challenged, and they are thus being asked to deepen their understanding of the underlying structures that permit a result to hold in one setting, but break down in another.
This example also ties directly into student misconceptions. For example, we teach students in high school that if the product of two polynomials is zero, then to solve we set each one separately
equal to zero. Yet this does not hold with nonzero numbers. For example, working in polynomials with real coefficients, we know that f(x) * g(x)=0 implies either f(x) = 0 or g(x) = 0. Yet it is not
the case that if f(x) * g(x) = 4, then either f(x) = 2 or g(x) = 2.
The three above examples represent just a few of the many connections between abstract algebra and secondary mathematics. There has been a longstanding debate in the mathematics and mathematics
education communities concerning the knowledge secondary mathematics teachers need to provide effective instruction. Central to this debate is what content knowledge secondary teachers should have in
order to communicate mathematics to their students, assess student thinking, and make curricular and instructional decisions. This debate has already led to many fruitful projects (e.g., Connecting
Middle School and College Mathematics [(CM)2] (Papick, n.d.); Mathematics Education for Teachers I (2001) and II (2012); Mathematical Understanding for Secondary Teaching: A Framework and
Classroom-Based Situations (Heid, Wilson, & Blume, in press). A common thread in these projects is the belief that mathematics teachers should have a strong mathematical foundation along with the
knowledge of how advanced mathematics is connected to secondary mathematics (Papick, 2011). But questions remain regarding what secondary content stems from connections to advanced mathematics, which
connections are important, and how might knowledge of such connections influence practice. Our working group hopes to continue to explore these connections and contribute to our collective
understanding of teacher education.
Baldinger, E. (2013). Connecting abstract algebra to high school algebra. In Martinez, M. & Castro Superfine, A. (Eds.). Proceedings of the 35th annual meeting of the North American Chapter of the
International Group for the Psychology of Mathematics Education (pp. 733–736). Chicago, IL: University of Illinois at Chicago.
Baldinger, E. (2014). Studying abstract algebra to teach high school algebra: Investigating future teachers’ development of mathematical knowledge for teaching (Unpublished doctoral dissertation).
Stanford University, Stanford, CA.
Conference Board of the Mathematical Sciences. (2001). The mathematical education of teachers (Issues in Mathematics Education, Vol. 11). Providence, RI: American Mathematical Society.
Conference Board of the Mathematical Sciences. (2012). The mathematical education of teachers II (Issues in Mathematics Education, Vol. 17). Providence, RI: American Mathematical Society.
Darling-Hammond, L. (2000). Teacher quality and student achievement: A review of state policy evidence. Educational Policy Analysis Archives, 8(1). Retrieved from http://epaa.asu.edu
Heid, M. K., Wilson, P., & Blume, G. W. (in press). Mathematical Understanding for Secondary Teaching: A Framework and Classroom-Based Situations. Charlotte, NC: Information Age Publishing.
Monk, D. H. (1994). Subject matter preparation of secondary mathematics and science teachers and student achievement. Economics of Education Review, 13(2), 125–145.
Nathan, M. J. & Koedinger, K. R. (2000). An investigation of teachers’ beliefs of students’ algebra development. Cognition and Instruction, 18, 209–237.
Nathan, M. J. & Petrosino, A. (2003). Expert blind spot among preservice teachers. American Education Research Journal, 40, 905–928.
Papick, I. (n.d.) Connecting Middle School and College Mathematics Project. Retrieved March 7, 2015 from http://www.teachmathmissouri.org/
Papick, I. J. (2011). Strengthening the mathematical content knowledge of middle and secondary mathematics teachers. Notices of the AMS, 58(3), 389-392.
Wasserman, N. (2014). Introducing algebraic structures through solving equations: Vertical content knowledge for K-12 mathematics teachers. PRIMUS: Problems, Resources, and Issues in Mathematics
Undergraduate Studies, 24, 191–214.
Wasserman, N. (2015). Abstract algebra for algebra teaching: Influencing school mathematics instruction. Canadian Journal of Science, Mathematics and Technology Education (online first). DOI: 10.1080
Wasserman, N. & Stockton, J. (2013). Horizon content knowledge in the work of teaching: A focus on planning. For the Learning of Mathematics, 33(3), pp. 20–22.
1 Response to Connections between Abstract Algebra and High School Algebra: A Few Connections Worth Exploring
1. As a former IB student who took HL mathematics I was thankful to have experienced and learned about abstract algebra in high school. As a math major when I got to linear algebra it was so much
easier for me than most of my classmates because I had that experience. Even when I got to modern algebra this year it was still not as difficult as people made it out to be because of that prior
experience. However, watching my classmates struggle with those classes simply because they didn’t have the prior experience was frustrating to me. Most high school math classes don’t cover
abstract algebra and it leaves the students who pursue mathematics in university at a disadvantage. I believe it should be a course anyone interested in pursuing mathematics in university should
have access to in high school. It makes those university courses which are required to graduate with a math degree (at least at my university) much easier as well as the fact that it expands the
mathematical framework a student has. This allows them to approach problems in different ways and find solutions in much easier and simpler ways. However, this would mean that schools would have
to higher teachers who are math majors that want to teach instead of having teachers who pursue a career in education and get forced to teach math courses. So I feel like the best solution would
be for some university to offer students who are interested in pursuing a career in or studying mathematics a course online that they can take to teach themselves the concepts of abstract algebra
so they are more prepared in university.
This entry was posted in K-12 Education and tagged abstract algebra, connections between advanced and secondary mathematics, high school algebra, teacher professional development, teacher training.
Bookmark the permalink. | {"url":"https://blogs.ams.org/matheducation/2015/12/10/connections-between-abstract-algebra-and-high-school-algebra-a-few-connections-worth-exploring/","timestamp":"2024-11-13T22:59:13Z","content_type":"text/html","content_length":"71736","record_id":"<urn:uuid:a266d6e2-9cd6-4d86-ab9e-e68dff8ab3e0>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00130.warc.gz"} |
Mimblewimble-Grin Blockchain Protocol Overview
Table of Contents
Depending on whom you ask, Mimblewimble is either a tongue-tying curse or a blockchain protocol designed to be private and scalable. The transactions in Mimblewimble are derived from confidential
transactions by Greg Maxwell [1], which in turn are based on the Pedersen commitment scheme. On 19 July 2016, Tom Elvis Jedusor left a white paper on the tor network describing how Mimblewimble could
work. As the potential for this was realized, work was done to make this a reality. One of these projects is Grin, which is a minimalistic implementation of Mimblewimble. Further information can be
found in [2] and [3].
Mimblewimble Protocol Overview
Mimblewimble publishes all transactions as confidential transactions. All inputs, outputs and change are expressed in the following form:
$ r \cdot G + v \cdot H $
where $ G $ and $ H $ are elliptic curves, $ r $ a private key used as a blinding factor, $ v $ the value and “$ \cdot $” is Elliptic-curve cryptography (ECC) multiplication.
An example transaction can be expressed as input = output + change.
$ (r_i \cdot G + v_i \cdot H) = (r_o \cdot G + v_o \cdot H) + (r_c \cdot G + v_c + \cdot H) $
But this requires that
$ r_i = r_o + r_c $
A more detailed explanation of how Mimblewimble works can be found in the Grin GitHub documents [4].
Cut-through and Pruning
Grin includes something called “cut-through” in the transaction pool. Cut-through removes outputs from the transaction pool, which have already been spent as new inputs, using the fact that every
transaction in a block should sum to zero. This is shown below:
$ output - inputs = kernel_-excess +(part \mspace{3mu} of)kernel_- offset $
The kernel offset is used to hide which kernel belongs to which transaction and we only have a summed kernel offset stored in the header of each block.
We don’t have to record these transactions inside the block, although we still have to record the kernel as the kernel proof transfer of ownership to make sure that the whole block sums to zero, as
expressed in the following formula:
$ sum(ouputs) - sum(inputs) = sum(kernel_-excess) + kernel_-offset $
An example of cut-through follows:
I1(x1) +---> O1
+---> O2
I2(x2,O2) +---> O3
I3(O3) +---> O4
+---> O5
After cut-through:
I1(x1) +---> O1
I2(x2) +---> O4
+---> O5
In the preceding examples, “I” represents new inputs, “X” represents inputs from previous blocks and “O” represents outputs.
This causes Mimblewimble blocks to be much smaller than normal bitcoin blocks, as the cut-through transactions are no longer listed under inputs and outputs. In practice, after this we can still see
there was a transaction, because the kernel excess still remains, but the actual hidden values are not recorded.
Pruning takes this same concept but goes into past blocks. Therefore, if an output in a previous block gets spent, it will be removed from that block. Pruning removes the leaves from the Merkle
Mountain Range (MMR) as well. Thus, it allows the ledger to be small and scalable. According to the Grin team [3], assuming 10 million transactions with 100,000 unspent outputs, the ledger will be
roughly 130GB, which can be divided into the following parts:
• 128GB transaction data (inputs and outputs);
• 1GB transaction proof data;
• 250MB block headers.
The total storage requirements can be reduced if cut-through and pruning are applied. The ledger will shrink to approximately 1.8GB and will result in the following:
• 1GB transaction proof data;
• UTXO size 520MB;
• 250MB block headers.
Grin Blocks
The grin block contains the following data:
1. Transaction outputs, which include for each output:
□ A Pedersen commitment (33 bytes).
□ A range proof (over 5KB at this time).
2. Transaction inputs, which are just output references (32 bytes).
3. Transaction fees, in clear text.
4. Transaction “proofs”, which include for each transaction:
□ The excess commitment sum for the transaction (33 bytes).
□ A signature generated with the excess (71 bytes average).
5. A block header that includes Merkle trees and proof of work (approximately 250 bytes).
The Grin header:
Header Field Description
Hash Unique hash of block
Version Grin version
Previous Block Unique hash of previous block
Age Time the block was mined
Cuckoo Solution The wining cuckoo solution
Difficulty Difficulty of the solved cuckoo
Target Difficulty Difficulty of this block
Total Difficulty Total difficulty of mined chain up to age
Total Kernel Offset Kernel offset
Nonce Random number for cuckoo
Block Reward Coinbase + fee reward for block
The rest of the block contains a list of kernels, inputs and outputs. Appendix A contains an example of a grin block.
Trustless Transactions
Schnorr signatures have been done in Tari Labs University (TLU). Please look here for a more detailed explanation [7].
Since Grin transactions are obscured by Pedersen Commitments, there is no proof that money was actually transferred. To solve this problem, we require the receiver to collaborate with the sender in
building a transaction and, more specifically, the kernel signature [4].
When Alice wants to pay Bob, the transaction will be performed using the following steps:
1. Alice selects her inputs and her change. The sum of all blinding factors (change output minus inputs) is $ r_s $.
2. Alice picks a random nonce ks and sends her partial transaction, $ k_s\cdot G $ and $ r_s\cdot G $ to Bob.
3. Bob picks his own random nonce $ k_r $ and the blinding factor for his output $ r_r $. Using $ r_r $, Bob adds his output to the transaction.
4. Bob computes the following:
□ message $ m = fee \Vert lock_-height $;
□ Schnorr challenge $ e = SHA256(m \Vert k_r \cdot G + k_s\cdot G \Vert r_r\cdot G + r_s\cdot G) $; and
□ his side of the signature, $ s_r = k_r + e\cdot G $.
5. Bob sends $ s_r $ and $ k_r\cdot G $ and $ r_r\cdot G $ to Alice.
6. Alice computes $ e $, just like Bob did, and can check that $ s_r\cdot G = k_r\cdot G + e\cdot r_r \cdot G $.
7. Alice sends her side of the signature, $ s_s = k_s + e\cdot r_s $, to Bob.
8. Bob validates $ s_s\cdot G $, just like Alice did for $ s_r\cdot G $ in step 5, and can produce the final signature $ sig = (s_s + s_r , \mspace{6mu} k_s\cdot G + k_s\cdot G) $ as well as the
final transaction kernel, including $ sig $ and the public key $ r_r\cdot G + r_s\cdot G$.
In a normal Grin transaction, the signature [4], just the normal fee, gets signed as the message. But to get an absolute time-locked transaction, the message can be modified taking the block height
and appending the fee to that. A block with a kernel that includes a lock height greater than the current block height is then rejected.
$ m = fee \Vert h $
Taking into account how an absolute time-locked transaction is constructed, the same idea can be extended by taking the relative block height and not the absolute height, but also adding a specific
kernel commitment. In this way, the signature references a specific block as height. The same principle counts as with absolute time-locked transactions in that a block with a kernel containing a
relative time-locked transaction that has not passed, is rejected.
$ m = fee \Vert h \Vert c $
Multi-signatures (multisigs) are also known as N-of-M signatures. This means that N amount out of M amount of peers need to agree before a transaction can be spent.
When Bob and Alice [6] want to do a 2‑of‑2 multisig contract, the contract can be done using the following steps:
1. Bob picks a blinding factor $ r_b $ and sends $ r_b\cdot G $ to Alice.
2. Alice picks a blinding factor $ r_a $ and builds the commitment $ C= r_a\cdot G + r_b\cdot G + v\cdot H $; she sends the commitment to Bob.
3. Bob creates a range proof for $ v $ using $ C $ and $ r_b $, and sends it to Alice.
4. Alice generates her own range proof and aggregates it with Bob, finalizing the multiparty output $ O_{ab} $.
5. The kernel is built following the same procedure as used with Trustless Transactions.
We observe that the output $ O_{ab} $ is unknown to both parties, because neither knows the whole blinding factor. To be able to build a transaction spending $ O_{ab} $, someone would need to know $
r_a + r_b $ to produce a kernel signature. To produce the original spending kernel, Alice and Bob need to collaborate.
Atomic Swaps
Atomic swaps can be used to exchange coins from different blockchains in a trustless environment. In the Grin documentation, this is handled in the contracts documentation [6] and in the contract
ideas documentation [8]. In practice, there has already been an atomic swap between Grin and Ethereum [9], but this only used the Grin testnet with a modified Grin implementation, as the release
version of Grin did not yet support the required contracts. TLU has a section about Atomic Swaps [7].
Atomic swaps work with 2‑of‑2 multisig contracts, one public key being Alice’s, the other being the hash of a preimage that Bob has to reveal. Consider public key derivation $ x\cdot G $ to be the
hash function and by Bob revealing $ x $, Alice can then produce an adequate signature, proving she knows $ x $ (in addition to her own private key).
Alice will swap Grin with Bob for Bitcoin. We assume Bob created an output on the Bitcoin blockchain that allows spending by Alice if she learns a hash preimage $ x $, or by Bob after time $ T_b $.
Alice is ready to send her Grin to Bob if he reveals $ x $.
Alice will send her Grin to a multiparty timelock contract with a refund time $ T_a < T_b $. To send the 2‑of‑2 output to Bob and execute the swap, Alice and Bob start as if they were building a
normal trustless transaction:
1. Alice picks a random nonce $ k_s $ and her blinding sum $ r_s $ and sends $ k_s\cdot G $ and $ r_s\cdot G $ to Bob.
2. Bob picks a random blinding factor $ r_r $ and a random nonce $ k_r $. However, this time, instead of simply sending $ s_r = k_r + e\cdot r_r $ with his $ r_r\cdot G $ and $ k_r\cdot G $, Bob
sends $ s_r’ = k_r + x + e\cdot r_r $ as well as $ x\cdot G $.
3. Alice can validate that $ s_r’\cdot G = k_r\cdot G + x\cdot G + r_r\cdot G $. She can also check that Bob has money locked with $ x\cdot G $ on the other chain.
4. Alice sends back her $ s_s = k_s + e\cdot x_s $ as she normally would, now that she can also compute $ e = SHA256(m \Vert k_s\cdot G+k_r\cdot G) $.
5. To complete the signature, Bob computes $ s_r = k_r + e\cdot r_r $ and the final signature is $ (s_r + s_s, \mspace{6mu} k_r\cdot G + k_s\cdot G) $.
6. As soon as Bob broadcasts the final transaction to get his Grin, Alice can compute $ s_r’ - s_r $ to get $ x $.
Prior to completing the atomic swap, Bob needs to know Alice’s public key. Bob would then create an output on the Bitcoin blockchain with a 2‑of‑2 multisig similar to alice_pubkey secret_pubkey 2
OP_CHECKMULTISIG. This should be wrapped in an OP_IF so Bob can get his money back after an agreed-upon time. All of this can even be wrapped in a Pays To Script Hash (P2SH). Here, secret_pubkey is
$x\cdot G$ from the previous section.
To verify the output, Alice would take $x\cdot G$, recreate the bitcoin script, hash it and check that her hash matches what’s in theP2SH (step 2 in the Multisig section). Once she gets $x$ (step 6),
she can build the two signatures necessary to spend the 2‑of‑2, having both private keys, and get her bitcoin.
[1] G. Maxwell, “Confidential Transactions”, 2017 [online]. Available: https://people.xiph.org/~greg/confidential_values.txt. Accessed: 2018‑10‑24.
[2] P. Robinson, H. Odendaal, S. W. van Heerden and A. Zaidelson, “Grin vs. BEAM, a Comparison”, 2018 [online]. Available: https://tari‑labs.github.io/tari-university/protocols/grin-beam-comparison/
MainReport.html#grin-vs-beam-a-comparison. Accessed: 2018‑10‑08.
[3] Y. Roodt, H. Odendaal, S. W. van Heerden, R. Robinson and A. Kamal, “Grin Design Choice Criticisms - Truth or Fiction”, 2018 [online]. Available: https://tari-labs.github.io/tari-university/
protocols/grin-design-choice-criticisms/MainReport.html. Accessed: 2018‑10‑08.
[4] B. Simon et al., “Grin Document Structure”, 2018 [online]. Available: https://github.com/mimblewimble/grin/blob/master/doc/table_of_contents.md. Accessed: 2018‑10‑24).
[5] I. Peverell et al., “Pruning Blockchain Data”, 2016 [online]. Available: https://github.com/mimblewimble/grin/blob/master/doc/pruning.md. Accessed: 2018‑10‑26.
[6] I. Peverell et al., “Contracts”, 2018 [online]. Available: https://github.com/mimblewimble/grin/blob/master/doc/contracts.md. Accessed: 2018‑10‑26.
[7] “Tari Labs University”. Tari Labs, 2018 [online]. Available: https://tari-labs.github.io/tari-university/. Accessed: 2018‑10‑27.
[8] Q. Le Sceller, “Contract Ideas”, 2018 [online]. Available: https://github.com/mimblewimble/grin/blob/master/doc/contract_ideas.md. (Accessed: 2018‑10‑27.)
[9] Jasper, “First Grin Atomic Swap!” (2018) [online]. Available: https://medium.com/grinswap/first-grin-atomic-swap-a16b4cc19196. Accessed: 2018‑10‑27.
Appendix A: Example of Grin Block
Hash 02cb5e810857266609511699c8d222ed4e02883c6b6d3405c05a3caea9bb0f64
Version 1
Previous Block 0343597fe7c69f497177248913e6e485f3f23bb03b07a0b8a5b54f68187dbc1d
Age 2018-10-23, 08:03:46 UTC
Cuckoo Solution Size
Difficulty 37,652
Target Difficulty 17,736
Total Difficulty 290,138,524
Total Kernel Offset b52ccdf119fe18d7bd12bcdf0642fcb479c6093dca566e0aed33eb538f410fb5
Nonce 7262225957146290317
Block Reward 60 grin
Fees 14 mg
4 x Inputs
No. Commit
1 0898a4b53964ada66aa16de3d44ff02228c168a23c0bd71b162f4366ce0dae24b0
2 09a173023e9c39c923e626317ffd384c7bce44109fea91a9c142723bfa700fce27
3 086e0d164fe92d837b5365465a6b37af496a4f8520a2c1fccbb9f736521631ba96
4 087a00d61f8ada399f170953c6cc7336c6a0b22518a8b02fd8dd3e28c01ee51fdb
5 x Outputs
No. Output Type Commit Spent
1 Transaction 09eac33dfdeb84da698c6c17329e4a06020238d9bb31435a4abd9d2ffc122f6879 False
2 Transaction 0860e9cf37a94668c842738a5acc8abd628c122608f48a50bbb7728f46a3d50673 False
3 Coinbase 08324cdbf7443b6253bb0cdf314fce39117dcafbddda36ed37f2c209fc651802d6 False
4 Transaction 0873f0da4ce164e2597800bf678946aad1cd2d7e2371c4eed471fecf9571942b4f False
5 Transaction 09774ee77edaaa81b3c6ee31f471f014db86c4b3345f739472cb12ecc8f40401df False
3 x Kernels
No. Features Fee Lock Height
1 DEFAULT_KERNEL 6 mg 7477
2 DEFAULT_KERNEL 8 mg 7477
3 COINBASE_KERNEL 0 grin 7482
Apart from the header information, we can only see that this block contains two transactions from the two kernels present. Between these two transactions, we only know that there were four inputs and
four outputs. Because of the way in which Mimblewimble obfuscates the transaction, we don’t know the values or which input and output belong to which transaction. | {"url":"https://tlu.tarilabs.com/protocols/grin-protocol-overview","timestamp":"2024-11-14T13:52:02Z","content_type":"text/html","content_length":"194446","record_id":"<urn:uuid:64637135-97df-4d0e-8df1-f1ee6a4d7568>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00454.warc.gz"} |
Optimizing the Regression Model: The Challenge of Intercept–Bias and Slope “Correction”
The archnemesis of calibration modeling and the routine use of multivariate models for quantitative analysis in spectroscopy is the confounded bias or slope adjustments that must be continually
implemented to maintain calibration prediction accuracy over time. A perfectly developed calibration model that predicted well on day one suddenly has to be bias adjusted on a regular basis to pass a
simple bias test when predicted values are compared to reference values at a later date. Why does this problem continue to plague researchers and users of chemometrics and spectroscopy?
The subject of bias and slope, also known as intercept and slope adjustments, following calibration transfer has been an integral part of the application of multivariate calibrations since the very
beginning. It is well understood, and widely accepted, that following the transfer of multivariate calibrations from one instrument to another there is a bias adjustment required for the predicted
results to conform to the reference values from a set of reference transfer samples. There have been many attempts to reduce this bias or intercept requirement, but the fact remains that intercept
and slope are routinely used for calibration transfer from one instrument to another.
One may derive a number of explanations for the requirement for intercept or slope adjustments following calibration transfer. One reason often given is that laboratory values differ. This may be
true in some cases, but is completely irrelevant to the issue that if one transfers a calibration from one identical instrument to another the predicted values should be identical for the same sample
set from one instrument to another (within normal statistical variation). This brings us to the key point of discussion for this column.
If one will engage in a gedankenexperiment with us, momentarily one can conceive of a time and technology where intercept and slope will not be required for one instrument to produce identical
results to another instrument following calibration transfer. In this thought experiment let us conceive of instrumentation that is precisely alike in spectral shape from one instrument to another
when measuring an identical sample. For such an instrument, the predicted values using the same calibration equation across instruments will give precisely the same results. We have discussed this
issue in the past and demonstrated that when using near-infrared (NIR) spectrophotometers for predictions, even if one uses random numbers for the dependent variables (representing the reference or
primary laboratory results), that different instruments will closely predict these random numbers across instruments following calibration transfer. Thus, the instrument could be considered agnostic
in terms of the numbers it generates using a spectral measurement when combined with multivariate calibration equations (1).
We have also pointed out in the past some of the various requirements for successful calibration transfer and the wide differences between commercial instruments in terms of agreement for wavelength
and photometric axes registration (2-7). Furthermore, we have reminded ourselves that spectrophotometers using Beer’s law (and even Raman spectrometers that “self determine” pathlength) measure moles
per unit volume or mass per unit volume and do not measure odd or contrived reference values simply by adding more terms to the regression model (8,9). Note that irrespective of the chemometric
approaches used to develop a calibration model, an inconsistent spectral shape with changing x- and y-axis registrations over time, and between instruments, will disrupt any model such that the
intercept and slope corrections remain a requirement following calibration transfer. Readers are also reminded that calibration models that accommodate too much wavelength and photometric variation
within the data lose significant predictive accuracy. So, to return to our thought experiment, we are imagining that all instruments are identical using a new form of manufacturing technology and
that we are able to measure the same set of samples on a series of such spectrophotometers and use the same identical calibration equation to obtain the same identical predicted results from each
instrument without the need for any bias and slope adjustments.
Note that the entire issue for relating the spectra from each instrument to specific reference laboratory results is immaterial for this discussion, since given any instrument the identical results
will be obtained. One must obviously decide what chemical parameters and reference laboratory to use to provide the correct reference values relative to any individual sample. As a reminder, this
should be reported as weight or mass per volume and not weight percent or other contrived values if one wants to better relate the spectroscopy to the reported predicted values.
Now, there are many issues associated with making instruments alike enough to eliminate the bias and slope problem. This column installment is intended to begin a discussion on the issues involved
and to surface some of the mathematical and engineering approaches intended to move toward correcting any instrument differences. The technology will certainly arise at some point to eliminate this
problem, so let us move in that direction and begin addressing the issues. We remind the reader that we will refer to no metaphysics; only standard physics and statistics are involved.
So, what are the main differences between instruments that cause the slope and intercept challenges related to calibration transfer? We discuss three of these differences here and hopefully more
fully address these and other issues in some future columns. For this article let us discuss: wavelength shift (x-axis), photometric shift (y-axis), and linewidth or lineshape (spectral resolution
and shape) as the primary causes of the intercept and slope variation between instruments following calibration transfer. Note that these authors have expounded on the intuitive problem of having a
fixed calibration model for use with a dynamic instrument condition in previous publications. Either the calibration must become dynamic to compensate for instrument changes or the instrument must be
corrected to a constant initial state if a fixed model is to be applied with consistent predictive results.
As we continue discussion with our thought experiment, one could envision a time when the spectrum of a pure compound could be defined in such a way that given a specific spectral resolution, one
would define a standard reference spectrum of a pure compound. The further expectation would be that all properly aligned and calibrated spectrophotometers would be expected to produce an identical
spectrum from the same sample of the pure compound. Such a future technology would require instrumentation to be precisely calibrated using first principles physical standards with a predefined
linewidth and lineshape. These aspects would be combined with a series of Procrustean algorithms to conform a measured result to a precise reference result. One may see this as a multistep process
resembling Figure 1.
So, along these lines of our thought experiment, let us now examine some of the issues associated with the challenge of instrumental differences that cause calibration transfer slope and intercept
woes. Such primary instrument differences include wavelength shift, photometric shift, and linewidth or lineshape changes. These are the primary causes for intercept and slope changes between
instruments used for calibration transfer.
To demonstrate the effects of changes in wavelength, photometric, and line-width or lineshape parameters in principle, and to demonstrate the mathematics used to compute the confidence limits
associated with comparing standard error, intercept (bias), and slope we present a simple univariate regression case with a single near Gaussian spectral band. For this demonstration, we examine the
effects of shifting or altering the spectral band on the predicted results when applying the original univariate calibration equation (10).
The Initial Spectra and Calibration Equation
A set of six spectra are derived having a single near symmetrical Gaussian band representing six concentrations of analyte. The spectra and data table are given as Figure 2, and Table I,
respectively. The linear regression, formed at 924.81 nm peak maximum, from this data results in the univariate linear regression equation 1. This equation will be applied to all spectra after they
have been altered to test the effect of the various alterations upon the predicted results. A summary of results and a discussion section will be presented at the conclusion of this column.
Figure 2: Simulated spectral data used for univariate calibration (six samples).
Changing the Wavelength Registration
In the first experiment, we shift wavelength registration by -1.0 nm, -0.5 nm, -0.25 nm, and +0.25 nm, +0.50 nm, and +1.0 nm. From testing commercial NIR instrumentation these differences would be
expected between manufacturers. When the wavelength registration is changed by these amounts the resulting spectra and prediction results are as follows in Figure 3, and Table II.
Figure 3: A single spectral band from one sample (sample 1) shifted by different wavelength amounts; the unshifted spectrum is represented by the dotted red line.
Changing the Photometric Registration
In the second experiment we shift photometric registration by -0.1 A, -0.05 A, -0.025 A, no shift, and +0.025 A, +0.05 A, and +0.1 A. These would be typical differences expected between manufacturers
or even within the same instrument over time. When the photometric registration is changed by these amounts the resulting spectra and prediction results are as follows in Figure 4, and Table III.
Figure 4: A single spectral band from one sample (sample 1) shifted by different photometric offset amounts; the unshifted spectrum is represented by the dotted red line.
Changing the Linewidth or Lineshape
In the third experiment, we change the linewidth as full width at half maximum (FWHM) to have values of 16.4, 16.8, 17.4, 17.6, 17.9, and 18.2 nm. These would be typical changes expected between
manufacturers or even within the same instrument design criteria over severe differences in manufacturing iterations with new model designs. When the linewidth is changed by these amounts the
resulting spectra and prediction results are as follows in Figure 5, and Table IV. Note that as the bandwidth changes, the amplitude also changes.
