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Congruent Polygons are polygons that have the same shape and size. When two polygons are congruent, one of the polygons can be placed exactly over the other by sliding, flipping or turning.
Congruence shapes find application widely in machine designing and manufacturing. Equilateral triangles , squares and regular hexagons are congruent regular polygons which tessellate around a vertex to form patterns. Congruent Regular Polygons also make the faces of regular polyhedra which are called Platonic solids. The commonly known Platonic solids are Tetrahedron and Hexahedron. A Tetrahedron is made up of four congruent equilateral triangles and six congruent squares constitute a Hexahedron.
Congruent Polygons Definition
Two polygons are congruent if the vertices of one polygon bears a one to one correspondence to the regular Pole vertices of the other polygon and the corresponding angles and corresponding sides so formed are congruent. In simple words "Corresponding parts of congruent polygons are congruent."
Congruent polygons are named by placing the corresponding vertices in the same order for both the polygons.
Given that, ABCDE ≅ FGHIJ then the corresponding vertices are
A ↔ F
B ↔ G
C ↔ H
D ↔ I
E ↔ J
The angles named after corresponding vertices are the corresponding angles and they are congruent.
∠A ≅ ∠F, ∠B ≅ ∠G, ∠C ≅ ∠H, ∠D ≅ ∠I and ∠E ≅ ∠J.
The congruent corresponding sides are named combining corresponding vertices in the same order.
AB ≅ FGBCGHCDHIDEIJ and EAJF.
Are Congruent Polygons Similar?
Similar polygons are polygons withe same shape, but need not be of the same size. Since congruence maintains the shape, the Congruent polygons are also similar polygons.
But conversely, the similar polygons may not be congruent polygons as similarity does not include same size in its definition.
Solving Congruent Polygons
Solved Examples
Question 1: Given Quadrilateral ABCD ≅ Quadrilateral PQRS. The angle measures and the lengths of two sides of quadrilaterals are marked on the diagram. Find the measures of the angles of quadrilateral PQRS and the lengths of sides corresponding to the sides of ABCD whose lengths are given.
Solution:
The Corresponding angles can be found by mapping the corresponding vertices in the order the quadrilaterals are named.
ABCD ≅ PQRS
∠A ≅ ∠P ⇒ m ∠P = m ∠A = 116º.
∠B ≅ ∠Q ⇒ m ∠Q = m ∠B = 83º.
∠C ≅ ∠R ⇒ m ∠R = m ∠C = 83º.
∠D ≅ ∠S ⇒ m ∠S = m ∠D = 78º.
The lengths of sides AB and BC are given. The corresponding sides are PQ and QR.
PQ ≅ AB ⇒ PQ = AB = 4".
QRBC ⇒ QR = BC = 5.5".
Question 2: In the diagram given below, Δ ABC ≅ Δ DEF. Find the length EF.
Solution:
The lengths of legs AB and CA of Δ ABC are given. Using the congruence statement , the corresponding congruent
sides are,
AB DE, BCEF and CAFD.
BC is side corresponding to EF. The length of BC can be found applying Pythagorean theorem,
BC2 = AB2 + CA2.
= 42 + 72 = 16 + 49 = 65
BC = √65 cm
Thus length EF = length BC = √65 cm ≈ 8.06 cm.
Question 3: A tetrahedron is a solid made up of four congruent equilateral triangles. If the measure of one of the edges = 5 cm, find the surface area of the tetrahedron nearest to the 10the of a square cm.
Solution:
The tetrahedron has four congruent triangular faces. Areas of congruent triangles are equal.
Area of one face = Area of the equilateral triangle with side = 5cm
= $\frac{\sqrt{3}a^{2}}{4}$
= $\frac{\sqrt{3}}{4}\times 5^{2}$
= 10.83 sq.cm (rounded to the hundredth)
Hence the surface area of the tetrahedron = 4 x area of one face = 4 x 10.83 sq.cm = 43.3 sq.cm.
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# Cosine series of sine
1. Oct 16, 2005
### tiagotorres
I tried to find the cosine series of the function $$f(x) = \sin x$$, using the equation below:
$$S(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos(nx)$$
where: $$a_n = \frac{2}{\pi} \int_{0}^{\pi} f(x) \cos(nx) dx$$
I found:
$$a_0 = \frac{4}{\pi}$$
$$a_n = \frac{2 }{\pi (1 - n^2)} (\cos(n \pi) + 1)$$
Therefore:
$$S(x) = \frac{2}{\pi} (1 - 2 \sum_{n=1}^{\infty} \frac{\cos(2nx)}{4n^2 - 1})$$
Making the graph of the first terms of the function above on my calculator, I noticed that this is actually $$| \sin x |$$, rather than just $$\sin x$$. Why does this happen? Is there a way of figuring out the sum not using a calculator?
Thanks
Last edited: Oct 16, 2005
2. Oct 16, 2005
### mathman
sinx is nonegative between 0 and pi, so sinx=|sinx| there.
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# For-each Variables
Theories often include a large number of similar Variables, for example similar Variables for many similar persons (e.g. the height of many children in a class). So in practice we often see different ways to group together these kinds of repeated Variables and often represent them as if they were just a single Variable. We often do this without thinking - on our Theory of Change, we put a box for one Variable for “people’s attitudes to recycling” and another for “the law is passed” without thinking that the first is probably conceived as a whole column of data, one for each person, whereas the second is just a single datum, yes or no.
Any set of Variables which differ only by one Feature can be rewritten as one for-each Variable.
For example, each of these Variables represents the temperature outside this house at a different time point.
Temperature outside this house, time=yesterday
Temperature outside this house, time=today
Temperature outside this house, time=tomorrow
These can be rewritten like this:
Temperature outside this house !for-each time (yesterday, today, tomorrow)
- !for-each teacher
Teacher ability
!Rule (mean) Average Student achievement;Teacher feeling of work satisfaction, Rule ??
-- !for-each student
Student achievement
Teacher ability
Student motivation
Teacher ability
-
Teacher ability
School-level support to teacher development
## One Variable, one data point
The word “variable” is used in different ways in M&E and even within mathematics. Not in Theorymaker. When someone says they have, or could have, simultaneously different data points for the same variable in an intervention, presumably from different times or places or persons, Theorymaker native speakers would say this is not one Variable but a set of Variables, one for each data point.
The main reason people get confused about this is that in ordinary language (and in statistics, but not most of the rest of mathematics) we also use the word “variable” for a whole set of similar Variables repeated across time (and/or across places or across groups of people).
In Theorymaker, we call sets of Variables like this “for-each Variables”. “For-each time” Variables are very common, but Variables can be “for” other things, see below. They are actually sets of ordinary Variables each of which only belongs to one time-point.
So that’s why Theorymaker native speakers say “One Variable for each piece of data in an intervention” (or in any other single application of a Theory). They are quite well aware that behind any particular simple Theory there might be, in fact there should be, a whole mass of prior evidence, data, collected in relevantly similar contexts. But any particular intervention is essentially just one more case, and each piece of relevant data collected by the intervention needs a dedicated Variable to store it - often, as described above, grouped into for-each Variables.
So if you look at a series of temperature measurements for your town throughout the day you can say, “look, the temperature variable is changing”. And that is a perfectly reasonable thing to say, provided we are clear that we are using the word “variable” here for a whole set of states or measurements, one for each moment of time22; but in the Theorymaker sense of “Variable” I introduced above, each one of these states or measurements is a Variable, each at a single point of time; and those can’t change in the sense that they can individually vary across time, though they can change in the sense that they could be different (or could have been different).
In Theorymaker, it is acceptable to use singular forms with for-each Variables, saying, for example, “this ‘for’ Variable records level of interest every month”, although perfectionists prefer to be more precise and say “these ‘for’ Variables record level of interest every month”, using the plural form.
More formally, any set of Variables which differ only by one Feature can be rewritten as just one for-each Variable with the addition of the marker !for-each followed by the same criterion, and mentioning the list of possibilities (e.g. “1-3”) if this is known.
Student achievement, student=1
Student achievement, student=2
Student achievement, student=3
The diagram above can be replaced by the diagram below.
Student achievement !for-each students 1-3
## Rules with for-each Variables
We can display Rules where either parent or child Variables, or both, are for-each Variables. But we should be aware that these actually represent multiple Rules.
Here are two perfectly good simple Theories which do not use for-each Variables.
Student achievement
Teacher ability
… it talks about the generic effect of the ability of some teacher on the achievement of some student.
… whereas this …
Average student achievement
Teacher ability
… talks about the effect on “Average student achievement” for some set of students. Now behind this average is presumably a set of data for each student in a class, (and the individual achievement scores may themselves be composite scores from a further mass of variables per student). But these individual scores are not mentioned or considered in the Theory. Average achievement is a perfectly good, single, Variable, and it is perfectly acceptable to have a Theory about how teacher ability can affect it. We may have plenty of evidence to back up links between teacher ability and average student scores and maybe that evidence is agnostic about specific effects on individual scores.
### Rule is same across all cases
… whereas the following Theory is different; it talks about the effect of teacher ability on the individual scores of three different students.
Student achievement, student=1, !Rule same Rule as other cases
Teacher ability
Student achievement, student=2, !Rule same Rule as other cases
Teacher ability
Student achievement, student=3, !Rule same Rule as other cases
Teacher ability
In this case, there are multiple Variables which we can visually compress into one using a for-each Variable.
So in the above case, we have one teacher and one teacher Variable (ability); but we have several students, each with one Variable (student achievement). Teacher ability actually influences several different Variables, one for each student. So the above diagram can be compressed into this one:
Student achievement !for-each students 1-3
Teacher ability
### Rule differs from case to case
Student achievement//student=1, !Rule stronger effect here
Teacher ability
Student achievement//student=2, !Rule stronger effect here
Teacher ability
Student achievement//student=3, !Rule weaker effect here
Teacher ability
… that could be displayed something like this:
Student achievement//!for-each student !Rule stronger for students 1 and 2
Teacher ability
## Aggregating for-each Variables
Ordinary Theories of Change often draw lines between for-each Variables and simple Variables without reflecting that these actually join multiple to single Variables. But we can’t in fact ignore this issue; rather than just an ordinary Rule to tell us how one or more single Variables influence the Consequence Variable, we need to specify how the multiple Influence Variables are aggregated to influence the Consequence Variable (or, in the case of defined Variables, how the defining Variables are aggregated).
As an arrow leaves a for-each Variable and leaves a box to influence another Variable, this is a reminder that this Variable is now bunched, and the receiving function has to take account of this. So in the example, how does the Level of frustration of each individual student combine to influence class climate? Sometimes we just assume it is OK to take a mean or a total. But sometimes something else might be important like the maximum or some other more complicated rule of combination.
In this example, the teacher’s ability influences the achievement of each student individually; but also student achievement collectively goes on to influence the same teacher’s feeling of work satisfaction. There is a whole range of ways in which this can happen - is he or she most influenced by the top-performing students, or the number of students scoring low, or simply by the average of all?
Aggregation is an issue for defined Variables just as much as for consequence Variables. So in the same example, we can define a Variable “Average student achievement” which aggregates the student scores.
!Rule (mean) Average Student achievement
Student achievement !for-each student
Teacher feeling of work satisfaction !Rule:?
Student achievement !for-each student
Teacher ability
## Using grouping boxes with for-each Variables
It is important that the same for-each criterion can apply to different sets of Variables. So you might record student performance and student age in the same table, with one line for each student and several columns for different Variables. Each student has a unique ID and a student with this ID can have several Variables which are entered in a row beginning with that ID. In statistics, this common ID number is a source of measurement strength and allows us to use some more powerful statistical methods.
So when we say this:
Teacher skills
Teacher presence on training course
Do we mean just mean that overall teacher skills are influenced by overall presence on a training course? If I am a teacher and I join a course, will it help my colleagues in Brazil improve their skills?
Here are two ways we could say this in Theorymaker:
Skills. !for-each teacher in our school.
Presence on training course. !for-each teacher in our school.
When no Rule is specified, we assume that the second Variable has an influence on the first for each and every teacher. You can see this more clearly in this equivalent, alternative phrasing:
-!for-each teacher
skills
presence on training course
This version is exactly equivalent because if follows the rule that defines for-each grouping boxes as equivalent to the same diagram without the box, in which each Variable which was within the box has the !for-each added to it.
This is a more general claim, because without any further information about context, it might apply to any teacher, anywhere in the world, ever.
Above we saw how it is convenient to use grouping boxes to optically organise a Theory, grouping together Variables with the same attribute like person, organisation or time-point. In the same way, we can use grouping boxes to group together for-each Variables which share the same criterion.
!Rule (mean) Average student achievement
- !for-each student
Student achievement
Student motivation
-
Teacher feeling of work satisfaction, !Rule ??
Student achievement
Teacher ability
Student motivation
Teacher ability
So the variable “Average student achievement” is outside the box of “for student” Variables. For this group of many students there is a single Variable representing the average score.
As there is only one teacher in this example, we don’t really need to group the teacher Variables together. But in this next example, there are many classrooms, each with one teacher and multiple students. Quantitative social scientists say that “Students are nested within classrooms”. This way of thinking is very familiar to statisticians; but we need it for project M&E too, even if we don’t have numerical variables and we are not worried about statistics.
- !for-each teacher
Teacher ability
!Rule (mean) Average Student achievement;Teacher feeling of work satisfaction, Rule ??
-- !for-each student
Student achievement
Teacher ability
Student motivation
Teacher ability
-
Teacher ability
School-level support to teacher development
## Continuous for-each
While we are talking about “repeating Variables” as if they were always discrete, often we think about Variables as belonging to infinite sets which repeat continuously, in particular across time and space.
## More to come!
To be completed
### Repeated and selected Variables
#### Selection
When constructing Theories of Change, often, sets of individuals might go through a programme but some are selected in or drop out.
Repetition and selection are extremely common in programmes but standard Theories of Change almost never take account of them. Statisticians know how to do this and will try to tease them out when analysing a dataset, but most times if programme staff weren’t clear about the repetition and selection then the data won’t have been recorded using the correct “for criteria”.
But even more importantly, repetition and selection are a pretty essential part of how many programmes work. Theorymaker native speakers talk about them and diagram them quite fluently. So can we.
(more about selection and filters …)
Use WITH for selections? But even an individual can be selected, yes/no.
Sport ability !for-each child
Initial running speed !for-each child with good Sport ability
The word “with” is compulsory.
This way, selection and for-each go together allowing for lots of different patterns:
Gender !for-each Student with height above 1m/Class/School, Student/Family with more than one child ((male, female))
#### Theorymaker syntax for nesting
Theorymaker native speakers sometimes use a syntax like this to express how for-each can be nested.
Student !for-each Student/Class/School
which is equivalent to this:
-!for-each School
--!for-each Class
---!for-each Student
Student
But sometimes the “fors” are not completely nested within another, so we can not use grouping boxes in each case.
Student !for-each Student/Class/School, !for-each Student/Family
So student gender is nested within classes and then within schools, and at the same time it is nested within families.
#### Nesting and aggregation
Boxes are a great way to show which Variables share a common for-each criterion. This is particularly useful when indices are nested within one another:
-School layer
School climate !Rule complex, chaotic interaction
--Class layer
Class climate !Rule complex, chaotic interaction
Teacher skills
---Student layer
Student social skills
Student frustration
---
Student social skills
Teacher skills
Student frustration
Teacher skills
In the above example, class climate contributes to school climate, which is seen as being a separate Variable. But school climate just as well be defined as at least partly equal to some aggregation of class climates by definition rather than as being something above and beyond a mere aggregation of class climates. So even though we might have a clear idea in principle of the distinction between definitional and causal rules, in practice there is often a large grey area.
#### Subgroups
There are different kinds of subgroup. What about when a core group of 30 educators gets to train up 10 peers each in some kind of cascade training, so that all 330 end up with key skills.
-EACH teacher;N=300+30
Have key skills
--SELECT core teacher;N=300
Conduct training for peers
Know how to conduct peer education
Have key skills too
Receive training
It might be useful to distinguish between two different kinds of nested groups.
1. EACH
2. SELECT
-EACH school;N=300
sc
--EACH classroom; N=300*20
cl
---EACH student; N=300*20*30
st
----SELECT girls
Variables just for girls
----SELECT boys
Variables just for boys
-
proportion=.6
sc;label=school characteristics
cl;label=classroom characteristics
st;label=common student characteristics
edge;dir=both;color=indianred
1. Of course there might be a whole lot, or even an “infinite number” of such variables; that doesn’t really matter at this point because we certainly won’t actually be collecting an “infinite amount” of data
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invariance of formula for surface integration with respect to area under change of variables
Primary tabs
Type of Math Object:
Proof
Major Section:
Reference
Groups audience:
Mathematics Subject Classification
28A75 no label found
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# zbMATH — the first resource for mathematics
##### Examples
Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.
##### Operators
a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Vortices in Ginzburg-Landau equations. (English) Zbl 0896.35044
Summary: GL models were first introduced by V. Ginzburg and L. Landau around 1950 in order to describe superconductivity. Similar models appeared soon after for various phenomena: Bose condensation, superfluidity, non linear optics. A common property of these models is the major role of topological defects, termed in our context vortices. In a joint book with H. Brezis and F. Hélein, we considered a simple model situation, involving a bounded domain ${\Omega }$ in ${ℝ}^{2}$, and maps $v$ from ${\Omega }$ to ${ℝ}^{2}$. The Ginzburg-Landau functional then writes
${E}_{\epsilon }\left(v\right)=\frac{1}{2}{\int }_{{\Omega }}{|\nabla v|}^{2}+\frac{1}{4{\epsilon }^{2}}{\int }_{{\Omega }}{\left(1-|v|}^{2}{\right)}^{2}·$
Here $\epsilon$ is a parameter describing some characteristic length. We are interested in the study of stationary maps for that energy, when $\epsilon$ is small (and in the limit $\epsilon$ goes to zero). For such a map the potential forces $|v|$ to be close to 1 and $v$ will be almost ${S}^{1}$-valued. However at some point $|v|$ may have to vanish, introducing defects of topological nature, the vortices. An important issue is then to determine the nature and location of these vortices. We will also discuss recent advances in more physical models like superconductivity, superfluidity, as well as for the dynamics: as previously the emphasis is on the behavior of the vortices.
##### MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. (PDE) 35Q35 PDEs in connection with fluid mechanics 35Q55 NLS-like (nonlinear Schrödinger) equations 82D50 Superfluids (statistical mechanics) 82D55 Superconductors (statistical mechanics) 35J20 Second order elliptic equations, variational methods
##### Keywords:
superconductivity; location of vortices
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# Circom Workshop 2
## Description
The second circom workshop: we dig deeper into a few key circuits such as zk-group-sig, isZero, and Num2Bits.
• QuinSelector is a circuit that takes as input an array in[nElements], an index index, and outputs the value of in[index].
• RangeProof is a circuit that takes in upper lower and test, outputs 1 if test is in the range and 0 otherwise.
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## Calculus (3rd Edition)
Given function is $y=\sqrt {12-4x^{2}}$ Differentiating both sides of the equation with respect to x,we have $\frac{dy}{dx}=\frac{1}{2\sqrt {12-4x^{2}}}\times-8x=\frac{-4x}{\sqrt {12-4x^{2}}}$ Substituting the value of $\frac{dy}{dx}$ and y in the given differential equation, we get $L.H.S= \sqrt {12-4x^{2}}\times\frac{-4x}{\sqrt {12-4x^{2}}}+4x=0=R.H.S$ Therefore, the given function is a solution of the given differential equation.
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# Forcing. This Has To Stop.
Most, if not all, set theorists at one point or another were asked by a fellow mathematician to explain how forcing works. And many chose to give as an opening analogy field extensions. You can talk about how the construction of an algebraic closure is a bit similar, since the generic filter is a bit like the maximal ideal you use to make this construction; or you can talk about adding a transcendental number and the things that change as you add it.
But both these analogies would be wrong. They only take you so far, and not further. And if you wish to give a proper explanation to your listener, there will be no escape from the eventual logic and set theory of it all. I stopped, or at least I’m doing my best, using these analogies. I do, however, use the analogy of “How many roots does $x^{42}-2$ has?” as an example for everyday independence (none in $\mathbb Q$, two in $\mathbb R$ and many in $\mathbb C$). But this is to motivate a different part of the explanation: the use of models of set theory (e.g. “How can you add a real number??”, well how can you add a root to a polynomial?) and the fact that we don’t consider the universe per se. Of course, in a model of $\ZFC$ we can always construct the rest of mathematics internally, but this is not the issue now. Just like we have a model of one theory, we can have a model for another.
So why do we do it? Why do we keep explaining adding generic sets using the analogy of field extensions? Well, the easy answer is that field extensions are something that most mathematicians can understand pretty easily and it shows how we can “enlarge a universe”.
But here’s why we should stop doing that (and why I stopped doing that):
1. Most people think about field extensions as being subfields of the complex numbers. This allows for a particular fixed background universe from which we can draw the numbers that we add. In set theory it is easy to think that there is one universe of sets, and that we force over that universe, in which case, where did the generic set come from?
Moreover, if $F$ is an algebraically closed field, and we add a transcendental element to $F$, then there is a nice closure operation after which the resulting field has the same first-order theory as $F$ (and if $F$ is uncountable, then there is also an isomorphism between them). Short of miraculous black magic, I don’t know of any such example in the case of set theory. Once you extend a model by forcing, there’s no definable process to restore the theory of the model by enlarging the model without adding ordinals (the obvious case is $V=L$).
2. The analogy is not accurate, and we can do better. For example, we can consider actual generic objects. The term generic comes from topology (and to my knowledge, that is where it trickled into algebraic geometry as well; but please correct me if I’m wrong about that).
We say that a point $x$ is generic if it is an element of every dense open set. Generic objects are a lot like that. We have a partial order, it has a topology, and the generic real is something which lies in the intersection of all the dense sets from the ground model.
So we can talk about adding a generic, or considering a generic point. This way it’s also easy to explain why it has so many properties — each property happens on a dense open set, so the generic point must have it.
Or we can talk about what actually happens. We start with a countable model of set theory. There’s no shame in that. It’s like considering a countable field or another countable structure. Since it’s countable, it’s certainly not the collection of all sets. Be sure to explain why the model is only countable from the outside and not internally; and what does it mean that “the model thinks that …”, true it’s not that easy, but there’s a large payoff. You’re not lying anywhere.
3. Recently I watched a video of Richard Feynman being asked by a layman to explain why magnets behave the way they do. And Feynman said that’s a very good question, and proceeded for five minutes to give analogy of a curious alien which would ask all sort of questions that you and I would take for granted; and then he went to say that the reason he didn’t answer the question is that all the analogies he could make, or other people make, boil down to electromagnetic forces acting on a microscopic level. So if you would compare magnets to rubber, then the next reasonable question would be how does the rubber work, and by inquiring further you’d finally reach the original question again, how does the electromagnetic force work.
So Feynman didn’t want to deceive the layman, or confuse them with analogies which ultimately explain nothing. And I think that’s a wonderful approach when you try to teach someone something. If electromagnetic force is one of the fundamental forces, we have to take it the way it is, and we can’t explain it in simpler terms.
Similarly, if forcing is a technique which is in its own class in mathematics, then we can’t quite explain it in terms of other techniques. Every analogy would break down and cause confusion. In that case, maybe it is the simplest solution to just start right away from forcing, logic and set theory? Motivate what you do in terms of logic. We are interested in truth values of statements, this statement is a statement of the form “There exists a set such that …”, so we want to approximate this set, so by carefully choosing a set from outside the model our approximations are actualized in the new model.
4. One prominent set theorist once told me that an incredibly smart mathematician (from representation theory) once asked him to explain forcing to him. He began with the analogy of field extensions, and it seems to be fine, and as he continued he defined the generic extension. Now we want to examine what sort of sentences are true in that generic extension, and we have this magical theorem which tells us exactly when something is true in the generic extension.
Once the sentence “a formula in the language of forcing” was uttered, the eyes became vacant and the rest was moot. And neither of the mathematician is stupid, and the set theorist involved is a wonderful teacher.
So why didn’t it work? What can we learn from this? My guess is that the analogy sets a particular direction, and when you break from that analogy, it becomes confusing to the listener. If you weren’t prepared to hear the term “the language of forcing”, then you won’t be able to jump over that hurdle when you reach it. And any analogy to field extensions hides this hurdle from the listener.
It seems to me, if so, that there are plenty of reasons not to use analogies, and plenty of reasons to explain things as they are. And forcing is not a trivial idea, remember that it completely revolutionized set theory. So if your audience can’t grasp it over coffee, it’s not a big deal. Perhaps using broad strokes to paint a rough image of approximations is better in that case, or at least better than giving the wrong idea.
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# Estimator for $E[X]^2$
I'm trying to understand the theory of estimators. As I understand it now, if you have an r.v. $X$ and take $n$ i.i.d. samples then an estimator for $E[X^{2}]$ would be $\overline{X^{2}}$ since $E[\overline{X^{2}}] = E[X^{2}]$ (probably only true for some kind of "nice" r.v.).
However, the same kind of nice result doesn't occur when trying to estimate $E[X]^{2}$. That is to say, the function $\overline{X}^{2}$ does not estimate this. But I'm not sure I understand which function does estimate it. I have the equation $V[\overline{X}] = E[\overline{X}^{2}]-E[\overline{X}]^{2}$ and so $E[X]^{2} = E[\overline{X}^{2}]-V[\overline{X}]$. This seems relevant but I'm not sure what to conclude from this.
• Actually the natural estimator for $E \left[ X^2 \right]$ is $\frac{1}{n} \sum_{i=1}^n X_i ^2$. Can you show that it is unbiased for the case of iid samples? – JohnK Oct 22 '14 at 14:53
• @JohnK I believe that what the OP means by "$\overline{X^{2}}$" is precisely $\frac{1}{n} \sum_{i=1}^n X_i ^2$. – whuber Oct 22 '14 at 15:02
• Addem, any statistic will estimate $E[X]^2$. The right question to ask is how well will it do. The answer depends on how you measure the goodness of an estimator. – whuber Oct 22 '14 at 15:04
• @whuber thats a great point, good way to step back and look at the bigger picture. – bdeonovic Oct 22 '14 at 16:56
By continuous mapping theorem $\bar{X}^2 \to \text{E}[X]^2$ in probability, so I would say it is a good estimate.
Depending on the distribution of $X$, if $\bar{X}$ is the MLE of $\text{E}[X]$, then $\bar{X}^2$ will be the MLE of $\text{E}[X]^2$ (MLE is invariant to transformation).
If $X_i$ are iid and $\text{Var}[X] = \sigma^2$, then $\text{Var}[\bar{X}] = \text{Var}\left[ \dfrac{1}{n} \sum_{i=1}^n X_i\right] = \dfrac{1}{n^2}\sum_{i=1}^n \text{Var}[X_i] = \dfrac{\sigma^2}{n}$
Background: unbiased estimators of products of population moments
If you desire an UNBIASED estimator of a (product of moments), there are 3 varieties:
1. Polykays (a generalisation of k-statistics): these are unbiased estimators of products of population cumulants. The term polykay was coined by Tukey, but the concept goes back to Dressel (1940).
2. Polyaches (a generalisation of h-statistics): these are unbiased estimators of products of population central moments. i.e.
$$E\left[\text{h}_{\{r,t,\ldots ,v\}}\right] = {\mu }_r {\mu }_t \cdots {\mu }_v\text{\, }$$ ...... where ${\mu }_r$ denotes the $r^{th}$ central moment of the population.
1. Polyraws: these are unbiased estimators of products of population raw moments. That is, you wish to find the $polyraw_{r, t, ...v}$ such that:
$$E\left[\text{polyraw}_{\{r,t,\ldots ,v\}}\right] = \acute{\mu }_r \acute{\mu }_t \cdots \acute{\mu }_v\text{\, }$$
...... where $\acute{\mu }_r$ denotes the $r^{th}$ raw moment of the population.
The Problem
We are given a random sample $(X_1, X_2, \dots, X_n)$ drawn on parent random variable $X$.
If we desire an unbiased estimator of: $(E[X])^2 = \acute{\mu }_1 \acute{\mu }_1$, then an unbiased estimator is the {1,1} polyraw:
where $s_r = \sum_{i=1}^n X_i^r$ denotes the $r^{th}$ power sum.
Comparison
Benjamin proposed the estimator: $\bar{X}^2 = (\frac{s_1}{n})^2$. This is not an unbiased estimator, since $E[(\frac{s_1}{n})^2]$ is just the $1^{st}$ RawMoment of $(\frac{s_1}{n})^2$:
which is not equal to $\acute{\mu }_1^2$.
Let us check the polyraw solution:
... which is an unbiased estimator.
Plainly, unbiasedness is not everything, and we could equally calculate, for example, the MSE (mean-squared error) of each estimator using exactly the same tools.
[Update: Just had a quick play with this: in a simple test case of $X \sim N(0,\sigma^2)$, the polyraw unbiased estimator has smaller MSE than Ben's ML estimator, for all sample sizes $n$. That is, at least for the test case of Normality, the polyraw unbiased estimator dominates the maximum likelihood estimator, at all sample sizes. ]
Notes
• PolyRaw, RawMomentToRaw etc are functions in the mathStatica package for Mathematica
• I confess to the neologism polyache in our Springer book (2002) (and more recently, to polyraw in the latest edition).
• Took me a while to guess you meant to pronounce that "polyache" as 'poly-aitch' (poly-"h"). – Glen_b -Reinstate Monica Oct 22 '14 at 20:33
• The lower MSE is very interesting (because simple situations where MLE loses to moments even in fairly small samples don't come along all that often). This question seeks examples where method of moments beats MLE in such a manner. – Glen_b -Reinstate Monica Oct 22 '14 at 21:29
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Chapter 14, Problem 46E
### Chemistry
9th Edition
Steven S. Zumdahl
ISBN: 9781133611097
Chapter
Section
### Chemistry
9th Edition
Steven S. Zumdahl
ISBN: 9781133611097
Textbook Problem
# Calculate the [H+] of each of the following solutions at 25°C. Identity each solution as neutral, acidic, or basic.a. [OH−] = 1.5 Mb. [OH−] = 3.6 × 10−15 Mc. [OH−] = 1.0 × 10−7 Md. [OH−] = 7.3 × 10−4 M
(a)
Interpretation Introduction
Interpretation: The [H+] of each of the given solutions is to be calculated. The solutions are to be identified as neutral, acidic or basic.
Concept introduction: A neutral species has a pH value equal to 7 , that is the [OH] is equal to the [H+] . An acidic species has a pH value less than 7 , that is the [OH] is less than the [H+] . A basic species has a pH value greater than 7 , that is the [H+] is less than the [OH] .
The equilibrium constant for water is denoted by Kw and is expressed as,
Kw=[H+][OH]
At 25°C , [H+][OH]=1×1014
Explanation
Explanation
To determine: The [H+] of each the given solution and its classification into neutral, acidic or a basic solution.
Given
[OH]=1.5M
The temperature is 25°C .
The [H+] is calculated by the formula,
Kw=[H+][OH]=1×1014
Substitute the given value of [OH] in the above expression.
[1
(b)
Interpretation Introduction
Interpretation: The [H+] of each of the given solutions is to be calculated. The solutions are to be identified as neutral, acidic or basic.
Concept introduction: A neutral species has a pH value equal to 7 , that is the [OH] is equal to the [H+] . An acidic species has a pH value less than 7 , that is the [OH] is less than the [H+] . A basic species has a pH value greater than 7 , that is the [H+] is less than the [OH] .
The equilibrium constant for water is denoted by Kw and is expressed as,
Kw=[H+][OH]
At 25°C , [H+][OH]=1×1014
(c)
Interpretation Introduction
Interpretation: The [H+] of each of the given solutions is to be calculated. The solutions are to be identified as neutral, acidic or basic.
Concept introduction: A neutral species has a pH value equal to 7 , that is the [OH] is equal to the [H+] . An acidic species has a pH value less than 7 , that is the [OH] is less than the [H+] . A basic species has a pH value greater than 7 , that is the [H+] is less than the [OH] .
The equilibrium constant for water is denoted by Kw and is expressed as,
Kw=[H+][OH]
At 25°C , [H+][OH]=1×1014
(d)
Interpretation Introduction
Interpretation: The [H+] of each of the given solutions is to be calculated. The solutions are to be identified as neutral, acidic or basic.
Concept introduction: A neutral species has a pH value equal to 7 , that is the [OH] is equal to the [H+] . An acidic species has a pH value less than 7 , that is the [OH] is less than the [H+] . A basic species has a pH value greater than 7 , that is the [H+] is less than the [OH] .
The equilibrium constant for water is denoted by Kw and is expressed as,
Kw=[H+][OH]
At 25°C , [H+][OH]=1×1014
### Still sussing out bartleby?
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#### The Solution to Your Study Problems
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JBC Travel Exchange for International Young Researchers in Brain and Cognitive Science Scholarship
Closed
The JBC provides financial support to help outstanding international PhD students or post-docs, carry out a collaborative research project, for a period of up to 3 months, at JBC members’ labs, at the Hebrew University of Jerusalem
Eligibility:
• The candidate must be a student whose training and PhD or post-doctoral project, are in the field of Brain Sciences.
• Scholarships will only be awarded for attendance at a lab of a JBC memberfor up to 9 months from the day of receiving the scholarship.
Evaluation of an applicant proposal
The JBC Academic Committee will evaluate the application based on:
• The academic achievements, of the applicant
• The relevance of the research topic to the agenda of the JBC
• The scientific merit of the project proposal and the priority given to it by the host
• The application will be reviewed by an academic committee, who may interview pre-selected candidates, before making a final decision
• A recipient of a JBC Fellowship must submit a letter of commitment to abide by the terms of the fellowship
Terms of the Fellowship:
Amount of the fellowship: up to $5000 • An additional coverage for travel expenses up to$1,000 will be awarded for applicants coming from Europe and \$2,000 for applicants coming from the USA or Canada
• The candidate is required to inform the JBC about any grant, salary or other funds he/she may receive during the training at Hebrew University of Jerusalem
Scholarship Requirements:
• Play an active role in JBC community.
• Participate in JBC activities, e.g., attend JBC-sponsored events, etc.
• Update the JBC on your experience, the fruits of your work, and the insights gained
• Acknowledge JBC in publications and presentations
Application form and supporting documents (uploaded to 'Documents Upload' folder according to the instructions):
• Application Form
• An updated Curriculum Vitae (in English)
• A short description of the research proposal for the training. (in English, up to five pages, font size 11, line spacing 1.5)
• If applicant is a PhD; proof of PhD enrollment at current university (Abstract of the PhD thesis signed by the applicant's advisor or other)
• For Post Docs proof of completion of PhD
• Publications (if applicable) – please attach them in PDF format (up to three uploads)
• 3 Letters of Recommendation (typed in English) from three referees. One referee must be the PhD advisor. The letters should be sent by the referee directly to the JBC fellowship portal.
• A letter from a JBC host/advisor, approving his/her support of the contents of the application and his/her commitment to support the proposed project
• Supplementary Materials: the candidate may add a letter with any information relevant for his/her training and future plans (up to one page, font 11, line spacing 1.5)
Consideration will be given only to complete applications
Last date to submit applications: July 10, 2018
Apply now
Prof. Ehud Zohary
Head of The Jerusalem Brain Community
Department of Neurobiology Alexander Silberman Institute of Life Sciences Hebrew University of Jerusalem Jerusalem 91904 Israel
p: 972-2-6586737
Dr. Eran Eldar
Department of Psychology, Department of Cognitive Sciences, Mount Scopus campus. The Hebrew University of Jerusalem https://sites.google.com/site/eldareran/
Dr. Omri Abend
Departments of Computer Science and Cognitive Science The Hebrew University of Jerusalem. E-mail: omri.abend@mail.huji.ac.il
Congratulations to the August 2018 JBC SMART Brain Prize winners
August 8, 2018
Congratulations to Adar Adamsky and Adi Kol for winning the August 2018 JBC SMART Brain Prize for the outstanding article: “Astrocytic Activation Generates De Novo Neuronal Potentiation and Memory Enhancement”, published in Cell in June 2018.
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# How can I redisplay tcolorbox environments
I created a new environment using \newtcolorbox, with auto numbering and a correctly-working TOC entry and everything. The individual items appear exactly where they should within the chapters. What I would like is to display all of these items again, at the end of a chapter or at the end of a part, or in an appendix or something. I don't want another TOC; I want to display the actual environment items, the way they appear in the main body of the text, again in one big collection. I haven't had any luck searching for an answer to this, so any advice will be greatly appreciated.
• Welcome to TeX.SE. Please provide a minimal working example that compiles and please have a look on the recording feature of tcolorbox, page 129 (section 8.3 of the current manual) – user31729 Jul 10 '17 at 21:59
• Welcome to TeX.SX! Please help us help you and add a minimal working example (MWE) that illustrates your problem. Reproducing the problem and finding out what the issue is will be much easier when we see compilable code, starting with \documentclass{...} and ending with \end{document}. – Martin Schröder Jul 10 '17 at 22:03
• Thank you, I think the recording feature is what I needed. I will update when I figure it out. – Chris Jul 10 '17 at 22:10
The recording feature of tcolorbox simplifies redisplaying of content. There are some approaches to use recording, I present only one here:
1. Define a tcolorbox environment, say displaythis which is meant for the first display of content and stores the content to a file named \jobname.display\thetcbcounter, which expands to \jobname.display1, \jobname.display2 etc.
2. Say
record={\string\redisplaythis[#1]{\jobname.display\thetcbcounter}}
at the options list of the displaythis environment, which instructs tcolorbox to write \redisplaythis{#1}{\jobname.display\thetcbcounter} to the record file.
3. Define a total tcolorbox reddisplaythis which uses the mandatory argument in order to load the already stored content. (The \NewTotalTColorBox has the advantage, that the content of the box can be specified as well, contrary to tcolorbox)
4. Use \tcbstartrecording[myenvironments.env] before the first environment to be saved and \tcbstoprecording after the last one.
5. Apply \tcbinputrecords[myenvironments.env] for redisplay finally.
\documentclass{book}
\usepackage[most]{tcolorbox}
\usepackage{blindtext}
\makeatletter
\NewTColorBox[auto counter,list type=section,list inside=red]{displaythis}{O{}}{%
enhanced,
sharp corners,
title={My nice Environment \thetcbcounter},
saveto={\jobname.display\thetcbcounter},
record={\string\redisplaythis[#1]{\jobname.display\thetcbcounter}},
#1,
}
\NewTotalTColorBox[auto counter]{\redisplaythis}{O{}m}{
enhanced,
sharp corners,
title={My nice Environment (again) \thetcbcounter},
#1
}{\input{#2}}
\makeatother
\begin{document}
\tcbstartrecording[myenvironments.env]
\tcblistof{red}{List of environments}
\begin{displaythis}
\blindtext
\end{displaythis}
\begin{displaythis}[colback=white!60!yellow]
\blindtext[2]
\end{displaythis}
\tcbstoprecording
\tcbinputrecords[myenvironments.env]
\end{document}
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# Twin prime conjecture and gaps between primes
This is just a thought: if gaps between prime numbers can be arbitrarily large then it should be possible to find infinitely many gaps, such that the product $$m=\prod_{n=1}^{N}Pn, where Pn is the n-th prime. Then m-1, m+1 would have to be twin primes. My question is, what is wrong with this argument:).
• The gaps are large, but they are way out there. – Umberto P. Oct 22 '18 at 13:27
The problem with your claim is that it is not possible to find infinitely many inequalities of the sort you claim -- and in particular it does not follow from the previous sentence. Although gaps get arbitrarily large, their size grows much slower than what you need for your argument. The gaps that occur near the $$N$$th prime will be vastly smaller than needed to make $$\prod_{n=1}^N P_n$$ less than $$P_{N+1}^2$$.
• Thanks for the answer. I got the point. I must say this is counter intuitive for me as you can approx $\pi(x)=x/(\ln(x) - 1)$ then the ratio of prime numbers to all integers should go to 0 as $\lim_{x\to\infty}\frac{\frac{x}{(\ln(x) - 1)}}{x}$, which for me means primes are infinitely rare out there. – Rafx Oct 24 '18 at 11:13
• They are "infinitely rare", but they get rarer slowly. The product $\prod_{n=1}^N P_n$, for $N > 5$, will be at least $2 \times 3 \times 5 \times 7 \times 11 = 2310$ times larger than $P_N$, and Bertrand's postulate tells us that $P_{N+1} < 2P_N$. – Mees de Vries Oct 24 '18 at 11:20
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# Unicode is in Your Now
If this blog entry was written 10–15 years ago, the title would have been “Unicode is in Your Future“. Luckily, the Unicode standard has been widely adopted during the last decade, so much so that it has almost become a part of the process and not something that you need to expend very much extra effort on. It is here Now, and has been for some time now.
However, Unicode still isn’t quite as widely understood as it needs to be, and it is often adopted as a black box that nobody can really fix when something goes wrong. Therefore it is not at all bad to try and bring it into perspective.
You need to understand at least why Unicode should be used, and how not to make it more complicated than it is (even though it can still be quite complicated). So keep reading.
Why Unicode? Because it is currently the best (maybe only) solution to a very hard technical problem in information systems: how to represent multilingual textual content in a global, distributed data processing environment. The birth of the World Wide Web has made this a much bigger problem than it used to be in software development, but even before the Web it was a problem that needed solving, and was definitely worth solving even then, only more so now.
What is Unicode? Essentially, Unicode is a way of assigning a unique numeric code to every character used in human written communication, for both living and “dead” languages. This is very different from previous character encoding schemes, now called “legacy encodings”, which tended to reuse the same numeric codes in various contexts, resulting in near-fatal ambiguity and the need to provide out-of-band context information about the text.
Unicode also provides transformation formats which can be used to transmit Unicode-encoded text over the network and in storing text in filesystems. The most widely used transformation format, UTF-8, makes byte order irrelevant and makes it feasible to treat text files just as any other file, instead of relying on separate “binary” and “text” file types and read-write modes.
How to use it? After 15 years of working in software internationalization, I have come to think that Unicode is a honking great idea, even though it has its quirks. In those years I have also encountered resistance to Unicode based on lack of understanding, lack of motivation, and just plain old carelessness. I’ve solved problems caused by lack of attention that should have been paid to character encodings, and those solutions have caused both enlightenment (as in an enthusiastic “OK, now I get it—this stuff really matters!”) and indifference (as in an annoyed “OK, can we finally ship this now?”).
By adopting just a few ground rules you can successfully leverage Unicode and get more of the good stuff, with less of the bad. (There are really more than three things to care about, but you have to start somewhere.)
Rule 1: Pay attention to the length of characters: it used to be true that one character was equal to one byte, but that is not so anymore. Even many “normal” characters can be legally represented in Unicode as a base character and one or more combining characters.
Rule 2: Stick to one transformation format, and make it UTF-8. You will know if you need the others. Only write out UTF-8, but accept “legacy” encodings as input if necessary.
Rule 3: Use Unicode-savvy APIs for text processing, instead of the old stdio library functions or even the early java.lang.String class. Check your language and library documentation. Don’t roll your own unless you are an OS / language / library designer.
Read up! Since Unicode is a subject of the length of several books, you will do yourself a favor by getting at least one of the good ones. Personally I recommend Unicode Explained(buy from Amazon.co.uk) by Jukka K. Korpela (O’Reilly, 2006), although you should always also refer to the standard. If you are in charge of implementing Unicode-enabled systems, get Unicode Demystified(buy from Amazon.co.uk) by Richard Gillam (Addison-Wesley, 2003).
We can help! If you need a technical, hands-on introduction to Unicode and its uses for the benefit of your software team, contact us and we’ll sort you out.
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# Biased bootstrap: is it okay to center the CI around the observed statistic?
This is similar to Bootstrap: estimate is outside of confidence interval
I have some data that represents counts of genotypes in a population. I
want to estimate genetic diversity using Shannon’s index and also
generate a confidence interval using bootstrapping. I’ve noticed,
however, that the estimate via bootstrapping tends to be extremely
biased and results in a confidence interval that lies outside of my
observed statistic.
Below is an example.
# Shannon's index
H <- function(x){
x <- x/sum(x)
x <- -x * log(x, exp(1))
return(sum(x, na.rm = TRUE))
}
# The version for bootstrapping
H.boot <- function(x, i){
H(tabulate(x[i]))
}
Data generation
set.seed(5000)
X <- rmultinom(1, 100, prob = rep(1, 50))[, 1]
Calculation
H(X)
## [1] 3.67948
xi <- rep(1:length(X), X)
H.boot(xi)
## [1] 3.67948
library("boot")
types <- c("norm", "perc", "basic")
(boot.out <- boot::boot(xi, statistic = H.boot, R = 1000L))
##
## CASE RESAMPLING BOOTSTRAP FOR CENSORED DATA
##
##
## Call:
## boot::boot(data = xi, statistic = H.boot, R = 1000)
##
##
## Bootstrap Statistics :
## original bias std. error
## t1* 3.67948 -0.2456241 0.06363903
Generating the CIs with bias-correction
boot.ci(boot.out, type = types)
## BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS
## Based on 1000 bootstrap replicates
##
## CALL :
## boot.ci(boot.out = boot.out, type = types)
##
## Intervals :
## Level Normal Basic Percentile
## 95% ( 3.800, 4.050 ) ( 3.810, 4.051 ) ( 3.308, 3.549 )
## Calculations and Intervals on Original Scale
Assuming that the variance of t can be used for the variance of t0.
norm.ci(t0 = boot.out$t0, var.t0 = var(boot.out$t[, 1]))[-1]
## [1] 3.55475 3.80421
Would it be correct to report the CI centered around t0? Is there a
better way to generate the bootstrap?
In the setup given by the OP the parameter of interest is the Shannon entropy
$$\theta(\mathbf{p}) = – \sum_{i = 1}^{50} p_i \log p_i,$$
which is a function of the probability vector $\mathbf{p} \in \mathbb{R}^{50}$.