Discussion and Review of Results
Table V summarizes results from these experiments. For each parameter tested (that is, the standard error of prediction, the slope, and the intercept) the confidence limits and significance decisions
are given in separate columns. The equations for computing the confidence limits for each statistical parameter (that is, predicted values, slope, and intercept) are given in equations 2 through 6.
Confidence Limits for Predicted Values from a Regression
To compute the upper and lower confidence limits for any predicted value, the following equations 2 and 3 are used. First compute the standard error of calibration (SEC)-this is 0.01 for our example.
The SEC is computed as shown in equation 2.
For this equation n is the number of calibration samples; k is the number of wavelengths (or factors) used for the initial calibration; y[i] is the reference laboratory results for each ith sample;
and Å·[i] is the predicted results for each ith sample.
The confidence limit for the predicted value is then computed as shown in equation 3 (11).
Where CL[Ŷ] indicates the confidence limits for the predicted value based on the standard error of calibration, the t value is for n - k - 1 degrees of freedom, and the Mahalanobis distance (D) of
the predicted sample is used. For this simulation one may substitute 0 or 1 for the D value. In our example case, t = 2.78 for n - 2 = 4 degrees of freedom and 95% confidence for a two-tailed α =
0.05 test. The predicted value confidence limit is posted within Table V, indicating that all changes had a significant effect on the predicted values. The confidence limit for any predicted value is
computed to be ±0.04 for our example.
Confidence Limits for Slope from a Regression
The confidence limits for the slope of a regression are computed as shown in equations 4 and 5. First we compute the standard error of the slope as shown in equation 4.
For this equation n is the number of calibration samples; x[i] is the reference laboratory results for each ith sample; and x[i] is the mean reference value for all samples, that is, the mean
concentration value. So yi is the dependent variable; x[i] the independent variable; and n is the number of samples. Note: y[i ]and Å·[i] are as in equation 2.
The confidence limit for the slope of the regression line is then computed as shown in equation 5.
Where CL[slope] gives the confidence limits for the slope value based on the standard error of the slope, the t value is for n - 2 degrees of freedom. For this example, t = 2.78 for n - 2 = 4 degrees
of freedom and 95% confidence for a two-tailed α = 0.05 test. The slope confidence limit is posted within Table V indicating that offset and a slight wavelength shift of -0.25 nm have no effect on
the slope value; all other changes had a significant effect on the slope values. The confidence limit for the slope is computed to be ±0.003 for our example.
Confidence Limits for Intercept or Bias from a Regression
The confidence limits for the intercept or bias of a regression are computed as shown in equations 2 and 6. First we compute the standard error of a validation set or alternatively the standard error
of the calibration as shown in equation 2. The confidence limit for the intercept or bias value is then computed as shown in equation 6.
Where CL[intercept] gives the confidence limits for the intercept value based on the standard error of calibration or prediction, the t value is for n -2 degrees of freedom. For our example case, t
= 2.78 for n - 2 = 4 degrees of freedom and 95% confidence for a two-tailed α = 0.05 test. The intercept confidence limit is posted within Table V indicating that all effects (other than a -0.25 nm
wavelength shift) had a significant effect on the intercept values. The confidence limit for the intercept is computed to be ±0.011 for our example.
Summary of Results
Note that in our univariate regression example that most changes to a spectrum had a serious effect on intercept and slope values, even though in this example only the peak height would change the
prediction results. In the multivariate case, not only peak height but also spectral shape would have a more profound effect on predicted values. Some spectral changes may be partially mitigated by
changing the intercept or bias for each prediction equation, although changing the bias does not bring the model into conformance with the spectral changes. We note that more complex multivariate
calibration is partially able to compensate for some instrument differences that cause spectral changes; however, the same prediction results will not arise from intercept and slope corrections when
spectra are significantly different; recalibration is a requirement when spectra are significantly different between instruments.
We note for the univariate example shown that we have demonstrated some of the issues and mathematics used to assess calibration transfer when spectral profiles are different from one instrument to
another. This instrument difference problem is magnified when the specific type of instrument design is not matched, such as dispersive grating versus Fourier transform-based interferometer
spectrophotometers versus diode array, and so on. This first column, demonstrating a univariate case on the subject of intercept and slope changes caused by typical instrumental differences,
demonstrates the concepts and mathematics that will be explored in future discussions of this topic.
(1) H. Mark and J. Workman Jr., Spectroscopy22(6), 14-22 (2007).
(2) H. Mark and J. Workman Jr., Spectroscopy 28(2), 24-37 (2013).
(3) J. Workman Jr. and H. Mark, Spectroscopy 28(5), 12-25 (2013).
(4) J. Workman Jr. and H. Mark, Spectroscopy 28(6), 28-35 (2013).
(5) J. Workman Jr. and H. Mark, Spectroscopy 28(10), 24-33 (2013).
(6) J. Workman Jr. and H. Mark, Spectroscopy 29(6), 18-27 (2014).
(7) J. Workman Jr. and H. Mark, Spectroscopy 29(11), 14-21 (2014).
(8) H. Mark and J. Workman Jr., Spectroscopy 27(10), 12-17 (2012).
(9) H. Mark and J. Workman Jr., Spectroscopy 29(2), 24-37 (2014).
(10) H. Mark, Principles and Practice of Spectroscopic Calibration, 1st Ed. (Wiley, New York, 1991), pp. 7, 14, 15.
(11) ASTM E1655-05(2012), “Standard Practice for Infrared Multivariate Quantitative Analysis,” (American Society for Testing and Materials [ASTM] International, West Conshohocken, Pennsylvania,
Jerome Workman Jr. serves on the Editorial Advisory Board of Spectroscopy and is the Executive Vice
President of Engineering at Unity Scientific, LLC, in Brookfield, Connecticut. He is also an adjunct professor at U.S. National University in La Jolla, California, and Liberty University in
Lynchburg, Virginia. His e-mail address is: JWorkman04@gsb.columbia.edu
Howard Mark serves on the Editorial Advisory Board of Spectroscopy and runs a consulting service, Mark Electronics, in Suffern, New York. He can be reached via e-mail: hlmark@nearinfrared.com | {"url":"https://www.spectroscopyonline.com/view/optimizing-regression-model-challenge-intercept-bias-and-slope-correction","timestamp":"2024-11-04T05:39:18Z","content_type":"text/html","content_length":"475158","record_id":"<urn:uuid:cbd6e9d0-dd7e-4d98-894a-ee2c3f1de390>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00334.warc.gz"} |
[Solved] A particle moving along the x axis in sim | SolutionInn
A particle moving along the x axis in simple harmonic motion starts from its equilibrium position, the
A particle moving along the x axis in simple harmonic motion starts from its equilibrium position, the origin, at t = 0 and moves to the right. The amplitude of its motion is 2.00 cm, and the
frequency is 1.50 Hz.
(a) Show that the position of the particle is given by x = (2.00cm) sin (3.00 t) Determine
(b) The maximum speed and the earliest time (t > 0) at which the particle has this speed,
(c) The maximum acceleration and the earliest time (t > 0) at which the particle has this acceleration, and
(d) The total distance traveled between t = 0 and t = 1.00 s.
Fantastic news! We've Found the answer you've been seeking! | {"url":"https://www.solutioninn.com/particle-moving-along-the-x-axis-in-simple","timestamp":"2024-11-04T21:38:24Z","content_type":"text/html","content_length":"81556","record_id":"<urn:uuid:7a964fa2-a466-41a3-9b51-5c6bbcb3a372>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00662.warc.gz"} |
Statement 1:sound waves cannot propagate through vacuum but lig-Turito
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Statement 1:Sound waves cannot propagate through vacuum but light waves can
Statement 2:Sound waves cannot be polarised but light waves can be polarized
A. Statement 1 is True, Statement 2 is True; Statement 2 is correct explanation for Statement 1
B. Statement 1 is True, Statement 2 is True; Statement 2 is not correct explanation for Statement 1
C. Statement 1 is True, Statement 2 is False
D. Statement 1 is False, Statement 2 is True
The correct answer is: Statement 1 is True, Statement 2 is True; Statement 2 is not correct explanation for Statement 1
Sound waves cannot propagate through vacuum because sound waves are mechanical waves. Light waves can propagate through vacuum because light waves are electromagnetic waves. Since sound waves are
longitudinal waves, the particles moves in the direction of propagation, therefore these waves cannot be polarised
Get an Expert Advice From Turito. | {"url":"https://www.turito.com/ask-a-doubt/physics--statement-1-is-false-statement-2-is-true-statement-1-is-true-statement-2-is-false-statement-1-is-true-stat-qd72390","timestamp":"2024-11-14T12:13:45Z","content_type":"application/xhtml+xml","content_length":"536753","record_id":"<urn:uuid:f77da84e-0949-41a6-9ba7-5076e4f757f8>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00801.warc.gz"} |
Only released in EOL distros:
Package Summary
Library for computing transformations in arbitrary graph structures.
Use GitHub to report bugs or submit feature requests. [View active issues]
transform_graph is a library for computing transformations between coordinate frames in an arbitrary graph structure.
• transform_graph is not a distributed system. Programmers simply create and use the transform_graph graph as an object in memory.
• transform_graph does not keep track of transformations over time.
• The transform graph does not have to be structured in a tree. Instead, frames can be arranged in an arbitrary graph (weakly connected, cyclic, disconnected, etc.)
• transform_graph does not depend on ROS except to convert from common message types. As a result, you do not need to run a ROS master to use transform_graph, and it is suitable for use in pure
unit tests.
Quick start
The library's generated documentation explains how to use transform_graph in detail. Below are a few quick examples illustrating how it can be used.
transform_graph::Graph maintains the graph of transformations and is the primary interface to transform_graph:
1 #include "transform_graph/transform_graph.h"
3 int main(int argc, char** argv) {
4 transform_graph::Graph graph;
5 return 0;
6 }
Add frames to the graph using transform_graph::Graph::Add:
1 transform_graph::Graph graph;
3 geometry_msgs::Pose torso_pose;
4 pose.position.z = 0.4;
5 pose.orientation.w = 1;
7 graph.Add("torso_lift_link", transform_graph::RefFrame("base_link"), torso_pose);
Get points in different frames using transform_graph::Graph::DescribePosition. In this example, we want to know what a point 10 cm in front of the robot's wrist is, expressed in the base frame:
1 geometry_msgs::Point pt;
2 pt.x = 0.1;
3 transform_graph::Transform pt_in_base;
4 graph.DescribePosition(pt, transform_graph::Source("wrist"), transform_graph::Target("base_link"), &pt_in_base);
5 Eigen::Vector3d v = pt_in_base.vector(); | {"url":"https://wiki.ros.org/transform_graph","timestamp":"2024-11-05T10:34:09Z","content_type":"text/html","content_length":"33300","record_id":"<urn:uuid:0c59a981-417b-4c17-a171-13df43a1d7b7>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00010.warc.gz"} |
10 Strategies for Crafting a First-Class Advanced Mathematics Assignment10 Strategies for Crafting a First-Class Advanced Mathematics Assignment
If you want to make an advanced mathematics assignment, thankyourself for landing on this guide. Advanced mathematics involves high-level courses including calculus, AP calculus, and statistics. You
can write in number theory, combinatorics and differential equations.
According to the University of Nottingham, almost 700014 students pursue undergraduate mathematics programs yearly. Another research report published by MEIstates that 48.0% of males and 48.5% of
females achieved A+ and A grades in advanced-level mathematics in 2022. So, this is the number that reflects students’ interest in this course.
Although mathematics is challenging, it holds good job opportunities in the emerging market. Nowadays, companies are looking for quantitative and analytical skill holders who can solve complex tasks
easily. Its practical implementation includes operational research, as a statistical analyst in metrology and engineering designs.
Hence, students learning advanced mathematics courses will hunt and secure excellent future opportunities. For this, they must attempt different assignments and get top grades.
Writing an assignment on this logical subject is a buzzing task. Students get stuck solving complex problems, outlining the structure and reviewing assignments before submission. To overcome this
pressure, they often look forward to seeking aid from experienced assignment writers. They better know how to structure, and synchronise the whole stuff to achieve high grades in a course.
Let us read more about advanced mathematics and learn effective strategies to write a brilliant assignment.
What Is Considered Advanced Mathematics?
Advanced mathematics involves advanced skills and knowledge such as:
• Complex and abstract concepts.
• Differential equations, calculus, derivation, integration.
• Understanding the quadratic, polynomials and nonlinear equations.
• Advanced subjects i.e., pre-calculus, calculus, AP statistics, IB Math SL, HL, and further math.
Making an advanced mathematics assignment requires a keen effort with a logical perspective. You can begin with the smaller scale problems to a larger scale. Write in an expository style and let the
readers understand quickly.
Top 10 Strategies to Write Successful Advanced Mathematic Assignments
You should be subject-specific in writing an advanced mathematics assignment. Always think out of the box and craft the analytical and logically ordered information. For making rewarding assignments,
see these proven ten strategies.
These strategies will help you impress your supervisor a lot. Let us read them for enhanced productivity.
1. Thoroughly Read the Instructions
First, must read the instructions for making an advanced mathematics assignment. The guidelines will give the right direction to write effectively. Every institute has its own assignment writing
criteria. You can see the below-mentioned guidelines provided by RIT (Rochester Institute of Technology):
• Mention the topic of your assignment. For example, write it like Differential Equations, Assignment # 2, due date 5, Feb 2024.
• Try to use the proper formatting. Don’t squeeze the problems together instead write one below the other. Don’t write the problems next to each other as it will cause ambiguity.
• Never invent your notions and abbreviations. Use the proper mathematical symbols which are meaningful. For example, for percentage use %, integration symbol use∫.
By carefully reading all the instructions, the writing process becomes easier. The little you need to do is put effort into understanding the task, instructions and institute criteria. After properly
understanding the task, proceed to the next step i.e., discussion with fellows.
2. Collaborate With Fellows
Mathematics is not a theoretical subject which requires only a literature review. Instead, it demands deep and critical thinking to understand the problems. From understanding to deriving solutions
you need to conduct a group study.
Collaboration opens new ways of thinking and analysing. Sit and discuss topic details with peers, especially for making an advanced mathematics assignment. Suppose you are making an assignment on
“Fundamental theorems of Calculus” then must discuss:
• The computing difference between antiderivative at the upper limit and lower limits of integration.
• Discuss how to calculate slopes.
• The area under the graph.
Get a clear understanding of the “fundamental theorem of calculus” by seeing the solved example.
3. Begin With Challenging Problems
Think about the challenging word problems for writing an advanced mathematics assignment. Working on a difficult problem aims to impress your supervisor, which awards excellent grades. For example,
applications of integration are a challenging problem to start. You can research and write about its practical implementation in different subjects including,
• Economics and business
• Physics and engineering
• Probability and statistics
• Computer graphics and animations.
• Differential equations.
Other than this, see the different topics that will help create a strong impression.
Advanced Mathematics Assignment Topics for 2024
Choosing a topic is time-consuming and requires extensive research. We care about the students and therefore crafted a list of topics. Some important and innovative topics for 2024 are mentioned
• Exploring the Linear Approximation of Vector-Valued Functions of Several Variables.
• Implementing Integration in Real-World Scenarios.
• Solving the Problems Using PDFs (Probability Density Functions).
• Using Integration to Understand the Demand and Supply Curves.
4. Research From Online Sources
Comprehend your advanced mathematics assignment ideas by researching them from online resources. There are different websites for conducting mathematics research, including,
1. MathSciNet
2. Scopus
3. Google Scholar
4. European Mathematical Society
5. Mathematics Matter
Explore different guidebooks and research journals to solve exercise and statement problems. Consume as much information as you can from the above-mentioned credible online resources. Besides this,
you can also see some advanced mathematics assignment examples proposed by Southern University of Baton Rouge,
• The derivative matrix in advanced calculus.
• Surface integrals and theorems of Green.
• Multiple integrals and line integrals.
• Stokes divergence theorem.
5. Review Class Material
For the advanced mathematics assignment writing, the best practice is to review your class material. Take the class notes, lectures, and modules and relate them to your assignment topic. The
advantage of this strategy is that it will refresh your mind. Moreover, it gives clues to the problem where you feel stuck.
For example, understanding the theories, formulas and methods will make your assignment more manageable. Therefore, keeping your lecture notes with you while writing an assignment is always advised.
6. Avoid Distractions
Mathematics is a tricky and critical subject which involves multi-step solving. The subject becomes more sensitive when it is studied on an advanced level. Therefore, you must eliminate all the
distractions while composing an advanced mathematics assignment.
Distractions could be the sources that restrain you from being focused. There are different sources of distraction i.e., internal and external. External sources may include text messages, calls,
music and social interactions. Internal sources that may also affect the writing process are fatigue, illness and hunger.
Besides this, mobile phones are the biggest distraction for students today. They have to stay in touch with the supervisor, internet and institute updates using it. You can manage to check the
relevant updates before you start writing. So, keeping your mobile phone aside while doing an advanced mathematics assignment is advised.
7. Use Tricks to Solve Problems
Never stress the length of the equation instead execute the asked question. For instance, see the sample below to understand the solution to the equation.
Image Source: CUEMATH
You will experience different problems while making assignments on advanced mathematics. At that stage, don’t panic just follow a good methodology to derive the exact solution. You can solve the
equations by following the step-wise procedure:
• Divide the equation into two halves.
• Analyse which half can be solved first.
• Think about the method of solving.
• Use the short trick formula to derive the answer.
• Then combine it with the other part of the equation.
• Relate the both sides and find the answer.
By implementing all these tricks, you can solve the complexities of questions. This will save you time and let you craft a polished assignment.
8. Cite the Resources
To avoid plagiarism in your advanced mathematics assignment, add proper referencing. Referring to the source of gathered information proves the authenticity of your assignment. Every university or
institute refers to an appropriate citation style.
Students have to adhere strictly to the advised citations. There are various citation styles for quoting the references such as:
• Vancouver.
• Chicago.
• APA.
• MLA.
• Harvard.
Adding references to your assignment strengthens your research and acknowledges other people’s work. Moreover, it helps the reader understand how you have used others’ ideas and arguments in your
research work. From here they get satisfied by the quality of information displayed in your advanced mathematics assignment.
9. Use Appropriate Fonts
Using appropriate fonts increases the readability of your assignment. It reflects the clarity of ideas and solving problems. Moreover, it also promotes the
• Legibility.
• Communicates the message.
• Informs about the tone.
The criteria regarding the usage of fonts in writing an advanced mathematics assignment differ. You have to use the fonts recommended by your institute.
You can see the prescribed fonts provided by the University of Exeter:
● Latin Modern (the LaTeX defaults in Xetex)
• Times font for text and math.
• Lucida
• Minion
• Cambria
• Palatino fonts for text and math.
10. Review Your Assignment
Students miss the reviewing part of completing assessments for advanced mathematics. In this section, you must identify your assignment’s weak areas. Spot out all the grammatical mistakes in the
text, and check the punctuation and sentence structuring. Other than this, match the solutions to the problems you have written in your assignment.
Check the fonts, margins, and formatting; they are well-organised. Ensure all information written in an assignment is updated, authentic and equally covered. Keep in mind that when reviewing your
assignment, you must take breaks. Taking breaks lets you refresh your mind and think more logically about the equations.
What Is the Difference Between Mathematics and Advanced Mathematics?
Mathematics Advanced Mathematics
Basic or standard mathematics involves standard mathematical ideas such as In contrast, further or advanced mathematics broadens the ideas in applied domains. The main courses in advanced
TrigonometryAlgebra Calculus. Fractions.ShapesPercentages, ratios and exponents. mathematics include Pre-calculus Decision mathematics. Differential equations Linear algebra with applications.
What Is the Highest Level of Mathematics?
The highest mathematics level is a doctoral degree, a PhD in Mathematics. This degree opens various areas such as,
1. Topology
2. Probability
3. Partial differential equations
4. Numerical analysis
5. Fluid dynamics
6. Algebraic geometry
7. Quantum dynamics
8. Statistics
9. Number theory.
10. Mathematical biology.
Crafting an advanced mathematics assignment is a prolonged process. Implementing the above-mentioned ten strategies can boost your assignment. Using tricky formulas, and referencing information
resources strengthens the credibility of your assignment. Make sure that readers will find your assignment valuable. Supervisors analyse the assignments and give excellent grades considering the
efforts of students.
When it comes to excelling in mathematics you need to write effective assignments. Many students struggle to visualise the concepts and relate problems to real-world scenarios. In this scenario,
students badly suffer from fear of getting poor grades. So, to manage all these things they tend to avail of advanced mathematics assignment writing help from experts. They have a team of highly
qualified and experienced writers providing exceptional services.
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Jean Tshimanga Ilunga
The method of conjugate gradients (CG) is widely used for the iterative solution of large sparse systems of equations $Ax=b$, where $A\in\Re^{n\times n}$ is symmetric positive definite. Let $x_k$
denote the $k$–th iterate of CG. In this paper we obtain an expression for $J_k$, the Jacobian matrix of $x_k$ with respect to $b$. We use … Read more | {"url":"https://optimization-online.org/author/jtshimanga/","timestamp":"2024-11-04T05:51:07Z","content_type":"text/html","content_length":"88384","record_id":"<urn:uuid:5ac1961e-7583-4930-b9fb-c6bb2e46f726>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00267.warc.gz"} |
gfscompare3D (1) manual
These programs follow the usual GNU command line syntax, with long options starting with two dashes (`-'). A summary of options is included below.
-x, --mixed
Compute error only in mixed cells.
-m V, --min=V
Set minimum of color scale to V (used with -S).
-M V, --max=V
Set maximum of color scale to V.
-a, --abs
Output the absolute value of the error field.
-C, --constant
Apply a constant shift to one of the field, minimizing the error between the two fields (useful for pressure).
-w, --not-weighted
Do not use area-weighted norm estimation.
-c, --centered
Use error estimation for cell-centered variables.
-p P, --period=P
Shifts FILE1 by P along the x axis.
-H, --histogram
Output (error,volume) pairs for each cell used to compute the error norms.
-o, --output
Output a GTS representation of the error field.
-S, --squares
Output an OOGL representation of the error field.
-G, --gnuplot
Output a gnuplot representation of the error field.
-t, --triangulate
Use center of mass triangulation.
-l, --log
Output the log10 of the absolute value of the error field.
-f L, --full=L
Compare only leaf cells descendants of a cell full at level L or all full leaf cells if L = -1.
-r, --refined
Display error norm on the finest grid.
-n, --nocheck
Do not check solid fractions.
-g C, --gradient=C
Use the C component of the gradient of VAR.
-v, --verbose
Display difference statistics and other info.
-h, --help
Display the help and exit. | {"url":"http://carta.tech/man-pages/man1/gerris-gfscompare3D.1.html","timestamp":"2024-11-07T09:50:41Z","content_type":"text/html","content_length":"12415","record_id":"<urn:uuid:ebc64899-f633-4a18-aeee-836d668ed889>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00598.warc.gz"} |
Information Bottleneck
In a [[induction-deduction framework]] Induction, Deduction, and Transduction , for a given training dataset
$$ \{X, Y\}, $$
a prediction Markov chain^1
$$ X \to \hat X \to Y, $$
where $\hat X$ is supposed to be the minimal sufficient statistics of $X$. $\hat X$ is the minimal data that can still represent the relation between $X$ and $Y$, i.e., $I(X;Y)$, the [[mutual
information]] Mutual Information Mutual information is defined as $$ I(X;Y) = \mathbb E_{p_{XY}} \ln \frac{P_{XY}}{P_X P_Y}. $$ In the case that $X$ and $Y$ are independent variables, we have $P_{XY}
= P_X P_Y$, thus $I(X;Y) = 0$. This makes sense as there would be no “mutual” information if the two variables are independent of each other. Entropy and Cross Entropy Mutual information is closely
related to entropy. A simple decomposition shows that $$ I(X;Y) = H(X) - H(X\mid Y), $$ which is the reduction of … between $X$ and $Y$. There are competing effects in this framework:
1. On one hand, as an induction process, the less complexity of the representation the better, i.e., smaller $R\equiv I(X;\hat X)$.
2. However, if we are too extreme and come up with a $\hat X$ that is too simple, we reach a very small $R$ but we lose the effectiveness in the deduction process. We can not make good predictions.
The deduction process requires the “preserved relevant information”, $I_Y\equiv hat X;Y$ to be large.
An optimal representation $\hat X$ should minimize the following Lagrangian^1
$$ \mathcal L &= R - \beta I_Y \\ &= I(X;\hat X) - \beta I(\hat X;Y), $$
where $\beta$ is Lagrange multiplier.
To see that this is an Lagrangian, this Lagrangian is equivalent to ^1
$$ \tilde{\mathcal L} = I(X;\hat X) + \beta I(X;Y\vert \hat X) $$
That is we are looking for a $\hat X$ that minimizes the mutual information between $X$ and $\hat X$, $I(X;\hat X)$, but under the constraint $I(X;Y\vert \hat X)=0$, where $I(X;Y\vert\hat X)$ is the
mutual information between $X$ and $Y$ but conditioned on $\hat X$. Then $\beta$ is our Lagrange multiplier (see this chart).
Planted: by L Ma;
L Ma (2022). 'Information Bottleneck', Datumorphism, 04 April. Available at: https://datumorphism.leima.is/wiki/learning-theory/information-bottleneck/. | {"url":"https://datumorphism.leima.is/wiki/learning-theory/information-bottleneck/?ref=footer","timestamp":"2024-11-12T03:47:42Z","content_type":"text/html","content_length":"115945","record_id":"<urn:uuid:240ef0bf-33d9-461c-9517-3bc10d1aa953>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00803.warc.gz"} |
1D-2D Hydraulic Modeling of a Diversion Channel on the Cavally River in Zouan-Hounien, Cote d’Ivoire
1D-2D Hydraulic Modeling of a Diversion Channel on the Cavally River in Zouan-Hounien, Cote d’Ivoire ()
1. Introduction
Intensification of human activities on watersheds in general is one of the factors that favor erosion phenomena and roughness modification as well as the section of rivers [1]. Among these
activities, mining activities and infrastructure construction (dams, bridges, diversion of waterways, etc.) occupy a prominent place. Therefore, an understanding of these developments through
hydrological and hydraulic studies is necessary to ensure the protection of the environment in general and hydro systems in particular [2] [3] [4] [5] [6]. To provide precise answers to their
understanding, several studies have already been conducted using different hydraulic models [7]. Among the main anthropogenic causes of the modification of the functioning of rivers, some authors
such as Alexeevsky et al. (2013) [8] and Maio et al. (2013) [9] underlined the preponderant role of the construction of infrastructures or their suppression on the modification of river geometry.
Côte d’Ivoire has four major rivers (Cavally, Sassandra, Bandama and Comoe). Their basins undergo anthropogenic activities of all kinds. The Cavally River, located on the border between Côte d’Ivoire
and Liberia, crosses the Ity Mining Company (SMI) exploitation zone in Zouan-Hounien. The Cavally River has a lot of meanders in this area and the stream bed is heavily disturbed by illegal miners.
Several hydraulic structures such as bridges and diversion channels are planned to be constructed on the watercourse namely the construction of a diversion channel on the river bed. Therefore, the
diversion of watercourses can have undesirable social and environmental consequences, namely the modification of social structures and the hydrological dysfunction of the watershed concerned. In
addition, whole villages may be engulfed or people may be forced to find new livelihoods or change their way of life [8] [10]. In view of all the consequences that such a project can generate,
in-depth studies must be carried out to size the diversion channel to ensure hydraulic conditions like the initial state of the watercourse. The main objective of this study is to establish a 1D-2D
hydraulic model to design a diversion channel capable of ensuring flow conditions hydraulically like the initial conditions of the watercourse.
2. Material and Methods
2.1. Study Area
The Cavally River is a lower cross border watershed between Guinea, Ivory Coast and Liberia. Located in the west of Côte d’Ivoire, the Cavally River begins in Guinea, in the North of Mount Nimba with
more than 1000 meters as approximate altitude (Figure 1). The lower watershed covers a complete area of 28,800 Sq. km at Tate hydrometric station located at 60 km from the mouth. Côte d’Ivoire
doesn’t possess but about 15,000 Sq. km of watershed [11]. In the framework of this study, the chosen outlet is the hydrometric station of Floleu located at downstream of the Ity station in the
Zouan-Hounien region. The low watershed has an area of 3647.53 Sq. km. The region of Zouan-Hounien is in the mountain region of Côte d’Ivoire; its relief is hilly. Zouan-Hounien is in the forest area
and its climate is the mountain climate with two seasons: one rainy season from May to October and one dry season from November to March. The annual average temperature is 25.6˚C. The annual average
precipitation is 1866 mm. The driest month is January with a precipitation of 15 mm. The most important precipitations are recorded in September and they are 357 mm in average [12].