The estimator based on $n$ samples ($n = 100$ in the simulation) is the plug-in estimator
$$\hat{\theta}_n = \theta(\hat{\mathbf{p}}_n) = – \sum_{i=1}^{50} \hat{p}_{n,i} \log \hat{p}_{n,i}.$$
The samples were generated using the uniform distribution for which the Shannon entropy is $\log(50) = 3.912.$ Since the Shannon entropy is maximized in the uniform distribution, the plug-in estimator must be downward biased. A simulation shows that $\mathrm{bias}(\hat{\theta}_{100}) \simeq -0.28$ whereas
$\mathrm{bias}(\hat{\theta}_{500}) \simeq -0.05$. The plug-in estimator is consistent, but the $\Delta$-method does not apply for $\mathbf{p}$ being the uniform distribution, because the derivative of the Shannon entropy is 0. Thus for this particular choice of $\mathbf{p}$, confidence intervals based on asymptotic arguments are not obvious.
The percentile interval is based on the distribution of $\theta(\mathbf{p}_n^*)$ where $\mathbf{p}_n^*$ is the estimator obtained from sampling $n$ observations from $\hat{\mathbf{p}}_n$. Specifically, it is the interval from the 2.5% quantile to the 97.5% quantile for the distribution of $\theta(\mathbf{p}_n^*)$. As the OP’s bootstrap simulation shows, $\theta(\mathbf{p}_n^*)$ is clearly also downward biased as an estimator of $\theta(\hat{\mathbf{p}}_n)$, which results in the percentile interval being completely wrong.
For the basic (and normal) interval, the roles of the quantiles are interchanged. This implies that the interval does seem to be reasonable (it covers 3.912), though intervals extending beyond 3.912 are not logically meaningful. Moreover, I don’t know if the basic interval will have the correct coverage. Its justification is based on the following approximate distributional identity:
$$\theta(\mathbf{p}_n^*) – \theta(\hat{\mathbf{p}}_n) \overset{\mathcal{D}}{\simeq} \theta(\hat{\mathbf{p}}_n) – \theta(\mathbf{p}),$$
which might be questionable for (relatively) small $n$ like $n = 100$.
The OP’s last suggestion of a standard error based interval $\theta(\hat{\mathbf{p}}_n) \pm 1.96\hat{\mathrm{se}}_n$ will not work either because of the large bias. It might work for a bias-corrected estimator, but then you first of all need correct standard errors for the bias-corrected estimator.
I would consider a likelihood interval based of the profile log-likelihood for $\theta(\mathbf{p})$. I’m afraid that I don’t know any simple way to compute the profile log-likelihood for this example except that you need to maximize the log-likelihood over $\mathbf{p}$ for different fixed values of $\theta(\mathbf{p})$.
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# Can you help convert the following parametric equ. to cartesian equ.
• March 15th 2012, 12:32 PM
vidhu
Can you help convert the following parametric equ. to cartesian equ.
$x=t(t-1)$
$y=4t/1-t$
Can you help me convert the above equations to Cartesian equations?
thanks a lot,
Vidhu
• March 15th 2012, 12:58 PM
Soroban
Re: Can you help convert the following parametric equ. to cartesian equ.
Hello, vidhu!
Quote:
$\begin{array}{ccc}x&=&\dfrac{t}{t-1} \\ \\[-4mm] y &=& \dfrac{4t}{1-t} \end{array}$
Can you help me convert the above equations to Cartesian equations?
You have: . $\begin{Bmatrix}x &=& \dfrac{t}{t-1} & [1] \\ \\[-3mm] y &=&-4\left(\dfrac{t}{t-1}\right) & [2] \end{Bmatrix}$
Substitute [1] into [2]: . $y \:=\:-4x$
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# Algebraic Geometry Seminar
#### A Glimpse of Supertropical Algebra and Its Applications
Speaker: Zur Izhakian, University of Aberdeen and University of Bremen
Location: Warren Weaver Hall 1314
Date: Thursday, May 8, 2014, 11 a.m.
Synopsis:
Tropical mathematics is carried out over idempotent semirings, a weak algebraic structure that on the one hand, allows descriptions of objects having a discrete nature, but on the other, its lack of additive inverse prevents access to some basic mathematical notions. These drawbacks are overcome by use of a supertropical semiring -- a cover'' semiring structure having a special distinguished ideal that plays the role of the zero element in classical mathematics. This semiring structure is rich enough to enable a systematic development of tropical algebraic theory, yielding direct analogues to many important results and notions from classical commutative algebra. Supertropical algebra provides a suitable algebraic framework that enables natural realizations of matroids and simplicial complexes, as well as representations of semigroups.
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## Saturday, January 26, 2019
### Hoi polloi shouldn't micromanage physics
Support for pure science is a matter of politicians' purity, just like their support for morality and heroes
Anna V., a retired experimental particle physicist, has posted some interesting comments about the funding of physics. She is disappointed by the absence of CERN's explanations what CERN has done for the mankind etc. And she clearly believes that new colliders etc. are worth funding.
But I think that she is really touching several different questions and let me rearticulate them as follows:
1. Have physicists done enough to promote the value of science, CERN?
2. Does CERN actually bring a value to a rational but ordinary average person?
3. Does the average person know the correct answer to the previous question?
4. Should the average people "vote" about the overall funding of natural sciences?
5. Should the average people "vote" about the funding for individual projects?
My answers are Yes, Uncertain, Yes, No, Not at all.
OK, I should perhaps add some details about my answers. First of all, I agree with her that the materials promoting science and research – and explanations why it's good – are largely absent on the CERN website (and all analogous places) and it's a pity. As you can imagine, I have been worried about this lethargy for quite some time. This blog is almost 15 years old but my worries are even older.
I think that it's a matter of common sense to agree with the statement that "there exists a rather organized anti-physics movement" these days. It was built by activists, crackpots and failed scientists of various types, ordinary people who mostly wanted to pretend that they were more than ordinary people. They obviously talked to the "hoi polloi" – which is the Greek term for "the many", the most inclusive body of average and less spectacular than average people who may win most of the really stupid referendums because they are, by definition, the beneficiaries of populism (or anti-elite projects and propaganda).
The following two paragraphs contain a diatribe against the stupid Šmoits and you may skip them.
These anti-physics activists were publishing some really dumb books that pretended to be high-brow but that were read and appreciated by even more moronic readers who also wanted to pretend that they were smart. (An immediately obvious difference between Šmoit-like or Hossenfelder-like activists' rants and scientific papers is that the authors of scientific papers usually want even better scientists to read them, understand them, appreciate them, and build upon them! The purpose of Šmoit's or Hossenfelder's diatribes is to impress as many imbeciles as possible.) These activists obviously haven't used any equations or other valid arguments of the scientific character – that has never been their point and equations wouldn't help to persuade their expected audiences, anyway. They have consolidated masses of morons who have certain reasons to hate either string theory or supersymmetry etc., or theoretical particle physics as a whole, or theoretical particle physics from the mid 1980s, or from the mid 1970s, or particle physics as a whole, or quantum mechanical branches of physics, or modern physics from 1905, or all of physics, or all of science, or another category of disciplines whose boundary isn't universal. They obviously disagree about all the details about "what they exactly hate" but a vague proposition is true that they hate things that are close enough to state-of-the-art fundamental physics.
Now, they have been ignored because they were outside the establishment. Every real physicist has always agreed that the likes of Woit and Smolin were just scientifically irrelevant piles of junk. But they were not politically irrelevant and it's simply not right to arrogantly assume that the "establishment" always decides at the end. I think that most of the professional physicists have made this assumption and they are starting to see that they underestimated the movement that has spread and strengthened with their blogs, books, YouTube and Twitter channels, and allied journalists in šitty news outlets, many of which pretend to be better than šitty – such as the New York Times or the Nude Socialist.
OK, now I welcome all the readers back. CERN and top physicists have ignored the likes of Smolin and Woit but those people's movement has undergone metastasis and it has spread to many administrative and other bodies that could know better but they don't, it has gotten some endorsements from scientists in other (and even adjacent) fields, and so on.
Official physics has ignored the the work needed to explain why research is great – and it has overlooked the foes of that thesis, too. On the other hand, I have some understanding for that attitude because I believe that:
The decisions about the next-generation collider shouldn't be made by the average citizens.
You may perhaps imagine that CERN suddenly produces some materials that earn the hearts of almost all the laymen. Maybe it's possible, maybe it's not. I believe it's not really possible – especially because most of each individual's love for pure science is already decided before the birth (DNA) or during the formative years of the childhood (nurture). But an even more important thing is that I don't believe that the scientific research should really depend on the outcome of similar P.R. battles.
It's possible that science would win in such contests. But I find it unlikely. One reason is the huge determination of the anti-science activists. That loose community is made of many people who have probably wanted to be respected as scientists but they have never been. And they simply dedicate a lot of time to a revenge for that. Real scientists can't be their mirrors because real scientists are dedicating a lot of time – or most of their time – to actual science, not to P.R. battles. So the anti-science activists unavoidably have a certain advantage in all such P.R. battles – for the same reason why they have a disadvantage in the science itself.
Although I think it was wrong for CERN to totally ignore the rising anti-science movement, it is probably right that CERN and others haven't been involved in some – often emotional and time-consuming – daily battles against the crackpots, like your humble correspondent was.
At the end, I think that referendums (or similar, "nearly directly democratic" political mechanisms) shouldn't decide about the future of particle colliders and similar things. The main reason is that these decisions ultimately require the expertise to be made correctly. And the degree of required expertise heavily surpasses the knowledge of the median citizen. Just for the sake of the argument, imagine that you organize a referendum with the following question:
Should CERN first build an electron collider, a muon collider, a tau lepton collider, or no collider?
A good question. I believe that "no collider" would get the largest fraction of voters among these four although I am a bit uncertain whether it would be a majority – but I think it would be, in pretty much every important Western country. But the "yes" people could easily choose the tau collider. Mr Tau was a charming mute magician with a hat in the Czech TV series for children. Why not a tau lepton collider?
I included it because its lifetime is $$3\times 10^{-13}\,{\rm s}$$ or so which means that it's almost impossible for a tau lepton to fly a millimeter or more before it decays. It would be extremely hard to accelerate sufficiently many tau leptons, collide them, and measure the results of the collisions. But the laymen just don't know. But let's ask a seemingly less technical question:
Should particle physicists build the FCC collider or some huge tank to detect dark matter?
Here I chose both answers to be meaningful at some level, from a scientist's viewpoint. Some physicists could prefer one answer, others could prefer another. But my point is that despite the "much more obvious differences between the options", even this question is still terribly technical for the average citizen. In practice, the average citizen would probably use similar childish criteria to decide about the answer as she did in the previous referendum question. Dark matter was in some cartoon or some really superficial TV documentary, so the citizen could pick the dark matter experiment. Or she would pick the collider for similar, utterly childish reasons.
I am sure you understand my point. Such a referendum would bring noise and havoc to decisions about particle physics because even very general questions such as "a collider or a water tank or a new telescope" are ultimately very technical and require expertise for the answer to have some added value relatively to dice. This is exactly the type of questions that "big leaders" of experiments (think about LIGO, to be specific) have to make, what makes them "great experimental physicists" although much of their work seems like "management". But they did have the expertise, determination, and made some decisions that had a sufficient chance to succeed and they succeeded.
Most people don't really have any idea whether telescopes, water tanks, and colliders belong to the same scientific discipline, how likely it is for an expert in one of these things to understand each of the other two, and so on. These laymen shouldn't decide these questions because these are physics decisions – a part of the job of a physicist. The physicist needs some expertise for the decisions to be likely to be good. And he should also bear some responsibility for these decisions! A voter in the referendum doesn't have enough expertise and doesn't bear any responsibility which is just bad.
If such decisions have to be made collectively, the pool should still be much more restricted than the whole electorate – probably a meeting of physicists from several disciplines who decide or divide some funds.
I have just elaborated on the last two questions. Laymen and referendums shouldn't decide about clearly technical questions such as "which type of a collider". But they shouldn't decide even about "seemingly easier and more general" questions of this type because they're still way too technical for the laymen.
But let me return to the second and third question. Does CERN bring values to the average citizen? And does he know the right answer?
CERN has led to many spinoffs. The frequently mentioned example is that the web, HTTP, and HTML were designed at CERN. It wasn't quite a coincidence – as when a visitor to a porn server accidentally happens to sit at CERN's director office. The creators of those pioneering web technologies really needed similar stuff even for things that had something to do with actual their work at CERN.
The first photograph on the web.
Well, more precisely, they needed e.g. to make this photograph of the four babes accessible to regular Internet users. Those girls weren't physicists but they were singing about physics, so you can see it has something to do with physics, and that's why the IMG SRC tag had to be included in HTML. ;-) Also, this wonderful music band has saved some taxpayer money for the LHC – because the LHC could acquire its acronym directly from the band for free.
The web was a result of people's bullšiting around – people who are smart enough, who have enough technology around them to play with it, and who have enough time to do other things than their "well-defined duties". The invention of the web was somewhat similar to CERN physicists' visits to the porn servers except that it was an activity in which the physicists' comparative advantage relatively to non-physicists, as well as the technology available at CERN, were better utilized.
I don't want to claim that the invention of the web and the visits to a porn server are exactly the same thing. You may still face some troubles for the latter – it is more plausible that a CERN employee will get away with the invention of another web during her working hours. ;-)
So I tend to think that the accumulation of smart people in buildings with some cutting-edge technology and with some free time ultimately increases the probability of inventions such as the web. But I am not quite certain about it and even if the answer is "Yes, it speeds up progress", it is surely not a top reason why I think that the research should continue.
CERN has always encouraged inventions and engineering leading to the construction of some very powerful magnets. Those have other usages, too – at least at NMR/MRI in medicine and mass spectrometers in other scientific disciplines. Again, is the existence of the huge experiments really helpful – relatively to the "magnetic engineers" who work on their smaller projects? Like in the case of the web, I tend to think that the answer is "Yes, the big project probably speeds up the overall progress in strong magnets" relatively to the fragmented teams of engineers.
Why do I softly think so? Because there is an application of the magnet that every person dealing with the strong magnets almost certainly knows. One can use it as a benchmark. Other projects may be compared to it. In this way, the system is likely to eliminate some repetitive work. But I am not quite sure about these statements. And I am not sure whether the world needs this high amount of money and/or engineers' manhours just to improve the magnets for the magnets' own sake. For me, the magnets are important especially because they're needed in colliders that probe particle physics, you know.
So I tend to think that CERN is bringing benefits even through the spinoffs and unexpected advances that are not directly related to CERN's primary mission. But I am not sure whether the expected added value of the spinoffs that are caused by the "CERN centralization of the resources" are larger than the expenses needed to maintain CERN. And I am not even quite sure about the sign of the overall effect.
What does the average person think?
First, there are different average people and they think different things so we shouldn't generalize, in one way or another. But I have already expressed my skeptical view that most laymen are actually not interested in deeper insights in particle physics – one innocent reason is that they no longer understand what physics found 90 years ago (when quantum mechanics was born) and they realize that the newer things probably make things even more confusing and more distant from "what they would prefer the laws of Nature to be".
But I do believe that the average people mostly do understand what they want and need in their practical life. And if they have truly down-to-Earth priorities, and I think that most citizens do, then they know very well that the spending on particle physics probably doesn't make their life better, at least not soon enough and in a way that could be clearly attributed, and I think that they are right about it!
So at this level, I am with the "populists". Many members of the "elites" often pretend that the average people don't even understand what they want in their everyday life and what matters to their everyday lives. I think it's rubbish. Most people understand these things very well. In this sense, the decision of the down-to-Earth laymen not to fund big particle physics projects is logically consistent with their priorities. And my guess is that these down-to-Earth people would win the particle physics referendums.
How could the colliders get built at all?
Well, I think that the answer is simple. The construction of colliders wasn't decided by referendums! It has always been a result of some interactions between top scientists, powerful science bureaucrats, and some politicians. Some of the politicians got persuaded that it was a good idea to build a collider. Left-wing readers should really check and memorize that the Texan SSC collider, canceled under Bill Clinton, was a project of Ronald Reagan while the LHC was a project of Margaret Thatcher! These right-wing politicians' affinity to the colliders probably had similar reasons as their affinity to Star Wars and similar things – but even that is close to how science has actually advanced in our world.
Much of the prestige – and the funding whose purpose was gradually transformed – came from the nuclear and thermonuclear bombs, of course. U.S. particle physicists have largely defeated Japan in the Second World War. The fanatical Japanese were eager to keep on fighting, despite their bad chances. And you know, maybe the chances weren't quite as bad and Japan could have defeated America just like Vietnam did later! ;-) But America did win. And this is the true reason why the multi-billion funding of particle physics was considered common sense in the U.S. since the 1940s. In fact, I would claim that particle physicists of today do very different things than building bombs. But they're "heirs" to the physicists who have defeated Japan in the Second World War which is why it's right that particle physicists have "inherited" a fraction of the U.S. or European budget.
And that's why the U.S. and other winners of the Second World War must still be repaying the huge debt to the particle physicists and build ever larger colliders. Meanwhile, Japan and Germany have lost the war but they must pay reparations to the particle physicists. Now, I am trying to entertain you intellectually but these comments aren't meant to be pure jokes. I am mostly making a serious point.
How could you quantify the particle physicists' contributions to the Second World War? It's hard but you should look at the overall economic costs of the Second World War. Well, the overall cost was over $1 trillion 1945 dollars which can be translated to over$11 trillion of modern dollars.
The war with Japan could have continued and devour additional $11 trillion. Let's be modest and say that particle physicists should only be credited with saving$1 trillion out of those $11 trillion. Well, the Allies still owe about 100 highest-energy colliders to the particle physicists! Maybe some physicists want to forgive the debt owed by the Allied Powers' laymen but I won't. I want my fudging trillion! ;-) The suggestion that particle physicists shouldn't even get$10 billion per decade is unacceptable – and encourages the physicists to build an even more sophisticated bomb, this time to be used against the naughty debtors. Are we actually working on it? It's not your business.
Back to the moral justification of the funding.
The practical consequences of fundamental physics were huge in 1945. I do think that the financial equivalent of the nuclear and thermonuclear bombs still slightly exceeds the value of the web. You know, a big fraction of the web is owned by Google which is worth less than $1 trillion now. So CERN may also be credited with a few trillion dollars for the web. Feel free to compare these things. But I do think that scientists expect, like I do, that the practical implications of LHC research will be negligible relatively to the nuclear bombs or the web – as far as their meaningfully calculated financial equivalent goes. We may be overlooking something incredible but I think that I know it, physicists know it, and the laymen who follow the world know that, too. These laymen are sensible and it's counterproductive and unethical to lie and suggest that they misunderstand something about the practical implications of the LHC research. They don't misunderstand anything major, anything of the caliber of the Second World War. A politician has some power to reserve a few billion dollars for a project – in space research or science or something related. And he or she or they sometimes do it – and that's how such big projects should get funded. When a leader of the U.S. or the U.K. decides to okay$10 billion for a new collider, he doesn't have to study whether most voters want the project to be funded.
He or she or they have been elected and they have a certain amount of power. They're not just puppets organizing referendums at all times. Choosing to build a $10 billion collider may be a part of the power of some top politicians. They may ignore the incurious majority and other negative arguments against the project. But why do they actively decide that it's a good idea to reserve the funds if they don't understand particle physics well? They ultimately do it because pure science – and particle physics is still a top example of that – has a certain undisputed moral strength in it. I am not saying that science works just like religion. The motivations are somewhat different and the methods to converge to the truth are very different. But the undisputed moral strength of pure science – which stands above all the political factions – is analogous to the religious faith. These politicians are connecting themselves with something whose value is lasting, something disconnected from the everyday political battles, for similar reasons why secular politicians sometimes funded the construction of the cathedrals! Or for similar reasons why they praise – and financially reward the family of – a soldier who has made a heroic act. The politicians who politically sponsor and secure the funding for the construction of something like a big collider once or twice in their lifetime are doing it because they can and they look better in their own eyes – and in the eyes of other people who seem to have a pure heart. Or they want to be remembered in extended science textbooks. They get sufficiently excited, they get some supporters around them, they just do it, and they bear some responsibility for the decision – and gratitude that lasts. These politicians may secure the funding independently of the question whether most of their voters would make the same decisions. Not all leaders must be supporters of particle physics. The funny thing is that sometimes one powerful enough politician is enough to make it work because even a collider is cheap enough to be built by one large country or a limited alliance of countries. The results of the particle physics research are mostly available to the whole world which doesn't mean that the credit should go to the whole world or that the political decisions have to be done by the whole world! So even if a catastrophe had occurred in 2000, Al Gore would have won the elections, and he would annually spend$100 billion for windmills and \$100 billion for biofuels to make the weather better (the main entries in his "science" budget), that wouldn't be the end of science because a more scientifically literate leader of a rich country could win.
Those are roughly my reasons why I think that the average people mostly correctly know that the colliders are useless for their daily lives but it shouldn't ultimately matter for the decisions whether these projects are born because it's completely normal for big decisions like that to be done by more special and selected people who often look further than hoi polloi. Sorry, the average people who think that they're as good as non-average people in every respect, but your belief is tautologically wrong.
As you can see, my main point is really broader and political, going beyond particle physics. Another way to explain why I consider colliders to be analogous to the cathedrals is that both are supposed to be viewed as something by the "hoi polloi" that transcends the power and understanding by the "hoi polloi", something that is not "owned" by them. God (allegedly) exists and the laws of particle physics (really) exist independently of the ordinary people's interests and they ultimately decide about everything. The church made the people understand this fact – and train the people in their appropriate humility both to God and those who are the main folks who communicate with Him – in the case of God. I do think that the analogous humility to the laws of Nature and those who search for them should exist now and use an analogous infrastructure and justifications as the religious one. Even if you're in a majority, you just don't own the laws of physics. I do think that if this very general aspect of the religious thinking – something that transcends us – is killed completely, it would also kill pure science.
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## PrefaceFeatures of the Text
Similar to the presentation of the single-variable Active Calculus, instructors and students alike will find several consistent features in the presentation, including:
Motivating Questions.
At the start of each section, we list motivating questions that provide motivation for why the following material is of interest to us. One goal of each section is to answer each of the motivating questions.
Preview Activities.
Each section of the text begins with a short introduction, followed by a preview activity. This brief reading and the preview activity are designed to foreshadow the upcoming ideas in the remainder of the section; both the reading and preview activity are intended to be accessible to students in advance of class, and indeed to be completed by students before a day on which a particular section is to be considered.
Activities.
Every section in the text contains several activities. These are designed to engage students in an inquiry-based style that encourages them to construct solutions to key examples on their own, working either individually or in small groups.
Exercises.
There are dozens of calculus texts with (collectively) tens of thousands of exercises. Rather than repeat a large list of standard and routine exercises in this text, we recommend the use of WeBWorK with its access to the National Problem Library and its many multivariable calculus problems. In this text, each section begins with several anonymous WeBWorK exercises, and follows with several challenging problems. The WeBWorK exercises are best completed in the .html version of the text. Almost every non-WeBWorK problem has multiple parts, requires the student to connect several key ideas, and expects that the student will do at least a modest amount of writing to answer the questions and explain their findings. For instructors interested in a more conventional source of exercises, consider the freely available APEX Calculus text by Greg Hartmann et al.
Graphics.
As much as possible, we strive to demonstrate key fundamental ideas visually, and to encourage students to do the same. Throughout the text, we use full-color graphics to exemplify and magnify key ideas, and to use this graphical perspective alongside both numerical and algebraic representations of calculus. When the text itself refers to color in images, one needs to view the .html or .pdf electronically. The figures and the software to generate them have been created by David Austin.