2.1.1. Characterization of Derivation Area
The meander of the Cavally River that will be cut is located in the vicinity of the Ity mining company between kilometer point PK7 + 400 and PK10 + 400. The meander is 3 km long (Figure 2).
Figure 1. Study area of sub-watershed of Cavally River.
Figure 2. Implementation area of the diversion channel on the Cavally River bed.
2.1.2. Data Acquisition and Analysis
The measurements of water flows and the profiles of water levels between Ity station (upstream) and Floleu station (downstream) of the diversion channel were respectively carried out from July to
October 2015, in October 2018 (rainy season) and February 2019 (dry season). The flows of the river were measured by means of an Acoustic Doppler current profiler (ADCP). The measurements of the
profiles of water levels were realized by means of a differential Global Positioning System (GPS) models STONEX S8 Plus. Displacements on the river were carried out using an outboard motor boat made
for 26 km along the Cavally River with a step of 200 m between 2 points of measurements. A digital elevation model (DEM) with 90 m resolution was used for the determination of altitudes and the
slopes (https://cgiarcsi.community/data/srtm-90m-digital-elevation-database-v4-1/). This DEM was corrected by combining with a Lidar image of 1.30 m resolution using the Global Mapper version 15
triangulation tools.
2.2. Methods
2.2.1. Diversion Channel Design Criteria
The diversion channel was designed for a return period of 2 years with a flow rate of 240 m^3/s, after flood estimation by frequency analysis. The Cavally River hydrology was studied using records
from three hydrometric stations: Flampleu, Ity and Toulepleu (see Figure 1). Long term daily flow (1955-2001) and water level measurements were used.
The diversion channel was designed to not to increase in the water flow elevation of the river during the floods. To this end the channel will have approximatively similar cross section area as the
river in natural condition.
In order to avoid sediment deposition and erosion of the slab the average flow velocity in the channel was maintained between 0.5 m/s and 1.5 m/s as recommended by several authors [4] [13]. The
hydraulic parameters determined are presented by Equations (1), (2) and (3) and (Figure 3).
• Wet section (S)
• Wet perimeter (P)
• Hydraulic radius (R[h])
where: b: width of the channel bottom; y: depth; m: slope coefficient.
2.2.2. Modeling Water Surface Elevation by HEC-RAS
Frequency Analysis was used to select the law that best fits the flood estimation of the Ity station to obtain the most appropriate return period for the derivation channel sizing. The model allowed
understanding the evolution of the distribution of the hydraulic parameters which are among others the velocities, the water levels and the flows before the realization of the diversion channel while
considering the results of the frequency analysis.
For the purposes of this study, flows are considered non-permanent given the variation in hydraulic parameters (flows, water levels, depths, velocities) as a function of time. The hydraulic design of
the diversion channel was based on the simulation the flow using HEC-RAS 1D-2D hydrodynamic software for natural and modified conditions. The equations used in the case of a non-stationary flow are
among others the conservation of the mass (Equation (4)) and the conservation of the momentum (Equation (5)) [14].
• Mass conservation
The unsteady differential form of the mass conservation equation is:
$\frac{\partial H}{\partial t}+\frac{\partial \left(hu\right)}{\partial x}+\frac{\partial \left(hv\right)}{\partial y}+q=0$(4)
• Momentum conservation
$\begin{array}{l}\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=-g\frac{\partial H}{\partial x}+{v}_{t}\left(\frac{{\partial }^{2}u}{\partial {x}^{2}}+\
frac{{\partial }^{2}u}{\partial {y}^{2}}\right)-{c}_{f}u+fv\\ \frac{\partial u}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}=-g\frac{\partial H}{\partial y}+{v}_{t}\left
(\frac{{\partial }^{2}v}{\partial {x}^{2}}+\frac{{\partial }^{2}v}{\partial {y}^{2}}\right)-{c}_{f}v+fu\end{array}\right\}$(5)
where t is time, h is the water surface elevation, u and v are the velocities components in the x and y direction respectively, q is a source/sink flux term, g is the gravitational acceleration, v[t]
is the horizontal eddy viscosity coefficient, c[f] is the bottom friction coefficient, and f is the Coriolis parameter.
The mesh of the 2D model was realized on 50 m by 50 m. The Digital Elevation Model (DEM) was created by the fusion of an airborne LiDAR survey provided by SMI and a SRTM 90 digital elevation model.
The SRTM 90 was adjusted by −3.85 m to obtain the same reference system as the LiDAR.
A flood hydrograph was imposed as a boundary condition upstream during the calibration phase of the model over the period from January 01 to April 30, 1988 with a time step of one hour (1 hour) and a
longitudinal slope of 0.000194 m/m downstream. The results were validated over a dry period (01 January to 26 March 1983) and a wet period (18 July to 10 October 2015) by comparing simulated results
with field observations. These periods were chosen to understand the behavior of the model regardless of the season and the consistency of the data obtained on the watershed. The fit between the
predicted and observed values was evaluated using two functions: the Nash coefficient and the correlation coefficient given respectively by the Equations (6) and (7):
$Nash=1-\frac{\sqrt{\underset{i=1}{\overset{n}{\sum }}{\left({q}_{ci}-{q}_{oi}\right)}^{2}}}{\sqrt{\underset{i=1}{\overset{n}{\sum }}{\left({q}_{oi}-{\stackrel{¯}{q}}_{o}\right)}^{2}}}$(6)
$r=\frac{\underset{i=1}{\overset{n}{\sum }}\left({q}_{ci}-\stackrel{¯}{{q}_{c}}\right)\left({q}_{oi}-\stackrel{¯}{{q}_{o}}\right)}{\sqrt{\underset{i=1}{\overset{n}{\sum }}{\left({q}_{ci}-\stackrel{¯}
{{q}_{c}}\right)}^{2}\underset{i=1}{\overset{n}{\sum }}{\left({q}_{oi}-\stackrel{¯}{{q}_{o}}\right)}^{2}}}$(7)
In these two expressions, n represents the height of the sequence, q[oi] represents the observed flow for calculated flow for the pace time i in m^3/s; q[ci] is the calculated flow for the pace time
i in m^3/s; q[0] is the observed average flow in m^3/s.
3. Results and Discussion
3.1. Results
3.1.1. Diversion Channel Design
The diversion channel has a length of 280 m, the manning coefficient retained is n = 0.06 m^1/3·s^−1. The average speed allowed in the canal is V = 1.1 m/s, a bank fruit m = 2.5 m for a longitudinal
slope of the diversion channel S[f] = 0.0036 m/m, a base b = 25 m a mirror width B = 65 m; depth y = 7 m; a wet perimeter P = 54.3 m; a wet section S = 222 Sq m. The side slope of the diversion
channel: 2.5H: 1.0V.
3.1.2. Hydraulic Modeling before and after the Diversion Channel Completion
Figure 4 and Figure 5 show the results of the calibration and the validation of the model after variation of the Manning coefficient. The calibration of the model lead to Manning coefficient values n
= 0.052 m^1/3·s^−1 for the minor bed and 0.06 m^1/3·s^−1 for the major bed. The flows used are those of the year 1983, 1988 and 2015. The graphical observation (Figure 4 and Figure 5) shows a good
synchronism between the simulated values and the observed values. This good similarity between observed and simulated flows is evidenced by the numerical results, which show a strong correlation of
0.94 and a Nash coefficient of 0.92 for the calibration period. Nash values are 0.95; 0.83 and correlation values are 0.97
Figure 4. Comparison between observed and simulated flow rates after model calibration from 01 January to 30 April 1988.
Figure 5. Comparison between flows observed and simulated after the model validation from January 1 to March 26, 1983(a) and from July 18 to October 10, 2015(b).
and 0.85 for validations periods. This good correlation between the simulated flow rates and observed flows shows that the hydraulic model reproduces well the dynamics of the flows of the
3.1.3. Comparison of Cavally River Velocities before and after the Diversion Channel Completion
The velocities before and after the realization of the diversion channel are relatively low on the level of banks and high in the minor bed (Figure 6). These values in the same order of magnitude and
lie between 0.10 and 1.6 m/s. Velocities decrease considerably by the upstream towards downstream. The highest velocities are in the zone of cut of the meander. It should be also noted that under the
natural conditions there is an acceleration of the flow in concave banks at the entry of the meander.
3.1.4. Comparison of Water Levels before and after Diversion Channel Completion
The maps in Figure 7 show the difference in water level before and after the installation of the diversion channel. It can be noted that downstream of the canal the water levels remain almost in
their natural state channel. On the other hand, just upstream of the canal, water levels have decreased by about 0.2 m, resulting in a slight increase in the Cavally River’s hydraulic capacity at
this location.
3.1.5. Flood Propagation Area Model before and after the Diversion Channel Construction
The maps in Figure 8 show the propagation area of the Cavally River before the construction of the bypass channel for floods with the 20- and 100-years return period. With reference to this figure,
the water level varies from 262.1 to 262.4 m for the 20-year flood, and from 262.4 to 262.7 m for the 100-year flood in the meander cut-off zone (diversion channel).
Flooding areas changed slightly in the meander cutting zone and the water level decreased immediately downstream of the road by about 20 cm for both cases (Figure 9). In general, the difference in
water levels of the floodplain is almost negligible.
With regard to the results of the flood propagation model the most important depths were observed at the upstream zone of the section of the watercourse. The water level increases of 20 cm upstream
the channel (Figure 10).
Figure 6. Velocities distribution before (a) and after (b) diversion channel.
Figure 7. Difference in water surface elevation (c) between existing conditions (a) and future conditions (b).
Figure 8. Flood propagation area before the diversion channel construction: 100 years (a) and 20 years (b).
Figure 9. Flood propagation area after the diversion channel construction: 100 years (a) and 20 years (b).
Figure 10. Water level difference after the diversion channel construction20 years (a) and 100 years (b).
3.2. Discussion
As part of the design of the Cavally River diversion channel using a 1D-2D hydraulic model, studies were carried out to impose flow conditions like the watercourse. The geometric characteristics of
the bypass channel are justified by the iteration method. The two-year return period estimated for the sizing of the diversion channel is between the maximum flood interval (238 - 300 m^3/s) recorded
on the catchment and respects the initial conditions of the flows of Cavally River. Wang et al. (2010) [15] showedthat the design flow must be estimated with the largest flood in the watershed in
order to ensurebetter flow conditions.
The results obtained by the HEC-RAS model showed in general that the model reproduces well the conditions of flow of the river according to Nash coefficients which vary from 0.83 to 0.95 [6] [16].
The Manning coefficient used for the calibration and validation of the hydraulic model is 0.052 m^1/3·s^−1 for the minor bed and 0.06 m^1/3.s^−1 for the major bed. This value is high, but this is
quite acceptable considering the composition of the soils, gold panning activities in the stream bed as well as the nature of the existing vegetation at the level of Cavally River. The roughness of
the watercourse is a highly variable parameter that depends on the number of factors such as surface roughness, vegetation cover, channel irregularities, channel alignment [17]. The values obtained
in this study confirm those of many authors who indicate that the Manning-Strickler coefficient is generally high for highly anthropogenic streams through material extraction activities [18] [19].
The velocities before and after the derivation of the diversion channel are relatively low at the banks and high in the minor bed. It is important to note that under natural conditions there is an
acceleration of the flow in the concave bank at the entrance of the meander. After completion of the bypass channel speeds become slightly high. This could be due to layer gradients that increase
velocity vectors due to shape profiles and non-uniform velocity distribution at depth [20] [21].
Comparison of the water levels before and after the diversion of the river at Ity station showed that the Cavally River will not be really disturbed by this channel because the reduction of the water
level by 0.2 m will cause an increase practically negligible flow velocity. Moreover the decrease in the water level (0.2 m) will be insignificant compared to the total fluctuations of the river
which vary in a year from 6 to 7 m. The reduction of the water level is due to the diversion channel which contributes to a reduction of the fluctuations of water levels by minimizing the phenomena
of flooding [22] [23] [24]. Therefore, the hydraulic conditions upstream and downstream of the diversion channel would not be significantly different from the existing natural conditions.
4. Conclusion
The work presented in this article focuses on the realization of a hydraulic model to design a diversion channel capable of ensuring a hydraulic operation like the initial conditions (water levels,
flow and speed) of the Cavally River. About the diversion channel, it has a length of 280 m and to avoid an oversize of it, it has been dimensioned for a return period of 2 years with a flow rate of
240 m^3/s, an average speed of 1.1 m/s and a slope of 0.0036 m/m. Flow velocities in the Cavally River range from 0.1 to 1.6 m/s from upstream to downstream. Hydraulic conditions (water levels, flow
and velocity) in the channel after diversion will remain substantially like the natural state of the watercourse. The diversion channel will therefore have no significant impact on the hydraulic
operation of the Cavally River.
The authors acknowledge the financial and logistical support of the Ity Mining Company (SMI). | {"url":"https://www.scirp.org/journal/paperinformation?paperid=94591","timestamp":"2024-11-12T16:24:43Z","content_type":"application/xhtml+xml","content_length":"142957","record_id":"<urn:uuid:e1601a90-38d1-4345-bb0f-73d75b8039aa>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00658.warc.gz"} |
Introduction to Real numbers (10th class )Chapter-1,Notes,Definition,concept,important pointsIntroduction to Real numbers (10th class )Chapter-1,Notes,Definition,concept,important points
Hi friends and my dear students! In this post, I have covered Andhra Pradesh class -10 Introduction to Real numbers (chapter-1) important points After Reading Mathematics Real numbers (10th class )
Notes With important points, please do share it with your friends. You can Learn Maths for All Classes here.
Key concepts:
*The collection of counting numbers are called natural numbers. Set of natural numbers are denoted by 'N'
* The collection of natural numbers includes zero are called whole numbers. Set of whole numbers is denoted by W'
W=(0} U { 1,2,3,.. }
= {0,1,2,3,. }
* The collection of whole numbers includes negative numbers are called integers.
Set of integers are denoted by Z' or I
Z={...3,-2,-1,} U {0,1,2,3,.... }
= {.. 3,-2,-1,0,1,2,3,... }
Note: 1) The number O' is neither positive nor negative
2) There is no integer between any two consecutive integers.
Lemma: Lema is a proven statement used for proving another statement Dividend = Divisor x quotient + remainder Euclid's division lemma: for a pair of given positive integers 'a' and b' there exist
unique pair of integers q' and 'r' such that a=b q+ r where o ≤ r<b.
Eg: Given positive integers 17 and 6
Let a=17 and b=6
17 = 6x2+5 where 0≤5< 6
Steps to obtain HCF by Euclid's division lemma.
Step 1: Consider two positive integers 'a' and b' such that a>b
Step 2: by Euclid's lemma. To find whole numbers 'q' and 'r' such that a=b q+ r
Step 3: If 'r'=0 then b' is the HCF of a and b Step 4 If r + 0 then consider the division of b with the remainder r, continue this process till the remainder becomes zero
Eg:- Find HCF of 972 and 21 let a=972, b=21
by using Euclid's division lemma,
972=21 x 46+6
Here remainder is 6
By Euclid's division lemma
21= 6x3+3
Here remainder is 3.
By Euclid's division lemma,
6 =3x2+0
Now remainder is 'O' .
HCF of 972 and 21 is 3
Note: Euclid's division lemma is stated for only positive integers but it can be extended for all integers except 'O'
Prime number: A natural number N>1have exactly two distinct factors (1 and itself) are called prime numbers.
Eg: 2,3,5,7,11,13,..
Composite number: Natural number N(>1) have more than two factors are called composite numbers
Eg:- 4,6,8,9,10,....
Note: 1) There are infinite primes and composite numbers
2) 1 is neither prime nor composite
3) 2' is even prime
Co-prime: Two numbers are said to be co-primes it their HCF in 1
Ex: (2,3), (4,5)....
Factor: If a number a divides another number b exactly then 'a' is a factor of b'
Eg:- 2 divides 6 exactly so 2 is a factor of 6
Also Check
Introduction to Knowing Our Numbers Key Points
SSC (10th class) Trigonometry Exercise - 11.1 Solution
SSC(10th class) Trigonometry Exercise - 11.1 Solutions
FUNDAMENTAL THEOREM OF ARITHMETIC: Every composite number can be expressed as a product of primes and this factorisation is unique.
Eg:- 24= 2 x 2 x 2 x 3.
To obtain LCM and HCF of given number by prime factorization method.
Step 1: Let us consider given numbers be 'a' and b'
Step 2: Express a and b as product of prime powers
Step 3: HCF=product of the smallest power of each common prime factors
LCM=product of the greatest power of each prime factors of the numbers
Rational numbers:
The collection of numbers which are in the form p/q where p and q are integers and q≠ 0 are called rational numbers set of rational numbers is denoted by Q.
Eg:- 3 /2 ,-5 ,1/2
Every rational number can be written in the form of terminating decimal or non-terminating repeating decimals.
Between any two rationals there exist infinite rational numbers
A rational number between any two rational number a and b is a+b/2
Terminating decimals in rational numbers: Let x=p/q be a rational number such that the prime factorisation of is of the form 2^n x 5^m where n and m are non-negative integers. Then x has decimal
expansion which terminates.
Eg:- x=3/20=p/q
Q=20=2^2 x 5
Which is the form of 2^n x 5^m
3/20 is terminating decimals
Non terminating, recuring decimals in rational numbers: Let x=p/q be a rational number, such that the prime factorisation of q is not in the form of 2^n x 5^m where n and m are non-negative integers.
Then x has a decimal expansion, which in non-terminating repeating.
Eg:-1) x=23 /35=p/q
q= 35 = 5' x7 which is not in the form of 2^n x 5m
23/35 is non-terminating repeating decimal
2) x= 1/7 =p/q
Q=7 which not in the form of 2^n x 5^m
1/7= 0.142857.... is a non-terminating repeating decimal.
Irrational number: Number which cannot be written in the form of p/q are called irrational numbers. the set of irrational numbers are denoted by Q or S.
Eg: 0.101001000...,1.256789124569....
Every non-perfect square number is an irrational number.
Eg:- V2, 3, V6,.
Note:-1) 7 is an irrational number
2) An irrational number between a and b is√ab
Ex: irrational number between 2 and 3 is 2x3 = √ 6
Properties:- 1) Sum of a rational number and an irrational number is an irrational number Eg:- 2+√3,5+√7.
2)Difference of a rational and an irrational number is an irrational number
Eg;4 -V5, 4-V11
3) Product of a non zero rational and an irrational numbers is an irrational number
Eg:- 5√3,6√7 ,.. ...
4) Quotient of a non zero rational and an irrational number is an irrational number
Eg:- 5/√3, √7/4
5) the sum of the two irrational numbers need not be irrational.
Eg:- a = 3- √2,b = 3+ V2
Then a+b=3-V2 +3+ V2
= 6 (rational)
6) The product of two irrational numbers need not be irrational,
Eg:- a = √5,b =2V125
Then a x b=√ 5 x 2√125
= 2x5x 5
= 50 (rational)
Real numbers: The set of rational and irrational numbers together are called real numbers. set of real numbers are denoted by R=QUQ'
Logarithm: Logarithmic form and exponent form are two sides of the same coin. ie) every logarithmic form can be written in the exponent form and vice versa. The theory of logarithm is obtained from
the theory of indices.
If a>1. Then a^x increases as x increases. and If a<1, then a^x decreases as x decreases.
Eg:- 2^x increases if x increases
(1/2)^x decreases if x decreases.
Hence log[a ]^x is an increasing function if a>1 and log[a ]^x is a decreasing function if a<1
Logarithms are used for all sorts of
calculations in engineering, science, business and economics.
Defiinition:- If a^x = N then x = log when (a # 1) > 0 and N>0 for some a,N € R.
Eg:- log[2]^5, log[10] ^100 There are two systems of logarithms.
1) Common logarithms (or) briggs logarithms
2) 2) Natural logarithms (or) Naperian logarithms.
Common Logarithms: Logarithms to the base '10' are called common logarithms.
Eg:- log[10]^50, log[10] ^100
Natural logarithms: Logarithms to the base 'e' are called as natural logarithms. Eg:- log[e]^x, log[e] ^5
Here e is an irrational number also called exponential number the value of e=2.718 (Approximately)
Laws of logarithms:
First law: log[a]^ xy = log[a]^ x+ log[a]^y where x,y and 'a' are positive real numbers and a≠1 Proof:- Let log[a]^x, = m
and log[a]^y, = n
By definition; x= a^n.. (1)
y = a^m.... (2)
By multiplying 1 & 2; xy = a^n x a^m
Xy = a^m+n
log[a]^ xy = m+n
log[a]^ xy = log[a]^ x+ log[a]^y
hence proved
2) log[a]^ x/y = log[a]^ x-log[a]^y
3) log[a]^ 1= 0^
4) a^log[a]^m = m
log[a ]x^m =mlog[a]^x
characteristic and Mantissa: Consider a number N>0. Then let the value of log[10]^N consist of two parts One a integral part, The other a proper fraction. The integral part in called the
characternstic and the fractional (or) decimal part is called the mantissa. The mantissa is always lie between 0 and 1
Eg:- log[10]^15 = 1.176 Here 1 is characteristic 0.176 is mantissa
Note:-1) Characteristic of 'n' digited number is n-1
2) If the characteristic is n then the number of digits is n+1
0 comments: | {"url":"http://maths.grammarknowledge.com/2020/10/introduction-to%20-real%20numbers-class10-Notes-concept-important-points.html","timestamp":"2024-11-09T19:10:16Z","content_type":"application/xhtml+xml","content_length":"174009","record_id":"<urn:uuid:569cd8e8-9a2c-4622-b43a-1b11ee17c8e4>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00398.warc.gz"} |
The Smiths took out a 30-year mortgage. They will have to make 360 monthly payments on this mortgage. Solution: 30-years x 12 months = 360 monthly payments
If John ran a 5 kilometer marathon, how long was that in meters?
If John ran a 5-kilometer marathon, it was 5,000 meters long.
What is the sum of 6 feet 10 inches and 8 feet 9 inches?
The sum of 6 feet 10 inches and 8 feet 9 inches is 15 feet 7 inches.
What’s the conversion factor used to convert miles to yards?
The conversion factor used to convert miles to yards is 1 mile = 1,760 yards.
Change 75 millimeters to decimeters?
75 Millimeters = 0.75 Decimeters.
If a thermometer indicates 40 degrees Celsius, what’s the temperature in degrees Fahrenheit?
If a thermometer indicates 40 degrees Celsius, the temperature in degrees Fahrenheit is 104?F. | {"url":"https://weegy.com/Home.aspx?Id=ArchivePage&SpModeType=1&SpAccountId=&SpRow=3113001&SpLevel=4","timestamp":"2024-11-12T12:03:07Z","content_type":"application/xhtml+xml","content_length":"176201","record_id":"<urn:uuid:60712f5a-79c0-4fad-bb57-8b09a3927a28>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00363.warc.gz"} |
Estimating the noise variance
Next: Colored noise Up: Solution by weighting functions Previous: Nonuniqueness and instability
The simplest method of choosing v^2 in the plane and then choose some arbitrary fraction of it, say 10% of the average. Although this method worked in Figure 2, I prefer another. I chose median value
of v^2. (In other words, we conceptually prepare a list of the numbers v^2; then we sort the list from smallest to largest; and finally we choose the value in the middle. In reality, median
calculation is quicker than sorting.)
Notice that Figure 2 uses more initial crosstalk than Figure 1. Without the extra crosstalk I found that the first iteration worked so well, the second one was not needed. Thus I could not illustrate
the utility of nonlinear estimation without more crosstalk.
Next: Colored noise Up: Solution by weighting functions Previous: Nonuniqueness and instability Stanford Exploration Project | {"url":"https://sep.stanford.edu/sep/prof/pvi/uni/paper_html/node12.html","timestamp":"2024-11-11T06:47:49Z","content_type":"text/html","content_length":"5300","record_id":"<urn:uuid:95e2bfef-8050-46b5-8852-e4072ff5081f>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00227.warc.gz"} |
Observation of B <sup>±</sup> → ρ <sup>±</sup>ρ <sup>0</sup> decays
An overview is given of the first observation of the vector-vector (VV) decay mode Β ^+→ρ ^+ρ ^0. These decays produce final states where both ρ mesons are either longitudinally or transversely
Original language English
Article number 221801
Pages (from-to) 221801/1-221801/5
Journal Physical review letters
Volume 91
Issue number 22
Publication status Published - 2003 Nov 28
ASJC Scopus subject areas
• General Physics and Astronomy
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Zhang, J., Nakao, M., Abe, K., Abe, K., Abe, T., Adachi, I., Aihara, H., Akatsu, M., Asano, Y., Aso, T., Aulchenko, V., Aushev, T., Bahinipati, S., Bakich, A. M., Ban, Y., Banas, E., Behera, P. K.,
Bizjak, I., Bondar, A., ... Zürcher, D. (2003).
Observation of B ^± → ρ ^±ρ ^0 decays
Physical review letters
(22), 221801/1-221801/5. Article 221801.
Observation of B ^± → ρ ^±ρ ^0 decays.
/ Zhang, J.; Nakao, M.; Abe, K. et al.
Physical review letters
, Vol. 91, No. 22, 221801, 28.11.2003, p. 221801/1-221801/5.
Research output: Contribution to journal › Article › peer-review
Zhang, J, Nakao, M, Abe, K, Abe, K, Abe, T, Adachi, I, Aihara, H, Akatsu, M, Asano, Y, Aso, T, Aulchenko, V, Aushev, T, Bahinipati, S, Bakich, AM, Ban, Y, Banas, E, Behera, PK, Bizjak, I, Bondar, A,
Bozek, A, Bračko, M, Browder, TE, Casey, BCK, Chang, P, Chao, Y, Cheon, BG, Chistov, R, Choi, SK, Choi, Y, Choi, YK, Danilov, M, Dong, LY, Dragic, J, Drutskoy, A, Eidelman, S, Eiges, V, Enari, Y,
Fukunaga, C, Gabyshev, N, Garmash, A, Gershon, T, Gordon, A, Guo, R, Handa, F, Hara, T, Hastings, NC, Hayashii, H, Hazumi, M, Hinz, L, Hokuue, T, Hoshi, Y, Hou, WS, Hsiung, YB, Huang, HC, Igarashi,
Y, Iijima, T, Inami, K, Ishikawa, A, Itoh, R, Iwasaki, H, Iwasaki, M, Iwasaki, Y, Jang, HK, Kang, JH, Kang, JS, Kataoka, SU, Katayama, N, Kawai, H, Kawamura, N, Kawasaki, T, Kim, DW, Kim, HJ, Kim, H,
Kim, JH, Kim, SK, Kinoshita, K, Koppenburg, P, Korpar, S, Križan, P, Krokovny, P, Kulasiri, R, Kumar, S, Kuzmin, A, Kwon, YJ, Leder, G, Lee, SH, Lesiak, T, Li, J, Limosani, A, Lin, SW, Liventsev, D,
MacNaughton, J, Majumder, G, Mandl, F, Marlow, D, Matsumoto, H, Matsumoto, T, Matyja, A, Mitaroff, W, Miyabayashi, K, Miyata, H, Mohapatra, D, Mori, T, Nagamine, T, Nagasaka, Y, Nakadaira, T, Nakano,
E, Nam, JW, Natkaniec, Z, Nishida, S, Nitoh, O, Nozaki, T, Ogawa, S, Ohshima, T, Okabe, T, Okuno, S, Olsen, SL, Ostrowicz, W, Ozaki, H, Park, H, Park, KS, Parslow, N, Perroud, JP, Piilonen, LE,
Rozanska, M, Sagawa, H, Saitoh, S, Sakai, Y, Sarangi, TR, Satpathy, A, Schneider, O, Schümann, J, Schwanda, C, Schwartz, AJ, Semenov, S, Senyo, K, Seuster, R, Sevior, ME, Shibata, T, Shibuya, H,
Sidorov, V, Singh, JB, Stanič, S, Starič, M, Sugi, A, Sumisawa, K, Sumiyoshi, T, Suzuki, S, Suzuki, SY, Swain, SK, Takahashi, T, Takasaki, F, Tamai, K, Tamura, N, Tanaka, M, Taylor, GN, Teramoto, Y,
Tomura, T, Tovey, SN, Trabelsi, K, Tsuboyama, T, Tsukamoto, T, Uehara, S, Uno, S, Varner, G, Varvell, KE, Wang, CC, Wang, CH, Wang, JG, Wang, MZ, Watanabe, Y
, Won, E
, Yabsley, BD, Yamada, Y, Yamaguchi, A, Yamashita, Y, Yamauchi, M, Yanai, H, Yang, H, Yusa, Y, Zhang, ZP, Zheng, Y, Zhilich, V, Žontar, D & Zürcher, D 2003, '
Observation of B ^± → ρ ^±ρ ^0 decays
Physical review letters
, vol. 91, no. 22, 221801, pp. 221801/1-221801/5.
title = "Observation of B ± → ρ ±ρ 0 decays",
abstract = "An overview is given of the first observation of the vector-vector (VV) decay mode Β +→ρ +ρ 0. These decays produce final states where both ρ mesons are either longitudinally or
transversely polarized.",
author = "J. Zhang and M. Nakao and K. Abe and K. Abe and T. Abe and I. Adachi and H. Aihara and M. Akatsu and Y. Asano and T. Aso and V. Aulchenko and T. Aushev and S. Bahinipati and Bakich, {A. M.}
and Y. Ban and E. Banas and Behera, {P. K.} and I. Bizjak and A. Bondar and A. Bozek and M. Bra{\v c}ko and Browder, {T. E.} and Casey, {B. C.K.} and P. Chang and Y. Chao and Cheon, {B. G.} and R.
Chistov and Choi, {S. K.} and Y. Choi and Choi, {Y. K.} and M. Danilov and Dong, {L. Y.} and J. Dragic and A. Drutskoy and S. Eidelman and V. Eiges and Y. Enari and C. Fukunaga and N. Gabyshev and A.
Garmash and T. Gershon and A. Gordon and R. Guo and F. Handa and T. Hara and Hastings, {N. C.} and H. Hayashii and M. Hazumi and L. Hinz and T. Hokuue and Y. Hoshi and Hou, {W. S.} and Hsiung, {Y.
B.} and Huang, {H. C.} and Y. Igarashi and T. Iijima and K. Inami and A. Ishikawa and R. Itoh and H. Iwasaki and M. Iwasaki and Y. Iwasaki and Jang, {H. K.} and Kang, {J. H.} and Kang, {J. S.} and
Kataoka, {S. U.} and N. Katayama and H. Kawai and N. Kawamura and T. Kawasaki and Kim, {D. W.} and Kim, {H. J.} and Hyunwoo Kim and Kim, {J. H.} and Kim, {S. K.} and K. Kinoshita and P. Koppenburg
and S. Korpar and P. Kri{\v z}an and P. Krokovny and R. Kulasiri and S. Kumar and A. Kuzmin and Kwon, {Y. J.} and G. Leder and Lee, {S. H.} and T. Lesiak and J. Li and A. Limosani and Lin, {S. W.}
and D. Liventsev and J. MacNaughton and G. Majumder and F. Mandl and D. Marlow and H. Matsumoto and T. Matsumoto and A. Matyja and W. Mitaroff and K. Miyabayashi and H. Miyata and D. Mohapatra and T.
Mori and T. Nagamine and Y. Nagasaka and T. Nakadaira and E. Nakano and Nam, {J. W.} and Z. Natkaniec and S. Nishida and O. Nitoh and T. Nozaki and S. Ogawa and T. Ohshima and T. Okabe and S. Okuno
and Olsen, {S. L.} and W. Ostrowicz and H. Ozaki and H. Park and Park, {K. S.} and N. Parslow and Perroud, {J. P.} and Piilonen, {L. E.} and M. Rozanska and H. Sagawa and S. Saitoh and Y. Sakai and
Sarangi, {T. R.} and A. Satpathy and O. Schneider and J. Sch{\"u}mann and C. Schwanda and Schwartz, {A. J.} and S. Semenov and K. Senyo and R. Seuster and Sevior, {M. E.} and T. Shibata and H.
Shibuya and V. Sidorov and Singh, {J. B.} and S. Stani{\v c} and M. Stari{\v c} and A. Sugi and K. Sumisawa and T. Sumiyoshi and S. Suzuki and Suzuki, {S. Y.} and Swain, {S. K.} and T. Takahashi and
F. Takasaki and K. Tamai and N. Tamura and M. Tanaka and Taylor, {G. N.} and Y. Teramoto and T. Tomura and Tovey, {S. N.} and K. Trabelsi and T. Tsuboyama and T. Tsukamoto and S. Uehara and S. Uno
and G. Varner and Varvell, {K. E.} and Wang, {C. C.} and Wang, {C. H.} and Wang, {J. G.} and Wang, {M. Z.} and Y. Watanabe and E. Won and Yabsley, {B. D.} and Y. Yamada and A. Yamaguchi and Y.
Yamashita and M. Yamauchi and H. Yanai and Heyoung Yang and Y. Yusa and Zhang, {Z. P.} and Y. Zheng and V. Zhilich and D. {\v Z}ontar and D. Z{\"u}rcher",
year = "2003",
month = nov,
day = "28",
language = "English",
volume = "91",
pages = "221801/1--221801/5",
journal = "Physical review letters",
issn = "0031-9007",
publisher = "American Physical Society",
number = "22",
TY - JOUR
T1 - Observation of B ± → ρ ±ρ 0 decays
AU - Zhang, J.
AU - Nakao, M.
AU - Abe, K.
AU - Abe, K.
AU - Abe, T.
AU - Adachi, I.
AU - Aihara, H.
AU - Akatsu, M.
AU - Asano, Y.
AU - Aso, T.
AU - Aulchenko, V.
AU - Aushev, T.
AU - Bahinipati, S.
AU - Bakich, A. M.
AU - Ban, Y.
AU - Banas, E.
AU - Behera, P. K.
AU - Bizjak, I.
AU - Bondar, A.
AU - Bozek, A.
AU - Bračko, M.
AU - Browder, T. E.
AU - Casey, B. C.K.
AU - Chang, P.
AU - Chao, Y.
AU - Cheon, B. G.
AU - Chistov, R.
AU - Choi, S. K.
AU - Choi, Y.
AU - Choi, Y. K.
AU - Danilov, M.
AU - Dong, L. Y.
AU - Dragic, J.
AU - Drutskoy, A.
AU - Eidelman, S.
AU - Eiges, V.
AU - Enari, Y.
AU - Fukunaga, C.
AU - Gabyshev, N.
AU - Garmash, A.
AU - Gershon, T.
AU - Gordon, A.
AU - Guo, R.
AU - Handa, F.
AU - Hara, T.
AU - Hastings, N. C.
AU - Hayashii, H.
AU - Hazumi, M.
AU - Hinz, L.
AU - Hokuue, T.
AU - Hoshi, Y.
AU - Hou, W. S.
AU - Hsiung, Y. B.
AU - Huang, H. C.
AU - Igarashi, Y.
AU - Iijima, T.
AU - Inami, K.
AU - Ishikawa, A.
AU - Itoh, R.
AU - Iwasaki, H.
AU - Iwasaki, M.
AU - Iwasaki, Y.
AU - Jang, H. K.
AU - Kang, J. H.
AU - Kang, J. S.
AU - Kataoka, S. U.
AU - Katayama, N.
AU - Kawai, H.
AU - Kawamura, N.
AU - Kawasaki, T.
AU - Kim, D. W.
AU - Kim, H. J.
AU - Kim, Hyunwoo
AU - Kim, J. H.
AU - Kim, S. K.
AU - Kinoshita, K.
AU - Koppenburg, P.
AU - Korpar, S.
AU - Križan, P.
AU - Krokovny, P.
AU - Kulasiri, R.
AU - Kumar, S.
AU - Kuzmin, A.
AU - Kwon, Y. J.
AU - Leder, G.
AU - Lee, S. H.
AU - Lesiak, T.
AU - Li, J.
AU - Limosani, A.
AU - Lin, S. W.
AU - Liventsev, D.
AU - MacNaughton, J.
AU - Majumder, G.
AU - Mandl, F.
AU - Marlow, D.
AU - Matsumoto, H.
AU - Matsumoto, T.
AU - Matyja, A.
AU - Mitaroff, W.
AU - Miyabayashi, K.
AU - Miyata, H.
AU - Mohapatra, D.
AU - Mori, T.
AU - Nagamine, T.
AU - Nagasaka, Y.
AU - Nakadaira, T.
AU - Nakano, E.
AU - Nam, J. W.
AU - Natkaniec, Z.
AU - Nishida, S.
AU - Nitoh, O.
AU - Nozaki, T.
AU - Ogawa, S.
AU - Ohshima, T.
AU - Okabe, T.
AU - Okuno, S.
AU - Olsen, S. L.
AU - Ostrowicz, W.
AU - Ozaki, H.
AU - Park, H.
AU - Park, K. S.
AU - Parslow, N.
AU - Perroud, J. P.
AU - Piilonen, L. E.
AU - Rozanska, M.
AU - Sagawa, H.
AU - Saitoh, S.
AU - Sakai, Y.
AU - Sarangi, T. R.
AU - Satpathy, A.
AU - Schneider, O.
AU - Schümann, J.
AU - Schwanda, C.
AU - Schwartz, A. J.
AU - Semenov, S.
AU - Senyo, K.
AU - Seuster, R.
AU - Sevior, M. E.
AU - Shibata, T.
AU - Shibuya, H.
AU - Sidorov, V.
AU - Singh, J. B.
AU - Stanič, S.
AU - Starič, M.
AU - Sugi, A.
AU - Sumisawa, K.
AU - Sumiyoshi, T.
AU - Suzuki, S.
AU - Suzuki, S. Y.
AU - Swain, S. K.
AU - Takahashi, T.
AU - Takasaki, F.
AU - Tamai, K.
AU - Tamura, N.
AU - Tanaka, M.
AU - Taylor, G. N.
AU - Teramoto, Y.
AU - Tomura, T.
AU - Tovey, S. N.
AU - Trabelsi, K.
AU - Tsuboyama, T.
AU - Tsukamoto, T.
AU - Uehara, S.
AU - Uno, S.
AU - Varner, G.
AU - Varvell, K. E.
AU - Wang, C. C.
AU - Wang, C. H.
AU - Wang, J. G.
AU - Wang, M. Z.
AU - Watanabe, Y.
AU - Won, E.
AU - Yabsley, B. D.
AU - Yamada, Y.
AU - Yamaguchi, A.
AU - Yamashita, Y.
AU - Yamauchi, M.
AU - Yanai, H.
AU - Yang, Heyoung
AU - Yusa, Y.
AU - Zhang, Z. P.
AU - Zheng, Y.
AU - Zhilich, V.
AU - Žontar, D.
AU - Zürcher, D.
PY - 2003/11/28
Y1 - 2003/11/28
N2 - An overview is given of the first observation of the vector-vector (VV) decay mode Β +→ρ +ρ 0. These decays produce final states where both ρ mesons are either longitudinally or transversely
AB - An overview is given of the first observation of the vector-vector (VV) decay mode Β +→ρ +ρ 0. These decays produce final states where both ρ mesons are either longitudinally or transversely
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M3 - Article
AN - SCOPUS:80051549138
SN - 0031-9007
VL - 91
SP - 221801/1-221801/5
JO - Physical review letters
JF - Physical review letters
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M1 - 221801
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LibGuides: Artificial Intelligence (AI) and Information Literacy: Why AIs Are Bad at Math
Why AIs Are Bad at Math
LLMs (Large Language Models) are often poor with numbers because they are primarily trained on text data, which lacks the structured understanding of mathematical operations and symbols needed to
accurately solve math problems; they struggle to interpret the meaning behind mathematical notation and often rely on statistical patterns in text that may not translate to correct calculations,
leading to inaccurate answers even for simple arithmetic tasks.
Using AI as a Math Study Partner
Fact-checking AI
Now that you know some common errors that AI text generators make, how do we go about fact-checking AI outputs? Go to the next page in this guide to learn about fact-checking using lateral reading. | {"url":"https://libguides.westsoundacademy.org/artificial-intelligence-ai-and-information-literacy/why-ais-are-bad-at-math","timestamp":"2024-11-02T10:30:49Z","content_type":"text/html","content_length":"32202","record_id":"<urn:uuid:0de05222-cd52-407f-a689-330a6b3f0a71>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00016.warc.gz"} |
Subtract Two Vectors in R - Data Science Parichay
In this tutorial, we will look at how to subtract two vectors in the R programming language with the help of some examples.
Vector Subtraction in R
You can use the - operator to subtract two vectors in R. Arithmetic operations on vectors are computed element-wise. That is when you subtract two vectors, the corresponding elements are subtracted
If the vectors are of the same length, corresponding elements (elements with the same index) are subtracted together. The following image illustrates this.
If the vectors are of different lengths, the shorter vector will be recycled to match the length of the longer vector. The following image shows how this is done.
Let’s look at some examples of subtracting two vectors in R.
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Subtract two numeric vectors of the same length in R
First, let’s subtract two numeric vectors having the same length using the - operator.
# create two vectors
v1 <- c(1, 2, 3)
v2 <- c(0, 1, 2)
# subtract v2 from v1
v <- v1 - v2
# display the resulting vector
[1] 1 1 1
Here, we subtract two vectors of length three. The resulting vector is also of length three with each element resulting from the corresponding elementwise subtraction of vector v2 from vector v1.
Subtract two numeric vectors with different lengths in R
Let’s now subtract two numeric vectors having different lengths.
# create two vectors
v1 <- c(1, 2, 3)
v2 <- c(1, 1, 1, 1, 1)
# subtract v2 from v1
v <- v1 - v2
# display the resulting vector
Warning message in v1 - v2:
“longer object length is not a multiple of shorter object length”
[1] 0 1 2 0 1
Here, we subtract vector v2 (of length 5) from vector v1 (of length 3). You can see that the resulting vector is of length 5.
The shorter vector is recycled in order to compute the difference with the longer vector. The values 1 and 2 are recycled to the index four and five respectively. Note that we get a warning that the
longer vector’s length is not a multiple of the shorter vector’s length. This is why the shorter vector was not completely recycled, only the first two values were recycled to match the length of the
longer vector.
Let’s look at another example. Here, the longer vector’s length is a multiple of the shorter vector’s length.
# create two vectors
v1 <- c(1, 2, 3)
v2 <- c(1, 1, 1, 1, 1, 1)
# subtract v2 from v1
v <- v1 - v2
# display the resulting vector
[1] 0 1 2 0 1 2
We get the resulting vector of length six (same as the length of the longer vector). Also, note that we don’t get any warnings here because the shorter vector is recycled once completely to compute
the difference with the longer vector.
You might also be interested in –
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General relativity
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General relativity
General relativity (GR, also known as the general theory of relativity or GTR) is the geometric theory of gravitation published by Albert Einstein in 1915 and the current description of gravitation
in modern physics. General relativity generalizes special relativity and refines Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and
time, or spacetime. In particular, the curvature of spacetime is directly related to the energy and momentum of whatever matter and radiation are present. The relation is specified by the Einstein
field equations, a system of partial differential equations.
Some predictions of general relativity differ significantly from those of classical physics, especially concerning the passage of time, the geometry of space, the motion of bodies in free fall, and
the propagation of light. Examples of such differences include gravitational time dilation, gravitational lensing, the gravitational redshift of light, and the gravitational time delay. The
predictions of general relativity in relation to classical physics have been confirmed in all observations and experiments to date. Although general relativity is not the only relativistic theory of
gravity, it is the simplest theory that is consistent with experimental data. However, unanswered questions remain, the most fundamental being how general relativity can be reconciled with the laws
of quantum physics to produce a complete and self-consistent theory of quantum gravity.
Einstein's theory has important astrophysical implications. For example, it implies the existence of black holes—regions of space in which space and time are distorted in such a way that nothing, not
even light, can escape—as an end-state for massive stars. There is ample evidence that the intense radiation emitted by certain kinds of astronomical objects is due to black holes. For example,
microquasars and active galactic nuclei result from the presence of stellar black holes and supermassive black holes, respectively. The bending of light by gravity can lead to the phenomenon of
gravitational lensing, in which multiple images of the same distant astronomical object are visible in the sky. General relativity also predicts the existence of gravitational waves, which have since
been observed directly by the physics collaboration LIGO. In addition, general relativity is the basis of current cosmological models of a consistently expanding universe.
Widely acknowledged as a theory of extraordinary beauty, general relativity has often been described as the most beautiful of all existing physical theories.^[2]
Soon after publishing the special theory of relativity in 1905, Einstein started thinking about how to incorporate gravity into his new relativistic framework. In 1907, beginning with a simple
thought experiment involving an observer in free fall, he embarked on what would be an eight-year search for a relativistic theory of gravity. After numerous detours and false starts, his work
culminated in the presentation to the Prussian Academy of Science in November 1915 of what are now known as the Einstein field equations.^[3] These equations specify how the geometry of space and
time is influenced by whatever matter and radiation are present, and form the core of Einstein's general theory of relativity.^[4] The 19th century mathematician Bernhard Riemann's non-Euclidean
geometry, called Riemannian Geometry, provided the key mathematical framework which Einstein fit his physical ideas of gravity on, and enabled him to develop general relativity.^[5]
The Einstein field equations are nonlinear and very difficult to solve. Einstein used approximation methods in working out initial predictions of the theory. But as early as 1916, the astrophysicist
Karl Schwarzschild found the first non-trivial exact solution to the Einstein field equations, the Schwarzschild metric. This solution laid the groundwork for the description of the final stages of
gravitational collapse, and the objects known today as black holes. In the same year, the first steps towards generalizing Schwarzschild's solution to electrically charged objects were taken, which
eventually resulted in the Reissner–Nordström solution, now associated with electrically charged black holes.^[6] In 1917, Einstein applied his theory to the universe as a whole, initiating the field
of relativistic cosmology. In line with contemporary thinking, he assumed a static universe, adding a new parameter to his original field equations—the cosmological constant—to match that
observational presumption.^[7] By 1929, however, the work of Hubble and others had shown that our universe is expanding. This is readily described by the expanding cosmological solutions found by
Friedmann in 1922, which do not require a cosmological constant. Lemaître used these solutions to formulate the earliest version of the Big Bang models, in which our universe has evolved from an
extremely hot and dense earlier state.^[8] Einstein later declared the cosmological constant the biggest blunder of his life.^[9]
During that period, general relativity remained something of a curiosity among physical theories. It was clearly superior to Newtonian gravity, being consistent with special relativity and accounting
for several effects unexplained by the Newtonian theory. Einstein himself had shown in 1915 how his theory explained the anomalous perihelion advance of the planet Mercury without any arbitrary
parameters ("fudge factors").^[10] Similarly, a 1919 expedition led by Eddington confirmed general relativity's prediction for the deflection of starlight by the Sun during the total solar eclipse of
May 29, 1919,^[11] making Einstein instantly famous.^[12] Yet the theory entered the mainstream of theoretical physics and astrophysics only with the developments between approximately 1960 and 1975,
now known as the golden age of general relativity.^[13] Physicists began to understand the concept of a black hole, and to identify quasars as one of these objects' astrophysical manifestations.^[14]
Ever more precise solar system tests confirmed the theory's predictive power,^[15] and relativistic cosmology, too, became amenable to direct observational tests.^[16]
Over the years, general relativity has acquired a reputation as a theory of extraordinary beauty.^[2]^[17]^[18] Subrahmanyan Chandrasekhar has noted that at multiple levels, general relativity
exhibits what Francis Bacon has termed, a "strangeness in the proportion" (i.e. elements that excite wonderment and surprise). It juxtaposes fundamental concepts (space and time versus matter and
motion) which had previously been considered as entirely independent. Chandrasekhar also noted that Einstein's only guides in his search for an exact theory were the principle of equivalence and his
sense that a proper description of gravity should be geometrical at its basis, so that there was an "element of revelation" in the manner in which Einstein arrived at his theory.^[19] Other elements
of beauty associated with the general theory of relativity are its simplicity, symmetry, the manner in which it incorporates invariance and unification, and its perfect logical consistency.^[20]
From classical mechanics to general relativity
General relativity can be understood by examining its similarities with and departures from classical physics. The first step is the realization that classical mechanics and Newton's law of gravity
admit a geometric description. The combination of this description with the laws of special relativity results in a heuristic derivation of general relativity.^[21]
Geometry of Newtonian gravity
At the base of classical mechanics is the notion that a body's motion can be described as a combination of free (or inertial) motion, and deviations from this free motion. Such deviations are caused
by external forces acting on a body in accordance with Newton's second law of motion, which states that the net force acting on a body is equal to that body's (inertial) mass multiplied by its
acceleration.^[22] The preferred inertial motions are related to the geometry of space and time: in the standard reference frames of classical mechanics, objects in free motion move along straight
lines at constant speed. In modern parlance, their paths are geodesics, straight world lines in curved spacetime.^[23]
Conversely, one might expect that inertial motions, once identified by observing the actual motions of bodies and making allowances for the external forces (such as electromagnetism or friction), can
be used to define the geometry of space, as well as a time coordinate. However, there is an ambiguity once gravity comes into play. According to Newton's law of gravity, and independently verified by
experiments such as that of Eötvös and its successors (see Eötvös experiment), there is a universality of free fall (also known as the weak equivalence principle, or the universal equality of
inertial and passive-gravitational mass): the trajectory of a test body in free fall depends only on its position and initial speed, but not on any of its material properties.^[24] A simplified
version of this is embodied in Einstein's elevator experiment, illustrated in the figure on the right: for an observer in a small enclosed room, it is impossible to decide, by mapping the trajectory
of bodies such as a dropped ball, whether the room is at rest in a gravitational field, or in free space aboard a rocket that is accelerating at a rate equal to that of the gravitational field.^[25]
Given the universality of free fall, there is no observable distinction between inertial motion and motion under the influence of the gravitational force. This suggests the definition of a new class
of inertial motion, namely that of objects in free fall under the influence of gravity. This new class of preferred motions, too, defines a geometry of space and time—in mathematical terms, it is the
geodesic motion associated with a specific connection which depends on the gradient of the gravitational potential. Space, in this construction, still has the ordinary Euclidean geometry. However,
spacetime as a whole is more complicated. As can be shown using simple thought experiments following the free-fall trajectories of different test particles, the result of transporting spacetime
vectors that can denote a particle's velocity (time-like vectors) will vary with the particle's trajectory; mathematically speaking, the Newtonian connection is not integrable. From this, one can
deduce that spacetime is curved. The resulting Newton–Cartan theory is a geometric formulation of Newtonian gravity using only covariant concepts, i.e. a description which is valid in any desired
coordinate system.^[26] In this geometric description, tidal effects—the relative acceleration of bodies in free fall—are related to the derivative of the connection, showing how the modified
geometry is caused by the presence of mass.^[27]
Relativistic generalization
As intriguing as geometric Newtonian gravity may be, its basis, classical mechanics, is merely a limiting case of (special) relativistic mechanics.^[28] In the language of symmetry: where gravity can
be neglected, physics is Lorentz invariant as in special relativity rather than Galilei invariant as in classical mechanics. (The defining symmetry of special relativity is the Poincaré group, which
includes translations, rotations and boosts.) The differences between the two become significant when dealing with speeds approaching the speed of light, and with high-energy phenomena.^[29]
With Lorentz symmetry, additional structures come into play. They are defined by the set of light cones (see image). The light-cones define a causal structure: for each event A, there is a set of
events that can, in principle, either influence or be influenced by A via signals or interactions that do not need to travel faster than light (such as event B in the image), and a set of events for
which such an influence is impossible (such as event C in the image). These sets are observer-independent.^[30] In conjunction with the world-lines of freely falling particles, the light-cones can be
used to reconstruct the space–time's semi-Riemannian metric, at least up to a positive scalar factor. In mathematical terms, this defines a conformal structure^[31] or conformal geometry.
Special relativity is defined in the absence of gravity, so for practical applications, it is a suitable model whenever gravity can be neglected. Bringing gravity into play, and assuming the
universality of free fall, an analogous reasoning as in the previous section applies: there are no global inertial frames. Instead there are approximate inertial frames moving alongside freely
falling particles. Translated into the language of spacetime: the straight time-like lines that define a gravity-free inertial frame are deformed to lines that are curved relative to each other,
suggesting that the inclusion of gravity necessitates a change in spacetime geometry.^[32]
A priori, it is not clear whether the new local frames in free fall coincide with the reference frames in which the laws of special relativity hold—that theory is based on the propagation of light,
and thus on electromagnetism, which could have a different set of preferred frames. But using different assumptions about the special-relativistic frames (such as their being earth-fixed, or in free
fall), one can derive different predictions for the gravitational redshift, that is, the way in which the frequency of light shifts as the light propagates through a gravitational field (cf. below).
The actual measurements show that free-falling frames are the ones in which light propagates as it does in special relativity.^[33] The generalization of this statement, namely that the laws of
special relativity hold to good approximation in freely falling (and non-rotating) reference frames, is known as the Einstein equivalence principle, a crucial guiding principle for generalizing
special-relativistic physics to include gravity.^[34]
The same experimental data shows that time as measured by clocks in a gravitational field—proper time, to give the technical term—does not follow the rules of special relativity. In the language of
spacetime geometry, it is not measured by the Minkowski metric. As in the Newtonian case, this is suggestive of a more general geometry. At small scales, all reference frames that are in free fall
are equivalent, and approximately Minkowskian. Consequently, we are now dealing with a curved generalization of Minkowski space. The metric tensor that defines the geometry—in particular, how lengths
and angles are measured—is not the Minkowski metric of special relativity, it is a generalization known as a semi- or pseudo-Riemannian metric. Furthermore, each Riemannian metric is naturally
associated with one particular kind of connection, the Levi-Civita connection, and this is, in fact, the connection that satisfies the equivalence principle and makes space locally Minkowskian (that
is, in suitable locally inertial coordinates, the metric is Minkowskian, and its first partial derivatives and the connection coefficients vanish).^[35]
Having formulated the relativistic, geometric version of the effects of gravity, the question of gravity's source remains. In Newtonian gravity, the source is mass. In special relativity, mass turns
out to be part of a more general quantity called the energy–momentum tensor, which includes both energy and momentum densities as well as stress: pressure and shear.^[36] Using the equivalence
principle, this tensor is readily generalized to curved spacetime. Drawing further upon the analogy with geometric Newtonian gravity, it is natural to assume that the field equation for gravity
relates this tensor and the Ricci tensor, which describes a particular class of tidal effects: the change in volume for a small cloud of test particles that are initially at rest, and then fall
freely. In special relativity, conservation of energy–momentum corresponds to the statement that the energy–momentum tensor is divergence-free. This formula, too, is readily generalized to curved
spacetime by replacing partial derivatives with their curved-manifold counterparts, covariant derivatives studied in differential geometry. With this additional condition—the covariant divergence of
the energy–momentum tensor, and hence of whatever is on the other side of the equation, is zero— the simplest set of equations are what are called Einstein's (field) equations:
On the left-hand side is theEinstein tensor, a specific divergence-free combination of the Ricci tensorand the metric. Whereis symmetric. In particular,
is the curvature scalar. The Ricci tensor itself is related to the more general Riemann curvature tensor as
On the right-hand side, *
• is the energy–momentum tensor. All tensors are written in
abstract index notation.^[37] Matching the theory's prediction to observational results forplanetaryorbitsor, equivalently, assuring that the weak-gravity, low-speed limit is Newtonian mechanics, the
proportionality constant can be fixed as, whereis thegravitational constantandthe speed of light in vacuum.^[38] When there is no matter present, so that the energy–momentum tensor vanishes, the
results are the vacuum Einstein equations,
In general relativity, the world line of a particle free from all external, non-gravitational force is a particular type of geodesic in curved spacetime. In other words, a freely moving or falling
particle always moves along a geodesic.
The geodesic equation is:
whereis a scalar parameter of motion (e.g. theproper time), andareChristoffel symbols(sometimes called theaffine connectioncoefficients orLevi-Civita connectioncoefficients) which is symmetric in the
two lower indices. Greek indices may take the values: 0, 1, 2, 3 and thesummation conventionis used for repeated indicesand. The quantity on the left-hand-side of this equation is the acceleration of
a particle, and so this equation is analogous toNewton's laws of motionwhich likewise provide formulae for the acceleration of a particle. This equation of motion employs theEinstein notation,
meaning that repeated indices are summed (i.e. from zero to three). The Christoffel symbols are functions of the four space-time coordinates, and so are independent of the velocity or acceleration or
other characteristics of atest particlewhose motion is described by the geodesic equation.
Alternatives to general relativity
There are alternatives to general relativity built upon the same premises, which include additional rules and/or constraints, leading to different field equations. Examples are Whitehead's theory,
Brans–Dicke theory, teleparallelism, f(R) gravity and Einstein–Cartan theory.^[39]
Definition and basic applications
The derivation outlined in the previous section contains all the information needed to define general relativity, describe its key properties, and address a question of crucial importance in physics,
namely how the theory can be used for model-building.