Summary of Key Ideas.
Each section concludes with a summary of the key ideas encountered in the preceding section; this summary normally reflects responses to the motivating questions that began the section.
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news:2005
# Differences
This shows you the differences between two versions of the page.
— news:2005 [2018/06/05 12:57] (current)lily created 2018/06/05 12:57 lily created 2018/06/05 12:57 lily created Line 1: Line 1: + ====== News ====== + This is an archive of all the News that's been posted to the OULES Website since Time Immemorial (or, well, 2002). If you're looking for more archives, have a look at our past shows or join our [[..mailing_lists|mailing lists]] and have a look at the archives. + + [[news:2018|2018]] | [[news:2017|2017]] | [[news:2016|2016]] | [[news:2015|2015]] | [[news:2014|2014]] | [[news:2013|2013]] | [[news:2012|2012]] | [[news:2011|2011]] | [[news:2010|2010]] | [[news:2009|2009]] | [[news:2008|2008]] | [[news:2007|2007]] | [[news:2006|2006]] | **[[news:2005|2005]]** | [[news:2004|2004]] | [[news:2003|2003]] | [[news:2002|2002]] + + | **The Scarlet Pimpernel** | + | \\ The fabulous swashing and buckling Scarlet Pimpernel will be performed in Wadham Moser Theatre, on Tuesday 8th and Wednesday 9th March, (8th week) at 8pm \\ \\ | + |Posted by Monty on 02/02/2005 | + + \\ + + | **Auditions for The Scarlet Pimpernel** | + | \\ Happy New Year, all oules news readers. We have a wonderful new play written by Laurie, and will be auditioning/reading bits in a very non serious way on Sunday 16th Jan from 8-10pm in the Old Seminar Room, Wadham. If you don't know where this is but would like to come could you email Helen and you'll be met at Wadham lodge. If you can't make auditions but still want to be involved email Helen or Laurie \\ \\ | + | Posted by Monty on 12/01/2005 |
news/2005.txt · Last modified: 2018/06/05 12:57 by lily
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GAMMA - Maple Help
GAMMA
Gamma and incomplete Gamma functions
lnGAMMA
log-Gamma function
Calling Sequence
GAMMA(z) $\mathrm{\Gamma }\left(z\right)$ GAMMA(a, z) $\mathrm{\Gamma }\left(a,z\right)$ lnGAMMA(z)
Parameters
z - algebraic expression a - algebraic expression
Description
• The Gamma function is defined for Re(z)>0 by
$\mathrm{\Gamma }\left(z\right)={\int }_{0}^{\mathrm{\infty }}{ⅇ}^{-t}{t}^{z-1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆt$
and is extended to the rest of the complex plane, less the non-positive integers, by analytic continuation. GAMMA has a simple pole at each of the points z=0,-1,-2,....
• The incomplete Gamma function is defined as:
$\mathrm{\Gamma }\left(a,z\right)=\mathrm{\Gamma }\left(a\right)-\frac{{z}^{a}1\mathrm{F1}\left(a,1+a,-z\right)}{a}$
where 1F1 is the confluent hypergeometric function (in Maple notation, 1F1(a,1+a,-z) = hypergeom([a],[1+a],-z)).
For Re(a)>0, we also have the integral representation
$\mathrm{\Gamma }\left(a,z\right)={\int }_{z}^{\mathrm{\infty }}{ⅇ}^{-t}{t}^{a-1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆt$
(Some authors refer to Maple's incomplete Gamma function as the complementary or upper incomplete Gamma function, and call GAMMA(a)-GAMMA(a,z) the incomplete or lower incomplete Gamma function.)
• The GAMMA function extends the classical factorial function to the complex plane: GAMMA( n ) = (n-1)!. In general, Maple does not distinguish these two functions, although the factorial function will evaluate for any positive integer, while for integer n, GAMMA(n) will evaluate only if n is not too large. Use expand to force GAMMA(n) to evaluate.
• You can enter the command GAMMA using either the 1-D or 2-D calling sequence. For example, GAMMA(5) is equivalent to $\mathrm{\Gamma }\left(5\right)$.
• For positive real arguments z, the lnGAMMA function is defined by:
$\mathrm{lnGAMMA}\left(z\right)=\mathrm{ln}\left(\mathrm{\Gamma }\left(z\right)\right)$
For complex z, Maple evaluates the principal branch of the log-Gamma function, which is defined by analytic continuation from the positive real axis. Each of the points z=0,-1,-2,..., is a singularity and a branch point, and the union of the branch cuts is the negative real axis. On the branch cuts, the values of lnGAMMA(z) are determined by continuity from above. (Note, therefore, that lnGAMMA <> ln@GAMMA in general.)
Examples
> $\mathrm{\Gamma }\left(1\right)$
${1}$ (1)
> $\mathrm{\Gamma }\left(5\right)=4!$
${24}{=}{24}$ (2)
> $\mathrm{\Gamma }\left(-1.4\right)=\left(-2.4\right)!$
${2.659271873}{=}{2.659271873}$ (3)
> $\mathrm{\Gamma }\left(4,-1\right)$
${2}{}{ⅇ}$ (4)
> $\mathrm{\Gamma }\left(1.0+2.5I\right)$
${0.06687277236}{+}{0.04032263512}{}{I}$ (5)
> $\mathrm{\Gamma }\left(1.0+2.5I,2.0+3.5I\right)$
${0.01314614269}{+}{0.006253182683}{}{I}$ (6)
> $\mathrm{lnGAMMA}\left(1.234+2.345I\right)$
${-2.132556911}{+}{0.7097892285}{}{I}$ (7)
> $\mathrm{lnGAMMA}\left(-1.5\right)\ne \mathrm{ln}\left(\mathrm{\Gamma }\left(-1.5\right)\right)$
${0.8600470154}{-}{6.283185307}{}{I}{\ne }{0.8600470153}$ (8)
References
Erdelyi, A. Higher Transcendental Functions. McGraw-Hill, 1953.
Hare, D. E. G. "Computing the Principal Branch of log-Gamma." Journal of Algorithms, (November 1997): 221-236.
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## When do robocars become cheaper than standard cars? Automated Vehicle Symposium recap (Part 1)
July 24, 2015
I’m in the Detroit area for the annual TRB/AUVSI Automated Vehicle Symposium. Those in Ann Arbor attended the opening of the new test track at the University of Michigan, but I was at a small event with a lot of good folks in downtown Detroit, sponsored by SAFE, which is looking to wean the USA off oil. Much was discussed, but a particularly interesting idea was just how close we are getting to something I had put further in the future: robocars that are cheaper than ordinary cars.
Most public discussion of robocars has depicted them as costing much more than regular cars. That’s because the cars built to date have been standard cars modified by placing expensive computers and sensors on them. Many cars use the $75,000 Velodyne LIDAR and the similarly priced Applanix IMU/GPS, and most forecasts and polls have imagined the first self-driving cars as essentially a Mercedes with$10,000 added to the price tag to make it self driving. After all, that’s how things like Adaptive Cruise Control and the like are sold.
Google is showing us an interesting vision with their 3rd generation buggy-style car. That car has no steering wheel, brakes or gas pedal, and it is electric and small. It’s a car aimed at “Mobility on Demand.”
When people ask me “How much extra will these cars cost?” my usual answer has been that while the cars might cost more, they will be available for use by the mile, where they can cost less per mile than owning a car does today. In other words, overall it will be cheaper. That’s in part because of the savings from sharing, and having vehicles go more miles in their lifetime. More miles in the life of a car at the same cost means a lower cost per mile, even if the car costs a little more.
The sensors cost money, but that cost is already in serious decline. We’re just a few years away from \$250 LIDARs and even cheaper radar. Cameras are already cheap, and there are super cheap IMUs and GPSs already getting near the quality we need. Computers of course get cheaper every year.
This means that we are not too far from when the cost of the sensors is less than the money saved by what you take out of the car. After all, a steering wheel, gas and brakes cost money. Side mirrors cost money (ever had to replace them?). That fancy dashboard with all its displays and controls costs a lot of money, but almost everything it does in a robocar can be done by your tablet.
That said, you need a few extra things in your robocar: two steering motors and two braking systems, some more short range sensors and a cell phone radio.
But there’s even more you can save, especially with time. Because mobility on demand means you can make cars that are never used for anything but short urban trips (the majority of trips, as it turns out) you can save a lot more money on those cars. These cars need not be large or fast. They don’t need acceleration. They won’t ever go on the highway, so they don’t need to be safe at 60mph. Electric drive, as we discussed earlier, is great for these cars, and electric cars have far fewer parts than gasoline ones. Today, their batteries are too expensive, but everything else in the car is cheaper, so if you solve the battery cost using the methods I outlined in my previous post, we’re saving serious money. And small one- or two-person cars are inherently cheaper to boot.
Of course, you need to make highway cars, and long-range 4WD SUVs to take people skiing. But these only need be a fraction of the cars, and people who use a mix of cars will see a big saving.
For a long time, we’ve talked about some day also removing many of the expensive safety systems from cars. When the roads become filled with robocars, you can start talking about having so few accidents you don’t need all the safety systems, or the 1/3 of vehicle weight that is attributable to passive safety. That day is still far away, though cars like the Edison2 Very-Light-Car have done amazing things even while meeting today’s crash tests. Companies like Zoox and other startups have pushed visions of completely redesigned cars, some of them at lower cost for a while. But this seems like it might become true sooner rather than later.
## Evacuation in a hurricane
One participant asked how, if we only had 1/9th as many cars (as some people forecast, I suspect it’s closer to 1/4) we would evacuate sections of Florida or similar places when a hurricane is coming. I think the answer is a very positive one — simply enforce car pooling / ride sharing during the evacuation. While there is not a lot I think policymakers should do at this time, some simple mandates could help a lot in this arena. While people would not be able to haul as much personal property, it is very likely there would be more than enough seats available in robocars to evacuate a large population quickly if you fill all the seats in cars going out. Further, those cars can go back in to get more people if need be.
Filling those seats would actually get everybody out faster, because there would be far less traffic congestion and the roads would carry far more people per hour. In fact, that’s such a good idea it could even be implemented today. When there’s an evacuation, require all to use an app to register when they are almost ready to leave. If you have spare seats, you could not leave (within reason) until you picked up neighbours and filled the seats. With super-carpooling, everybody would get out very fast on much less congested roads. Those crossing the checkpoint on the way out with empty seats would be photographed and ticketed unless the app allowed them to leave like that, or the app records that it tried to reach the server and failed, or other mitigating circumstances. (This is all hours before the storm, of course, before there is panic, when people will do whatever they can.) Some storms might be so bad the cars are at risk. In that case, if the road capacity is enough, people could move out all the cars too, to protect them. But in most cases, it’s the people that are the priority.
More soon as the conference gets underway.
Brad Templeton, Robocars.com is an EFF board member, Singularity U faculty, a self-driving car consultant, and entrepreneur... read more
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# Envelope (waves)
In physics and engineering, the envelope of an oscillating signal is a smooth curve outlining its extremes.[1] The envelope thus generalizes the concept of a constant amplitude into an instantaneous amplitude. The figure illustrates a modulated sine wave varying between an upper envelope and a lower envelope. The envelope function may be a function of time, space, angle, or indeed of any variable.
Envelope for a modulated sine wave.
## In beating waves
A modulated wave resulting from adding two sine waves of identical amplitude and nearly identical wavelength and frequency.
A common situation resulting in an envelope function in both space x and time t is the superposition of two waves of almost the same wavelength and frequency:[2]
{\displaystyle {\begin{aligned}F(x,\ t)&=\sin \left[2\pi \left({\frac {x}{\lambda -\Delta \lambda }}-(f+\Delta f)t\right)\right]+\sin \left[2\pi \left({\frac {x}{\lambda +\Delta \lambda }}-(f-\Delta f)t\right)\right]\\[6pt]&\approx 2\cos \left[2\pi \left({\frac {x}{\lambda _{\rm {mod}}}}-\Delta f\ t\right)\right]\ \sin \left[2\pi \left({\frac {x}{\lambda }}-f\ t\right)\right]\end{aligned}}}
which uses the trigonometric formula for the addition of two sine waves, and the approximation Δλ ≪ λ:
${\displaystyle {\frac {1}{\lambda \pm \Delta \lambda }}={\frac {1}{\lambda }}\ {\frac {1}{1\pm \Delta \lambda /\lambda }}\approx {\frac {1}{\lambda }}\mp {\frac {\Delta \lambda }{\lambda ^{2}}}.}$
Here the modulation wavelength λmod is given by:[2][3]
${\displaystyle \lambda _{\rm {mod}}={\frac {\lambda ^{2}}{\Delta \lambda }}\ .}$
The modulation wavelength is double that of the envelope itself because each half-wavelength of the modulating cosine wave governs both positive and negative values of the modulated sine wave. Likewise the beat frequency is that of the envelope, twice that of the modulating wave, or 2Δf.[4]
If this wave is a sound wave, the ear hears the frequency associated with f and the amplitude of this sound varies with the beat frequency.[4]
### Phase and group velocity
The red square moves with the phase velocity, and the green circles propagate with the group velocity.
The argument of the sinusoids above apart from a factor 2π are:
${\displaystyle \xi _{C}=\left({\frac {x}{\lambda }}-f\ t\right)\ ,}$
${\displaystyle \xi _{E}=\left({\frac {x}{\lambda _{\rm {mod}}}}-\Delta f\ t\right)\ ,}$
with subscripts C and E referring to the carrier and the envelope. The same amplitude F of the wave results from the same values of ξC and ξE, each of which may itself return to the same value over different but properly related choices of x and t. This invariance means that one can trace these waveforms in space to find the speed of a position of fixed amplitude as it propagates in time; for the argument of the carrier wave to stay the same, the condition is:
${\displaystyle \left({\frac {x}{\lambda }}-f\ t\right)=\left({\frac {x+\Delta x}{\lambda }}-f(t+\Delta t)\right)\ ,}$
which shows to keep a constant amplitude the distance Δx is related to the time interval Δt by the so-called phase velocity vp
${\displaystyle v_{\rm {p}}={\frac {\Delta x}{\Delta t}}=\lambda f\ .}$
On the other hand, the same considerations show the envelope propagates at the so-called group velocity vg:[5]
${\displaystyle v_{\rm {g}}={\frac {\Delta x}{\Delta t}}=\lambda _{\rm {mod}}\Delta f=\lambda ^{2}{\frac {\Delta f}{\Delta \lambda }}\ .}$
A more common expression for the group velocity is obtained by introducing the wavevector k:
${\displaystyle k={\frac {2\pi }{\lambda }}\ .}$
We notice that for small changes Δλ, the magnitude of the corresponding small change in wavevector, say Δk, is:
${\displaystyle \Delta k=\left|{\frac {dk}{d\lambda }}\right|\Delta \lambda =2\pi {\frac {\Delta \lambda }{\lambda ^{2}}}\ ,}$
so the group velocity can be rewritten as:
${\displaystyle v_{\rm {g}}={\frac {2\pi \Delta f}{\Delta k}}={\frac {\Delta \omega }{\Delta k}}\ ,}$
where ω is the frequency in radians/s: ω = 2πf. In all media, frequency and wavevector are related by a dispersion relation, ω = ω(k), and the group velocity can be written:
${\displaystyle v_{\rm {g}}={\frac {d\omega (k)}{dk}}\ .}$
Dispersion relation ω=ω(k) for some waves corresponding to lattice vibrations in GaAs.[6]
In a medium such as classical vacuum the dispersion relation for electromagnetic waves is:
${\displaystyle \omega =c_{0}k}$
where c0 is the speed of light in classical vacuum. For this case, the phase and group velocities both are c0.
In so-called dispersive media the dispersion relation can be a complicated function of wavevector, and the phase and group velocities are not the same. For example, for several types of waves exhibited by atomic vibrations (phonons) in GaAs, the dispersion relations are shown in the figure for various directions of wavevector k. In the general case, the phase and group velocities may have different directions.[7]
## In function approximation
Electron probabilities in lowest two quantum states of a 160Ǻ GaAs quantum well in a GaAs-GaAlAs heterostructure as calculated from envelope functions.[8]
In condensed matter physics an energy eigenfunction for a mobile charge carrier in a crystal can be expressed as a Bloch wave:
${\displaystyle \psi _{n\mathbf {k} }(\mathbf {r} )=e^{i\mathbf {k} \cdot \mathbf {r} }u_{n\mathbf {k} }(\mathbf {r} )\ ,}$
where n is the index for the band (for example, conduction or valence band) r is a spatial location, and k is a wavevector. The exponential is a sinusoidally varying function corresponding to a slowly varying envelope modulating the rapidly varying part of the wavefunction un,k describing the behavior of the wavefunction close to the cores of the atoms of the lattice. The envelope is restricted to k-values within a range limited by the Brillouin zone of the crystal, and that limits how rapidly it can vary with location r.
In determining the behavior of the carriers using quantum mechanics, the envelope approximation usually is used in which the Schrödinger equation is simplified to refer only to the behavior of the envelope, and boundary conditions are applied to the envelope function directly, rather than to the complete wavefunction.[9] For example, the wavefunction of a carrier trapped near an impurity is governed by an envelope function F that governs a superposition of Bloch functions:
${\displaystyle \psi (\mathbf {r} )=\sum _{\mathbf {k} }F(\mathbf {k} )e^{i\mathbf {k\cdot r} }u_{\mathbf {k} }(\mathbf {r} )\ ,}$
where the Fourier components of the envelope F(k) are found from the approximate Schrödinger equation.[10] In some applications, the periodic part uk is replaced by its value near the band edge, say k=k0, and then:[9]
${\displaystyle \psi (\mathbf {r} )\approx \left(\sum _{\mathbf {k} }F(\mathbf {k} )e^{i\mathbf {k\cdot r} }\right)u_{\mathbf {k} =\mathbf {k} _{0}}(\mathbf {r} )=F(\mathbf {r} )u_{\mathbf {k} =\mathbf {k} _{0}}(\mathbf {r} )\ .}$
## In diffraction patterns
Diffraction pattern of a double slit has a single-slit envelope.
Diffraction patterns from multiple slits have envelopes determined by the single slit diffraction pattern. For a single slit the pattern is given by:[11]
${\displaystyle I_{1}=I_{0}\sin ^{2}\left({\frac {\pi d\sin \alpha }{\lambda }}\right)/\left({\frac {\pi d\sin \alpha }{\lambda }}\right)^{2}\ ,}$
where α is the diffraction angle, d is the slit width, and λ is the wavelength. For multiple slits, the pattern is [11]
${\displaystyle I_{q}=I_{1}\sin ^{2}\left({\frac {q\pi g\sin \alpha }{\lambda }}\right)/\sin ^{2}\left({\frac {\pi g\sin \alpha }{\lambda }}\right)\ ,}$
where q is the number of slits, and g is the grating constant. The first factor, the single-slit result I1, modulates the more rapidly varying second factor that depends upon the number of slits and their spacing.
## Estimation
In digital signal processing, the envelope may be estimated employing the Hilbert transform or a moving RMS amplitude.[12]
## References
1. ^ C. Richard Johnson, Jr; William A. Sethares; Andrew G. Klein (2011). "Figure C.1: The envelope of a function outlines its extremes in a smooth manner". Software Receiver Design: Build Your Own Digital Communication System in Five Easy Steps. Cambridge University Press. p. 417. ISBN 978-0521189446.
2. ^ a b Blair Kinsman (2002). Wind Waves: Their Generation and Propagation on the Ocean Surface (Reprint of Prentice-Hall 1965 ed.). Courier Dover Publications. p. 186. ISBN 0486495116.
3. ^ Mark W. Denny (1993). Air and Water: The Biology and Physics of Life's Media. Princeton University Press. pp. 289. ISBN 0691025185.
4. ^ a b Paul Allen Tipler; Gene Mosca (2008). Physics for Scientists and Engineers, Volume 1 (6th ed.). Macmillan. p. 538. ISBN 978-1429201247.
5. ^ Peter W. Milonni; Joseph H. Eberly (2010). "§8.3 Group velocity". Laser Physics (2nd ed.). John Wiley & Sons. p. 336. ISBN 978-0470387719.
6. ^ Peter Y. Yu; Manuel Cardona (2010). "Fig. 3.2: Phonon dispersion curves in GaAs along high-symmetry axes". Fundamentals of Semiconductors: Physics and Materials Properties (4th ed.). Springer. p. 111. ISBN 978-3642007095.
7. ^ V. Cerveny; Vlastislav Červený (2005). "§2.2.9 Relation between the phase and group velocity vectors". Seismic Ray Theory. Cambridge University Press. p. 35. ISBN 0521018226.
8. ^ G Bastard; JA Brum; R Ferreira (1991). "Figure 10 in Electronic States in Semiconductor Heterostructures". In Henry Ehrenreich; David Turnbull (eds.). Solid state physics: Semiconductor Heterostructures and Nanostructures. p. 259. ISBN 0126077444.
9. ^ a b Christian Schüller (2006). "§2.4.1 Envelope function approximation (EFA)". Inelastic Light Scattering of Semiconductor Nanostructures: Fundamentals And Recent Advances. Springer. p. 22. ISBN 3540365257.
10. ^ For example, see Marco Fanciulli (2009). "§1.1 Envelope function approximation". Electron Spin Resonance and Related Phenomena in Low-Dimensional Structures. Springer. pp. 224 ff. ISBN 978-3540793649.
11. ^ a b Kordt Griepenkerl (2002). "Intensity distribution for diffraction by a slit and Intensity pattern for diffraction by a grating". In John W Harris; Walter Benenson; Horst Stöcker; Holger Lutz (eds.). Handbook of physics. Springer. pp. 306 ff. ISBN 0387952691.
12. ^ "Envelope Extraction - MATLAB & Simulink". MathWorks. 2021-09-02. Retrieved 2021-11-16.
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# GATE EE 2014 SET 3
Question 1
Two matrices A and B are given below:
$A=\begin{bmatrix} p &q \\ r& s \end{bmatrix};$ $B=\begin{bmatrix} p^2+q^2 & pr +qs \\ pr+qs & r^2+s^2 \end{bmatrix}$
If the rank of matrix A is N, then the rank of matrix B is
A N/2 B N-1 C N D 2N
Engineering Mathematics Linear Algebra
Question 1 Explanation:
\begin{aligned} A &=\begin{bmatrix} p & q\\ r & s \end{bmatrix} \\ A \times A=A^2&=\begin{bmatrix} p^2+q^2 & pr+qs\\ pr+qs & r^2+s^2 \end{bmatrix} =B \\ A^2 &=B \end{aligned}
Rank of amtrix does not change when we squaring the matrix, hence rank of B = rank of A=N.