Definition and basic properties
General relativity is a metric theory of gravitation. At its core are Einstein's equations, which describe the relation between the geometry of a four-dimensional pseudo-Riemannian manifold
representing spacetime, and the energy–momentum contained in that spacetime.^[40] Phenomena that in classical mechanics are ascribed to the action of the force of gravity (such as free-fall, orbital
motion, and spacecraft trajectories), correspond to inertial motion within a curved geometry of spacetime in general relativity; there is no gravitational force deflecting objects from their natural,
straight paths. Instead, gravity corresponds to changes in the properties of space and time, which in turn changes the straightest-possible paths that objects will naturally follow.^[41] The
curvature is, in turn, caused by the energy–momentum of matter. Paraphrasing the relativist John Archibald Wheeler, spacetime tells matter how to move; matter tells spacetime how to curve.^[42]
While general relativity replaces the scalar gravitational potential of classical physics by a symmetric rank-two tensor, the latter reduces to the former in certain limiting cases. For weak
gravitational fields and slow speed relative to the speed of light, the theory's predictions converge on those of Newton's law of universal gravitation.^[43]
As it is constructed using tensors, general relativity exhibits general covariance: its laws—and further laws formulated within the general relativistic framework—take on the same form in all
coordinate systems.^[44] Furthermore, the theory does not contain any invariant geometric background structures, i.e. it is background independent. It thus satisfies a more stringent general
principle of relativity, namely that the laws of physics are the same for all observers.^[45] Locally, as expressed in the equivalence principle, spacetime is Minkowskian, and the laws of physics
exhibit local Lorentz invariance.^[46]
The core concept of general-relativistic model-building is that of a solution of Einstein's equations. Given both Einstein's equations and suitable equations for the properties of matter, such a
solution consists of a specific semi-Riemannian manifold (usually defined by giving the metric in specific coordinates), and specific matter fields defined on that manifold. Matter and geometry must
satisfy Einstein's equations, so in particular, the matter's energy–momentum tensor must be divergence-free. The matter must, of course, also satisfy whatever additional equations were imposed on its
properties. In short, such a solution is a model universe that satisfies the laws of general relativity, and possibly additional laws governing whatever matter might be present.^[47]
Einstein's equations are nonlinear partial differential equations and, as such, difficult to solve exactly.^[48] Nevertheless, a number of exact solutions are known, although only a few have direct
physical applications.^[49] The best-known exact solutions, and also those most interesting from a physics point of view, are the Schwarzschild solution, the Reissner–Nordström solution and the Kerr
metric, each corresponding to a certain type of black hole in an otherwise empty universe,^[50] and the Friedmann–Lemaître–Robertson–Walker and de Sitter universes, each describing an expanding
cosmos.^[51] Exact solutions of great theoretical interest include the Gödel universe (which opens up the intriguing possibility of time travel in curved spacetimes), the Taub-NUT solution (a model
universe that is homogeneous, but anisotropic), and anti-de Sitter space (which has recently come to prominence in the context of what is called the Maldacena conjecture).^[52]
Given the difficulty of finding exact solutions, Einstein's field equations are also solved frequently by numerical integration on a computer, or by considering small perturbations of exact
solutions. In the field of numerical relativity, powerful computers are employed to simulate the geometry of spacetime and to solve Einstein's equations for interesting situations such as two
colliding black holes.^[53] In principle, such methods may be applied to any system, given sufficient computer resources, and may address fundamental questions such as naked singularities.
Approximate solutions may also be found by perturbation theories such as linearized gravity^[54] and its generalization, the post-Newtonian expansion, both of which were developed by Einstein. The
latter provides a systematic approach to solving for the geometry of a spacetime that contains a distribution of matter that moves slowly compared with the speed of light. The expansion involves a
series of terms; the first terms represent Newtonian gravity, whereas the later terms represent ever smaller corrections to Newton's theory due to general relativity.^[55] An extension of this
expansion is the parametrized post-Newtonian (PPN) formalism, which allows quantitative comparisons between the predictions of general relativity and alternative theories.^[56]
Consequences of Einstein's theory
General relativity has a number of physical consequences. Some follow directly from the theory's axioms, whereas others have become clear only in the course of many years of research that followed
Einstein's initial publication.
Gravitational time dilation and frequency shift
Assuming that the equivalence principle holds,^[57] gravity influences the passage of time. Light sent down into a gravity well is blueshifted, whereas light sent in the opposite direction (i.e.,
climbing out of the gravity well) is redshifted; collectively, these two effects are known as the gravitational frequency shift. More generally, processes close to a massive body run more slowly when
compared with processes taking place farther away; this effect is known as gravitational time dilation.^[58]
Gravitational redshift has been measured in the laboratory^[59] and using astronomical observations.^[60] Gravitational time dilation in the Earth's gravitational field has been measured numerous
times using atomic clocks,^[61] while ongoing validation is provided as a side effect of the operation of the Global Positioning System (GPS).^[62] Tests in stronger gravitational fields are provided
by the observation of binary pulsars.^[63] All results are in agreement with general relativity.^[64] However, at the current level of accuracy, these observations cannot distinguish between general
relativity and other theories in which the equivalence principle is valid.^[65]
Light deflection and gravitational time delay
General relativity predicts that the path of light will follow the curvature of spacetime as it passes near a star. This effect was initially confirmed by observing the light of stars or distant
quasars being deflected as it passes the Sun.^[66]
This and related predictions follow from the fact that light follows what is called a light-like or null geodesic—a generalization of the straight lines along which light travels in classical
physics. Such geodesics are the generalization of the invariance of lightspeed in special relativity.^[67] As one examines suitable model spacetimes (either the exterior Schwarzschild solution or,
for more than a single mass, the post-Newtonian expansion),^[68] several effects of gravity on light propagation emerge. Although the bending of light can also be derived by extending the
universality of free fall to light,^[69] the angle of deflection resulting from such calculations is only half the value given by general relativity.^[70]
Closely related to light deflection is the gravitational time delay (or Shapiro delay), the phenomenon that light signals take longer to move through a gravitational field than they would in the
absence of that field. There have been numerous successful tests of this prediction.^[71] In the parameterized post-Newtonian formalism (PPN), measurements of both the deflection of light and the
gravitational time delay determine a parameter called γ, which encodes the influence of gravity on the geometry of space.^[72]
Predicted in 1916^[73]^[74] by Albert Einstein, there are gravitational waves: ripples in the metric of spacetime that propagate at the speed of light. These are one of several analogies between
weak-field gravity and electromagnetism in that, they are analogous to electromagnetic waves. On February 11, 2016, the Advanced LIGO team announced that they had directly detected gravitational
waves from a pair of black holes merging.^[75]^[76]^[77]
The simplest type of such a wave can be visualized by its action on a ring of freely floating particles. A sine wave propagating through such a ring towards the reader distorts the ring in a
characteristic, rhythmic fashion (animated image to the right).^[78] Since Einstein's equations arenon-linear, arbitrarily strong gravitational waves do not obeylinear superposition, making their
description difficult. However, for weak fields, a linear approximation can be made. Such linearized gravitational waves are sufficiently accurate to describe the exceedingly weak waves that are
expected to arrive here on Earth from far-off cosmic events, which typically result in relative distances increasing and decreasing byor less. Data analysis methods routinely make use of the fact
that these linearized waves can beFourier decomposed.^[79]
Some exact solutions describe gravitational waves without any approximation, e.g., a wave train traveling through empty space^[80] or Gowdy universes, varieties of an expanding cosmos filled with
gravitational waves.^[81] But for gravitational waves produced in astrophysically relevant situations, such as the merger of two black holes, numerical methods are presently the only way to construct
appropriate models.^[82]
Orbital effects and the relativity of direction
General relativity differs from classical mechanics in a number of predictions concerning orbiting bodies. It predicts an overall rotation (precession) of planetary orbits, as well as orbital decay
caused by the emission of gravitational waves and effects related to the relativity of direction.
In general relativity, the apsides of any orbit (the point of the orbiting body's closest approach to the system's center of mass) will precess; the orbit is not an ellipse, but akin to an ellipse
that rotates on its focus, resulting in a rose curve-like shape (see image). Einstein first derived this result by using an approximate metric representing the Newtonian limit and treating the
orbiting body as a test particle. For him, the fact that his theory gave a straightforward explanation of Mercury's anomalous perihelion shift, discovered earlier by Urbain Le Verrier in 1859, was
important evidence that he had at last identified the correct form of the gravitational field equations.^[83]
The effect can also be derived by using either the exact Schwarzschild metric (describing spacetime around a spherical mass)^[84] or the much more general post-Newtonian formalism.^[85] It is due to
the influence of gravity on the geometry of space and to the contribution of self-energy to a body's gravity (encoded in the nonlinearity of Einstein's equations).^[86] Relativistic precession has
been observed for all planets that allow for accurate precession measurements (Mercury, Venus, and Earth),^[87] as well as in binary pulsar systems, where it is larger by five orders of magnitude.^
In general relativity the perihelion shift, expressed in radians per revolution, is approximately given by^[89]
• is the semi-major axis
• is the orbital period
• is the speed of light in vacuum
• is the orbital eccentricity
According to general relativity, a binary system will emit gravitational waves, thereby losing energy. Due to this loss, the distance between the two orbiting bodies decreases, and so does their
orbital period. Within the Solar System or for ordinary double stars, the effect is too small to be observable. This is not the case for a close binary pulsar, a system of two orbiting neutron stars,
one of which is a pulsar: from the pulsar, observers on Earth receive a regular series of radio pulses that can serve as a highly accurate clock, which allows precise measurements of the orbital
period. Because neutron stars are immensely compact, significant amounts of energy are emitted in the form of gravitational radiation.^[91]
The first observation of a decrease in orbital period due to the emission of gravitational waves was made by Hulse and Taylor, using the binary pulsar PSR1913+16 they had discovered in 1974. This was
the first detection of gravitational waves, albeit indirect, for which they were awarded the 1993 Nobel Prize in physics.^[92] Since then, several other binary pulsars have been found, in particular
the double pulsar PSR J0737-3039, in which both stars are pulsars.^[93]
Geodetic precession and frame-dragging
Several relativistic effects are directly related to the relativity of direction.^[94] One is geodetic precession: the axis direction of a gyroscope in free fall in curved spacetime will change when
compared, for instance, with the direction of light received from distant stars—even though such a gyroscope represents the way of keeping a direction as stable as possible ("parallel transport").^
[95] For the Moon–Earth system, this effect has been measured with the help of lunar laser ranging.^[96] More recently, it has been measured for test masses aboard the satellite Gravity Probe B to a
precision of better than 0.3%.^[97]^[98]
Near a rotating mass, there are gravitomagnetic or frame-dragging effects. A distant observer will determine that objects close to the mass get "dragged around". This is most extreme for rotating
black holes where, for any object entering a zone known as the ergosphere, rotation is inevitable.^[99] Such effects can again be tested through their influence on the orientation of gyroscopes in
free fall.^[100] Somewhat controversial tests have been performed using the LAGEOS satellites, confirming the relativistic prediction.^[101] Also the Mars Global Surveyor probe around Mars has been
Astrophysical applications
The deflection of light by gravity is responsible for a new class of astronomical phenomena. If a massive object is situated between the astronomer and a distant target object with appropriate mass
and relative distances, the astronomer will see multiple distorted images of the target. Such effects are known as gravitational lensing.^[104] Depending on the configuration, scale, and mass
distribution, there can be two or more images, a bright ring known as an Einstein ring, or partial rings called arcs.^[105] The earliest example was discovered in 1979;^[106] since then, more than a
hundred gravitational lenses have been observed.^[107] Even if the multiple images are too close to each other to be resolved, the effect can still be measured, e.g., as an overall brightening of the
target object; a number of such "microlensing events" have been observed.^[108]
Gravitational lensing has developed into a tool of observational astronomy. It is used to detect the presence and distribution of dark matter, provide a "natural telescope" for observing distant
galaxies, and to obtain an independent estimate of the Hubble constant. Statistical evaluations of lensing data provide valuable insight into the structural evolution of galaxies.^[109]
Gravitational wave astronomy
Observations of binary pulsars provide strong indirect evidence for the existence of gravitational waves (see Orbital decay, above). Detection of these waves is a major goal of current
relativity-related research.^[110] Several land-based gravitational wave detectors are currently in operation, most notably the interferometric detectors GEO 600, LIGO (two detectors), TAMA 300 and
VIRGO.^[111] Various pulsar timing arrays are using millisecond pulsars to detect gravitational waves in the 10−9 to 10−6 Hertz frequency range, which originate from binary supermassive blackholes.^
[112] A European space-based detector, eLISA / NGO, is currently under development,^[113] with a precursor mission (LISA Pathfinder) having launched in December 2015.^[114]
Observations of gravitational waves promise to complement observations in the electromagnetic spectrum.^[115] They are expected to yield information about black holes and other dense objects such as
neutron stars and white dwarfs, about certain kinds of supernova implosions, and about processes in the very early universe, including the signature of certain types of hypothetical cosmic string.^
[116] In February 2016, the Advanced LIGO team announced that they had detected gravitational waves from a black hole merger.^[75]^[76]^[77]
Black holes and other compact objects
Whenever the ratio of an object's mass to its radius becomes sufficiently large, general relativity predicts the formation of a black hole, a region of space from which nothing, not even light, can
escape. In the currently accepted models of stellar evolution, neutron stars of around 1.4 solar masses, and stellar black holes with a few to a few dozen solar masses, are thought to be the final
state for the evolution of massive stars.^[117] Usually a galaxy has one supermassive black hole with a few million to a few billion solar masses in its center,^[118] and its presence is thought to
have played an important role in the formation of the galaxy and larger cosmic structures.^[119]
Astronomically, the most important property of compact objects is that they provide a supremely efficient mechanism for converting gravitational energy into electromagnetic radiation.^[120]
Accretion, the falling of dust or gaseous matter onto stellar or supermassive black holes, is thought to be responsible for some spectacularly luminous astronomical objects, notably diverse kinds of
active galactic nuclei on galactic scales and stellar-size objects such as microquasars.^[121] In particular, accretion can lead to relativistic jets, focused beams of highly energetic particles that
are being flung into space at almost light speed.^[122] General relativity plays a central role in modelling all these phenomena,^[123] and observations provide strong evidence for the existence of
black holes with the properties predicted by the theory.^[124]
Black holes are also sought-after targets in the search for gravitational waves (cf. Gravitational waves, above). Merging black hole binaries should lead to some of the strongest gravitational wave
signals reaching detectors here on Earth, and the phase directly before the merger ("chirp") could be used as a "standard candle" to deduce the distance to the merger events–and hence serve as a
probe of cosmic expansion at large distances.^[125] The gravitational waves produced as a stellar black hole plunges into a supermassive one should provide direct information about the supermassive
black hole's geometry.^[126]
The current models of cosmology are based onEinstein's field equations, which include the cosmological constantsince it has important influence on the large-scale dynamics of the cosmos,
where *
• is the spacetime metric.^[127]
Isotropicand homogeneous solutions of these enhanced equations, theFriedmann–Lemaître–Robertson–Walker solutions,^[128] allow physicists to model a universe that has evolved over the past 14 billion
years from a hot, early Big Bang phase.^[129] Once a small number of parameters (for example the universe's mean matter density) have been fixed by astronomical observation,^[130] further
observational data can be used to put the models to the test.^[131] Predictions, all successful, include the initial abundance of chemical elements formed in a period ofprimordial nucleosynthesis,^
[132] the large-scale structure of the universe,^[133] and the existence and properties of a "thermalecho" from the early cosmos, thecosmic background radiation.^[134]
Astronomical observations of the cosmological expansion rate allow the total amount of matter in the universe to be estimated, although the nature of that matter remains mysterious in part. About 90%
of all matter appears to be dark matter, which has mass (or, equivalently, gravitational influence), but does not interact electromagnetically and, hence, cannot be observed directly.^[135] There is
no generally accepted description of this new kind of matter, within the framework of known particle physics^[136] or otherwise.^[137] Observational evidence from redshift surveys of distant
supernovae and measurements of the cosmic background radiation also show that the evolution of our universe is significantly influenced by a cosmological constant resulting in an acceleration of
cosmic expansion or, equivalently, by a form of energy with an unusual equation of state, known as dark energy, the nature of which remains unclear.^[138]
An inflationary phase,^[139] an additional phase of strongly accelerated expansion at cosmic times of around 10−33 seconds, was hypothesized in 1980 to account for several puzzling observations that
were unexplained by classical cosmological models, such as the nearly perfect homogeneity of the cosmic background radiation.^[140] Recent measurements of the cosmic background radiation have
resulted in the first evidence for this scenario.^[141] However, there is a bewildering variety of possible inflationary scenarios, which cannot be restricted by current observations.^[142] An even
larger question is the physics of the earliest universe, prior to the inflationary phase and close to where the classical models predict the big bang singularity. An authoritative answer would
require a complete theory of quantum gravity, which has not yet been developed^[143] (cf. the section on quantum gravity, below).
Kurt Gödel showed^[144] that solutions to Einstein's equations exist that contain closed timelike curves (CTCs), which allow for loops in time. The solutions require extreme physical conditions
unlikely ever to occur in practice, and it remains an open question whether further laws of physics will eliminate them completely. Since then, other—similarly impractical—GR solutions containing
CTCs have been found, such as the Tipler cylinder and traversable wormholes.
Causal structure and global geometry
In general relativity, no material body can catch up with or overtake a light pulse. No influence from an event A can reach any other location X before light sent out at A to X. In consequence, an
exploration of all light worldlines (null geodesics) yields key information about the spacetime's causal structure. This structure can be displayed using Penrose–Carter diagrams in which infinitely
large regions of space and infinite time intervals are shrunk ("compactified") so as to fit onto a finite map, while light still travels along diagonals as in standard spacetime diagrams.^[145]
Aware of the importance of causal structure, Roger Penrose and others developed what is known as global geometry. In global geometry, the object of study is not one particular solution (or family of
solutions) to Einstein's equations. Rather, relations that hold true for all geodesics, such as the Raychaudhuri equation, and additional non-specific assumptions about the nature of matter (usually
in the form of energy conditions) are used to derive general results.^[146]
Using global geometry, some spacetimes can be shown to contain boundaries called horizons, which demarcate one region from the rest of spacetime. The best-known examples are black holes: if mass is
compressed into a sufficiently compact region of space (as specified in the hoop conjecture, the relevant length scale is the Schwarzschild radius^[147]), no light from inside can escape to the
outside. Since no object can overtake a light pulse, all interior matter is imprisoned as well. Passage from the exterior to the interior is still possible, showing that the boundary, the black
hole's horizon, is not a physical barrier.^[148]
Early studies of black holes relied on explicit solutions of Einstein's equations, notably the spherically symmetric Schwarzschild solution (used to describe a static black hole) and the axisymmetric
Kerr solution (used to describe a rotating, stationary black hole, and introducing interesting features such as the ergosphere). Using global geometry, later studies have revealed more general
properties of black holes. With time they become rather simple objects characterized by eleven parameters specifying: electric charge, mass-energy, linear momentum, angular momentum, and location at
a specified time. This is stated by the black hole uniqueness theorem: "black holes have no hair", that is, no distinguishing marks like the hairstyles of humans. Irrespective of the complexity of a
gravitating object collapsing to form a black hole, the object that results (having emitted gravitational waves) is very simple.^[149]
Even more remarkably, there is a general set of laws known as black hole mechanics, which is analogous to the laws of thermodynamics. For instance, by the second law of black hole mechanics, the area
of the event horizon of a general black hole will never decrease with time, analogous to the entropy of a thermodynamic system. This limits the energy that can be extracted by classical means from a
rotating black hole (e.g. by the Penrose process).^[150] There is strong evidence that the laws of black hole mechanics are, in fact, a subset of the laws of thermodynamics, and that the black hole
area is proportional to its entropy.^[151] This leads to a modification of the original laws of black hole mechanics: for instance, as the second law of black hole mechanics becomes part of the
second law of thermodynamics, it is possible for black hole area to decrease—as long as other processes ensure that, overall, entropy increases. As thermodynamical objects with non-zero temperature,
black holes should emit thermal radiation. Semi-classical calculations indicate that indeed they do, with the surface gravity playing the role of temperature in Planck's law. This radiation is known
as Hawking radiation (cf. the quantum theory section, below).^[152]
There are other types of horizons. In an expanding universe, an observer may find that some regions of the past cannot be observed ("particle horizon"), and some regions of the future cannot be
influenced (event horizon).^[153] Even in flat Minkowski space, when described by an accelerated observer (Rindler space), there will be horizons associated with a semi-classical radiation known as
Unruh radiation.^[154]
Another general feature of general relativity is the appearance of spacetime boundaries known as singularities. Spacetime can be explored by following up on timelike and lightlike geodesics—all
possible ways that light and particles in free fall can travel. But some solutions of Einstein's equations have "ragged edges"—regions known as spacetime singularities, where the paths of light and
falling particles come to an abrupt end, and geometry becomes ill-defined. In the more interesting cases, these are "curvature singularities", where geometrical quantities characterizing spacetime
curvature, such as the Ricci scalar, take on infinite values.^[155] Well-known examples of spacetimes with future singularities—where worldlines end—are the Schwarzschild solution, which describes a
singularity inside an eternal static black hole,^[156] or the Kerr solution with its ring-shaped singularity inside an eternal rotating black hole.^[157] The Friedmann–Lemaître–Robertson–Walker
solutions and other spacetimes describing universes have past singularities on which worldlines begin, namely Big Bang singularities, and some have future singularities (Big Crunch) as well.^[158]
Given that these examples are all highly symmetric—and thus simplified—it is tempting to conclude that the occurrence of singularities is an artifact of idealization.^[159] The famous singularity
theorems, proved using the methods of global geometry, say otherwise: singularities are a generic feature of general relativity, and unavoidable once the collapse of an object with realistic matter
properties has proceeded beyond a certain stage^[160] and also at the beginning of a wide class of expanding universes.^[161] However, the theorems say little about the properties of singularities,
and much of current research is devoted to characterizing these entities' generic structure (hypothesized e.g. by the BKL conjecture).^[162] The cosmic censorship hypothesis states that all realistic
future singularities (no perfect symmetries, matter with realistic properties) are safely hidden away behind a horizon, and thus invisible to all distant observers. While no formal proof yet exists,
numerical simulations offer supporting evidence of its validity.^[163]
Each solution of Einstein's equation encompasses the whole history of a universe — it is not just some snapshot of how things are, but a whole, possibly matter-filled, spacetime. It describes the
state of matter and geometry everywhere and at every moment in that particular universe. Due to its general covariance, Einstein's theory is not sufficient by itself to determine the time evolution
of the metric tensor. It must be combined with a coordinate condition, which is analogous to gauge fixing in other field theories.^[164]
To understand Einstein's equations as partial differential equations, it is helpful to formulate them in a way that describes the evolution of the universe over time. This is done in "3+1"
formulations, where spacetime is split into three space dimensions and one time dimension. The best-known example is the ADM formalism.^[165] These decompositions show that the spacetime evolution
equations of general relativity are well-behaved: solutions always exist, and are uniquely defined, once suitable initial conditions have been specified.^[166] Such formulations of Einstein's field
equations are the basis of numerical relativity.^[167]
Global and quasi-local quantities
The notion of evolution equations is intimately tied in with another aspect of general relativistic physics. In Einstein's theory, it turns out to be impossible to find a general definition for a
seemingly simple property such as a system's total mass (or energy). The main reason is that the gravitational field—like any physical field—must be ascribed a certain energy, but that it proves to
be fundamentally impossible to localize that energy.^[168]
Nevertheless, there are possibilities to define a system's total mass, either using a hypothetical "infinitely distant observer" (ADM mass)^[169] or suitable symmetries (Komar mass).^[170] If one
excludes from the system's total mass the energy being carried away to infinity by gravitational waves, the result is the Bondi mass at null infinity.^[171] Just as in classical physics, it can be
shown that these masses are positive.^[172] Corresponding global definitions exist for momentum and angular momentum.^[173] There have also been a number of attempts to define quasi-local quantities,
such as the mass of an isolated system formulated using only quantities defined within a finite region of space containing that system. The hope is to obtain a quantity useful for general statements
about isolated systems, such as a more precise formulation of the hoop conjecture.^[174]
Relationship with quantum theory
If general relativity were considered to be one of the two pillars of modern physics, then quantum theory, the basis of understanding matter from elementary particles to solid state physics, would be
the other.^[175] However, how to reconcile quantum theory with general relativity is still an open question.
Quantum field theory in curved spacetime
Ordinary quantum field theories, which form the basis of modern elementary particle physics, are defined in flat Minkowski space, which is an excellent approximation when it comes to describing the
behavior of microscopic particles in weak gravitational fields like those found on Earth.^[176] In order to describe situations in which gravity is strong enough to influence (quantum) matter, yet
not strong enough to require quantization itself, physicists have formulated quantum field theories in curved spacetime. These theories rely on general relativity to describe a curved background
spacetime, and define a generalized quantum field theory to describe the behavior of quantum matter within that spacetime.^[177] Using this formalism, it can be shown that black holes emit a
blackbody spectrum of particles known as Hawking radiation leading to the possibility that they evaporate over time.^[178] As briefly mentioned above, this radiation plays an important role for the
thermodynamics of black holes.^[179]
The demand for consistency between a quantum description of matter and a geometric description of spacetime,^[180] as well as the appearance of singularities (where curvature length scales become
microscopic), indicate the need for a full theory of quantum gravity: for an adequate description of the interior of black holes, and of the very early universe, a theory is required in which gravity
and the associated geometry of spacetime are described in the language of quantum physics.^[181] Despite major efforts, no complete and consistent theory of quantum gravity is currently known, even
though a number of promising candidates exist.^[182]^[183]
Attempts to generalize ordinary quantum field theories, used in elementary particle physics to describe fundamental interactions, so as to include gravity have led to serious problems.^[184] Some
have argued that at low energies, this approach proves successful, in that it results in an acceptable effective (quantum) field theory of gravity.^[185] At very high energies, however, the
perturbative results are badly divergent and lead to models devoid of predictive power ("perturbative non-renormalizability").^[186]
One attempt to overcome these limitations is string theory, a quantum theory not of point particles, but of minute one-dimensional extended objects.^[187] The theory promises to be a unified
description of all particles and interactions, including gravity;^[188] the price to pay is unusual features such as six extra dimensions of space in addition to the usual three.^[189] In what is
called the second superstring revolution, it was conjectured that both string theory and a unification of general relativity and supersymmetry known as supergravity^[190] form part of a hypothesized
eleven-dimensional model known as M-theory, which would constitute a uniquely defined and consistent theory of quantum gravity.^[191]
Another approach starts with the canonical quantization procedures of quantum theory. Using the initial-value-formulation of general relativity (cf. evolution equations above), the result is the
Wheeler–deWitt equation (an analogue of the Schrödinger equation) which, regrettably, turns out to be ill-defined without a proper ultraviolet (lattice) cutoff.^[192] However, with the introduction
of what are now known as Ashtekar variables,^[193] this leads to a promising model known as loop quantum gravity. Space is represented by a web-like structure called a spin network, evolving over
time in discrete steps.^[194]
Depending on which features of general relativity and quantum theory are accepted unchanged, and on what level changes are introduced,^[195] there are numerous other attempts to arrive at a viable
theory of quantum gravity, some examples being the lattice theory of gravity based on the Feynman Path Integral approach and Regge Calculus,^[182] dynamical triangulations,^[196] causal sets,^[197]
twistor models^[198] or the path integral based models of quantum cosmology.^[199]
All candidate theories still have major formal and conceptual problems to overcome. They also face the common problem that, as yet, there is no way to put quantum gravity predictions to experimental
tests (and thus to decide between the candidates where their predictions vary), although there is hope for this to change as future data from cosmological observations and particle physics
experiments becomes available.^[200]
General relativity has emerged as a highly successful model of gravitation and cosmology, which has so far passed many unambiguous observational and experimental tests. However, there are strong
indications the theory is incomplete.^[201] The problem of quantum gravity and the question of the reality of spacetime singularities remain open.^[202] Observational data that is taken as evidence
for dark energy and dark matter could indicate the need for new physics.^[203] Even taken as is, general relativity is rich with possibilities for further exploration. Mathematical relativists seek
to understand the nature of singularities and the fundamental properties of Einstein's equations,^[204] while numerical relativists run increasingly powerful computer simulations (such as those
describing merging black holes).^[205] In February 2016, it was announced that the existence of gravitational waves was directly detected by the Advanced LIGO team on September 14, 2015.^[77]^[206]^
[207] A century after its introduction, general relativity remains a highly active area of research.^[208]
• Alcubierre drive (warp drive)
• Alternatives to general relativity
• Center of mass (relativistic)
• Contributors to general relativity
• Derivations of the Lorentz transformations
• Ehrenfest paradox
• Einstein–Hilbert action
• Einstein's thought experiments
• Introduction to the mathematics of general relativity
• Relativity priority dispute
• Ricci calculus
• Tests of general relativity
• Timeline of gravitational physics and relativity
• Two-body problem in general relativity
• Weak Gravity Conjecture
Citation Linkwww.black-holes.org"GW150914: LIGO Detects Gravitational Waves". Black-holes.org. Retrieved 18 April 2016.
Sep 24, 2019, 12:00 AM
Citation Linkopenlibrary.orgLandau, L. D.; Lifshitz, E. M. (1975), The Classical Theory of Fields, v. 2, Elsevier Science, Ltd., ISBN 978-0-08-018176-9, p. 228 "...the general theory of relativity
...was established by Einstein, and represents probably the most beautiful of all existing physical theories."
Sep 24, 2019, 12:00 AM
Citation Linkwww.st-andrews.ac.ukO'Connor, J.J. and Robertson, E.F. (1996), General relativity. Mathematical Physics index, School of Mathematics and Statistics, University of St. Andrews, Scotland.
Retrieved 2015-02-04.
Sep 24, 2019, 12:00 AM
Citation Linkopenlibrary.orgPais, Abraham (1982), 'Subtle is the Lord ...' The Science and life of Albert Einstein, Oxford University Press, ISBN 978-0-19-853907-0, ch. 9 to 15, Janssen, Michel
(2005), "Of pots and holes: Einstein's bumpy road to general relativity" (PDF), Annalen der Physik, 14 (S1): 58–85, Bibcode:2005AnP...517S..58J, doi:10.1002/andp.200410130; an up-to-date collection
of current research, including reprints of many of the original articles, is Renn, Jürgen, ed. (2007), The Genesis of General Relativity (4 Volumes), Dordrecht: Springer, ISBN 978-1-4020-3999-7; an
accessible overview can be found in Renn, Jürgen, ed. (2005), Albert Einstein—Chief Engineer of the Universe: Einstein's Life and Work in Context, Berlin: Wiley-VCH, ISBN 978-3-527-40571-8,
pp. 110ff. Einstein's original papers are found in Digital Einstein, volumes 4 and 6. An early key article is Einstein, Albert (1907), "Über das Relativitätsprinzip und die aus demselben gezogene
Folgerungen", Jahrbuch der Radioaktivität und Elektronik, 4: 411, cf. , ch. 9. The publication featuring the field equations is Einstein, Albert (1915), "Die Feldgleichungen der Gravitation",
Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin: 844–847, cf. , ch. 11–15
Sep 24, 2019, 12:00 AM
Citation Linkopenlibrary.orgMoshe Carmeli (2008).Relativity: Modern Large-Scale Structures of the Cosmos. pp.92, 93.World Scientific Publishing
Sep 24, 2019, 12:00 AM
Citation Linkopenlibrary.orgSchwarzschild, Karl (1916a), "Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie", Sitzungsber. Preuss. Akad. D. Wiss.: 189–196,
Bibcode:1916SPAW.......189S, Schwarzschild, Karl (1916b), "Über das Gravitationsfeld einer Kugel aus inkompressibler Flüssigkeit nach der Einsteinschen Theorie", Sitzungsber. Preuss. Akad. D. Wiss.:
424–434, Bibcode:1916skpa.conf..424S and Reissner, H. (1916), "Über die Eigengravitation des elektrischen Feldes nach der Einsteinschen Theorie", Annalen der Physik, 355 (9): 106–120,
Bibcode:1916AnP...355..106R, doi:10.1002/andp.19163550905 (later complemented in Nordström, Gunnar (1918), "On the Energy of the Gravitational Field in Einstein's Theory", Verhandl. Koninkl. Ned.
Akad. Wetenschap., 26: 1238–1245, Bibcode:1918KNAB...20.1238N)
Sep 24, 2019, 12:00 AM
Citation Linkopenlibrary.orgEinstein, Albert (1917), "Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie", Sitzungsberichte der Preußischen Akademie der Wissenschaften: 142, cf. , ch.
Sep 24, 2019, 12:00 AM
Citation Linkopenlibrary.orgHubble's original article is Hubble, Edwin (1929), "A Relation between Distance and Radial Velocity among Extra-Galactic Nebulae" (PDF), Proc. Natl. Acad. Sci., 15 (3):
168–173, Bibcode:1929PNAS...15..168H, doi:10.1073/pnas.15.3.168, PMC 522427, PMID 16577160; an accessible overview is given in Singh, Simon (2004), Big Bang: The Origin of the Universe, Fourth
Estate, Bibcode:2004biba.book.....S, ISBN 978-0-00-715251-3, ch. 2–4
Sep 24, 2019, 12:00 AM
Citation Linkopenlibrary.orgAs reported in Gamow, George (1970), My World Line, Viking Press, ISBN 978-0-670-50376-6. Einstein's condemnation would prove to be premature, cf. the section Cosmology,
Sep 24, 2019, 12:00 AM
Citation Linkopenlibrary.orgKennefick, Daniel (2005), "Astronomers Test General Relativity: Light-bending and the Solar Redshift", in Renn, Jürgen (ed.), One hundred authors for Einstein, Wiley-VCH,
pp. 178–181, ISBN 978-3-527-40574-9, Kennefick, Daniel (2007), "Not Only Because of Theory: Dyson, Eddington and the Competing Myths of the 1919 Eclipse Expedition", Proceedings of the 7th Conference
on the History of General Relativity, Tenerife, 2005, 0709, p. 685, arXiv:0709.0685, Bibcode:2007arXiv0709.0685K, doi:10.1016/j.shpsa.2012.07.010
Sep 24, 2019, 12:00 AM
Citation Linkbooks.google.comThorne, Kip (2003). The future of theoretical physics and cosmology: celebrating Stephen Hawking's 60th birthday. Cambridge University Press. p. 74.
ISBN 978-0-521-82081-3. Extract of page 74
Sep 24, 2019, 12:00 AM
Citation Linkopenlibrary.orgIsrael, Werner (1987), "Dark stars: the evolution of an idea", in Hawking, Stephen W.; Israel, Werner (eds.), 300 Years of Gravitation, Cambridge University Press,
pp. 199–276, ISBN 978-0-521-37976-2, ch. 7.8–7.10, Thorne, Kip S. (1994), Black Holes and Time Warps: Einstein's Outrageous Legacy, W W Norton & Company, ISBN 978-0-393-31276-8, ch. 3–9
Sep 24, 2019, 12:00 AM
Citation Linkopenlibrary.orgSections Orbital effects and the relativity of direction, Gravitational time dilation and frequency shift and Light deflection and gravitational time delay, and references
Sep 24, 2019, 12:00 AM
Citation Linkopenlibrary.orgSection Cosmology and references therein; the historical development is in Overbye, Dennis (1999), Lonely Hearts of the Cosmos: the story of the scientific quest for the
secret of the Universe, Back Bay, ISBN 978-0-316-64896-7
Sep 24, 2019, 12:00 AM
Citation Linkopenlibrary.orgWald, Robert M. (1984), General Relativity, University of Chicago Press, ISBN 978-0-226-87033-5Wald, Robert M. (1984), General Relativity, University of Chicago Press,
ISBN 978-0-226-87033-5, p. 3
Sep 24, 2019, 12:00 AM
Citation Linkopenlibrary.orgRovelli, Carlo (ed.) (2015), General Relativity: The most beautiful of theories (de Gruyter Studies in Mathematical Physics), Boston: Walter de Gruyter GmbH,
ISBN 978-3110340426, pp. 1–6 "General relativity is not just an extraordinarily beautiful physical theory providing the best description of the gravitational interaction we have so far. It is more."
Sep 24, 2019, 12:00 AM
Citation Linkopenlibrary.orgChandrasekhar, Subrahmanyan (1984), "The general theory of relativity - Why 'It is probably the most beautiful of all existing theories'", Journal of Astrophysics and
Astronomy, 5: 3–11, Bibcode:1984JApA....5....3C, doi:10.1007/BF02714967, p. 6
Sep 24, 2019, 12:00 AM
Citation Linkopenlibrary.orgEngler, Gideon (2002), "Einstein and the most beautiful theories in physics", International Studies in the Philosophy of Science, 16 (1): 27–37, doi:10.1080/
Sep 24, 2019, 12:00 AM | {"url":"https://everipedia.org/wiki/lang_en/General_relativity","timestamp":"2024-11-03T21:34:48Z","content_type":"text/html","content_length":"1049741","record_id":"<urn:uuid:7e03d133-46c6-428e-b4c1-18e80bb02f39>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00855.warc.gz"} |
Seminar Series: Topics in Special Functions and Number Theory
Dear all,
Welcome to 2022. We begin the year with a Ramanujan Special talk by Alan Sokal. The talk announcement is below.
We encourage you to distribute this announcement to friends and colleagues in your department or otherwise, so that they come to know of our seminar.
Talk Announcement:
Title: Coefficientwise Hankel-total positivity
Speaker: Alan Sokal (University College London and New York)
When: Thursday, January 27, 2022 - 4:00 PM - 5:00 PM (IST)
Where: Zoom: Write to sfandnt@gmail.com for the link
Tea or Coffee: Please bring your own.
A matrix $M$ of real numbers is called totally positive
if every minor of $M$ is nonnegative. Gantmakher and Krein showed
in 1937 that a Hankel matrix $H = (a_{i+j})_{i,j \ge 0}$
of real numbers is totally positive if and only if the underlying
sequence $(a_n)_{n \ge 0}$ is a Stieltjes moment sequence.
Moreover, this holds if and only if the ordinary generating function
$\sum_{n=0}^\infty a_n t^n$ can be expanded as a Stieltjes-type
continued fraction with nonnegative coefficients:
\sum_{n=0}^{\infty} a_n t^n
\cfrac{\alpha_0}{1 - \cfrac{\alpha_1 t}{1 - \cfrac{\alpha_2 t}{1 - \cfrac{\alpha_3 t}{1- \cdots}}}}
(in the sense of formal power series) with all $\alpha_i \ge 0$.
So totally positive Hankel matrices are closely connected with
the Stieltjes moment problem and with continued fractions.
Here I will introduce a generalization: a matrix $M$ of polynomials
(in some set of indeterminates) will be called
coefficientwise totally positive if every minor of $M$
is a polynomial with nonnegative coefficients. And a sequence
$(a_n)_{n \ge 0}$ of polynomials will be called
coefficientwise Hankel-totally positive if the Hankel matrix
$H = (a_{i+j})_{i,j \ge 0}$ associated to $(a_n)$ is coefficientwise
totally positive. It turns out that many sequences of polynomials
arising naturally in enumerative combinatorics are (empirically)
coefficientwise Hankel-totally positive. In some cases this can
be proven using continued fractions, by either combinatorial or
algebraic methods; I will sketch how this is done. In many other
cases it remains an open problem.
One of the more recent advances in this research is perhaps of
independent interest to special-functions workers:
we have found branched continued fractions for ratios of contiguous
hypergeometric series ${}_r \! F_s$ for arbitrary $r$ and $s$,
which generalize Gauss' continued fraction for ratios of contiguous
${}_2 \! F_1$. For the cases $s=0$ we can use these to prove
coefficientwise Hankel-total positivity.
Reference: Mathias P\'etr\'eolle, Alan D.~Sokal and Bao-Xuan Zhu, | {"url":"https://www.sfnt.org/2022/01/","timestamp":"2024-11-14T21:09:15Z","content_type":"application/xhtml+xml","content_length":"78771","record_id":"<urn:uuid:e0303a3f-4fea-4b1e-857b-d10771a33020>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00734.warc.gz"} |
Approximation and Estimation | Shiken
Approximation and Estimation
The Power of Approximation and Estimation in Mathematics
Mathematics can be challenging, especially when long and tedious calculations require a quick answer. Whether faced with a non-calculator exam or trying to estimate a restaurant bill, the techniques
of approximation and estimation prove to be useful. In this article, we will delve into the definitions and examples of these powerful tools.
Approximation and Estimation Defined
An approximation is a value close to the true value, but not an exact match. It is denoted by the symbol "≈". For example, we can use the approximation ≈ 3.14 for the irrational number pi, which
represents the ratio of a circle's circumference to its diameter.
On the other hand, estimation involves guessing or making a rough calculation to obtain a value close to the true value. For instance, we can estimate pi by measuring the circumference and diameter
of a circle and dividing them. So, if a circle has a diameter of 10 cm and a circumference of 31.4 cm, our estimation for pi would be ≈ 3.14.
Rounding Numbers
Before delving further, it is important to know how to round numbers - a crucial aspect of approximation and estimation. Rounding a number involves finding another number that is close to it but
easier to work with. Let's review the process with an example.
• Round 3728 to the nearest 10, 100, and 1000
• Solution:
• When rounding to the nearest 10, we look at the digits from the 10s column onwards - in this case, we have 28. Now, we ask ourselves, is 28 closer to 20 or 30? It is closer to 30, so 3728 rounds
to 2730.
• When rounding to the nearest 100, we consider the digits from the 100s column - in this case, 728. Is 728 closer to 700 or 800? It's closer to 700, so 3728 rounds to 3700.
• Lastly, when rounding to the nearest 1000, we look at the digits from the 1000s column - in this case, 3728. Is 3728 closer to 3000 or 4000? It's closer to 4000, so we round up to 4000.
Using Approximation and Estimation - Examples
Estimating Calculations
To estimate a calculation, round all the numbers involved to a "friendly" value. For instance, if we want to multiply 72 by 91, it is easier to work with both numbers rounded to the nearest 10. This
process is a form of approximation, where we say that 72 ≈ 70 and 91 ≈ 90. We can then use these numbers to estimate the multiplication result.
What is an estimate for 7.28 x 2.91?
This calculation may seem daunting without a calculator, but if we round both numbers to the nearest 10, we get 7 ≈ 10 and 2.91 ≈ 3. Thus, we can estimate the answer to be ≈ 30.
We can also calculate the percentage error between the estimated value and the actual value to see how close our estimation is to the real answer. In this case, the percentage error is only 0.026%,
indicating a good estimation.
Estimating Total Cost
Sometimes, we may need to estimate the total cost of a purchase. Let's see how we can do this using approximation and estimation.
You buy 32 packets of crisps for a party, with each packet costing 21p. Estimate the total cost of the crisps.
The total cost is 32 x 21p, so we need to multiply the values to get the cost in pence. To make this easier, we can round both numbers to the nearest 10, giving us ≈ 30 x 20p = 600p. Converting this
to pounds, we get a total estimate of £6.00.
Using approximation and estimation in our daily lives can save us time and effort, and with practice, we can hone our skills to make quick and accurate estimations. So, next time you are faced with a
lengthy calculation, remember the power of approximation and estimation.
The Importance of Estimation and Approximation
In mathematics, the ability to estimate or approximate values plays a crucial role in simplifying calculations. While these terms may seem interchangeable, they have distinct meanings and
applications. Let's explore the differences between estimation and approximation, and why these skills are essential in our everyday lives.
Solving Problems with Estimation and Approximation
To better understand the concept, let's take an example. If we know that 32 packets of crisps cost a total of £6, we can estimate that each packet costs around 19p.
The Power of Estimation and Approximation in Mathematics
Making quick and accurate calculations is essential in solving complex math problems, but it doesn't always have to be a lengthy and detailed process. Utilizing estimation and approximation
techniques can give us a quick idea of the cost or value without going through tedious calculations.
We can apply these skills not only in math but also in our daily lives, such as estimating the total cost of items in a shopping basket or approximating the bill at a restaurant. Even mathematicians
use these techniques in finding solutions to higher-order equations, using iterative methods to approximate the answers.
The Distinction Between Estimation and Approximation
While often used interchangeably, estimation and approximation have different meanings and uses. Estimation is an educated guess based on limited information, while approximation involves altering
the true value slightly to make it more manageable. For example, instead of calculating the exact cost at 19p per packet of crisps, we can approximate it to 20p for easier calculation.
The Relevance of Estimation and Approximation in Everyday Life
Apart from being useful in mathematics, estimation and approximation are essential skills in our daily lives. Property evaluators use estimation to determine the value of a property based on various
factors such as size, location, and amenities.
These skills also aid in making predictions and planning for the future. We can estimate the cost of a purchase or approximate the time needed for a task, enabling us to budget our resources and time
Remember, every time we make an estimate or approximation, we are honing our abilities to make quick calculations, solve intricate problems, and plan effectively for the future. So, keep practicing
and improving these skills!
Understanding Approximation and Estimation in Mathematics
In the world of mathematics, precision is crucial. However, there are instances where our estimations are sufficient for our calculations. This is where approximation and estimation come into play.
While they may seem similar, they have distinct meanings and uses.
What is Approximation in Math?
Approximation involves using a value close to the true value to make calculations more manageable or understandable. It can include rounding numbers or using simpler values to represent complex ones.
It is a valuable tool in math, making it easier and faster to solve problems.
What is Estimation in Math?
Estimation is the process of either guessing or roughly calculating a value to find an approximate answer. This is typically used when the true value is unknown, but we still want to get close to it.
The key difference between approximation and estimation is that estimation is used to find the true value, while approximation is used when we already know it.
How to Estimate in Math:
• First, round all numbers involved to an "easy" value, such as a whole number or decimal with fewer digits.
• Then, use mental math to perform the calculation with the rounded values to get an approximate answer.
For example, to estimate the cost of two televisions priced at £133.99 each, we can round the price to £135 and then multiply it by 2, giving us an estimated cost of £270.
The Similarities and Differences Between Approximation and Estimation:
Although both approximation and estimation involve finding values close to the true value, the main difference lies in whether we know the true value or need to find it. Approximation is used when we
already know the true value but need to simplify it, while estimation is used to find an answer close to the unknown true value. By understanding these concepts and when to apply them, we can enhance
our problem-solving skills and make our calculations more efficient. | {"url":"https://shiken.ai/math-topics/approximation-and-estimation","timestamp":"2024-11-10T17:20:58Z","content_type":"text/html","content_length":"80917","record_id":"<urn:uuid:094ecc6b-0f7d-4a46-9760-e73068ccb214>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00482.warc.gz"} |
fibration sequence
Hm ,wait, actually you are right and I am wrong in that the expression $Hom(X, A \times_K B)$ is needed in the following, and $Hom(X,A) \times_{Hom(X,K)} Hom(X,B)$ is not. That latter is needed to
see things like $\mathbf{H}(X, \Omega A) \simeq \Omega \mathbf{H}(X,A)$ and the like.
Allright, so thanks for catching that.
Hm, my impression is that you changed one correct expression to an equivalent correct expression.
But for the argument to follow the first correct expression is the one needed! On the other hand, this follows indeed using your correct expression.
So I guess we want both expressions. I have now implemented that. I have also renamed that object C into K, since the ambient category is already called C and so there was a bad notation clash.
Probably that didn't help to clarify the situation. Please have another look and see if it is better now.
fibration sequence
, changed the second diagram after "But the hom-functor has the crucial property..."
please someone check with the previos version to see if my correction is correct.
what I meant is that if one writes the diagram with in the upper right corner, then this is by definition an homotopy pullback, so it is not immediately clear what one means by saying that is exact.
I think now it looks better, but I edited it once again in order to make it even mor explicit: exactness of is ubiquously used in npov on cohomology, so let us state in it the clearest possible
way... :-)
please when you have time have a look | {"url":"https://nforum.ncatlab.org/discussion/900/fibration-sequence/","timestamp":"2024-11-08T06:12:12Z","content_type":"application/xhtml+xml","content_length":"43187","record_id":"<urn:uuid:167fb23f-7914-479e-b0ff-32643a019a8f>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00885.warc.gz"} |
Noise in Complex Systems and Stochastic Dynamics
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• Stochastic Ratchets
• Noise in Dynamical Systems: General Aspects
• Path Integrals Method
• Self-Organizing Systems
• Diffusion-Limited Reactions
• Quantum Stochastic Processes
• Periodic Driven Systems and Stochastic Resonance
• Noise-Induced Phase Transitions
• Thresholds, Signals, and Synchronization
• Noise in Systems with Delay
• Poster Session
Microscopic models of Brownian ratchets
Show abstract
A hard disk microscopic ratchet is introduced and studied with molecular dynamics. The properties of the systematic motion that appears when its two compartments are at different temperatures is
Applications of Brownian motors
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Brownian motors combine asymmetry (such as a “ratchet” potential) and stochastic (thermal) motion with non-equilibrium processes to generate directed particle flow. A brief general introduction to
Brownian motors is given, and the relevance of the ratchet model for biological motor proteins is highlighted. However, the impact of research on Brownian motors goes far beyond biophysics. A wealth
of novel phenomena has been predicted, and some of these phenomena have been observed in areas as diverse as synthetic chemistry, bio-molecular colloids, self-organizing systems, quantum electronics,
micro-fluidics, and materials science. Applications, such as novel actuators and molecular separation techniques, are evolving quickly. In the oral presentation, I will attempt to give an overview on
applications of ratchets and Brownian motors. In the present paper, I give a short overview and review then a recent experimental realization of a tunneling ratchet for electrons. Such electron
tunnelling ratchets can not only be used to generate particle currents, but also to pump heat. Using a realistic model, the heat pumping properties of the experimental electron ratchet are analysed.
Walking on ratchets: a model of two Brownian motors with bistable coupling
Show abstract
We propose a model for a walker moving on an asymmetric periodic ratchet potential. The walker has two 'feet' represented as two finite-size particles coupled nonlinearly through a double-well
potential. In contrast to linear coupling, the bistable potential admits a richer dynamics where the ordering of the particles can alternate. The transitions between the two stable points on the
bistable potential, correspond to a walking with alternating particles. In our model, each particle is acted upon by independent white noises, modeling thermal noise, and additionally we have an
external time-dependent force that drives the system out of equilibrium, allowing directed transport. This force can be common colored noise, periodic deterministic driving or fluctuations on the
bistable potential. In the equilibrium case, where only white noise is present, we perform a bifurcation analysis which reveals different walking patterns available for various parameter settings.
Numerical simulations showed the existence of current reversals and significant changes in the effective diffusion constant and in the synchronization index. We obtained an optimal coherent
transport, characterized by a maximum dimensionless ratio of the current and the effective diffusion (Peclet number), when the periodicity of the ratchet potential coincides with the equilibrium
distance between the two particles.
Noise in Dynamical Systems: General Aspects
Fluctuation-dissipation-dispersion relation for slowly varying processes
Show abstract
The famous Callen-Welton formula [1] is generalized to the systems with slowly varying parameters. Using the momentum method and the time multiscale technique, developed for a nonlocal plasma in [2]
it is shown that not only the dissipation but also the time derivatives of the dispersion determine the amplitude and the width of the spectrum lines of the fluctuations. In the general case, the
contribution of the second one may be comparable with the first one. This contribution is affected by a new nonlocal dispersive term which is not related to Joule dissipation and appears because of
an additional phase shift between the force and response of the system. The general formalism is illustrated by applications to several particular types of system. The influence of the dispersion
contributions on the quality factor of the system is discussed.
Solvability of dichotomous flows, dichotomous diffusion, and generalizations
Show abstract
We first consider the one-dimensional stochastic flow dx/dt = f(x) + g(x) xi(t), where xi(t) is a dichotomous Markov noise. A procedure involving the algebra of the relevant differential operators is
used to identify the conditions under which the integro-differential equation satisfied by the total probability density P(x,t) of the driven variable can be reduced to a differential equation of
finite order. This systematizes the enumeration of the "solvable" cases, of which the case of linear drift and additive noise is a notable one. We then revisit the known formula for the stationary
density that exists under suitable conditions in dichotomous flow, and indicate how this expression may be derived and interpreted on direct physical grounds. Finally, we consider a diffusion process
driven by an N-level extension of dichotomous noise, and explicitly derive the higher-order partial differential equation satisfied by P(x,t) in this case. This multi-level noise driven diffusion is
a process that interpolates between the usual extremes of dichotomous diffusion and Brownian motion. We comment on the possible use of certain algebraic techniques to solve the master equation for
this generalized diffusion.
Escape times and diffusion coefficients in fluctuating potentials
Show abstract
We investigate an overdamped Brownian particle moving in: (a) a dichotomously fluctuating metastable potential; (b) a random fluctuating periodic potential. For piece-wise linear potential we obtain
for case (a) the exact average lifetime and the mean first passage time as a function of the potential parameters, the noise intensity and the mean frequency of switchings of the dichotomous noise.
We find noise enhanced stability (NES) in the system investigated. The parameter regions of the fluctuating potential where NES effect can be observed are analytically derived. For case (b) we
consider a symmetric periodic potential modulated by white noise. We obtain for such a potential the same relationship between effective diffusion coefficient of Brownian particles and the mean
first-passage time, discovered previously for fixed periodic potential (see ref. 3). The phenomenon of diffusion acceleration in comparison with free particle case has been found for arbitrary
potential profile. The effective diffusion coefficients for sawtooth, sinusoidal and piecewise parabolic potentials are calculated in closed analytical form.
Different ways of stabilization of metastable states
Show abstract
Different ways to stabilize a classical particle located at metastable or unstable state include an application of a periodic field or noise. We consider discrete and space-extended double-well
potentials. Exact calculations are performed for the piecewise potentials.
Effects of weak spatiotemporal noise on a bistable one-dimensional system
Show abstract
We treat analytically a model that captures several features of the phenomenon of spatially inhomogeneous reversal of an order parameter. The model is a classical Ginzburg-Landau field theory
restricted to a bounded one-dimensional spatial domain, perturbed by weak spatiotemporal noise having a flat power spectrum in time and space. Our analysis extends the Kramers theory of noise-induced
transitions to the case when the system acted on by the noise has nonzero spatial extent, and the noise itself is spatially dependent. By extending the Langer-Coleman theory of the noise-induced
decay of a metastable state, we determine the dependence of the activation barrier and the Kramers reversal rate prefactor on the size of the spatial domain. As this is increased from zero and passes
through a certain critical value, a transition between activation regimes occurs, at which the rate prefactor diverges. Beyond the transition, reversal preferentially takes place in a spatially
inhomogeneous rather than in a homogeneous way. Transitions of this sort were not discovered by Langer or Coleman, since they treated only the infinite-volume limit. Our analysis uses higher
ranscendental functions to handle the case of finite volume. Similar transitions between activation regimes should occur in other models of metastable systems with nonzero spatial extent, perturbed
by weak noise, as the size of the spatial domain is varied.
Throwing ropes in the dark: the case of oscillating barriers
Show abstract
We present a novel path-integral method for the determination of time-dependent and time-averaged reaction rates in multidimensional, periodically driven escape problems at weak thermal noise. The so
obtained general expressions are evaluated explicitly for the situation of a sinusoidally driven, damped particle with inertia moving in a metastable, piecewise parabolic potential. A comparison with
data from Monte-Carlo simulations yields a very good agreement with analytic results over a wide parameter range.
Nonequilibrium distribution at finite noise intensity
Andriy Bandrivskyy, Stefano Beri, Dmitry G Luchinsky
Show abstract
The non-equilibrium distribution in dissipative dynamical systems with unstable limit cycle is analyzed in the next-to-leading order of the small-noise approximation of the Fokker-Planck equation.
The noise-induced variations of the non-equilibrium distribution are described in terms of topological changes in the pattern of optimal paths. It is predicted that singularities in the pattern of
optimal paths are shifted and cross the basin boundary in the presence of finite noise. As a result the probability distribution oscillates at the basin boundary. Theoretical predictions are in good
agreement with the results of numerical solution of the Fokker-Planck equation and Monte Carlo simulations.
Escape from a chaotic attractor with fractal basin boundaries
Show abstract
We study fluctuational transitions in a discrete dynamical system between two co-existing chaotic attractors separated by a fractal basin boundary. It is shown that there is a generic mechanism of
fluctuational transition through a fractal boundary determined by a hierarchy of homoclinic original saddles. The most probable escape path from a chaotic attractors to the fractal boundary is found
using both statistical analysis of fluctuational trajectories and Hamiltonian theory of fluctuations.
Quantifying self-organization in cyclic cellular automata
Cosma Rohilla Shalizi, Kristina Lisa Shalizi
Show abstract
Cyclic cellular automata (CCA) are models of excitable media. Started from random initial conditions, they produce several different kinds of spatial structure, depending on their control parameters.
We introduce new tools from information theory that let us calculate the dynamical information content of spatial random processes. This complexity measure allows us to quantitatively determine the
rate of self-organization of these cellular automata, and establish the relationship between parameter values and self-organization in CCA. The method is very general and can easily be applied to
other cellular automata or even digitized experimental data.