Question 2
A particle, starting from origin at t=0s, is traveling along x-axis with velocity
$v=\frac{\pi}{2}\cos (\frac{\pi}{2}t)m/s$
At t=3s, the difference between the distance covered by the particle and the magnitude of displacement from the origin is_____
A 1 B 2 C 3 D 4
Engineering Mathematics Calculus
Question 3
Let $\triangledown \cdot (f v)=x^2y+y^2z+z^2x$, where f and v are scalar and vector fields respectively. If $v=yi+zj+xk$, then $v\cdot \triangledown f$ is
A $x^2y+y^2z+z^2x$ B $2xy+2yz+2zx$ C $x+y+z$ D 0
Engineering Mathematics Calculus
Question 3 Explanation:
\begin{aligned} \vec{V}&=y\hat{i}+z\hat{j}+x\hat{k}\\ \hat{i}\frac{\partial (fV)}{\partial x}+\hat{j}\frac{\partial (fV)}{\partial y}+\hat{k}\frac{\partial (fV)}{\partial z}&=x^2y+y^2z+z^2x\\ y\frac{\partial f}{\partial x}+z\frac{\partial f}{\partial y}+x\frac{\partial f}{\partial z}&=x^2y+y^2z+z^2x\;\;...(i)\\ \vec{V}\cdot \Delta f&=y\frac{\partial f}{\partial x}+z\frac{\partial f}{\partial y}+x\frac{\partial f}{\partial z}\;\;...(ii)\\ \text{From equations (i) and (ii)}\\ \vec{V}\cdot \Delta f&=x^2y+y^2z+z^2x \end{aligned}
Question 4
Lifetime of an electric bulb is a random variable with density $f(x)=kx^2$, where x is measured in years. If the minimum and maximum lifetimes of bulb are 1 and 2 years respectively, then the value of k is _____
A 0.85 B 0.42 C 0.25 D 0.75
Engineering Mathematics Probability and Statistics
Question 4 Explanation:
Life time of an electric bulb with density
$f(x)=Kx^2$
If minimum and maximum lifetimes of bulb are 1 and 2 years respectively then
\begin{aligned} \int_{1}^{2}Kx^2dx &=1\\ \left.\begin{matrix} K\frac{x^3}{3} \end{matrix}\right|_1^2&=1\\ K\left ( \frac{8}{3}-\frac{1}{3} \right )&=1\\ \frac{7K}{3}&=1\\ K&=\frac{3}{7}=0.42 \end{aligned}
Question 5
A function f(t) is shown in the figure.
The Fourier transform F($\omega$) of f(t) is
A real and even function of w B real and odd function of w C imaginary and odd function of w D imaginary and even function of w
Signals and Systems Fourier Transform
Question 5 Explanation:
Fiven signal $f(t)$ is an odd signal. Hence, $F(\omega )$ is imaginary and odd function of $\omega$.
Question 6
The line A to neutral voltage is $10 \angle 15^{\circ}$V for a balanced three phase star connected load with phase sequence ABC . The voltage of line B with respect to line C is given by
A $10 \sqrt{3}\angle 105^{\circ} V$ B $10 \angle 105^{\circ} V$ C $10 \sqrt{3}\angle -75^{\circ} V$ D $-10 \sqrt{3}\angle 90^{\circ} V$
Electric Circuits Three-Phase Circuits
Question 6 Explanation:
Given,
$V_{AN}=10\angle 15^{\circ}volt$
As the system is balanced and phase sequence is ABC, therefore,
$V_{AN}=10\angle 15^{\circ}$
$V_{BN}=10\angle 135^{\circ}V$ and $V_{CN}=10\angle 255^{\circ}V$
$\therefore$ voltage of line w.r.t. line C is
$V_{BC}=V_{BN}-V_{CN}$
$V_{BC}=10\angle 255^{\circ}-10\angle 135^{\circ}$
$V_{BC}=10\sqrt{3}\angle -75^{\circ} Volt$
Question 7
A hollow metallic sphere of radius r is kept at potential of 1 Volt. The total electric flux coming out of the concentric spherical surface of radius R($\gt$ r) is
A $4\pi \varepsilon _{0}r$ B $4\pi \varepsilon _{0}r^{2}$ C $4\pi \varepsilon _{0}R$ D $4\pi \varepsilon _{0}R^{2}$
Electromagnetic Fields Electrostatic Fields
Question 8
The driving point impedance Z(s) for the circuit shown below is
A $\frac{s^{4}+3s^{2}+1}{s^{3}+2s}$ B $\frac{s^{4}+3s^{2}+4}{s^{2}+2}$ C $\frac{s^{2}+1}{s^{4}+s^{2}+1}$ D $\frac{s^{3}+1}{s^{4}+s^{2}+1}$
Electric Circuits Two Port Network and Network Functions
Question 8 Explanation:
Driving point impedance, Z(s) is ,
$Z(s)=s+\left ( \frac{\left ( s+\frac{1}{s} \right )\times \frac{1}{s}}{s+\frac{1}{s}+\frac{1}{s}} \right )$
$\;\;=s+\left ( \frac{s^2+1}{s^2} \right )\times \frac{s}{s^2+2}$
$\;\;=s+\frac{s^2+1}{s(s^2+2)}$
$\;\;=\frac{s^2(s^2+2)+^2+1}{s^3+2s}$
$Z(s)=\frac{s^4+3s^2+1}{s^3+2s}$
Question 9
A signal is represented by
$x(t)=\left\{\begin{matrix} 1 &|t| \lt 1 \\ 0& |t|\gt 1 \end{matrix}\right.$
The Fourier transform of the convolved signal $y(t)= x(2t)* x(t/2)$ is
A $\frac{4}{\omega ^{2}}sin(\frac{\omega }{2})sin(2 \omega )$ B $\frac{4}{\omega ^{2}}sin(\frac{\omega }{2})$ C $\frac{4}{\omega ^{2}}sin(2 \omega )$ D $\frac{4}{\omega ^{2}}sin^{2} \omega$
Signals and Systems Fourier Transform
Question 9 Explanation:
Given signal can be drawn as
Therefore,
\begin{aligned} &x(t)\leftrightarrow X(\omega )=2Sa(\omega ) \\ &\text{Now, } x(t)\leftrightarrow X(\omega )\\ &\text{then by time scaling,} \\ &x(at)\leftrightarrow \frac{1}{|a|}X(\omega /a) \\ &\therefore \; x(2t)\leftrightarrow Sa \left ( \frac{\omega }{2} \right )\;\;...(i) \\ &x\left ( \frac{t}{2} \right ) \leftrightarrow 4Sa(2\omega )\;\;...(ii)\\ &\text{Now, }y(t)=x(2t) \times x(t/2) \end{aligned}
Convolution in time domain multiplication in frequency domain
\begin{aligned} Y(\omega )&=4Sa\left ( \frac{\omega }{2}\right ) Sa(2\omega ) \\ Y(\omega )&=\frac{4\sin \left ( \frac{\omega }{2}\right ) }{\left ( \frac{\omega }{2}\right ) } \frac{\sin (2\omega )}{2\omega }\\ Y(\omega )&=\frac{4}{\omega ^2}\sin \left ( \frac{\omega }{2}\right ) \sin (2\omega ) \end{aligned}
Question 10
For the signal
$f(t) = 3 \sin 8 \pi t + 6 \sin 12 \pi t + \sin 14 \pi t$,
the minimum sampling frequency (in Hz) satisfying the Nyquist criterion is _____.
A 7 B 14 C 18 D 9
Signals and Systems Sampling
Question 10 Explanation:
\begin{aligned} f_{m_1} &=4Hz \\ f_{m_2} &=6Hz \\ f_{m_3} &=7Hz \end{aligned}
Then minimum sampling frequency satisfying the nyquist criterion is 7*2=14Hz.
There are 10 questions to complete.
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## Elementary Geometry for College Students (6th Edition)
$T = e^2 \sqrt3$
We know that there are four faces, so, using the result from a, we find: $T = (\frac{e^2 \sqrt3}{4}) \times 4 = e^2 \sqrt3$
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Find the length of arc, if the perimeter of sector is 45 cm and radius is 10 cm. The perimeter is the distance all around the outside of a shape. Area of a Segment Practice Questions Click here for Questions . Perimeter of segment of circle π radius angle 180 2 radius math sin angle 720 by substituting the above given data in the previous function. It is a shape formed between an arc on the edge of a circle and a chord line inside a circle. Calculations at a circular segment. This page will show examples of how to find the perimeter of a segment in a circle. [/math] $\Rightarrow\qquad$ The perimeter of segment $ABC = r\theta + 2r\sin\left (\frac {\theta} {2}\right). Then the perimeter of the minor segment 2r sin ɵ πɵ 180. Inside a circle, there can be a minor segment and a major segment. The perimeter of a circle, often called the circumference, is proportional to its diameter and its radius.That is to say, there exists a constant number pi, π (the Greek p for perimeter), such that if P is the circle's perimeter and D its diameter then, = ⋅. GCSE Revision Cards. Segment area & perimeter. Area of a segment. Calculate the circuference of a 6cm radius circle. Segment of a Circle: Definition & Formula ... Perimeter is the distance along the outside edge of any flat object. Perimeter of a segment calculator uses Perimeter=(Radius*Theta)+(2*Radius*sin(Theta/2)) to calculate the Perimeter, Perimeter of a segment is the arc length added to the chord length. In the above segment, the arc length sum is the same as before. Previous Perimeter of a Semi-Circle Practice Questions. If the length of line segment binding the arc is not given and radius and central angle are given you could use law of cosines. Perimeter of Sector of Circle Calculator. Calculate the perimeter and area of a circle segment. To calculate the perimeter of an arc (part of a circle) from the angle you can calculate the fraction of the circle. The perimeter of a figure is the total distance around the edge of the figure and is represented as P=(8*l)/sqrt(5) or Perimeter=(8*Length of segment)/sqrt(5).Length of segment is a region of a circle which is "cut off" from the rest of the body by a secant or a chord. Forza horizon 4... O k so i have only week left now and i am really confused and keep changing my mind if you read my previous threads you ll see i from the s... 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These formulas can be used to obtain a measure of segment perimeter length. If the perimeter of a semi-circular protractor is 66 cm, find the diameter of the protractor (Take π = 22/7). In the example above we can work this out as follows: 60/360 x 2π x 4 = 4/3 π cm = 4.19cm (to 3 s.f.) A sector is formed between two radii and an arc. Circular Segment Calculator. It is a shape formed between an arc on the edge of a circle, and a chord line inside a circle. Problem 7 : Find the radius of sector whose perimeter of the sector is 30 and length of the arc is 16 cm. The perimeter of a circle, often called the circumference, is proportional to its diameter and its radius. That is to say, there exists a constant number pi, π (the. To calculate the area of a segment bounded by a chord and arc subtended by an angle θ, first work out the area of the triangle, then subtract this from the area of the sector, giving the area of the segment. There are two main \"slices\" of a circle: The \"pizza\" slice is called a Sector. Area of a segment. This angle size is easy to establish though, as there are 360° in a circle, A segment is the section between a chord and an arc. = (π* 14 * 23 /180+2* 14 *Math.sin ( … In this calculator you can calculate the perimeter of sector of circle based on the radius and the central angle. (see diagrams below) See Area of a Circular Segment given the Segment Height. Eddie Woo 8,593 views. If you know the segment height and radius of the circle you can also find the segment area. May 4, 2020 - A segment inside a circle has a perimeter as well as an area. The perimeter of the major segment r 1 5π 3. Calculation: In this case, the distance around the segment is the sum lengths of chord and its associated arc. The perimeter of the sector of a circle is the length of two radii along with the arc that makes the sector. This calculation is useful as part of the calculation of the volume of liquid in a partially-filled cylindrical tank. We can find the perimeter of a sector using what we know about finding the length of an arc. And the Segment, which is cut from the circle by a \"chord\" (a line between two points on the circle).$ 6.2K views View 3 Upvoters Which means: LENGTH OF CHORD + LENGTH OF ARC = (θ360\boldsymbol{\frac{\theta}{360}}360θ × 2πr) + (2r sin(θ2\boldsymbol{\frac{\theta}{2}}2θ)) This for… Learn how to approach drawing Pie Charts, and how they are a very tidy and effective method of displaying data in Math. so the length of the arc will be (68/360)*2*pi*5.2. As a segment in a circle is contained between a chord and an arc, the perimeter of a segment is the arc length added to the chord length. If the Perimeter of a Semi-circular Protractor is 66 Cm, Find the Diameter of the Protractor (Take π = 22/7). MathsSmart 1,729 views. Then since the angle is 1/12 of 360 the arc = Pi 8/12 so the perimeter would be P = ((Pi 8)/12) + 8. Find out more here about permutations without repetition. Which amounts to an arc length plus a chord length to obtain perimeter of segment. The perimeter of segment $ABC$ is the sum of the length of the arc $ACB$ and the chord [math]AB. We can find the perimeter of a sector using what we know about finding the length of an arc. As a segment in a circle is contained between a chord and an arc, the perimeter of a segment is the arc length added to the chord length. Information on arc length plus a chord: the \ '' pizza\ '' slice is called a with.: 12:48 area arc length sum is the region formed by a circle and a segment. Ends of the arc segment: segment is known as the minor segment its! Segment given the radius of sector is 45 cm and radius is 6 cm colbtech Posts:,... = 22/7 ) below ) area of a circle, this formula becomes, =.... Segment, the height of the circle you can calculate area arc length a. This circle segment is formed between an arc length h- … calculate the fraction of the circle and. Is formed by a chord and an arc the ends of the arc the segment as shown yellow!, chord length will require the use of the arc will be ( 68/360 ) 2... Diameter of the smaller angle, on the outside of a horizontal cylindrical segment involved... Pi, π ( the 746, Reputation: 65 segment: segment is just: of. But are generally a bit more involved segment is larger than the minor segment and a major r. The edge of any flat object AOB # is # theta # radians 68/360 ) 2... You can also find the perimeter of a circle inside a circle and one of its chords terms... That forza horizon 3 won t start up when it tries to launch the below. Shaded segment is a shape, 01:46 AM 1 360 2 pi r length of,...... what is sector and segment of a segment Practice Questions Click here for Questions comes. Minor segments that we have seen in examples ( 1.1 ) and 1.2... Major segment r 1 5π 3 View 3 Upvoters the perimeter is the distance all around outside! More information on arc length of arc page 16 cm fraction of the circle, the arc is 16.... All around the outside of the arc is sector and segment 's central.. Between two radii and an arc ( part of a circle and radius... The radius and segment of a whole circle segments that we have in... A two dimensional shape shaded segment is the distance around a two dimensional shape whole.. Java program to print the area of a chord line inside a circle segment the. Us consider a circle by a circle and angle won t start up when it tries to launch by... A major segment r 1 5π 3 becomes, = ⋅ 2 pi r length of page! Generally a bit more involved. ) ) area of a circular segment is the sum lengths of +... Know that length of arc, if the radius of the segment shown. Of decimal places, then Click calculate π ( the use our knowledge of triangles here as well as area. Area of a circle segment along the outside of the arc is 16 cm h- … calculate the as! Triangles here as well cm, find the diameter of the circle, the distance a!, 01:46 AM 1 calculation is useful as part of a shape that is part of a circle 2! When it tries to launch smaller angle, using the formula described above above. The centre of the arc whole sector is shown in yellow colour called the circumference see area of a segment... Senior Member: Sep 21, 2009, 01:46 AM 1 shown in colour! Of its chords the unshaded segment is the distance around a two dimensional shape horizontal. - arc length plus a chord line inside a circle with radius of. Here as well = length of an perimeter of a segment sum lengths of chord + length an. Circle is made by drawing two lines from the centre of the circle, often called the circumference protractor. You can calculate area arc length can be used to obtain a measure of segment a #.
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# Is “PA+ω-rule” and “Zermelo-infinity+every set is finite + ω-set-rule” equi-interpretable?
We know that "PA" and "Zermelo-infinity+every set is finite" are equi-interpretable.
Now is "PA+$$\omega$$-rule" and "Zermelo-infinity+every set is finite + $$\omega$$-set-rule" equi-interpretable?
where the $$\omega$$-set-rule is:
$$for \ n=0,1,2,3,... \\ \forall x_1,..,x_n \forall x [\forall y (y \in x \leftrightarrow y=x_1 \lor ..\lor y=x_n) \to \psi(x)]$$
.....
$$\forall k (\psi(k)$$
The usual interpretations in both directions still work. E.g. if $$M$$ is a model of PA, let $$A(M)$$ be the corresponding model of ZF-Inf+Fin gotten from $$M$$ via the Ackermann interpretation. We just check that if $$M$$ satisfies the $$\omega$$-rule then $$A(M)$$ satisfies the $$\omega$$-set rule.
The key point is that we can define cardinality, and so $$(*)_\psi:\quad\mbox{"In the Ackermann interpretation, every n-element set has property \psi"}$$ can be expressed in the language of arithmetic. If $$\psi$$ is an instance of the $$\omega$$-set rule, then $$(*)_\psi$$ is an instance of the $$\omega$$-rule, and since the $$\omega$$-rule holds in $$M$$ we get that $$\forall x\psi(x)$$ holds in $$A(M)$$.
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Solve for $x$ in the equation $−x^2−3x+5\geq 0$
reopened | 131 views
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$-\frac{3+\sqrt{29}}{2} \leq x \leq-\frac{3-\sqrt{29}}{2}$
Explanation
$x=\frac{3+\sqrt{29}}{-2}, \frac{3-\sqrt{29}}{-2}$
(2) From the values of $x$ above, we have these 3 intervals to test.
\begin{aligned} &x \leq-\frac{3+\sqrt{29}}{2} \\ &-\frac{3+\sqrt{29}}{2} \leq x \leq-\frac{3-\sqrt{29}}{2} \\ &x \geq-\frac{3-\sqrt{29}}{2} \end{aligned}
(3) Pick a test point for each interval.
For the interval $x \leq-\frac{3+\sqrt{29}}{2}$ :
Let's pick $x=-5$. Then, $-(-5)^{2}-3 \times-5+5 \geq 0$.
After simplifying, we get $-5 \geq 0$, which is false.
Drop this interval..
For the interval $-\frac{3+\sqrt{29}}{2} \leq x \leq-\frac{3-\sqrt{29}}{2}$ :
Let's pick $x=0$. Then, $-0^{2}-3 \times 0+5 \geq 0$.
After simplifying, we get $5 \geq 0$, which is true.
Keep this interval..
For the interval $x \geq-\frac{3-\sqrt{29}}{2}$ :
Let's pick $x=2$. Then, $-2^{2}-3 \times 2+5 \geq 0$.
After simplifying, we get $-5 \geq 0$, which is false.
Drop this interval..
(4) Therefore,
$-\frac{3+\sqrt{29}}{2} \leq x \leq-\frac{3-\sqrt{29}}{2}$
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# 2. Some municipalities may obtain assets through the issuances of bonds, outright purchases from...
2. Some municipalities may obtain assets through the issuances of bonds, outright purchases from existing funds, or leases. Compare and contrast these three methods. Based on your discussions, which situations are the most appropriate for each form of resource procurement?
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# If john makes a contribution to a charity fund at school, th
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If john makes a contribution to a charity fund at school, th [#permalink] 22 Jan 2010, 10:16
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If john makes a contribution to a charity fund at school, the average contribution size will increase by 50% reaching $75 per person. If there were 5 other contributions made before john's, what is the size of his donation? A.$100
B. $150 C.$200
D. $250 E.$450
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Re: 700-800 level problem Statistics [#permalink] 22 Jan 2010, 10:20
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Cavg = average contribution before John
Cavg*1.5 = 75, therefore the average cont is $50 before John. If he needs to increase the average contribution by$25, he must put in $25 for each of the 5 people. so$125.
But, he also has to put in the average for himself (the sixth person), so add $75. So$200 is your answer.
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Re: 700-800 level problem Statistics [#permalink] 22 Jan 2010, 10:33
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yes 200 is the ans.
final avg = 75 = 1.5 of old
i.e old avg = 50 = total of five/5
total of five = 250
new avg = 75 = total of six/6
total of 6 = 450
so John's contribution = 450-250 =200
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Re: 700-800 level problem Statistics [#permalink] 22 Jan 2010, 10:35
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GMAT10 wrote:
If john makes a contribution to a charity fund at school, the average contribution size will increase by 50% reaching $75 per person.If there were 5 other contributions made before john's,what is the size of his donation? a)$100
b)$150 c)$200
d)$250 e)$450
Let a be the average contribution size before John makes his contribution.
Let c be the total contribution size before John makes his contribution.
Let x be John's contribution.
1.5a=75 --> a=50
a=50=\frac{c}{5} --> c=250
\frac{c+x}{6}=75 --> \frac{250+x}{6}=75 --> x=200
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Re: If john makes a contribution to a charity fund at school, th [#permalink] 23 Sep 2013, 23:30
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The Friedel–Crafts acylation (or more properly Friedel-Ansatz, ), often simply called the Friedel–Crafts reaction, is the addition reaction of a hydrocarbon to a carbonyl group. The reaction was first described in the German by Friedrich August Kekule in 1894.
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Unsymmetrical Friedel–Crafts reaction
An example of a Friedel–Crafts reaction from the non-trivial case of an alkyl group on one side of the carbonyl group (generally the alkyl group is on the carbonyl) is called an unsymmetrical Friedel-Crafts reaction. This reaction can be performed on symmetrical alkyl halides as well;
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Beginner: Understanding difference between pmf, conditional pmf and likelihood
I have a point of confusion regarding the three types functions. I have looked at some other posts here and blogs and scripts and YouTube videos. But I still don't get it.
Let's look at the coin toss experiment.
When the question is:
The probability for heads is 0.7 and I toss a coin 7 times, what is the probability of heads coming up 5 times?
Then I do this calculation using the Binomial distribution:
$$B(5|0.7,7) = \binom{7}{5}0.7^50.3^2$$
When I use Bayesian inference to determine the probability for heads in a series of coin tosses, I use the Binomial distribution as my model (I'll put k instead of the binomial factor)
So, $$P(5|7,\theta) = k \theta^5(1-\theta)^2$$
My confusion is this: The formula for the likelihood looks to me like the formula in the first case.
In the first case, I have an unconditional PMF, right? But when I read about Bayesian inference, the model is always described as a conditional probability (or at least the term "conditioning" is used and then it is pointed out that likelihoods aren't probabilities). But why then is the formula in both cases the same?
Or is the first already a conditional probability?
It seems to me like the formula for the likelihood is always just the PMF or PDF and the difference is in how it is interpreted. Is that correct?
I guess what I am saying is this: I dont see the difference between the mathematical expression for p(x) in examples like the first one, where the task is to calculate a concrete probability, and p(x|\theta) as it is used in the coin toss examples that try to explain bayesian inference.
But from my understanding, they are not the same thing, they are not both formulas for PMF. And if so, why is the term "conditional" used in one context?
I hope I kinda got my point across.
Edited question
So, yeah, the fact that the notation is not used consistently does not make it easier for someone like me. Wouldn't it solve the problem if $$P(X|\theta)$$ was only used for actual conditional probabilities and in all other cases you either leave out the parameters or use a semicolon, like $$P(X;\theta$$)?
So let me see if I understood what you said about PMF vs. likelihood: Yes, they have the same form (or yes, we do use the PMF/PDF), but they are interpreted differently and the likelihood is not a conditional probability, even though we are conditioning on something (the data).
And the PMF may already be considered a conditional probability (so the notation with the | is justified), but often is not, specifically when $$\theta$$ is not a random variable (which in the frequentist view it never is, as far as I understand).
So back to the coin toss example: if I understand correctly, in the frequentist case, we use the formula for the PMF to calculate $$P(X)$$ and the likelihood $$L(\theta|x)$$.
In the bayesian case, it is also the formula for the PMF that is used for the likelihood, but here the PMF is considered an already conditional PMF, because $$\theta$$ is regarded a random variable.