Diffusion-Limited Reactions
Reaction-diffusion processes in scale-free networks
Lazaros K. Gallos, Panos Argyrakis
Show abstract
In this work we investigate the dynamics of reaction-diffusion processes on scale-free networks. Particles of two types, A and B, are randomly distributed on such a network and diffuse using random
walk models by hopping to nearest neighbor nodes only. Here we treat the case where one species is immobile and the other is mobile. The immobile species acts as a trap, i.e. when particles of the
other species encounter a trap node they are immediately annihilated. We numerically compute Φ(n,c), the survival probability of mobile species at time n, as a function of the concentration of trap
nodes, c. We compare our results to the mean-field result (Rosenstock approximation), and the exact result for lattices of Donsker-Varadhan. We find that for high connectivity networks and high trap
concentrations the mean-field result of a simple exponential decay is also valid here. But for low connectivity networks and low c the behavior is much more complicated. We explain these trends in
terms of the number of sites visited, S([n]), the system size, and the concentration of traps.
Stochastic description of traffic breakdown
Show abstract
We present a comparison of nucleation in an isothermal-isochoric container with traffic congestion on a one-lane freeway. The analysis is based, in both cases, on the probabilistic description by
stochastic master equations. Further we analyze the characteristic features of traffic breakdowns. To describe this phenomenon we apply the stochastic model regarding the jam emergence to the
formation of a large car cluster on the highway.
Quantum Stochastic Processes
Numerical solution methods for quantum stochastic processes
Show abstract
The study of quantum stochastic processes presents severe difficulties, both on the theory level as well as on technical grounds. The numerically exact solution remains prohibitive even today. In
this paper we review and present new results for three different methods used for the modelling of quantum stochastic processes. These include a mixed quantum classical approach, semiclassical
initial value representations of the quantum propagator and the reduced density matrix approach as typified by the quantum Wigner-Fokker-Planck equation. A new semiclassical initial value
representation that does away with cumbersome prefactors which depend on the monodromy matrix elements but is exact for a harmonic oscillator is presented and its properties analysed. A recently
proposed systematic method for improving semiclassical initial value representations is reviewed. The generalization of the Wigner-Fokker-Planck equation to stochastic processes with memory is
obtained by using a novel integral equation representation.
Periodic Driven Systems and Stochastic Resonance
Critical exponents for escape of a driven particle near a bifurcation point
Show abstract
We study the rate of activated escape W in periodically modulated systems close to the saddle-node bifurcation point where the metastable state disappears. The escape rate displays scaling behavior
versus modulation amplitude A as A approaches the bifurcational value A[c], with 1nW ∝(A[c]-A)μ. For adiabatic modulation, the critical exponent is μ=3/2. Even if the modulation is slow far from the
bifurcation point, the adiabatic approximation breaks down close to A[c]. In the weakly nonadiabatic regime we predict a crossover to μ = 2 scaling. For higher driving frequencies, as A[c] is
approached there occurs another crossover, from Αμ=2 to μ=3/2. The general results are illustrated using a simple model system.
On stochastic resonance-like effect in detection
Show abstract
One of the most common characteristic of a system exhibiting stochastic resonance is the existence of a maximum in the output signal-to-noise ratio when plotted against the power of the input noise.
This property is at the root of the use of stochastic resonance in detection, since it is generally admitted that performance of detection increases with the signal-to-noise ratio. We show in this
paper that this statement is not always true by examining the key index of performance in detection: the probability of detection. Furthermore, when the probability of detection can be increased by
an increase of the power of the noise, we address the practical problem of adding noise. We consider in particular the alpha-stable case for which addition does not change the probability density
function of the noise.
Noise-Induced Phase Transitions
Macroscopic limit cycle via a noise-induced phase transition
Show abstract
Bistability generated via a noise-induced phase transition is reexamined from the view of macroscopic dynamical systems, which clarifies the role of fluctuation better than the conventional
Fokker-Plank or Langevin equation approach. Using this approach, we investigated the spatially-extended systems with two degrees of freedom per site. The model systems undergo a noise-induced phase
transition through a Hopf bifurcation, leading to a macroscopic limit cycle motion similar to the deterministic relaxation oscillation.
Twofold role of noise in doubly stochastic effects
Show abstract
We study nonlinear systems under two noisy sources to demonstrate the concept of doubly stochastic effects. In such effects noise plays a twofold role: first it induces a special feature in the
system, and second it interplays with this feature leading to noise-induced order. For this effect one needs to optimize both noisy sources, hence we call these phenomena doubly stochastic effects.
To show the generality of this approach we apply this concept to several basic noise-induced phenomena: stochastic resonance, noise-induced propagation and coherence resonance. Additionally, we
discuss an application of this concept to noise-induced transitions and ratchets. In all these noise-induced effects ordering occurs due to the joint action of two noisy sources.
Coupled Brownian motors: anomalous-to-normal hysteresis transition and noise-induced limit cycle
Show abstract
We study a model consisting of N nonlinear oscillators with global periodic coupling and local multiplicative and additive noises. The model was shown to undergo a nonequilibrium phase transition
towards a broken-symmetry phase exhibiting noise-induced ``ratchet" behavior. Here we review some aspects leading to an "anomalous--to--normal" transition in the ratchet's hysteretic behavior and
also show -as suggested by the absence of stable solutions when the load force is beyond a critical value- the existence of a limit cycle induced by both: multiplicative noise and global periodic
Thresholds, Signals, and Synchronization
Thresholds and noise
Show abstract
Random processes acting through dynamical systems with thresholds lie at the heart of many natural and man-made phenomena. The thresholds here considered are general including not only sharp or
“hard” boundaries but also a class of dynamical, nonlinear system functions some of which are themselves mediated by the noise. Processes include noise-induced transitions, postponed and advanced
bifurcations, noise enhanced propagation of coherent structures, and stochastic resonance and synchronization. Examples of these processes are found in a wide range of disciplines from physics and
chemistry to neuroscience and even human and animal behavior and perception. I will discuss some of these examples connecting them with their fundamental dynamical origins.
Nonrenewal spike trains generated by stochastic neuron models
Show abstract
Many of the stochastic neuron models employed in the neurobiological literature generate renewal point processes, i.e., successive intervals between spikes are statistically uncorrelated. Recently,
however, much experimental evidence for positive and negative correlations in the interspike interval (ISI) sequence of real neurons has been accumulated. It has been shown that these correlations
can have implications for neuronal functions. We study a leaky integrate-and-fire (LIF) model with a dynamical threshold or an adaptation current both of which lead to negative correlations.
Conditions are identified where these models are equivalent. The ISI statistics, the serial correlation coefficient, and the power spectrum of the spike train, are numerically investigated for
various parameter sets.
Catastrophes in locking systems driven by green noise
Show abstract
We consider a phase-locked loop for the case of an external signal with a stationary fluctuating phase. The problem reduces to the problem of a Brownian particle in a periodic potential driven by
“green” noises. We numerically simulate the case in which the random phase is the Ornstain-Uhlenbeck process. The rapid irreversible transition from stationary random motion (a locked state) to a
nonstationary one at a high near-constant rate (a running state) is shown to be possible for the case of the massive particle. We found that transition moments change suddenly for small variations of
external parameters. We call this phenomenon the “catastrophe”. The numerical results are compared with those obtained by the Krylov-Bogoliubov averaging method. The first approximation of the method
is found to be sufficiently accurate if the states coexist and the direct and backward transitions occur frequently enough.
Noise-assisted propagation of signals through the chain of level-crossing detectors
Nobuko Fuchikami, Toshifumi Sakaguchi
Show abstract
Noise-assisted propagation of periodic signals is investigated for one-dimensional arrays composed of one-way coupled level-crossing detectors (LCD). Analytical expressions are obtained for the
signal decay length through chains and the signal decay time through rings, where noise is uncorrelated so that the signal transmission from a LCD to the neighboring one is Markovian. Recent
numerical simulations for one-dimensional arrays of one-way coupled bistable oscillators are discussed in comparison to the present analytical results.
Stochastic synchronization: applications to oscillatory electroreceptors
Alexander Neiman, David Frank Russell, Frank E Moss, et al.
Show abstract
Classical notion of synchronization, introduced originally for periodical self-sustained oscillators, can be extended to stochastic systems. This can be done even in the case when the characteristic
times of a system are fully controlled by noise. Stochastic synchronization is then defined by imposing certain conditions to various statistical measures of the process. We review various approaches
to stochastic synchronization and apply them to study synchronization in the electrosensory system of paddlefish.
The data processing inequality and stochastic resonance
Mark D. McDonnell, Nigel G. Stocks, Charles E. M. Pearce, et al.
Show abstract
The data processing inequality of information theory states that given random variables X, Y and Z which form a Markov chain in the order X-->Y-->Z, then the mutual information between X and Y is
greater than or equal to the mutual information between X and Z. That is I(X) >= I(X;Z) . In practice, this means that no more information can be obtained out of a set of data then was there to begin
with, or in other words, there is a bound on how much can be accomplished with signal processing. However, in the field of stochastic resonance, it has been reported that a signal to noise ratio gain
can occur in some nonlinear systems due to the addition of noise. Such an observation appears to contradict the data processing inequality. In this paper, we investigate this question by using an
example model system.
Noise in Systems with Delay
Anticipated synchronization in neuronal systems subject to noise
Show abstract
We report the observation of synchrony in two unidirectionally coupled (master-slave) model neurons (implemented by electronic circuits) in a noisy environment. Both neurons are subjected to the same
random stimulus, and there is a recurrent inhibitory delayed connection in the slave neuron. We observe that synchrony occurs shifted tin time, such that the slave neuron anticipates, i.e.,
forecasts, the response of the master neuron. By incorporating the effects of unidirectional coupling, delayed feedback and common noise into models of two spiking neurons, we are able to simulate
successfully the experimental observations.
Feedback coupling in dynamical systems
Steffen Trimper, Knud Zabrocki
Show abstract
Different evolution models are considered with feedback-couplings. In particular, we study the Lotka-Volterra system under the influence of a cumulative term, the Ginzburg-Landau model with a
convolution memory term and chemical rate equations with time delay. The memory leads to a modified dynamical behavior. In case of a positive coupling the generalized Lotka-Volterra system exhibits a
maximum gain achieved after a finite time, but the population will die out in the long time limit. In the opposite case, the time evolution is terminated in a crash. Due to the nonlinear feedback
coupling the two branches of a bistable model are controlled by the the strength and the sign of the memory. For a negative coupling the system is able to switch over between both branches of the
stationary solution. The dynamics of the system is further controlled by the initial condition. The diffusion-limited reaction is likewise studied in case the reacting entities are not available
simultaneously. Whereas for an external feedback the dynamics is altered, but the stationary solution remain unchanged, a self-organized internal feedback leads to a time persistent solution.
Phase synchronization in noisy oscillators with nonisochronicity
Bernd Blasius, Gustavo C. Rodrigues
Show abstract
We study the synchronization of two nonidentical oscillators with nonvanishing nonisochronicity under the presence of uncorrelated Gaussian noise. To measure the amount of synchronization we
calculate the evolution of the phase difference. Without coupling both oscillators rotate with different natural frequencies. Due to the action of coupling this frequency difference is reduced until
finally, at a critical coupling strength, synchronization sets in. Under the presence of uncorrelated noise the observed frequency differences is still a monotonically decreasing function of coupling
strength but can never become zero due to noise induced phase slips. Here, we show that this usual picture of the transition to synchronization is strongly modified when the oscillators are
nonisochronous. In this case the onset of coupling can have different effects and may enlarge or even invert the natural frequency difference of the uncoupled oscillators. Our results can be
explained in terms of a noisy particle in a tilted potential.
Noise-induced transitions in overdamped systems: short times
Slanislav M. Soskin, Valentin I. Sheka, Tatiana L. Linnik, et al.
Show abstract
In the problem of the activation energy for a noise-induced transition over a finite given time in an arbitrary overdamped one-dimensional potential system, we find and classify all extremal paths
and provide a simple algorithm to explicitly select which is the most probable transition path (MPTP). The activation energy is explicitly expressed in quadratures. For the transition beyond the top
of the barrier, the MPTP does not possess turning points and the activation energy is a monotonously decreasing function of the transition time. For transitions between points lying on one and the
same slope of the potential well, which may be relevant e.g. for the problem of the tails of the prehistory probability density, the situation is more complicated: the activation energy is a
non-monotonous function of time and, most important, may possess bends corresponding to jump-wise switches in the topology of the MPTP; it can be proved also that the number of turning points in the
MPTP is necessarily less than two. The prefactor is calculated numerically using the scheme suggested by Lehmann, Reimann and Hanggi, PRE 55, 419 (1998). The theory is compared with simulations.
Stochastic excitation and synchronization in coupled FitzHugh-Nagumo elements
Show abstract
We investigate theoretically and numerically the activation process in a single-out and coupled FitzHugh-Nagumo elements. Two qualitatively different types of the dependence of the mean activation
time and of the mean cycling time on the coupling strength monotonic and non-monotonic have been found for identical elements. The influence of coupling strength, noise intensity and firing threshold
on the synchronization regimes and its characteristics is analyzed
A Fokker-Planck description for Parrondo's games
Show abstract
We discuss in detail two recently proposed relations between the Parrondo's games and the Fokker-Planck equation describing the flashing ratchet as the overdamped motion of a particle in a potential
landscape. In both cases it is possible to relate exactly the probabilities of the games to the potential in which the overdamped particle moves. We will discuss under which conditions current-less
potentials correspond to fair games and vie versa.
Experimental study of a noisy dissipative-driven ring lattice with Morse interactions
Show abstract
An experimental study has been carried out on a noisy dissipative-driven ring lattice of units coupled via Morse potentials. An electronic circuit mimicking the lattice dynamics and noise sources is
used. We show that inclusion of long range attractive forces facilitates clustering (at variance with the repulsive Toda ring) and van der Waals-like transition phenomena.
Mechanism of signal-to-noise ratio gain in a monostable threshold stochastic resonator
Show abstract
In the last few years, several papers have been published that reported high signal-to-noise ratio (SNR) gains in systems showing stochastic resonance. In the present work, we consider a
level-crossing detector driven by a periodic pulse train plus Gaussian band-limited white noise, and provide analytical formulae for the dependence of the SNR gain on the relevant parameters of the
input (the amplitude and the cut-off frequency of noise, the duty cycle of the deterministic signal and the distance between the threshold and the amplitude of the signal). Our results are valid in
the input parameter range wherein high gains are expected, that is, wherein the probabilities of missing and, especially, extra output peaks are very low. We also include numerical simulation results
that support the theory, along with illustrations of cases which are outside the validity of our theory.
From theory of infinitely divisible distributions to derivation of generalized master equation for Markov process
Show abstract
We show that the increment of generalized Wiener process (random process with stationary and independent increments) has the properties of a random value with infinitely divisible distribution. This
enables us to write the characteristic function of increments and then to obtain the new formula for correlation of the derivative of generalized Wiener process (non-Gaussian white noise) and its
arbitrary functional. IN the context of well-known functional approach to analysis of nonlinear dynamical systems based on a correlation formulae for nonlinear stochastic functionals, we apply this
result for derivation of generalized Fokker-Planck equation for probability density. We demonstrate that the equation obtained takes the form of ordinary Fokker-Planck equation for Gaussian white
noise and, at the same time, transforms in the fractional diffusion equation in the case of non-Gaussian white noise with stable distribution.
Drastic facilitation of the onset of global chaos due to an extremum in the dependence of eigenfrequency on energy
Slanislav M. Soskin, Oleg M. Yevtushenko, Riccardo Mannella
Show abstract
The Chirikov resonance-overlap criterion predicts the onset of global chaos if nonlinear resonances overlap in energy, which is conventionally assumed to require a non-small magnitude of
perturbation. We show that, for a time-periodic perturbation, the onset of global chaos may occur at unusually small magnitudes of perturbation if the unperturbed system possesses more than one
separatrix. The relevant scenario is the combination of the overlap in the phase space between resonances of the same order and their overlap in energy with chaotic layers associated with
separatrices of the unperturbed system. One of the most important manifestations of this effect is a drastic increase of the energy range involved into the unbounded chaotic transport in spatially
periodic system driven by a rather weak time-periodic force, which results in turn in the drastic increase either of the dc conductivity, if the system carries an electric charge, or the escape rate,
if the system is subject to noise. We develop the asymptotic theory and verify it in simulations. Various generalizations are delineated, in particular for the case of a time-independent
Pulse propagation in a model for the photosensitive Belousov-Zhabotinsky reaction with external noise
Show abstract
We study the dynamics of excitation pulses in a modified Oregonator model for the light-sensitive Belousov-Zhabotinsky (BZ)reaction assuming that the intensity of the applied illumination is a
spatiotemporal stochastic field with finite correlation time and correlation length. For a two-component version of the model we discuss the dependence of the pulse speed on the characteristic
parameters of the noise in the framework of a small noise approximation up to the first order in the correlation time. In the full three-component model we find enhancement of coherence resonance for
suitable chosen correlation time. Based on this observation, we propose a mechanism for noise-enhanced propagation of pulse trains in excitable media subjected to external fluctuations.
Discrete games of chance as models for continuous stochastic transport processes
Show abstract
Discrete games of chance can be used to illustrate principles of stochastic processes. For example, most readers are familiar with the use of discrete random walks to model the microscopic phenomenon
of Brownian motion. We show that discrete games of chance, such as those of Parrondo and Astumian, can be used to quantitatively model stochastic transport processes. Discrete games can be used as
“toy” models for pedagogic purposes but they can be much more than “toys”. In principle we could perform accurate simulations and we could reduce the errors of approximation to any desired level,
provided that we were prepared to pay the computational cost. We consider some different approaches to discrete games, in the literature, and we use partial differential equations to model the
particle densities inside a Brownian Ratchet. We apply a finite difference approach and obtain finite difference equations, which are equivalent to the games of Parrondo. The new games generalize
Parrondo's original games, in the context of stochastic transport problems. We provide a practical method for constructing sets of discrete games, which can be used to simulate stochastic transport
processes. We also attempt to place discrete games, such as those of Parrondo and Astumian, on a more sound philosophical basis.
Solution of the boundary value problem for nonlinear flows and maps
Show abstract
Fluctuational escape via an unstable limit cycle is investigated in stochastic flows and maps. A new topological method is suggested for analysis of the corresponding boundary value problems when the
action functional has multiple local minima along the escape trajectories and the search for the global minimum is otherwise impossible. The method is applied to the analysis of the escape problem in
the inverted Van der Pol oscillator and in the Henon map. An application of this technique to solution of the escape problem in chaotic maps with fractal boundaries, and in maps with chaotic saddles
embedded within the basin of attraction, is discussed.
Method for detecting the signature of noise-induced structures in spatiotemporal data sets: an application to excitable media
Show abstract
We formulate mathematical tools for analyzing spatiotemporal data sets. The tools are based on nearest-neighbor considerations similar to cellular automata. One of the analysis tools allows for
reconstructing the noise intensity in a data set and is an appropriate method for detecting a variety of noise-induced phenomena in spatiotemporal data. The functioning of these methods is
illustrated on sample data generated with the forest fire model and with networks of nonlinear oscillators. It is seen that these methods allow the characterization of spatiotemporal stochastic
resonance (STSR) in experimental data. Application of these tools to biological spatiotemporal patterns is discussed. For one specific example, the slime mold Dictyostelium discoideum, it is seen,
how transitions between different patterns are clearly marked by changes in the spatiotemporal observables.
Oscillatory electrochemical reactions at corroding silicon surface
Vitali Parkhutik, Junji Sasano, Yukio Ogata, et al.
Show abstract
The paper analyses the nature of chaotic and well-ordered oscillations of the anodic potential and open circuit potential of silicon immersed in aqueous electrolytes. These oscillations are observed
when experimental conditions are fine tuned in what corresponds to the current flowing through the system, composition of electrolyte, its viscosity, etc. It is assumed that the oscillations are due
to the accumulation of mechanical stress in the thin (50-80 nm) oxide film formed at the surface of silicon as a result of electrochemical anodic reaction. The stress is released by local etching of
the oxide and its lifting-on from the Si surface. The process repeats again and again yielding long-lasting oscillations of the anodic potential value (amplitude around 1-15 V, period 20-150 s) or of
the open circuit potential (several hundreds milli-volts). Along with temporal ordering of the process (oscillations of potential) there occurs a spatial ordering in the system - the surface of
corroding Si sample is covered with hexagonally ordered semi-spherical cells (diameter about 700 nm). The effect is well-fit by the general phenomenology of chaos-order transitions in che4mical
systems (bifurcations), strange attractors are the intrinsic features of these oscillations) and its kinetics is very similar to that of the Belousov-Zabotinsky reaction. However, oscillatory
processes on the corroding Si surface are caused by quite specific physical and chemical mechanisms, which are not well understood presently. We present the microscopic model for the oscillatory
behavior which involves, generation of local mechanical stress at the Si/electrolyte interface, non-linear electrochemical etching of Si, localization of the electric field at the etched surface,
Scaling properties of long-range correlated noisy signals: application to financial markets
Anna Carbone, Giuliano Castelli
Show abstract
Long-range correlation properties of financial stochastic time series y have been investigated with the main aim to demonstrate the ability of a recently proposed method to extract the scaling
parameters of a stochastic series. According to this technique, the Hurst coefficient H is calculated by means of the following function: EQUATION where y[n](i)is the moving average of y(i), defined
as EQUATION the moving average window and N[max] is the dimension of the stochastic series. The method is called Detrending Moving Average Analysis (DMA) on account of the several analogies with the
well-known Detrended Fluctuation Analysis (DFA). The DMA technique has been widely tested on stochastic series with assigned H generated by suitable algorithms. It has been demonstrated that the
ability of the proposed technique relies on very general grounds: the function EQUATION generates indeed a sequence of clusters with power-law distribution of amplitudes and lifetimes. In particular
the exponent of the distribution of cluster lifetime varies as the fractal dimension 2 - H of the series, as expected on the basis of the box-counting method. In the present paper we will report on
the scaling coefficients of real data series (the BOBL and DAX German future) calculated by the DMA technique.
Derivation of 1/f noise from resonances
Show abstract
We study non-interacting particles in a small subsystem which is weakly coupled to a reservoir. We show that this class of systems can be mapped into an extended form of the Friedrichs model. We
derive from the Hamiltonian dynamics that the number fluctuation in a subsystem is 1/f or 1/f^β noise. We show that this effect comes from the sum of resonances.
Dynamical formulation of Gaussian white noise
Show abstract
We study the connection between Hamiltonian dynamics and irreversible, stochastic equations, such as the Langevin equation. We consider a simple model of a harmonic oscillator (Brownian particle)
coupled to a field (heat bath). We introduce an invertible transformation operator Λ that brings us to a new representation where dynamics is decomposed into independent Markovian components,
including Brownian motion. The effects of Gaussian white noise are obtained by the non-distributive property of Λ with respect to products of dynamical variables. In this way we obtain an exact
formulation of white noise effects. Our method leads to a direct link between dynamics of Poincaré nonintegrable systems, probability and stochasticity.
Dynamical model of foreign exchange markets leading to Tsallis distribution
Show abstract
We present a model of financial markets originally proposed for a turbulent flow, as a dynamic basis of its intermittent behavior. Time evolution of the price change is assumed to be described by
Brownian motion in a power-law potential, where the 'temperature' fluctuates slowly. The model generally yields a fat-tailed distribution of the price change. Specifically a Tsallis distribution is
obtained if the inverse temperature is χ^2-distributed, which qualitatively agrees with intraday data of foreign exchange market. The so-called 'volatility', a quantity indicating the risk or
activity in financial markets, corresponds to the temperature of markets and its fluctuation leads to intermittency.
Directed current in ratchets by pseudo-Gaussian white noise: a seeming paradox from a Langevin equation and an alternative master equation
Nobuko Fuchikami, Shunya Ishioka
Show abstract
Symmetric white Poissonian shot noise with Gaussian distributed amplitudes is shown from an overdamped Langevin equation to induce directed motion of ratchets. The noise becomes white Gaussian in the
limit λ→∞, where λ is the average number of the delta pulses per unit time. The current tends to zero as 1/λ→0, which agrees with the fact that the directed motion cannot be induced by the thermal
noise alone. However, a finite value of 1/λ yields a finite value of the current no matter how small the former may be. Since 1/λ cannot be zero physically, this is a seeming contradiction of the
second law of thermodynamics, which originates from the Langevin equation. We discuss this point from an alternative master equation without assuming the Langevin equation.
Information essence of chaotic surface structures
Anna B. Solovieva, Serge F. Timashev, Grigory V. Vstovsky, et al.
Show abstract
A general phenomenological approach - a Flicker Noise Spectroscopy (FNS)- to revelation of information valuable parameters characterizing the arbitrary chaotic surfaces was develop to distinguish
their patterns and describe quantitatively their functional properties. The consideration was carried out in terms of correlation lengths and additional parameters characterizing the rate of
correlation links lost in the sequences of surface irregularities. The parameters are obtained by fitting the Fourier spectra and structural functions (difference moments of different orders)
calculated for the digitized surface profiles using the approximations derived on the base of model representation of the profiles as the sequences of irregularities of different types (“bursts”,
“jumps”, etc.). The method developed was applied to revelation of effects of a shungit filling agent in polypropylen matrix on the composite properties, revelation of hydrogen treatment effects on
the cleavage surfaces of LiF monocrystals after their dissolution in water with quantitative evaluations of their anisotropy, analysis of activity of vacuum deposited porphyrins layers in a
photosensibilized gnenration of singlet oxygen into gaseous phase. The approach elaborated can be used for developing the new control tools in nano-technologies, microelectronics, production of
polymeric material with the specific surface properties, and others.
Influence of spatiotemporal 1/f alpha-noise on structure formation in excitable media
Show abstract
The influence of spatiotemporally correlated power-law, i.e. 1/f^\alpha$, noise on pattern formation in a two dimensional excitable medium consisting of coupled FitzHugh-Nagumo (FHN) oscillators is
discussed. The signature of Spatiotemporal Stochastic Resonance (STSR) is investigated using the mutual information. It is found that optimal noise variance for STSR is minimal, if both the spatial
and temporal power spectral densities of the noise decay with a characteristic exponent of \alpha$=1. This effect is related to the band-pass frequency filtering characteristic of the FHN oscillator.
Inhomogeneity-enhanced coherence resonance in assemblies of uncoupled chaotic elements
Show abstract
We study the dynamics of assemblies of “uncoupled” identical chaotic elements under the influence of external noisy filed. It is numerically demonstrated that in the case where each chaotic element
exhibits type-I intermittency, the degree of the temporal regularity of the mean-field dynamics of the system reaches a maximum at a certain optimal noise intensity. Moreover, we also report that
inhomogeneous noise which drives each element partly independently enhances the coherence of the mean-field more than that of the case where all elements of the system receive a completely identical
noisy input, and the degree of the coherence as a function against the degree of inhomogeneity of noise shows a convex curve. In noisy uncoupled systems, the common part of noise which drives each
element can be regarded as the interaction among elements which corresponds to the coupling term in the case of coupled systems, so our finding that some degree of inhomogeneity enhances the
coherence of the dynamics is not trivial.
Deterministic stochastic resonance in chaotic diffusion
Show abstract
We show deterministic stochastic resonance (DSR) in chaotic diffusion when the diffusion map is modulated by a sinusoid. In chaotic diffusion, the map parameter determines the state transition rate
and the diffusion coefficient. The transition rate shows the diffusion intensity. Therefore, the parameter represents the intensity of the internal fluctuation. By this fact, increase of the
parameter maximizes the response of DSR as in standard stochastic resonance (SR) where the external noise intensity optimizes the response. Sinusoidally modulated diffusion is regarded as a
stochastic process whose transition rate is modulated by the sinusoid. Therefore, the transition dynamics can be approximated by a time-dependent random walk process. Using the mean transition rate
function against the map parameter, we can derive the DSR response depending on the parameter. Our approach is based on the rate modulation theory for SR. Even when the diffusion map is modulated by
the sinusoid and noise from an external environment, the increasing parameter can also maximize the DSR response. We can calculate the DSR response depending on the external noise intensity and the
map parameter. DSR takes advantage of applications to signal detection because the system has the control parameter corresponding to the internal fluctuation intensity.
Synchronization patterns in cerebral blood flow and peripheral blood pressure under minor stroke
Show abstract
Stroke is a leading cause of death and disability in the United States. The autoregulation of cerebral blood flow that adapts to changes in systemic blood pressure is impaired after stroke. We
investigate blood flow velocities (BFV) from right and left middle cerebral arteries (MCA) and beat-to-beat blood pressure (BP) simultaneously measured from the finger, in 13 stroke and 11 healthy
subjects using the mean value statistics and phase synchronization method. We find an increase in the vascular resistance and a much stronger cross-correlation with a time lag up to 20 seconds with
the instantaneous phase increment of the BFV and BP signals for the subjects with stroke compared to healthy subjects.
Stochastic processes with finite size scale invariance
Pierre-Olivier Amblard, Pierre Borgnat, Patrick Flandrin
Show abstract
We present a theory of stochastic processes that are finite size scale invariant. Such processes are invariant under generalized dilations that operate on bounded ranges of scales and amplitudes. We
recall here the theory of deterministic finite size scale invariance, and introduce an operator called Lamperti transform that makes equivalent generalized dilations and translations. This operator
is then used to defined finite size scale invariant processes as image of stationary processes. The example of the Brownian motion is presented is some details to illustrate the definitions. We
further extend the theory to the case of finite size scale invariant processes with stationary increments.
Data modeling of 1/f noise sets
Holger M. Jaenisch, James W. Handley
Show abstract
A novel method is presented for solving the inverse fractal problem for 1/f noise sets. The performance of this method is compared with classical data modeling methods. Applicability to different
distributions of noise is presented, along with an overview of important applications including data and image compression. | {"url":"https://spie.org/Publications/Proceedings/Volume/5114","timestamp":"2024-11-14T07:25:21Z","content_type":"text/html","content_length":"350058","record_id":"<urn:uuid:bfd49560-0b97-46aa-bc15-422f9224168b>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00180.warc.gz"} |
If XYZ invests $3,300 today and $3,300 in 1 year in an account
that has an...
Answer #1
She will have $ 9,704.42 in her account in 3 years.
Future value of money is calculated as follows:
Quarter Investment Future value of 1 Future value of investment
a b c=((1+(0.156/4))^(12-a)) d=b*c
0 $ 3,300 1.582656154 $ 5,222.77
4 $ 3,300 1.358076957 $ 4,481.65
Total $ 9,704.42
Similar Homework Help Questions
If XYZ invests $3,300 today and $3,300 in 1 year in an account that has an...
If XYZ invests $3,300 today and $3,300 in 1 year in an account that has an expected annual return of 15.6 percent, compounded quarterly, then how much money will she have in her account in 3 years?
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Hundredweight to Grams Converter
โ Switch toGrams to Hundredweight Converter
How to use this Hundredweight to Grams Converter ๐ ค
Follow these steps to convert given weight from the units of Hundredweight to the units of Grams.
1. Enter the input Hundredweight value in the text field.
2. The calculator converts the given Hundredweight into Grams in realtime โ using the conversion formula, and displays under the Grams label. You do not need to click any button. If the input
changes, Grams value is re-calculated, just like that.
3. You may copy the resulting Grams value using the Copy button.
4. To view a detailed step by step calculation of the conversion, click on the View Calculation button.
5. You can also reset the input by clicking on button present below the input field.
What is the Formula to convert Hundredweight to Grams?
The formula to convert given weight from Hundredweight to Grams is:
Weight[(Grams)] = Weight[(Hundredweight)] × 50802.34544
Substitute the given value of weight in hundredweight, i.e., Weight[(Hundredweight)] in the above formula and simplify the right-hand side value. The resulting value is the weight in grams, i.e.,
Calculation will be done after you enter a valid input.
Consider that a batch of premium tea leaves is weighed at 1 hundredweight.
Convert this weight from hundredweight to Grams.
The weight of tea leaves in hundredweight is:
Weight[(Hundredweight)] = 1
The formula to convert weight from hundredweight to grams is:
Weight[(Grams)] = Weight[(Hundredweight)] × 50802.34544
Substitute given weight of tea leaves, Weight[(Hundredweight)] = 1 in the above formula.
Weight[(Grams)] = 1 × 50802.34544
Weight[(Grams)] = 50802.3454
Final Answer:
Therefore, 1 cwt is equal to 50802.3454 g.
The weight of tea leaves is 50802.3454 g, in grams.
Consider that a shipment of luxury fabric weighs 2 hundredweight.
Convert this weight from hundredweight to Grams.
The weight of luxury fabric in hundredweight is:
Weight[(Hundredweight)] = 2
The formula to convert weight from hundredweight to grams is:
Weight[(Grams)] = Weight[(Hundredweight)] × 50802.34544
Substitute given weight of luxury fabric, Weight[(Hundredweight)] = 2 in the above formula.
Weight[(Grams)] = 2 × 50802.34544
Weight[(Grams)] = 101604.6909
Final Answer:
Therefore, 2 cwt is equal to 101604.6909 g.
The weight of luxury fabric is 101604.6909 g, in grams.
Hundredweight to Grams Conversion Table
The following table gives some of the most used conversions from Hundredweight to Grams.
Hundredweight (cwt) Grams (g)
0.01 cwt 508.0235 g
0.1 cwt 5080.2345 g
1 cwt 50802.3454 g
2 cwt 101604.6909 g
3 cwt 152407.0363 g
4 cwt 203209.3818 g
5 cwt 254011.7272 g
6 cwt 304814.0726 g
7 cwt 355616.4181 g
8 cwt 406418.7635 g
9 cwt 457221.109 g
10 cwt 508023.4544 g
20 cwt 1016046.9088 g
50 cwt 2540117.272 g
100 cwt 5080234.544 g
1000 cwt 50802345.44 g
The hundredweight is a unit of mass used in both the British imperial and US customary systems, with variations. In the US, the short hundredweight equals 100 pounds, while the British long
hundredweight equals 112 pounds. It is often used in agriculture and industry for measuring bulk commodities.
The gram is a metric unit of mass. It is equal to one thousandth of a kilogram. Grams are commonly used for small measurements of mass, especially in scientific and everyday contexts.
Frequently Asked Questions (FAQs)
1. What is the formula for converting Hundredweight to Grams in Weight?
The formula to convert Hundredweight to Grams in Weight is:
Hundredweight * 50802.34544
2. Is this tool free or paid?
This Weight conversion tool, which converts Hundredweight to Grams, is completely free to use.
3. How do I convert Weight from Hundredweight to Grams?
To convert Weight from Hundredweight to Grams, you can use the following formula:
Hundredweight * 50802.34544
For example, if you have a value in Hundredweight, you substitute that value in place of Hundredweight in the above formula, and solve the mathematical expression to get the equivalent value in
Weight Converter Android Application
We have developed an Android application that converts weight between kilograms, grams, pounds, ounces, metric tons, and stones.
Click on the following button to see the application listing in Google Play Store, please install it, and it may be helpful in your Android mobile for conversions offline. | {"url":"https://convertonline.org/unit/?convert=hundredweight-gram","timestamp":"2024-11-03T09:04:18Z","content_type":"text/html","content_length":"86798","record_id":"<urn:uuid:1562ee58-2e33-4612-8ad7-a97228f9c843>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00367.warc.gz"} |
Pascal Gollin gave a talk on the existence of a decomposition of an infinite graph into spanning trees at the discrete math seminar - Discrete Mathematics Group
Pascal Gollin gave a talk on the existence of a decomposition of an infinite graph into spanning trees at the discrete math seminar
On October 29, 2019, Pascal Gollin from IBS discrete mathematics group gave a talk on the existence of a decomposition of an infinite graph into spanning trees in terms of the existence of packing
and covering of spanning trees at the discrete math seminar. The title of his talk was “A Cantor-Bernstein-type theorem for spanning trees in infinite graphs“. | {"url":"https://dimag.ibs.re.kr/2019/pascal-gollin-seminar/","timestamp":"2024-11-07T17:35:24Z","content_type":"text/html","content_length":"141413","record_id":"<urn:uuid:04dfb8bc-fca7-48b1-883b-a4829de68405>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00517.warc.gz"} |
Good Putting Numbers - It's About 3 Putts Not Putts per Round
Good Putting Numbers – It’s About 3 Putts Not Putts per Round
All golfers have heard the well-worn phrase about putting – “You drive for show but you putt for dough.”
The importance of putting in the game of golf is drilled into the mind of all players from the moment they take it up and it’s easy to see why.
As a rule of thumb, at all levels of the game, the number of putts a golfer takes during a round will make up approximately 40% of their total score.
This percentage of course varies depending on how you are playing but however you slice it, the number of putts you take per round is going to be a key determining factor in what your score will be.
But how many putts per round is good?
As a general rule a total of between 31 and 34 putts per round is good putting. 36 putts per round or 2 or more putts per hole is considered poor putting while 30 putts per round or less is excellent
putting. The PGA Tour average is 29 putts per round with the best putter averaging 27.76 putts per round and the worst 30.36.
If we then consider how many putts per round you should have for your handicap or to break key scoring barriers the table below lists the averages.
HANDICAP AVG. PUTTS PER ROUND AVG. SCORE
Best pro 27.7 69.1
Scratch 31.5 75.7
1-5 32.6 79.8
6-10 33.7 84.8
11-15 34.8 90
16-20 35.8 95.1
21-25 36.8 100.5
25+ 38.6 108.8
All 35 90.8
Sources: PGA Tour, MyGolfSpy
The data shows that to break 100 you need to have 36 putts per round on average. To break 90 you will need to take only 34 putts a round while to break 80 takes 32 putts per round. Pros breaking 70
will only take 28 putts a round. 34 putts or less per round must therefore be judged good putting for the average golfer.
However, to assess whether you are putting well or not compared to your peers you should not now simply compare the number of putts you take per round to these averages.
It is vital to understand that putting is not an independent metric and that whether you are above or below average in terms of the number of putts you take per round can be explained in a variety of
different ways.
It may indeed mean you are a good putter but it may also mean you do not hit many greens in regulation or you consistently hit the ball close with your approach shots.
And forget about what you see on the TV.
Just because you see professionals holing a bunch of putts on the highlights reel does not necessarily mean they are all great putters or indeed are aiming to hole every putt!
Putts per Round is a Poor Measure of Putting Skill
Measuring the number of putts you take in any given round is a statistic that has been used by golfers for a long time to assess their putting performance.
And there is one simple reason for this – it is the easiest putting stat to measure.
But that does not mean it is the best statistic to measure whether you are putting well or have taken a good number of putts for the round you have just played.
And the reason the number of putts you have taken in a round is not the best stat for measuring whether you have putted well on any given day is that it takes no account of the distance you have hit
your putts from or the number of greens in regulation you hit.
Let’s take an example of two players playing together.
One hits all 18 greens in regulation and 2 putts every green but none of his first putts was from closer than 30 feet.
The second player also hits all 18 greens in regulation but is within 10 feet of the hole for every birdie putt and as a result makes 6 birdies.
Old habits die hard, but using putts per round instead of strokes gained putting is like driving a horse and buggy when a car is parked out front.
Professor Mark Broadie, author of ‘Every Shot Counts‘ (Amazon link)
The first player takes 36 putts and scores 72 compared to the second player who takes 30 putts and scores 66.
So the second player is clearly the better putter and that is why he had the better round?
The short answer is no they are not and just by measuring the average number of putts per round you will not get a true representation of how good a putter a golfer is.
As shown by this example the player who hits the ball farther away from the hole but is a fantastic putter will be hidden by the number of putts per round statistic.
They are better on the greens than the player who hits a lot of great iron shots closer to the hole, and therefore makes more putts, but you can’t tell that by simply looking at the total number of
putts in the round.
That is the reason why tour professionals now measure their putting performance using the metric of ‘Strokes Gained Putting’ rather than average putts per round.
And that is because this metric takes account of the distance of each putt.
Putting vs Scoring fact: In 2019, Rory McIlroy had the best average score (69.1) on the PGA Tour but was 14th in the list of average putts per round and tied only 24th on the list of ‘Strokes
Gained Putting’.
PGA Tour
To calculate ‘strokes gained putting’ the major golfing tours calculate the average number of putts a pro takes to hole out from every distance.
This gives each player what is called their “putting benchmark”.
During a tournament a player’s putts are then compared to their benchmark to determine whether their putting has caused them to gain or lose strokes on individual holes.
For example, if a pros ‘putting benchmark’ from 10 feet is 1.5 putts and they one putt from that distance on the first hole they will ‘gain’ 0.5 strokes.
If they two putt from the same distance on the second hole they lose 0.5 strokes.
And therefore over the course of a round a player can gain an insight into whether they putted well or badly by measuring how many putting strokes they gained or lost over 18 holes.
There is an additional calculation done on the pro tours to give a final strokes gained putting figure, which is based on an assessment of how easy or hard the greens are on a particular day, but the
basics of the measurement are based on the individual’s putting benchmark as explained above.
For the average golfer reducing the number of times they three putt from 11-30 feet is the quickest way to take strokes off their score.
Now that’s fine for the pros I hear you say but what about the average amateur player who doesn’t have the time or tools to work out their putting benchmark and measure how far each putt they hit
every round is.
I agree and clearly this is not a very practical measure for the amateur player.
I am simply highlighting it to show that you should not look at the average number of putts you take in a round as the true arbiter of how you are putting.
As we’ve seen it really depends on how long your putts are and how many greens you have hit.
So what’s an alternative stat that amateur players can more easily use to assess their putting performance better than the average number of putts per round?
Mark Sweeney, the founder of AimPoint Golf, teacher of five world #1 ranked players and named one of golf’s Top Innovators by Golf Digest Magazine, proposes a rough and ready formula to answer the
question – “Did I putt well today or not?”
Based on tour players’ putting and scrambling stats he proposes adding 22.5 to the number which results from dividing the number of greens in regulation you have hit by 2.
In formula terms this equates to the following:
Good number of putts = (GIR divided by 2) plus 22.5 OR (GIR / 2) + 22.5
This would mean if you hit 8 greens your target putts per round number would be 26.5 or (8 / 2) + 22.5.
If you hit 18 greens by comparison a good putting number would be 31.5.
This statistic will still not give you a good representation of how good your putting is from different distances as ‘strokes gained putting’ does but it will give a better representation of pure
putting performance during your round than the total number of putts in your round.
[Editor’s note – If you want to know what golf stats are the best to keep track of and which of the traditional ones are actually misleading check out our post on the best 10 stats to keep track of
Less 3 Putts is the Fast Way to Good Putting Stats
The number of stats in golf can be overwhelming and working out which ones actually matter most for your own game can feel that it requires a college degree in itself.
One quick glance at the stats pages of the major professional tours can lead you to get lost in a sea of numbers and more than a hundred metrics and measures.
Taking account of these is fine if you are a professional player and have teams of people to help you but what about the average recreational golfer who barely has time to get to the first tee in
time never mind pore over all their stats before, during and after a round to work out how well each aspect of their game is going.
The good news when it comes to putting though is that one statistic is the main key to measure for the vast majority of average golfers.
And that is the number of times you three putt.
Reducing your number of 3 putts has been proven time and again to be the quickest way to shave strokes off your score for the vast majority of amateur golfers.
And the simple reason the average golfer with a handicap of over 15 will 3 putt between 3-4 times every round (that is almost 6 times more than the average PGA tour player) – is because they leave
their first putt too far away from the hole.
Shotscope has found that the average distance the average 20 handicapper leaves their first putt away from the hole is almost 9ft.
The bad news with that according to Mark Broadie, Columbia Business School professor and author of Every Shot Counts, is that tour professionals make less than half (only 48%) of their putts from 9
[Note – Interested in learning more about your golf stats? Check out our review of Professor Broadie’s great book here.]
Reduce the length of your second putt and you will quickly make less 3 putts and score better
So what chance does the average golfer have of making that 2nd putt of about 9 feet if the pros can’t do it half the time?
Peter Sanders, stats guru to numerous PGA Tour players, including two-time major winner Zach Johnson, also highlights just how bad the average golfers’ mid-range putting can be.
While on the PGA Tour the pros’ average 2-putt range is 35 feet – in other words only when putts reach a length of 35 feet will pros start 3 putting more than they 1 putt – the average 2-putt range
for the average golfer with a handicap of between 15-19 reduces to just 16 feet.
As he points out 16 feet is a relatively short distance for the average golfer to start regularly three-putting.
So his view is that all recreational golfers should stop trying to hole putts from over 16 feet and simply start to aim to get the ball closer to the hole to give themselves a better chance of
avoiding those damaging 3 putts.
Remember, that despite what you see on the TV with professionals seemingly endlessly holing long puts, the statistics show that they actually hole very few long putts.
From over 20 feet they will on average only make 15% of them – in other words they are only, in reality, holing those putts once in nearly 7 attempts!
And despite what they may say in interviews afterwards from that distance they are very rarely aiming to make those putts.
At that distance they are focusing more on getting down in 2 by getting their first putt close enough.
And if it goes in then it is a bonus rather than being their primary intention.
So if you are looking to shave a number of strokes off your score get focusing on those mid to longer-range putts.
Mark suggests that for amateurs the ideal practice distance is 11-30 feet for 3 reasons:
1. Over half (that’s 9 greens a round) of the average golfer’s first putt distance takes place from between 11-30 feet.
2. From the 11-30 feet distance amateur players are 7-times more likely to three putt than a PGA Tour Player.
3. From more than 30 feet the difference between average players and tour pros is much less. They only 3-putt four times as frequently than pros at those greater distances.
When over 16 feet from the hole your goal should therefore be to give yourself a realistic chance of making a 2 putt rather than making the putt.
Speed much more than line matters on these occasions and aiming to get your first putt within a 2-3 feet imaginary circle around the hole is ideal.
If you do that chances are you will go a long way to reducing those number of 3-putts.
Stopping yourself from 3-putting is not easy but the rewards are high – a 25 handicapper would shave 4.25 shots off every round by eliminating three-putts.
Good scoring is still possible without a lot of 1 putt holes but there is absolutely no way you will score well with a lot of three putts.
Final Thought
Putts per round is not the best statistic to measure your putting on but the old adage of ‘drive for show and putt for dough’ is also not the correct one.
As statistics become an ever-increasing part of the game of golf there is evidence now to show that despite the old sayings about how important putting is other parts of your game are as important,
if not more so.
However, every score in golf ends on the putting green and whatever standard of player you are putting will still make up approximately 40% of your total score.
So it still does, and always will, remain a critical component of the game and taking fewer putts, particularly 3-putts, will lower your score and sometimes quite significantly.
In other words, when it comes to the greens and what stat is most important, you need to reduce those 3 putts more than anything else.
[Note – Just so you know, and we are upfront as an affiliate program participant, Golfing Focus, at no cost to you, earns from qualifying purchases made through links on this page.]
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KSEEB Solutions for Class 9 Maths Chapter 5 Triangles Ex 5.3
KSEEB Solutions for Class 9 Maths Chapter 5 Triangles Ex 5.3 are part of KSEEB Solutions for Class 9 Maths. Here we have given Karnataka Board Class 9 Maths Chapter 5 Triangles Exercise 5.3.
Karnataka Board Class 9 Maths Chapter 5 Triangles Ex 5.3
Question 1.
∆ABC and ∆DBC are two isosceles triangles on the same base BC and vertices A and D are on the same side of BC. If AD is extended to intersect BC at P, show that
(i) ∆ABD ≅ ∆ACD
(ii) ∆ABP ≅ ∆ACP
(iii) AP bisects ∠A as well as ∠D.
(iv) AP is the perpendicular bisector of BC.
Data : ∆ABC and ∆DBC are two isosceles triangles on the same base BC and vertices A and D are on the same side of BC. AD is extended to intersect BC at P.
To Prove:
(i) ∆ABD ≅ ∆ACD
(ii) ∆ABP ≅ ∆ACP
(iii) AP bisects ∠A as well as ∠D.
(iv) AP is the perpendicular bisector of BC
(v) AD is the angular bisector of ∠A.
(i) In ∆ABD and ∆ACD,
AB = AC (data)
BD = DC (data)
AD is common.
S.S.S. Congruence rule.
∴ ∆ABD ≅ ∆ACD
(ii) In ∆ABP and ∆ACP,
AB = AC (data)
∠ABP = ∠ACP (Opposite angles)
∠BAP = ∠CAP (∵ ∆ABD ≅ ∆ACD proved)
Now ASA postulate.
∆ABP ≅ ∆ACP.
(iii) ∆BAD ≅ ∆CAD proved.
AP bisects ∠A.
In ∆BDP and ∆CDP,
BD = DC (data)
BP = PC (proved)
DP is common.
∴ ∆BDP ≅ ∆CDP (SSS postulate)
∴ ∠BDP = ∠CDP
∴ DP bisects ∠D.
∴ AP bisects ∠D.
(iv) Now, ∠APB + ∠APC = 180° (Linear pair)
∠APB + ∠APB = 180°
2 ∠APB = 180
∴ ∠APB = \(\frac{180}{2}\)
∴∠APB = 90°
∠APB = ∠APC = 90°
BP = PC (proved)
∴ AP is the perpendicular bisector BC.
(v) AP is the angular bisector of ∠A.
Angular bisector of ∠A is aD, because AD, AP is in one line.
Question 2.
AD is an altitude of an isosceles triangle ABC in which AB = AC. Show that A
(i) AD bisects BC
(ii) AD bisects ∠A.
Data: AD is an altitude of an isosceles triangle ABC in which AB = AC.
To Prove:
(i) AD bisects BC.
(ii) AD bisects ∠A.
Proof: i) In ∆ABD and ∆ACD,
∠ADB = ∠ADC (∵ AD ⊥ BC)
AB = AC (data)
AD is common.
∴ ∆ABD ≅ ∆ACD
∴ BD = DC
∴ AD bisects BC.
(ii) ∠BAD = ∠CAD (∵ ∆ADB ≅ ∆ADC)
∴ AD bisects ∠A.
Question 3.
Two sides AB and BC and median AM of one triangle ABC are respectively equal to sides PQ and QR and median PN of ∆PQR. Show that :
(i) ∆ABM ≅ ∆PQN
(ii) ∆ABC ≅ ∆PQR.
Data: Two sides AB and BC and median AM of one triangle ABC are respectively equal to sides PQ and QR and median PN of ∆PQR.
To Prove:
(i) ∆ABM ≅ ∆PQN
(ii) ∆ABC ≅ ∆PQR.
Proof: (i) In ∆ABC,
AM is the median drawn to BC.
∴ BM = \(\frac{1}{2} \) BC
Similarly, in ∆PQR,
QN = \(\frac{1}{2}\) QR
But, BC = QR
\(\frac{1}{2} \) BC = \(\frac{1}{2}\) QR
∴ BM = QN
In ∆ABM and ∆PQN,
AB = PQ (data)
BM = QN (data)
AM = PN (proved)
∴ ∆ABM ≅ ∆PQN (SSS postulate)
(ii) In ∆ABC and ∆PQR,
AB = PQ (data)
∠ABC = ∠PQR (proved)
BC = QR (data)
∴ ∆ABC ≅ ∆PQR (SSS postulate)
Question 4.
BE and CF are two equal altitudes of a triangle ABC. Using the RHS congruence rule, prove that the triangle ABC is isosceles.
Data: BE and CF are two equal altitudes of a triangle ABC.
To Prove: ABC is an isosceles triangle.
Proof : BE = CF (data)
In ∆BCF and ∆CBE,
∠BFC = ∠CEB = 90° (data)
BC is a common hypotenuse.
As per Right angle, hypotenuse, side postulate,
∴ ∆BCF ≅ ∆CBE
∴ ∠CBF = ∠BCE
∴ ∠CBA = ∠BCA
∴ AB = AC
∴ ∆ABC is an isosceles triangle.
Question 5.
ABC is an isosceles triangle with AB = AC. Draw AP ⊥ BC to show that ∠B = ∠C.
Data: ABC is an isosceles triangle with AB = AC.
To Prove : ∠B = ∠C
Construction: Draw AP ⊥ BC.
Proof: In ∆ABC, AP ⊥ BC and AB = BC.
∴ In ∆ABP and ∆ACP
∠APB = ∠APC = 90° ( ∵ AP ⊥ BC)
Hypotenuse AB = Hypotenuse AC
AP is common.
As per RHS Postulate,
∆ABP ≅ ∆ACP
∴ ∠ABP = ∠ACP
∴ ∠ABC = ∠ACB
∴∠B = ∠C.
We hope the KSEEB Solutions for Class 9 Maths Chapter 5 Triangles Ex 5.3 helps you. If you have any query regarding Karnataka Board Class 9 Maths Chapter 5 Triangles Exercise 5.3, drop a comment
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Arrow Kinetic Energy Calculator - Calculator Doc
Arrow Kinetic Energy Calculator
The kinetic energy of an arrow is an essential factor in archery, hunting, and various other applications. It is directly proportional to the mass of the arrow and the square of its velocity. A
heavier and faster arrow will have more kinetic energy, resulting in better performance, deeper penetration, and more impact on the target. Understanding the kinetic energy of an arrow helps in
choosing the right equipment and optimizing performance.
The formula for calculating the kinetic energy of an arrow is:
Kinetic Energy (KEa) = 0.5 ∗ m ∗ AV²
• m = Mass of the arrow (in kilograms)
• AV = Velocity of the arrow (in meters per second)
How to Use
1. Enter the mass of the arrow in kilograms.
2. Enter the velocity of the arrow in meters per second.
3. Click the “Calculate” button to get the kinetic energy of the arrow in Joules.
4. The result will be displayed in the output field below the button.
Suppose you have an arrow with a mass of 0.03 kg (30 grams) and a velocity of 50 meters per second. Using the formula:
KEa = 0.5 ∗ 0.03 ∗ 50²
KEa = 0.5 ∗ 0.03 ∗ 2500
KEa = 0.5 ∗ 75
KEa = 37.5 Joules
So, the kinetic energy of the arrow would be 37.5 Joules.
1. What is kinetic energy?
Kinetic energy is the energy an object possesses due to its motion. For an arrow, this is the energy transferred to the target upon impact.
2. Why is the mass of the arrow important?
The mass of the arrow affects its kinetic energy. A heavier arrow will carry more energy, assuming the velocity remains constant.
3. How does arrow velocity affect kinetic energy?
The velocity of the arrow has a significant impact on kinetic energy because kinetic energy increases with the square of the velocity.
4. What unit is used for kinetic energy?
Kinetic energy is measured in Joules (J), which is the standard unit of energy in the International System of Units (SI).
5. Can I use this calculator for any arrow size?
Yes, you can use this calculator for any arrow as long as you know its mass and velocity.
6. Does air resistance affect the kinetic energy?
Air resistance reduces the velocity of the arrow, and hence, it reduces the kinetic energy during flight. However, this calculator assumes ideal conditions without considering air resistance.
7. Is kinetic energy the same as momentum?
No, kinetic energy and momentum are different. Kinetic energy is a scalar quantity, while momentum is a vector quantity and depends on both mass and velocity.
8. Can I increase kinetic energy by using a faster bow?
Yes, a bow that can shoot arrows at higher speeds will increase the kinetic energy, provided the mass of the arrow remains the same.
9. Why do I need to know the kinetic energy of an arrow?
Knowing the kinetic energy helps in understanding the potential impact and penetration capabilities of an arrow, which is crucial for hunting and target shooting.
10. What happens if I use a lighter arrow?
A lighter arrow will have less kinetic energy at the same velocity. However, lighter arrows often travel faster but with less momentum, impacting penetration.
11. How does kinetic energy affect arrow penetration?
Higher kinetic energy typically results in better penetration, especially when hunting. However, other factors like arrowhead design and target material also affect penetration.
12. Can I calculate kinetic energy for other projectiles with this formula?
Yes, the same formula applies to any projectile, such as bullets, rocks, or thrown objects, as long as you know the mass and velocity.
13. Does the type of arrow material affect kinetic energy?
The material affects the arrow’s mass, which in turn affects kinetic energy. Heavier materials like carbon fiber or metal will result in higher kinetic energy.
14. What is a good kinetic energy range for hunting?
For hunting larger animals, a kinetic energy range between 50-70 Joules is recommended. For smaller animals, 25-50 Joules might be sufficient.
15. Can I use this calculator for crossbow arrows?
Yes, the same principles apply to crossbow bolts or arrows. You need the mass and velocity to calculate kinetic energy.
16. Is kinetic energy the only factor in arrow performance?
No, other factors like arrow stability, flight dynamics, and accuracy also contribute to overall performance.
17. Does increasing arrow length affect kinetic energy?
Increasing arrow length can affect the mass of the arrow, which may indirectly impact kinetic energy.
18. What is the difference between arrow kinetic energy and draw weight?
Draw weight is the force needed to pull the bowstring, while kinetic energy is the energy the arrow carries in motion. Both are related but are not the same.
19. Can I calculate the kinetic energy of a moving arrow in the air?
Yes, but the velocity will change due to air resistance, which would need to be accounted for in more advanced calculations.
20. How does draw length affect arrow kinetic energy?
A longer draw length allows for more stored energy in the bow, which can transfer to the arrow as kinetic energy.
Understanding the kinetic energy of an arrow is crucial for archers, hunters, and sports enthusiasts. It not only helps in selecting the right equipment but also aids in optimizing performance. Using
this Arrow Kinetic Energy Calculator, you can quickly determine the energy output based on the arrow’s mass and velocity, enabling you to make informed decisions on the field. | {"url":"https://calculatordoc.com/arrow-kinetic-energy-calculator/","timestamp":"2024-11-06T09:11:03Z","content_type":"text/html","content_length":"87878","record_id":"<urn:uuid:dc650892-5964-4722-a369-e8b3523d9a6a>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00126.warc.gz"} |