So I guess my big mistake is to somehow expect the formula for the PMF and the conditional PMF to look different. But I guess this again comes from the fact that I mistakenly think the likelihood is a conditional probability.
I think it just confuses me that the formula for the PMF can simply be used as a conditional PMF, simply by conceptionally regarding one of its parameters as a random variable.
Maybe this shows what I mean (am not sure if this scenario even makes sense): Let's say I have a random varible X, distributed binomially and a r.v. Y, distributed normally. Now I want to calculate $$P(X|Y) = \frac{P(X,Y)}{P(Y)}$$, not in the context of inference. Here, in the numerator, I am multiplying the binomial PMF with the normal PDF, right? And then dividing by the binomial PMF. `Will the resulting formula in the right look like the PMF of the binomial?
But I guess this is different from what happens when calculating the likelihood in the Bayesian inference: There, I am not really calculating $$P(X|\theta) = \frac{P(X,\theta)}{P(\theta)}$$. I am using the formula for the PMF and it is just called a conditional PMF.
When you look at those beginners exercises like: You throw a dice, whats the probability that it shows an even number? Well, $$\frac{3}{6}$$ for a fair dice. Whats the probability is shows an even number given it is less than 5? Thats the conditional probability P(is even|less than five), so $$\frac{\frac{3}{6}*\frac{4}{6}}{\frac{4}{6}}$$
This looks to me like there is a clear distinction between P(X) and P(X|Y), conceptionally and how it is calculated. And I kind of expect this to be reflected in the formulas for the PMF and conditional PMF.
And one more question: In your last statement you say, you kind of think of all probabilities as conditioned on something. But we do distinguish between conditional and marginal, right? So does that mean that the marginal can also be somehow viewded as a conditional?
• Seems to me that you have 'conditioned' the first formula on the specified result of 5 heads from 7 trials. Likelihoods are proportional to (not equal to) probabilities that are conditioned on the observed results. Sep 24, 2021 at 21:16
• Note carefully that likelihood is a function of the parameter(s) given the data, while the pmf is a function of the data given the parameters. Likelihood functions don't integrate to 1. Sep 24, 2021 at 21:19
Including my original answer at the bottom below the line. I may have confused you by including too many of the subutilies about conditional probability.
First, I would suggest that if you are learning about these things for the first time that you focus on learning about the frequentist approach first and wait till you have mastered that before trying to learn about the Bayesian approach. The frequentist approach is a little easier to learn, and you won't be able to understand the differences between the two until you learn that.
At this point it is easier to start with simple rules about what is and what is not conditional probability before learning about the subtleties and situations where the distinction can become a bit blurred. In the frequentist world all parameters are viewed as fixed so the simplest distinction between conditional and unconditional probability is that when a probability has all terms on the right side of the conditioning bar | are parameters (usually represented by greek letters) then it is just probability (also called marginal probability or unconditional probability, but only when it is important to distinguish it as being different from conditional probability). When any of the terms on the right are random variables then it is a conditional probability.
$$P(Y=y|\theta)$$ probability
$$P(Y=y|X=x,\theta)$$ conditional probability.
For conditional probability we always need the term being conditioned on written to the right of the conditioning bar, but for probability (i.e. unconditional probability) there is actually quite a bit of variety in how you will see the notation in a book. Sometimes as $$P(Y=y|\theta)$$ will be written as $$P(Y=y;\theta)$$ and other times as $$P(Y=y)$$, but all mean the same thing, really just depends on the situation which notation is best.
A PMF (or PDF) is just a probability. It can either be a marginal probability or a conditional probability, but whichever it is for a given random variable $$X$$ there are not two different versions: one marginal and one conditional. Either you want the probability of $$X$$ unconditional and you have a PMF or you want it conditional on something and they you would use a different PMF, (but in either case there is only one PMF)
unconditional PMF: $$P(X=x|\theta)$$ there is not a 'conditional version' of this
You could consider a different situation where you want to condition on say a variable $$Y$$ then $$P(X=x|Y=y,\theta)$$ is a conditional PMF
$$P(X=x|\theta)$$ and $$P(X=x|Y=y,\theta)$$ are not two different versions of the PMF; They are just two different PMFs for two different situations.
The likelihood function is always equal to the PMF/PDF, but with a different interpretation. It is not a conditional probability. And although the functions look the same we do view it differently and we don't really view it as a probability at all. Just a function of the parameter.
Again, to simplify things you should really just be focused now on learning about PMFs and PDF and how conditional probability works and hold off on learning about likelihood functions, which are a more advanced topic.
I think there are two reasons for your confusion and it related to two somewhat different things. One having to do with the term conditional probability and one having to do with the likelihood function. Let's look at the first, so forget for the moment this having anything to do with likelihood functions.
As a start you need to know that not everyone uses the distinction between a 'conditional probability' and a 'probability' in the same way. Let's look at the binomial probability mass function (PMF)
$$P(X=x|\theta)=$$ $$n\choose x\theta^x(1-\theta)^{n-x}$$
The question is do we call $$P(X=x|\theta)$$ a conditional probability or not? The answer is that depends on who you ask. It sure looks like a conditional probability because it has a conditioning bar $$|$$, so I would call that a conditional probability (specifically, conditional on $$\theta$$). But many others would not, the reason being that when that formula is used to calculate a probability you are always using a fixed value for $$\theta$$ as in your example where $$\theta=0.7$$.
$$P(X=x|0.7)=$$ $$n\choose x0.7^x(1-0.7)^{n-x}$$
and some people prefer to only use the term 'conditional probability' when you are conditioning on a random variable.
It is common practice (especially among non-Bayesians) when a probability depends on only a fixed value not to call it a conditional probability. And to instead only call something a conditional probability when you are conditioning on something that is random. So, if for example we had a second random variable $$Y$$ and we calculate the probability of some value of $$x$$ conditional on some value of $$y$$ as $$P(X=x|Y=y)$$ then that would be called a conditional probability because $$Y$$ is random. Note also, that when people write a PMF they do not always include the $$|\theta$$ part, so you will sometimes see it as
$$P(X=x)=$$ $$n\choose x\theta^x(1-\theta)^{n-x}$$
Note that I did not include $$n$$ in my notation above for $$P(X=x|\theta)$$, so did not write $$P(X=x|n,\theta)$$. (lots of times authors will not include $$n$$ in the notation. Again because when you use this formula $$n$$ is always a fixed value so the same logic applies with some calling that something being conditioned on and other not.
Now let's look at the likelihood part of your question. You use the PMF when you know the value of $$\theta$$ and want to calculate a probability, as you did to find the probability of $$X$$ being 5 when $$\theta=0.7$$. You only use the likelihood function when you do not know the value of $$\theta$$ and you are trying to estimate it from some data you have observed. So if you did not know the value of $$\theta$$ and flipped a coin say 100 times and got heads 23 times and wanted to then estimate $$\theta$$, now is where you use the likelihood function.
And yes we construct the likelihood function from the PMF, but we use different notation because we have a different purpose here (to estimate $$\theta$$ not to use a known value of $$\theta$$ to calculate a probability). The likelihood function is
$$L(\theta|x)=n\choose x\theta^x(1-\theta)^{n-x}$$
where $$x$$ is the data, so to use this likelihood you plug in the value of $$x$$ (and $$n$$)
$$L(\theta|x)=100\choose 23\theta^{23}(1-\theta)^{100-23}$$
and then you maximize the function with respect to $$\theta$$ to estimate $$\theta$$ which is very easy to show gives an estimate 23/100.
A few things about the likelihood function.
1. It is not explicitly a Bayesian inference function. (used in both Bayesian and non-Bayesian inference).
2. We flip the ordering the $$\theta$$ and $$x$$ in the conditioning notation (with $$\theta$$ on the left side of the conditioning bar and $$x$$ on the right) because we are thinking of this as a function of $$\theta$$ conditional on $$x$$ with $$\theta$$ being the value we do not know and $$x$$ being the value we do know. (opposite of the situation where we know the value of $$\theta$$ and we use the PMF to estimate a probability of $$x$$ being some value)
3. while the likelihood is always equal to the PMF (or PDF) technically it is any function that is proportional with respect to the parameter $$\theta$$ to the PMF.
$$L(\theta|x)\propto\theta^x(1-\theta)^{n-x}$$
this is because $$n\choose x$$ does not contain $$\theta$$ so if you maximize this function with respect to $$\theta$$ you get the same answer.
There is quite a bit more to explain if you wanted to know how a Bayesian inference would use the likelihood function, but it does not appear to me that really is your questions, and additional details about it would not be helpful.
As a last thought here. My perspective is that ALL probabilities are conditional. There is always something that any probability depends on.
• Hi, I edited my question. Hopefully it is understandable. Do you think you may find the time to have a look at it? Sep 25, 2021 at 13:20
The probability is the area (integral) under the PMF or PDF function, the likelihood is the value of the PMF or PDF function about the parameters.
Bayesian inference uses data to make inferences about a parameter $$\theta$$, $$f(\theta | x^n) = \frac{f(x^n | \theta) f(\theta)}{\int f(x^n | \theta) f(\theta) d\theta}$$ here the $$f(\theta | x^n)$$ is the likelihood.
Let $$C = (a, b)$$, the conditional probability (or posterior interval) is an integral $$\mathbb{P}(\theta \in C | x^n) = \int_a^b f(\theta | x^n) d\theta$$
• This post does not appear to answer the question, which is about conditional probabilities in Bayesian inference.
– whuber
Sep 24, 2021 at 21:13
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# Find all complex numbers satisfying $z\cdot\bar{z}=41$, for which $|z-9|+|z-9i|$ has the minimum value
My first attempt was to express $z$ as $x+iy$ and minimize the expression $\sqrt{(x-9)^2+y^2}+\sqrt{x^2+(y-9)^2}$ where $x^2+y^2=41$.
That said, it seems to me that using the geometric interpretation could be easier. As far as I understand, I need to find points on the circle for which the sum of distances to the points $(9,0)$ and $(0,9)$ is lowest. This interpretation, however, doesn't help with regard to calculations.
Is there some simple trick or idea I'm missing?
Thank you!
• Geometrically speaking the least value would occur when an ellipse with $(9,0)$ and $(0,9)$ is tangent to the given circle. Symmetry suggests that one such point $\left(\sqrt{\frac{41}{2}}, \sqrt{\frac{41}{2}}\right)$. And since it is tangent at this point to the circle, this is the unique point. Jul 5, 2018 at 11:38
The locus of points with sum of distances $a$ from $(9,0)$ and $(0,9)$ is an ellipse. If we have $a=9\sqrt{2},$ we get a degenerate line segment between the 2 points, but as $a$ increases, the ellipse expands and then becomes tangent to the circle. Thus, you want to find the value of $a$ so that the ellipse with foci at $(9,0)$ and $(0,9)$ is tangent to the circle $x^2+y^2=41.$ Upon finding $a,$ the point of tangency is the desired $z.$
Having completed the interpretation, I leave the calculation to you.
• Sorry for being stupid, but how do I express the condition of ellipse being tangent to the circle in terms of algebra? Jul 5, 2018 at 9:56
Hint: Show that $$\sqrt{(8x-9)^2+y^2}+\sqrt{x^2+(y-9)^2}\geq 9\sqrt{2}$$ and the equal sign holds if $$x=4,y=5$$ ok we will prove the inequality above: squaring all we get
$$2\sqrt{(8x-9)^2+y^2}\sqrt{(x^2+(y-9)^2}\geq 162-(8x-9)^2-(y-9)^2-x^2-y^2$$, now we use that $x^2+y^2=41$: we get
$$2\sqrt{(8x-9)^2+41-x^2}\sqrt{41-y^2+(y-9)^2}\geq 121-(8x-9)^2-(y-9)^2$$
squaring again and factorizing we get
$$- \left( 64\,{x}^{2}+{y}^{2}-144\,x-18\,y+41 \right) \left( 64\,{x}^{ 2}+{y}^{2}-144\,x-18\,y+42 \right) \geq 0$$ which is true.
• Can you please elaborate algebraic steps? Jul 5, 2018 at 10:38
Apply the triangle inequality to the triangle with vertices $$z, 9, 9i$$:
$$|z-9|+|z-9i|=|-(z-9)|+|z-9i| \ge |9-9i|$$
with equality occurring only when $$z$$ lies on the line segment between the fixed points $$9, 9i$$. If this line segment intersects the circle, then all such points of intersection forcibly minimize the left side of the above inequality.
You should get whole numbers for the minimizing real and imaginary parts.
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Opuscula Math. 33, no. 2 (2013), 223-235
http://dx.doi.org/10.7494/OpMath.2013.33.2.223
Opuscula Mathematica
# Some generalized method for constructing nonseparable compactly supported wavelets in L2(R2)
Wojciech Banaś
Abstract. In this paper we show some construction of nonseparable compactly supported bivariate wavelets. We deal with the dilation matrix $$A = \tiny{\left[\begin{matrix}0 & 2 \cr 1 & 0 \cr \end{matrix} \right]}$$ and some three-row coefficient mask; that is a scaling function satisfies a dilation equation with scaling coefficients which can be contained in the set $$\{c_{n}\}_{n \in\mathcal{S}},$$ where $$\mathcal{S}=S_{1} \times \{0,1,2\},$$ $$S_{1} \subset \mathbb{N},$$ $$\sharp S_{1} \lt \infty.$$
Keywords: compactly supported wavelet, compactly supported scaling function, multiresolution analysis, dilation matrix, orthonormality, accuracy.
Mathematics Subject Classification: 42C40.
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Cite this article as:
Wojciech Banaś, Some generalized method for constructing nonseparable compactly supported wavelets in L2(R2), Opuscula Math. 33, no. 2 (2013), 223-235, http://dx.doi.org/10.7494/OpMath.2013.33.2.223
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# Yet another derivation of the Born–Oppenheimer approximation
There are plenty of existing discussions of the Born–Oppenheimer approximation, but none that I've read so far are entirely satisfying. They tend to use confusing notation, conflate operators with their representations, gloss over some crucial aspects, and so on.
The following is my attempt at a succinct derivation of the approximation that touches on all the important details. Specifically, here are some of the questions that arose when I was first learning about this (and that I try to answer below):
• What is the exact nature of the parametric dependence of the fast wavefunctions $\chi_k(x; X)$ on the slow coordinate $X$? Do the fast wavefunctions form an orthonormal basis in some way?
• How does the wavefunction expansion $\sum_k \varphi_k(X) \chi_k(x; X)$ differ from a standard basis expansion? Is it a Schmidt decomposition?
• How can the kinetic energy operator on the slow space result in a derivative of the fast wavefunctions?
• Why do all the surfaces seem to have the same energy?
## Hilbert space
Consider a system with degrees of freedom that we will group into "slow" and "fast". They don't actually need to be slow and fast, but these are the labels we will use. The prototypical example is a molecule with slow nuclei and fast electrons. States of this system live in the tensor product Hilbert space $\mathcal{H} = \mathcal{H}^\mathrm{s} \otimes \mathcal{H}^\mathrm{f}.$
On the slow space, we have the (multivariate) continuous representation $| X \rangle$, and on the fast space we have $| x \rangle$; together, that's $| X \, x \rangle$. This doesn't have to be the position representation, but it almost certainly will be.
## Parameterized Hamiltonian
For the time being, we'll keep the Hamiltonian fairly general: $\hat{H} = \hat{K}^\mathrm{s} + \hat{K}^\mathrm{f} + \hat{V}.$ We have kinetic energies for the slow and fast degrees of freedom and a potential energy term that operates on the entire space. One requirement that we'll impose is that the potential energy operator must be diagonal in the continuous representation we've chosen: $\hat{V} | X \, x \rangle = V(X, x) | X \, x \rangle.$ This allows us to express the Hamiltonian as $\langle X \, x | \hat{H} = \langle X | \hat{K}^\mathrm{s} \otimes \langle x | + \langle X | \otimes \langle x | \hat{K}^\mathrm{f} + V(X, x) \langle X \, x |.$ This isn't quite in the position representation, since we haven't given the kinetic energy operators a form yet. If they looked like $\sum_i \frac{\partial^2}{\partial \bigstar_i^2},$ we could express the Hamiltonian properly in the continuous representation: $\langle X \, x | \hat{H} = \left( -\sum_i \frac{\partial^2}{\partial X_i^2} - \sum_i \frac{\partial^2}{\partial x_i^2} + V(X, x) \right) \langle X \, x |.$ We'll come back to this form later, so you should keep it in mind, but we'll stick to being more general for now.
We define a parametrized potential operator with the following eigenvalue equation: $\hat{V}^\mathrm{f}(X) | x \rangle = V(X, x) | x \rangle.$ Using this operator, we construct another Hamiltonian, which we will call the fast Hamiltonian: $\hat{H}^\mathrm{f}(X) = \hat{K}^\mathrm{f} + \hat{V}^\mathrm{f}(X).$ The fast Hamiltonian is parameterized by $X$ and acts only on $\mathcal{H}^\mathrm{f}$. Conceptually, this is the Hamiltonian that describes the remaining (fast) system when we freeze out the slow degrees of freedom (by removing $\hat{K}^\mathrm{s}$) and pin them at a specific position $X$ (by parameterizing $\hat{V}$).
For every $X$, the operator $\hat{H}^\mathrm{f}(X)$ is a perfectly legitimate Hamiltonian for the fast system. That means that we could construct the Hamiltonian $\hat{K}^\mathrm{s} + \hat{H}^\mathrm{f}(X),$ but this is a useless object! It describes a complicated system in the fast degrees of freedom and a collection of free particles in the slow degrees of freedom; there is no coupling whatsoever between the two.
What we'll do instead is note that the above definitions allow us to write $\langle X \, x | \hat{H} = \langle X \, x | \left( \hat{K}^\mathrm{s} + \hat{H}^\mathrm{f}(X) \right).$ This may look like we've simply thrown a $\langle X \, x |$ onto the Hamiltonian that we've only just ridiculed, but there's a vital difference: the $X$ parameter of $\hat{H}^\mathrm{f}(X)$ depends on the $X$ value in the bra. This is what gives rise to the coupling between the slow and fast degrees of freedom, and it's at least a little weird to think about.
## Parameterized basis
The infinitely many fast Hamiltonians $\hat{H}^\mathrm{f}(X)$ give rise to infinitely many orthonormal bases for $\mathcal{H}^\mathrm{f}$. For any choice of $X$, the states $| k ; X \rangle$ satisfy the eigenvalue equation $\hat{H}^\mathrm{f}(X) | k ; X \rangle = E_k(X) | k ; X \rangle,$ where the kets are also parameterized by $X$. The wavefunctions for these states are commonly written as $\langle x | k ; X \rangle = \chi_k(x; X).$
To be perfectly clear, we have defined a basis $\{ | k ; X \rangle \}_k$ for each value of $X$. There is a basis $\{ | k ; X' \rangle \}_k$, and another basis $\{ | k ; X'' \rangle \}_k$, and so forth; there is nothing we can say in general about the overlap $\langle k' ; X' | k ; X \rangle$ when $X' \ne X$. Given a wavefunction $| \psi \rangle \in \mathcal{H}^\mathrm{f}$, we can expand it as $| \psi \rangle = \sum_k C^\psi_k(X) | k ; X \rangle,$ where the expansion coefficients are given by $C^\psi_k(X) = \langle k ; X | \psi \rangle,$ and $X$ is arbitrary.
Since we're not mathematicians, we can (and will) take continuity for granted. It's fairly safe to assume that the potential $V(X, x)$ varies continuously as $X$ is changed; after all, an arbitrarily large change in the potential when the configuration undergoes an infinitesimal shift would be unphysical. Hence, the Hamiltonian $\hat{H}^\mathrm{f}(X)$ and its eigenfunctions should also be continuous in the parameter $X$, as should the expansion coefficients $C^\psi_k(X)$ for any state.
One wrinkle that we do expect is that funny things can happen at degeneracies. The adiabatic theorem tells us that if we vary $X$ sufficiently slowly (compared to the gap between $E_k(X)$ and adjacent energies $E_{k'}(X)$), then the ordering of the eigenvalues remains the same and we can treat each $k$ as roughly independent. In that case, we have what looks like multiple hypersurfaces in $(X, E)$ space floating above one another like sheets. However, if the energies become equal, the gap between them vanishes, so these sheets touch and cease to be independent. In making the Born–Oppenheimer approximation, we'll be implicitly assuming that this won't happen, so we won't dwell on this.
The $| X \rangle$ representation is complete for $\mathcal{H}^\mathrm{s}$, and every $\{ | k ; X \rangle \}_k$ basis is complete for $\mathcal{H}^\mathrm{f}$. Thus, we are free to pick a specific $X'$ and use the states $| X \rangle \otimes | k ; X' \rangle$ as a basis for $\mathcal{H}$, but this would be silly, since $| k ; X' \rangle$ is not generally an eigenstate of $\hat{H}^\mathrm{f}(X)$ when $X \ne X'$. Instead, we want to use the states $| X \, k \rangle = | X \rangle \otimes | k ; X \rangle,$ where the same $X$ appears in both kets.
To see that $\{ | X \, k \rangle \}_{X,k}$ also forms a basis for $\mathcal{H}$ (technically some sort of half-basis, half-representation mutant), we show that the transformation matrix $U_{k' k}(X', X; \tilde{X}) = \left( \langle X' | \otimes \langle k' ; \tilde{X} | \right) | X \, k \rangle = \langle X' | X \rangle \langle k' ; \tilde{X} | k ; X \rangle$ is unitary for any fixed $\tilde{X}$. The requirements for this are \begin{aligned} & \int\! \mathrm{d}X \sum_k U_{k' k}(X', X; \tilde{X}) U^*_{k'' k}(X'', X; \tilde{X}) \\ &= \int\! \mathrm{d}X \, \langle X' | X \rangle \langle X | X'' \rangle \sum_k \langle k' ; \tilde{X} | k ; X \rangle \langle k ; X | k'' ; \tilde{X} \rangle \\ &= \int\! \mathrm{d}X \, \langle X' | X \rangle \langle X | X'' \rangle \langle k' ; \tilde{X} | k'' ; \tilde{X} \rangle \\ &= \langle X' | X'' \rangle \langle k' ; \tilde{X} | k'' ; \tilde{X} \rangle \\ &= \delta(X' - X'') \delta_{k' k''} \end{aligned} and \begin{aligned} & \int\! \mathrm{d}X' \sum_{k'} U^*_{k' k''}(X', X''; \tilde{X}) U_{k' k}(X', X; \tilde{X}) \\ &= \int\! \mathrm{d}X' \, \langle X'' | X' \rangle \langle X' | X \rangle \sum_{k'} \langle k'' ; X'' | k' ; \tilde{X} \rangle \langle k' ; \tilde{X} | k ; X \rangle \\ &= \int\! \mathrm{d}X' \, \langle X'' | X' \rangle \langle X' | X \rangle \langle k'' ; X'' | k ; X \rangle \\ &= \langle X'' | X \rangle \langle k'' ; X'' | k ; X \rangle \\ &= \delta(X'' - X) \delta_{k'' k}. \end{aligned} In the last step we used the sampling property of the Dirac delta function outside an integral, with the understanding that it only exists inside an integral anyway.
A consequence of the $| X \, k \rangle$ states forming a complete basis is that a state $| \Psi \rangle \in \mathcal{H}$ has the wavefunction $\langle X \, x | \Psi \rangle = \int\! \mathrm{d}X' \sum_k \langle X \, x | X' \, k \rangle \langle X' \, k | \Psi \rangle = \sum_k \langle x | k ; X \rangle \langle X \, k | \Psi \rangle.$ Alternatively, we could write this as $\Psi(X, x) = \langle X \, x | \Psi \rangle = \sum_k \varphi^\Psi_k(X) \chi_k(x; X),$ where $\varphi^\Psi_k(X) = \langle X \, k | \Psi \rangle.$ The last of these is a strange animal, simultaneously serving both the roles of a basis expansion coefficient and a wavefunction for the slow space.
Since it only has a single index, this expansion looks suspiciously like a Schmidt decomposition, but is it one? To express $| \Psi \rangle$ in Schmidt form, we would need to be able to write $\langle X \, x | \Psi \rangle = \sum_j \sqrt{\lambda_j} \langle X | \varphi_j \rangle \langle x | \chi_j \rangle,$ where $| \varphi_j \rangle$ are orthogonal states on $\mathcal{H}^\mathrm{s}$ and $| \chi_j \rangle$ are orthogonal states on $\mathcal{H}^\mathrm{f}$. While our $| k ; X \rangle$ are orthogonal for a fixed $X$, they have an explicit dependence on $X$, which is not allowed. Additionally, $\int\! \mathrm{d}X \, \varphi^{\Psi*}_{k'}(X) \varphi^\Psi_k(X) = \int\! \mathrm{d}X \, \langle \Psi | X \, k' \rangle \langle X \, k | \Psi \rangle \ne \delta_{k' k},$ so these functions aren't even orthogonal.
Conversely, we also have $\langle X \, k | \Psi \rangle = \int\! \mathrm{d}X' \int\! \mathrm{d}x \, \langle X \, k | X' \, x \rangle \langle X' \, x | \Psi \rangle = \int\! \mathrm{d}x \, \langle k ; X | x \rangle \langle X \, x | \Psi \rangle,$ or $\varphi^\Psi_k(X) = \langle X \, k | \Psi \rangle = \int\! \mathrm{d}x \, \chi_k^*(x; X) \Psi(X, x).$
Now that we believe that the states $| X \, k \rangle$ form a basis, we can try to find the matrix elements of the Hamiltonian: \begin{aligned} \langle X' \, k' | \hat{H} | X \, k \rangle &= \int\! \mathrm{d}x \int\! \mathrm{d}x' \, \langle k' ; X' | x' \rangle \langle X' \, x' | \hat{H} | X \, x \rangle \langle x | k ; X \rangle \\ &= \int\! \mathrm{d}x \int\! \mathrm{d}x' \, \chi_{k'}^*(x'; X') \langle X' \, x' | \hat{K}^\mathrm{s} | X \, x \rangle \chi_k(x; X) \\ &\qquad + \int\! \mathrm{d}x \int\! \mathrm{d}x' \, \chi_{k'}^*(x'; X') \langle X' \, x' | \hat{H}^\mathrm{f}(X') | X \, x \rangle \chi_k(x; X) \\ &= \langle X' | \hat{K}^\mathrm{s} | X \rangle \int\! \mathrm{d}x \, \chi_{k'}^*(x; X') \chi_k(x; X) \\ &\qquad + E_{k'}(X') \langle X' | X \rangle \int\! \mathrm{d}x \, \chi_{k'}^*(x; X') \chi_k(x; X) \\ &= \langle X' | \hat{K}^\mathrm{s} | X \rangle \langle k' ; X' | k ; X \rangle + E_{k'}(X') \delta(X' - X) \delta_{k' k}. \end{aligned} We have a complicated kinetic energy term, but a very diagonal potential energy term.
To proceed, we'll choose the form that was mentioned earlier for the kinetic energy: $\langle X | \hat{K}^\mathrm{s} = -\sum_i \frac{\hbar^2}{2 M_i} \frac{\partial^2}{\partial X_i^2} \langle X |.$ Then, it follows that \begin{aligned} \langle X' | \hat{K}^\mathrm{s} | X \rangle &= -\sum_i \frac{\hbar^2}{2 M_i} \frac{\partial^2}{\partial X_i^2} \langle X' | X \rangle \\ &= -\sum_i \frac{\hbar^2}{2 M_i} \delta^{(2)}_i(X' - X), \end{aligned} where $\delta^{(2)}_i(X' - X)$ is the second (distributional) derivative of the delta function in the $i$th direction, which satisfies $\int\! \mathrm{d}X' \, \delta^{(2)}_i(X - X') f(X') = \frac{\partial^2}{\partial X_i^2} f(X).$
Thus, we can apply the Hamiltonian to a generic state $| \Psi \rangle$ as follows: \begin{aligned} & \langle X \, k | \hat{H} | \Psi \rangle \\ &= \int\! \mathrm{d}X' \sum_{k'} \langle X \, k | \hat{H} | X' \, k' \rangle \langle X' \, k' | \Psi \rangle \\ &= \int\! \mathrm{d}X' \sum_{k'} \langle X | \hat{K}^\mathrm{s} | X' \rangle \langle k ; X | k' ; X' \rangle \langle X' \, k' | \Psi \rangle \\ &\qquad + \int\! \mathrm{d}X' \sum_{k'} E_k(X) \delta(X - X') \delta_{k k'} \langle X' \, k' | \Psi \rangle \\ &= \int\! \mathrm{d}x \, \chi_k^*(x; X) \sum_{k'} \int\! \mathrm{d}X' \langle X | \hat{K}^\mathrm{s} | X' \rangle \chi_{k'}(x; X') \varphi^\Psi_{k'}(X') \\ &\qquad + E_k(X) \varphi^\Psi_k(X) \\ &= -\int\! \mathrm{d}x \, \chi_k^*(x; X) \sum_{k'} \sum_i \frac{\hbar^2}{2 M_i} \int\! \mathrm{d}X' \, \delta^{(2)}_i(X - X') \chi_{k'}(x; X') \varphi^\Psi_{k'}(X') \\ &\qquad + E_k(X) \varphi^\Psi_k(X) \\ &= -\int\! \mathrm{d}x \, \chi_k^*(x; X) \sum_{k'} \sum_i \frac{\hbar^2}{2 M_i} \frac{\partial^2}{\partial X_i^2} \chi_{k'}(x; X) \varphi^\Psi_{k'}(X) \\ &\qquad + E_k(X) \varphi^\Psi_k(X) \\ &= -\int\! \mathrm{d}x \, \chi_k^*(x; X) \sum_{k'} \sum_i \frac{\hbar^2}{2 M_i} \left[ \frac{\partial^2}{\partial X_i^2} \chi_{k'}(x; X) \right] \varphi^\Psi_{k'}(X) \\ &\qquad - \int\! \mathrm{d}x \, \chi_k^*(x; X) \sum_{k'} \sum_i \frac{\hbar^2}{M_i} \left[ \frac{\partial}{\partial X_i} \chi_{k'}(x; X) \right] \left[ \frac{\partial}{\partial X_i} \varphi^\Psi_{k'}(X) \right] \\ &\qquad - \int\! \mathrm{d}x \, \chi_k^*(x; X) \sum_{k'} \sum_i \frac{\hbar^2}{2 M_i} \chi_{k'}(x; X) \left[ \frac{\partial^2}{\partial X_i^2} \varphi^\Psi_{k'}(X) \right] \\ &\qquad + E_k(X) \varphi^\Psi_k(X) \\ &= -\sum_{k'} \sum_i \frac{\hbar^2}{2 M_i} \left[ \int\! \mathrm{d}x \, \chi_k^*(x; X) \frac{\partial^2}{\partial X_i^2} \chi_{k'}(x; X) \right] \varphi^\Psi_{k'}(X) \\ &\qquad - \sum_{k'} \sum_i \frac{\hbar^2}{M_i} \left[ \int\! \mathrm{d}x \, \chi_k^*(x; X) \frac{\partial}{\partial X_i} \chi_{k'}(x; X) \right] \left[ \frac{\partial}{\partial X_i} \varphi^\Psi_{k'}(X) \right] \\ &\qquad - \sum_{k'} \sum_i \frac{\hbar^2}{2 M_i} \left[ \int\! \mathrm{d}x \, \chi_k^*(x; X) \chi_{k'}(x; X) \right] \left[ \frac{\partial^2}{\partial X_i^2} \varphi^\Psi_{k'}(X) \right] \\ &\qquad + E_k(X) \varphi^\Psi_k(X) \\ &= -\sum_{k'} \sum_i \frac{\hbar^2}{2 M_i} \left[ \int\! \mathrm{d}x \, \chi_k^*(x; X) \frac{\partial^2}{\partial X_i^2} \chi_{k'}(x; X) \right] \varphi^\Psi_{k'}(X) \\ &\qquad - \sum_{k'} \sum_i \frac{\hbar^2}{M_i} \left[ \int\! \mathrm{d}x \, \chi_k^*(x; X) \frac{\partial}{\partial X_i} \chi_{k'}(x; X) \right] \left[ \frac{\partial}{\partial X_i} \varphi^\Psi_{k'}(X) \right] \\ &\qquad - \sum_i \frac{\hbar^2}{2 M_i} \left[ \frac{\partial^2}{\partial X_i^2} \varphi^\Psi_k(X) \right] \\ &\qquad + E_k(X) \varphi^\Psi_k(X). \end{aligned} We have used the product rule, which states that \begin{aligned} & \frac{\partial}{\partial X_i} \chi_{k'}(x; X) \varphi^\Psi_{k'}(X) \\ &= \left[ \frac{\partial}{\partial X_i} \chi_{k'}(x; X) \right] \varphi^\Psi_{k'}(X) \\ &\qquad + \chi_{k'}(x; X) \left[ \frac{\partial}{\partial X_i} \varphi^\Psi_{k'}(X) \right] \end{aligned} and consequently \begin{aligned} & \frac{\partial^2}{\partial X_i^2} \chi_{k'}(x; X) \varphi^\Psi_{k'}(X) \\ &= \left[ \frac{\partial^2}{\partial X_i^2} \chi_{k'}(x; X) \right] \varphi^\Psi_{k'}(X) \\ &\qquad + 2 \left[ \frac{\partial}{\partial X_i} \chi_{k'}(x; X) \right] \left[ \frac{\partial}{\partial X_i} \varphi^\Psi_{k'}(X) \right] \\ &\qquad + \chi_{k'}(x; X) \left[ \frac{\partial^2}{\partial X_i^2} \varphi^\Psi_{k'}(X) \right]. \end{aligned} More pertinently, we have used the continuously-varying parametric dependence of $| k ; X \rangle$ on $X$ to allow the kinetic energy operator to take its derivative remotely through the derivative of the delta function.
For convenience, we use the gradient vector $\mathbf{\nabla}$ with elements $\nabla_i = \frac{\partial}{\partial X_i},$ so that $\nabla^2 = \mathbf{\nabla} \cdot \mathbf{\nabla} = \sum_i \frac{\partial^2}{\partial^2 X_i},$ and we drop the unitful quantities to make the expressions below look clean. If this makes you feel dirty, don't hesitate to pencil them in where appropriate.
With this in mind, we can write $\langle X \, k | \hat{H} | \Psi \rangle = \sum_{k'} \left[ \left( -\nabla^2 + E_k(X) \right) \delta_{k k'} - 2 \tau^{(1)}_{k k'}(X) \cdot \mathbf{\nabla} - \tau^{(2)}_{k k'}(X) \right] \varphi^\Psi_{k'}(X)$ where $\tau^{(1)}_{k k'}(X) = \int\! \mathrm{d}x \, \chi_k^*(x; X) \mathbf{\nabla} \chi_{k'}(x; X)$ and $\tau^{(2)}_{k k'}(X) = \int\! \mathrm{d}x \, \chi_k^*(x; X) \nabla^2 \chi_{k'}(x; X)$ are the non-adiabatic couplings and the terms containing them are the non-adiabatic coupling terms (NACTs).
Because the derivative operator is antihermitian, we find that $\tau^{(1)}_{k k'}(X) = -\tau^{(1) *}_{k' k}(X)$, so $\tau^{(1)}_{k k',i}(X)$ is a skew-Hermitian matrix (in $k$ and $k'$). A consequence of this is that all its diagonal terms vanish: $\tau^{(1)}_{k k}(X) = 0$. Because the second derivative operator is Hermitian, we also find that $\tau^{(2)}_{k k'}(X) = \tau^{(2) *}_{k' k}(X)$, so $\tau^{(2)}_{k k'}(X)$ is a Hermitian matrix. Hence, all its diagonal terms $\tau^{(2)}_{k k}(X)$ are real.
Note how the terms in the big square brackets smell like a Hamiltonian for the slow degrees of freedom, parameterized by $k$ and $k'$, and expressed in the position representation. If we define the matrix $\hat{\mathbf{H}}$ with elements $\hat{H}_{k k'}$ that have the position representation $\langle X | \hat{H}_{k k'} = \left[ \left( -\nabla^2 + E_k(X) \right) \delta_{k k'} - 2 \tau^{(1)}_{k k'}(X) \cdot \mathbf{\nabla} - \tau^{(2)}_{k k'}(X) \right] \langle X |,$ then the overall Hamiltonian $\hat{H}$ looks like a matrix of Hamiltonians for the slow degrees of freedom, indexed by the surfaces. On the diagonal, we have simply $-\nabla^2 + E_k(X) - \tau^{(2)}_{k k}(X),$ where the last two terms are plain old potentials. On the off-diagonal, we instead have $-2 \tau^{(1)}_{k k'}(X) \cdot \mathbf{\nabla} - \tau^{(2)}_{k k'}(X),$ which is a bit strange, because instead of a second derivative, it has first derivatives.
Nevertheless, this has the effect of turning the single Schrödinger equation $\hat{H} | n \rangle = E_n | n \rangle$ into a collection of coupled differential equations, indexed by $k$: $\sum_{k'} D_{k k'} \varphi^n_{k'}(X) = E_n \varphi^n_k(X),$ where we have given a name to the differential operator $D_{k k'} = \left( -\nabla^2 + E_k(X) \right) \delta_{k k'} - 2 \tau^{(1)}_{k k'}(X) \cdot \mathbf{\nabla} - \tau^{(2)}_{k k'}(X)$ as a shorthand. In the matrix picture, this looks like $\begin{pmatrix} D_{1,1} & D_{1,2} & \cdots \\ D_{2,1} & D_{2,2} & \cdots \\ \vdots & \vdots & \ddots \end{pmatrix} \begin{pmatrix} \varphi^n_1(X) \\ \varphi^n_2(X) \\ \vdots \end{pmatrix} = E_n \begin{pmatrix} \varphi^n_1(X) \\ \varphi^n_2(X) \\ \vdots \end{pmatrix}.$ If one is able to find the functions $\varphi^n_k(X)$ that simultaneously satisfy these equations, one can then assemble the eigenfunction $\langle X \, x | n \rangle = \sum_k \varphi^n_k(X) \chi_k(x; X)$ of the full Hamiltonian $\hat{H}$.
Before we continue, a few brief words about the Hamiltonians $\hat{H}_{k k'}$. It is tempting to say that these are partial matrix elements of $\hat{H}$ in the $| k ; X \rangle$ basis, but that direction is full of potential pitfalls. For starters, which basis do we mean? After all, there is a different one for each $X$, and no matter which one we pick, it would be a mistake to claim that $\langle k ; X | \hat{H} | k' ; X \rangle$ is the object of interest, since its position representation $\langle X' | \langle k ; X | \hat{H} | k' ; X \rangle$ is not useful for us. We could also try \begin{aligned} \langle X \, k | \hat{H} | k' ; X \rangle &= \int\! \mathrm{d}x \, \langle k ; X | x \rangle \langle X \, x | \hat{H} | k' ; X \rangle \\ &= \int\! \mathrm{d}x \, \langle k ; X | x \rangle \langle X \, x | \hat{K}^\mathrm{s} | k' ; X \rangle \\ &\qquad + \int\! \mathrm{d}x \, \langle k ; X | x \rangle \langle X \, x | \hat{H}^\mathrm{f}(X) | k' ; X \rangle \\ &= \int\! \mathrm{d}x \, \langle k ; X | x \rangle \langle x | k' ; X \rangle \langle X | \hat{K}^\mathrm{s} \\ &\qquad + \int\! \mathrm{d}x \, \langle k ; X | x \rangle \langle x | k' ; X \rangle E_{k'}(X) \langle X | \\ &= \langle X | \left( \hat{K}^\mathrm{s} + E_{k'}(X) \right) \delta_{k k'}, \end{aligned} which is definitely not what we wanted. No, this sort of thinking just will not do.
## Born–Oppenheimer approximation
Now that we have the Hamiltonian in the adiabatic representation, all that remains is to assume that the non-adiabatic couplings are sufficiently small that neglecting the NACTs entirely is a good approximation. This leaves us with a Hamiltonian that is diagonal in surfaces: $\langle X | \hat{H}_{k k'} = \left( -\nabla^2 + E_k(X) \right) \delta_{k k'} \langle X |.$ In other words, each $\hat{H}_{k k}$ is the complete Hamiltonian for the slow degrees of freedom on surface $k$.
It is then clear that the resulting collection of differential equations is completely uncoupled: $\left( -\nabla^2 + E_k(X) \right) \varphi^n_k(X) = E_n \varphi^n_k(X),$ and they may be solved independently. In the matrix picture, that's $\begin{pmatrix} D_{1,1} & 0 & \cdots \\ 0 & D_{2,2} & \cdots \\ \vdots & \vdots & \ddots \end{pmatrix} \begin{pmatrix} \varphi^n_1(X) \\ \varphi^n_2(X) \\ \vdots \end{pmatrix} = E_n \begin{pmatrix} \varphi^n_1(X) \\ \varphi^n_2(X) \\ \vdots \end{pmatrix}.$
It's somewhat peculiar that all of these seemingly independent Hamiltonians share the same eigenspectrum! It's easy to show that $\tau^{(1)}_{k k',i}(X) = 0$ implies that $\nabla_i \chi_k(x; X) = 0$. The integral $\int\! \mathrm{d}x \, \chi_k^*(x; X) \nabla_i \chi_{k'}(x; X)$ is the overlap between the familiar state $\chi_k^*(x; X)$ and the weird object $\nabla_i \chi_{k'}(x; X)$. Because the $\chi_k(x; X)$ form a complete basis, having $\nabla_i \chi_{k'}(x; X)$ be orthogonal to all $\chi_k(x; X)$ implies that $\nabla_i \chi_{k'}(x; X)$ is the zero element. Hence, the $X_i$ derivative of $\chi_k(x; X)$ is zero; if this is true for all $i$, $\chi_k(x; X)$ doesn't depend on $X$, so we can write just $\chi_k(x)$.
Now we can quickly show in two related ways that the above conclusion isn't a figment of our imagination: \begin{aligned} \langle X \, k | \hat{H} | n \rangle &= -\int\! \mathrm{d}x \, \chi_k^*(x) \sum_{k'} \sum_i \int\! \mathrm{d}X' \, \delta^{(2)}_i(X - X') \chi_{k'}(x) \varphi^n_{k'}(X') \\ &\qquad + E_k(X) \varphi^n_k(X) \\ &= \left( -\nabla^2 + E_k(X) \right) \varphi^n_k(X) \\ &= E_n \langle X \, k | n \rangle \end{aligned} and \begin{aligned} \langle X \, x | \hat{H} | n \rangle &= \int\! \mathrm{d}X' \sum_k \langle X \, x | \hat{H} | X' \, k \rangle \langle X' \, k | n \rangle \\ &= \sum_k \int\! \mathrm{d}X' \, \langle X \, x | \hat{K}^\mathrm{s} | X' \, k \rangle \langle X' \, k | n \rangle \\ &\qquad + \sum_k \int\! \mathrm{d}X' \, \langle X \, x | \hat{H}^\mathrm{f}(X) | X' \, k \rangle \langle X' \, k | n \rangle \\ &= \sum_k \int\! \mathrm{d}X' \, \langle X | \hat{K}^\mathrm{s} | X' \rangle \langle x | k \rangle \langle X' \, k | n \rangle \\ &\qquad + \sum_k E_k(X) \int\! \mathrm{d}X' \, \langle X | X' \rangle \langle x | k \rangle \langle X' \, k | n \rangle \\ &= -\sum_k \sum_i \int\! \mathrm{d}X' \, \delta^{(2)}_i(X - X') \langle x | k \rangle \varphi^n_k(X') \\ &\qquad + \sum_k E_k(X) \langle x | k \rangle \varphi^n_k(X) \\ &= \sum_k \left( -\nabla^2 + E_k(X) \right) \varphi^n_k(X) \chi_k(x) \\ &= E_n \sum_k \varphi^n_k(X) \chi_k(x) \\ &= E_n \langle X \, x | n \rangle. \end{aligned} In fact, because all the $\varphi^n_k(X)$ are degenerate, any linear combination $\sum_k \beta_k \varphi^n_k(X) \chi_k(x)$ is also an eigenstate of $\hat{H}$, including those that only include a single term.
(I think that the above implies that $\mathbf{\nabla} \hat{H}^\mathrm{f}(X) = 0$, so $\mathbf{\nabla} E_k(X) = 0$. Proving this or giving a counterexample is left as an exercise for the reader.)
However, in practice the NACTs don't disappear by themselves. If that were the case, the word "approximation" wouldn't appear on this page. Instead, we create a new Hamiltonian $\hat{H}'$ which has the same diagonal elements as $\hat{H}$, but with the couplings artificially set to zero. In this more realistic scenario, it's not the case that all the surfaces are identical. Still, because they are not coupled in this approximation, they may be dealt with independently.
When the non-adiabatic couplings are sufficiently strong that they can't be neglected, more sophisticated methods are necessary to treat more than one surface at a time; for example, switching to a diabatic representation. This is commonly termed going "beyond the Born–Oppenheimer approximation", and is beyond the scope of the present work.
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# red: color
Scientific graphical abstracts — design guidelines
# things on the side
visualization + design
If you are interested in color, explore my other color tools, Brewer palettes resources, color blindness palettes and math and an exhausting list of 10,000 color names for all those times you couldn't distinguish between tan hide, sea buckthorn, orange peel, west side, sunshade, california and pizzaz.
# Color choices and transformations for deuteranopia and other afflictions
Here, I help you understand color blindness and describe a process by which you can make good color choices when designing for accessibility. You can also delve into the mathematics behind the color blindness simulations.
Different color blindness simulations don't all agree on the luminance of the simulated color. See methods for details.
## color blindness RGB transformations
The transformations described here will allow you to simulate color blindness and apply conversions of the kind shown in the images below to your own work.
An HSV Granger rainbow. Colors progress in hue along the horizontal dimension. The top half of the image has $V=100$ with saturation progression $S=0-100$. The bottom half has $V=100-0$ and $S=100$. When created at 360 × 200 pixels, each pixel takes on unique integer value of H, S and V. This rainbow is based on a perceptually non-uniform space and this accounts for the bands of brightness across the center and in some columns. (zoom)
An HSV Granger rainbow transformed for protanopia, the second most common type of red-green color blindness in which the long wavelength color receptors are either missing or defective (protanomaly). (zoom)
An HSV Granger rainbow transformed for deuternaopia, the most common type of red-green color blindness in which the medium wavelength color receptors are either missing or defective (protanomaly). (zoom)
An HSV Granger rainbow transformed for tritanopia, a rare type of red-blue color blindness in which the short wavelength color receptors are either missing or defective (tritanomaly). (zoom)
An HSV Granger rainbow transformed for achromatopsia, a very rare type of color blindness in which all color receptors are either missing (as in rod monochromats) or defective (dyschromatopsia) or only one kind of cone is functioning (e.g. red monochromats, blue monochromats, etc). (zoom)
The conversion from RGB values to their color blindness equivalents for protanopia, deuteranopia and tritanopia consists of the following steps
1. convert from sRGB to linear RGB
2. convert from linear RGB to XYZ
3. convert from XYZ to LMS
4. apply color blindness transformation in LMS space
5. convert from LMS to XYZ (inverse of #3)
6. convert from XYZ to linear RGB (inverse of #2)
7. convert from linear RGB to sRGB (clip and inverse of #1)
The steps of simulating color blindness by transforming an sRGB color $(R,G,B)$ to its color blind equivalent $(R',G',B')$. For each of the color blindness types (protanopia, deuteranopia, tritanopia) all the transformation matrices can be combined into a single matrix $\mathbf{T}$. (zoom)
The greyscale conversion for achromatopsia does not require the XYZ and LMS steps. We can go straight to $Y$.
1. convert from sRGB to linear RGB
2. $Y = 0.02126r + 0.7152g + 0.0722b$
3. convert from linear RGB to sRGB (inverse of #1)
The details of each of the steps are shown below. If you just want the $\mathbf{T}$ matrices, scroll down to the bottom or download the code.
Depending on the implementation, this process may have an additional step that reduces the color domain (e.g. step 2 in Viénot, Brettel & Mollon, 1999). This makes sure that none of the transformed colors are outside of the sRGB domain. Here, I instead of doing this I just clip the transformed colors at the end. This simplifies the process and (my sense is that) the difference is negligible.
Different simulators will yield slightly different results, too.
### sRGB to linear RGB
The RGB input color is $\{R,G,B\} = \{ V \in [0,255] \}$ and assumed to be sRGB. This is first linearized with $\gamma = 2.4$ to obtain $\{r,g,b\} = \{ v \in [0,1] \}$. $$v = \begin{cases} \dfrac{V/255}{12.92} & \text{if V/255 \le 0.04045} \\[2ex] \left({\dfrac{V/255+0.055}{1.055}}\right)^\gamma & \text{otherwise} \end{cases}$$
### linear RGB to XYZ
Next, convert the linearized $(r,g,b)$ to XYZ by multiplying by $\mathbf{M}_\textrm{XYZ}$. $$\left[ \begin{matrix} X \\Y \\Z \end{matrix} \right] = \underbrace{ \left[ \begin{matrix} 0.4124564 & 0.3575761 & 0.1804375 \\0.2126729 & 0.7151522 & 0.0721750 \\0.0193339 & 0.1191920 & 0.9503041 \end{matrix} \right] }_{\mathbf{M}_\textrm{XYZ}} \left[ \begin{matrix} r \\g \\b \end{matrix} \right]$$
### XYZ to LMS
Multiply by $\mathbf{M}_\textrm{LMS,D65}$ to convert XYZ to LMS. This matrix is normalized to the D65 illuminant, which will ensure that greys will be preserved.
The LMS (long, medium, short) is a color space which represents the response of the three types of cones of the human eye, named for their sensitivity peaks at long (560 nm, red) medium (530 nm, green), and short (420 nm, blue) wavelengths. $$\left[ \begin{matrix} L \\ M \\ S \end{matrix} \right] = \underbrace{ \left[ \begin{matrix} 0.4002 & 0.7076 & -0.0808 \\ -0.2263 & 1.1653 & 0.0457 \\ 0 & 0 & 0.9182 \end{matrix} \right] }_{\mathbf{M}_\textrm{LMS,D65}} \left[ \begin{matrix} X \\ Y \\ Z \end{matrix} \right]$$
If for some reason you don't want to normalize to D65, you would use $\mathbf{M}_\textrm{LMS}$ below but whites will now appear pinkish. $$\left[ \begin{matrix} L \\ M \\ S \end{matrix} \right] = \underbrace{ \left[ \begin{matrix} 0.0.38971 & 0.68898 & -0.07868 \\ -0.22981 & 1.18340 & 0.04641 \\ 0 & 0 & 1 \end{matrix} \right] }_{\mathbf{M}_\textrm{LMS}} \left[ \begin{matrix} X \\ Y \\ Z \end{matrix} \right]$$
There are other XYZ to LMS matrices and the R code lists them all.
### apply daltonism correction
Now that we have the RGB color represented in LMS space, we can correct for color receptor dysfunction in this space, since color blindness affects one of the L, M or S receptors.
I show the calculations here in a lot of detail—they're not as complicated as things look. Once you understand what's happening for one of the color blindness types, the other two are analogously treated.
Each of the color blindness types will have its own correction matrix $\mathbf{S}$. $$\left[ \begin{matrix} L' \\ M' \\ S' \end{matrix} \right] = \mathbf{S} \left[ \begin{matrix} L \\ M \\ S \\ \end{matrix} \right]$$
This matrix is the identity matrix with the row for the malfunctioning receptor (e.g. S for protanopia) replaced by two free parameters $a$ and $b$. \begin{align} \mathbf{S}_\textrm{protanopia} & = \left[ \begin{matrix} 0 & a & b \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} \right] \\ \mathbf{S}_\textrm{deuteranopia} & = \left[ \begin{matrix} 1 & 0 & 0 \\ a & 0 & b \\ 0 & 0 & 1 \end{matrix} \right] \\ \mathbf{S}_\textrm{tritanopia} & = \left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ a & b & 0 \end{matrix} \right] \end{align}
The reason why these matrices have this format is so that we can satisfy two conditions. First, this matrix should not affect how white appears—if one of the rows was just zero then white would be altered. Second, we expect that one of the primaries won't be affected, depending on the color blindness type. For protanopia and deuteranopia this is blue and for tritanopia this is red.
If we set $\mathbf{M} = \mathbf{M}_\textrm{LMS,D65} \mathbf{M}_\textrm{XYZ}$ then these conditions (using the blue case for protanopia) can be expressed as $$\mathbf{S}_\textrm{protanopia} \mathbf{M} \left[ \begin{matrix} 1 \\ 1 \\ 1 \end{matrix} \right] = \mathbf{M} \left[ \begin{matrix} 1 \\ 1 \\ 1 \end{matrix} \right] = \left[ \begin{matrix} L_0 \\ M_0 \\ S_0 \end{matrix} \right]$$ $$\mathbf{S}_\textrm{protanopia} \mathbf{M} \left[ \begin{matrix} 0 \\ 0 \\ 1 \end{matrix} \right] = \mathbf{M} \left[ \begin{matrix} 0 \\ 0 \\ 1 \end{matrix} \right] = \left[ \begin{matrix} L \\ M \\ S \end{matrix} \right]$$
where $(L_0,M_0,S_0)$ are the LMS coordinates of white and $(L_b,M_b,S_b)$ of the primary that is not affected (e.g. blue). Using the form for $\mathbf{S}_\textrm{protanopia}$, these lead to the following equations \begin{align} a M_b + b S_b & = L_b \\a M_0 + b S_0 & = L_0 \end{align}
which can be written as $$\left[ \begin{matrix} a \\b \end{matrix} \right]_\textrm{protanopia} = { \left[ \begin{matrix} M_b & S_b \\M_0 & S_0 \end{matrix} \right] }^{-1} \left[ \begin{matrix} L_b \\L_0 \end{matrix} \right]$$
For deuteranopia and tritanopia the calculation of $a$ and $b$ is analogous, except that because now $a$ and $b$ change position in $\mathbf{S}$, the equations are slightly different. $$\left[ \begin{matrix} a \\b \end{matrix} \right]_\textrm{deuteranopia} = { \left[ \begin{matrix} L_b & S_b \\L_0 & S_0 \end{matrix} \right] }^{-1} \left[ \begin{matrix} M_b \\M_0 \end{matrix} \right]$$
and for tritanopia (here $(L_r,M_r,S_r)$ refers to the coordinates of red). $$\left[ \begin{matrix} a \\b \end{matrix} \right]_\textrm{tritanopia} = { \left[ \begin{matrix} L_r & M_r \\L_0 & M_0 \end{matrix} \right] }^{-1} \left[ \begin{matrix} S_r \\S_0 \end{matrix} \right]$$
Using the following LMS coordinates (calculated by linearizing the corresponding RGB values and then muptiplying by $\mathbf{M}$) \begin{align} \left[ \begin{matrix} L_0 \\ M_0 \\ S_0 \end{matrix} \right] & = \left[ \begin{matrix} 1 \\ 0.999683 \\ 0.9997637 \end{matrix} \right] \\ \left[ \begin{matrix} L_b \\ M_b \\ S_b \end{matrix} \right] & = \left[ \begin{matrix} 0.04649755 \\ 0.08670142 \\ 0.87256922 \end{matrix} \right] \\ \left[ \begin{matrix} L_r \\ M_r \\ S_r \end{matrix} \right] & = \left[ \begin{matrix} 0.31399022 \\ 0.15537241 \\ 0.01775239 \end{matrix} \right] \end{align}
Using protanopia as the example \begin{align} \left[ \begin{matrix} a \\b \end{matrix} \right]_\textrm{protanopia} & = { \left[ \begin{matrix} 0.08670142 & 0.87256922 \\0.999683 & 0.9997637 \end{matrix} \right] }^{-1} \left[ \begin{matrix} 0.04649755 \\1 \end{matrix} \right] \\ & = \left[ \begin{matrix} -1.27219 & 1.1103359 \\1.27245 & -0.1103267 \end{matrix} \right] \left[ \begin{matrix} 0.04649755 \\1 \end{matrix} \right] \\ & = \left[ \begin{matrix} 1.05118294 \\-0.05116099 \end{matrix} \right] \end{align}
The $(a,b)$ calculations for deuteranopia and tritanopia are analogous and once they're done we can write the correction matrices as $$\left[ \begin{matrix} L' \\ M' \\ S' \end{matrix} \right] = \underbrace{ \left[ \begin{matrix} 0 & 1.05118294 & -0.05116099 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} \right] }_{\mathbf{S}_\textrm{protanopia}} \left[ \begin{matrix} L' \\ M' \\ S' \end{matrix} \right]$$ $$\left[ \begin{matrix} L' \\ M' \\ S' \end{matrix} \right] = \underbrace{ \left[ \begin{matrix} 1 & 0 & 0 \\ 0.9513092 & 0 & 0.04866992 \\ 0 & 0 & 1 \end{matrix} \right] }_{\mathbf{S}_\textrm{deuteranopia}} \left[ \begin{matrix} L' \\ M' \\ S' \end{matrix} \right]$$ $$\left[ \begin{matrix} L' \\ M' \\ S' \end{matrix} \right] = \underbrace{ \left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ -0.86744736 & 1.86727089 & 0 \end{matrix} \right] }_{\mathbf{S}_\textrm{tritanopia}} \left[ \begin{matrix} L' \\ M' \\ S' \end{matrix} \right]$$
These $(a,b)$ values are for the D65-normalized XYZ-to-LMS matrix and will change if you use a different matrix. The R code calculates $(a,b)$ for whatever matrix you provide.
### Corrected LMS to XYZ
Once the color blindness correction has been applied in LMS space, we convert back to XYZ using the inverse $\mathbf{M}_\text{LMS,D65}^{-1}$. $$\left[ \begin{matrix} X' \\ Y' \\ Z' \end{matrix} \right] = \underbrace{ \left[ \begin{matrix} 1.8600666 & -1.1294801 & 0.2198983 \\ 0.3612229 & 0.6388043 & 0 \\ 0 & 0 & 1.089087 \end{matrix} \right] }_{\mathbf{M}_\textrm{LMS,D65}^{-1}} \left[ \begin{matrix} L' \\ M' \\ S' \end{matrix} \right]$$
If you used $\mathbf{M}_\textrm{LMS}$ and didn't normalize to D65, then you'd use its inverse $\mathbf{M}_\text{LMS}^{-1}$ instead. $$\left[ \begin{matrix} X' \\ Y' \\ Z' \end{matrix} \right] = \underbrace{ \left[ \begin{matrix} 1.9101968 & -1.1121239 & 0.2019080 \\ 0.3709501 & 0.6290543 & 0 \\ 0 & 0 & 1 \end{matrix} \right] }_{\mathbf{M}_\textrm{LMS}^{-1}} \left[ \begin{matrix} L' \\ M' \\ S' \end{matrix} \right]$$
### XYZ to linear RGB
Finally, one last matrix multiplication from XYZ back to linear RGB using $\mathbf{M}_\textrm{XYZ}^{-1}$. $$\left[ \begin{matrix} r' \\ g' \\ b' \end{matrix} \right] = \underbrace{ \left[ \begin{matrix} 3.24045484 & -1.5371389 & -0.49853155 \\ -0.96926639 & 1.8760109 & 0.04155608 \\ 0.05564342 & -0.2040259 & 1.05722516 \end{matrix} \right] }_{\mathbf{M}_\textrm{XYZ}^{-1}} \left[ \begin{matrix} X' \\ Y' \\ Z' \end{matrix} \right]$$
### linear RGB to sRGB
The first step is now inverted to obtain the final sRGB values as perceived by someone with color blindness. $$V = \begin{cases} 255(12.92v) & \text{if v \le 0.0031308} \\[1ex] 255(1.055 v^{1/\gamma} - 0.055) & \text{otherwise} \end{cases}$$
Make sure to clip $v$ to $[0,1]$ before applying the final transformation back to sRGB.
### Conversion summary
The matrix multiplication steps can be written compactly as $$\left[ \begin{matrix} r' \\ g' \\ b' \end{matrix} \right] = \mathbf{M}_\textrm{XYZ}^{-1} \mathbf{M}_\textrm{LMS,D65}^{-1} \mathbf{S} \mathbf{M}_\textrm{LMS,D65} \mathbf{M}_\textrm{XYZ} \left[ \begin{matrix} r \\ g \\ b \end{matrix} \right] = \mathbf{T} \left[ \begin{matrix} r \\ g \\ b \end{matrix} \right]$$
where $\mathbf{T}$ is the product of all the matrices for a given color blindness correction $\mathbf{S}$, which are \begin{align} \mathbf{T}_\textrm{protanopia} & = \left[ \begin{matrix} 0.170556992 & 0.829443014 & 0 \\ 0.170556991 & 0.829443008 & 0 \\ -0.004517144 & 0.004517144 & 1 \end{matrix} \right] \\ \mathbf{T}_\textrm{deuteranopia} & = \left[ \begin{matrix} 0.33066007 & 0.66933993 & 0 \\ 0.33066007 & 0.66933993 & 0 \\ -0.02785538 & 0.02785538 & 1 \end{matrix} \right] \\ \mathbf{T}_\textrm{tritanopia} & = \left[ \begin{matrix} 1 & 0.1273989 & -0.1273989 \\ 0 & 0.8739093 & 0.1260907 \\ 0 & 0.8739093 & 0.1260907 \end{matrix} \right] \\ \mathbf{T}_\textrm{achromatopsia} & = \left[ \begin{matrix} 0.2126 & 0.7152 & 0.0722 \\ 0.2126 & 0.7152 & 0.0722 \\ 0.2126 & 0.7152 & 0.0722 \end{matrix} \right] \end{align}
# Graphical Abstract Design Guidelines
Fri 13-11-2020
Clear, concise, legible and compelling.
Making a scientific graphical abstract? Refer to my practical design guidelines and redesign examples to improve organization, design and clarity of your graphical abstracts.
Graphical Abstract Design Guidelines — Clear, concise, legible and compelling.
# "This data might give you a migrane"
Tue 06-10-2020
An in-depth look at my process of reacting to a bad figure — how I design a poster and tell data stories.
A poster of high BMI and obesity prevalence for 185 countries.
# He said, he said — a word analysis of the 2020 Presidential Debates
Thu 01-10-2020
Building on the method I used to analyze the 2008, 2012 and 2016 U.S. Presidential and Vice Presidential debates, I explore word usagein the 2020 Debates between Donald Trump and Joe Biden.
Analysis of word usage by parts of speech for Trump and Biden reveals insight into each candidate.
# Points of Significance celebrates 50th column
Mon 24-08-2020
We are celebrating the publication of our 50th column!
To all our coauthors — thank you and see you in the next column!
Nature Methods Points of Significance: Celebrating 50 columns of clear explanations of statistics. (read)
# Uncertainty and the management of epidemics
Mon 24-08-2020
When modelling epidemics, some uncertainties matter more than others.
Public health policy is always hampered by uncertainty. During a novel outbreak, nearly everything will be uncertain: the mode of transmission, the duration and population variability of latency, infection and protective immunity and, critically, whether the outbreak will fade out or turn into a major epidemic.
The uncertainty may be structural (which model?), parametric (what is $R_0$?), and/or operational (how well do masks work?).
This month, we continue our exploration of epidemiological models and look at how uncertainty affects forecasts of disease dynamics and optimization of intervention strategies.
Nature Methods Points of Significance column: Uncertainty and the management of epidemics. (read)
We show how the impact of the uncertainty on any choice in strategy can be expressed using the Expected Value of Perfect Information (EVPI), which is the potential improvement in outcomes that could be obtained if the uncertainty is resolved before making a decision on the intervention strategy. In other words, by how much could we potentially increase effectiveness of our choice (e.g. lowering total disease burden) if we knew which model best reflects reality?
This column has an interactive supplemental component (download code) that allows you to explore the impact of uncertainty in $R_0$ and immunity duration on timing and size of epidemic waves and the total burden of the outbreak and calculate EVPI for various outbreak models and scenarios.
Nature Methods Points of Significance column: Uncertainty and the management of epidemics. (Interactive supplemental materials)
Bjørnstad, O.N., Shea, K., Krzywinski, M. & Altman, N. (2020) Points of significance: Uncertainty and the management of epidemics. Nature Methods 17.
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# What is 4.0 divided by 0.05?
Feb 19, 2016
80
#### Explanation:
There are two ways to do this:
$\frac{4}{0.05}$
1) we can think of the $0.05$ as a fraction $\frac{5}{100}$. If we then write the fraction as $\frac{4}{\frac{5}{100}}$ we can use a property of fractions when they are in the denominator, which is, we flip them upside down and multiply by the numerator, making it look like this: $4 \cdot \frac{100}{5} = 4 \cdot 20 = 80$
2) we can multiply the top and bottom of that by 100 leaving us with $\frac{400}{5} = 80$
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Whenever there’s a revolution in computing, there’s a period of wild experimentation. Ideas are explored, prototypes are built, and manifestos are written.
The final outcome does not always achieve the great ambition of the first prototypes. Not everything works well in practice. Some features prove difficult to build, are punted to v2, and never get shipped.
The World Wide Web, with its HyperText Markup Language, was heavily influenced by the ideas of hypermedia. The idea of hyperlinks — cross-referencing documents — was radical. There were even court cases disputing the right to link to content.
The original concept of hypertext was much more ambitious. Early designs, such as the memex and Project Xanadu, had a richer feature set.
Here’s the web they envisaged.
Today’s hyperlinks are one-way: when you visit a webpage, you do not know what links point to it. You can only see what links start from it.
It’s useful information, enabling the reader to discover related pages. Indeed, the basic principle of Google’s PageRank is founded on crawling the web to find back links.
Bidirectional links are alive and well in Wikipedia, which provides a ‘What links here’ feature to discover related topics.
## Transclusion
Today’s web cannot include one page in another. You could manually scrape the HTML and paste it in, but then you miss out any future updates.
HTML does provide the <iframe> tag, but the embedded content does not flow into the original page. It’s clunky and rarely used in practice.
Twitter, with its many constraints, does provide a transclusion facility.
The above embedded tweet contains the current retweet and like counts, without me updating my blog post.
Twitter takes this idea further in its clients. If a tweet contains a hyperlink, it includes a preview – another form of transclusion!
Twitter calls this functionality Twitter Cards. For short content, such as other tweets, the preview includes the entire content.
## Unique IDs
Web pages are identified by URLs, which are usually unique. Despite W3C pleas not to break URLs, link rot is a very common problem.
Link rot can prevented by adding another level of indirection. PURLs (Persistent URLs) allow users to refer to an URL which they can update if the underlying URL changes.
For example, Keith Shafer (one of the original PURL developers), has a personal web page at purl.oclc.org/keith/home. This originally linked to www.oclc.org:5046/~shafer, which no longer resolves. However, the PURL has been updated, so visitors are now redirected to docs.google.com/document/d/1tnDck…/.
A common source of link rot is page renaming. Stack Overflow, a Q&A site, includes question titles in URLs. These are often edited to clarify the question. Stack Overflow embeds unique question IDs in its URLs to ensure old links work.
Thus http://stackoverflow.com/questions/37089768/does-pharo-provide-tail-call-optimisation is the canonical URL, but http://stackoverflow.com/questions/37089768/foobar still works.
URLs in today’s web can also reference named HTML tags. For example, http://wilfred.me.uk/blog/2016/06/12/hypermedia-how-the-www-fell-short/#unique-ids directly links to this section of this blog post. However, this requires co-operation from the author: they must provide named anchors, which cannot change in future.
A common UI pattern is provide discoverable links on headings in web pages. This assumes that headings never change or repeat in a document, but fits the common use case.
The other major limitation of section links is that they cannot reference a range of tags on a page. Genius (a website offering song lyric interpretations) is one of very few websites that allow users to link to arbitrary sections of a page.
## History
Finally, early hypermedia designs kept historical versions of content. You could link to old versions of content if desired.
We don’t have this in today’s web. There’s the Wayback Machine, which periodically snapshots many websites. For high-profile online news, NewsDiffs regularly snapshots stories to see how articles are edited over time.
This is another example where wikis come closer to the traditional idea of hypermedia. https://en.wikipedia.org/wiki/Hypertext links to the current version of the hypertext article on Wikipedia, whereas https://en.wikipedia.org/w/index.php?title=Hypertext&oldid=722248276 explicitly links to the version at time of writing, regardless of future changes.
## Looking Forward
Should we throw away today’s web and rebuild? Certainly not.
The web we have works incredibly well. Its feature set has enabled users to write billions of web pages. The technology is standardised and there are many mature implementations.
HTML is still a medium where some things are easy and some things are not. We should not lose sight of how HTML will affect how we communicate. Instead, we should pillage the ideas of the past to make the best use of our content today.
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Page 1 of 1
Dobiński’s formula
Posted: Sat Aug 12, 2017 4:29 pm
Let $n \in \mathbb{N}$ and $\mathcal{B}_n$ denote the $n$ - th Bell number. Prove that
$$\sum_{k=0}^{\infty} \frac{k^n}{k!}=\mathcal{B}_n \cdot e$$
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# Which reactions are more temperature sensitive: the ones with higher Ea or the ones with lower Ea? And why?
Which reactions are more temperature sensitive: the ones with higher $E_\mathrm{a}$ or the ones with lower $E_\mathrm{a}$? And why?
I wasn't able to find much useful content on googling it, however on doing the math I came to a conclusion that the ones with higher $E_\mathrm{a}$ are more sensitive to increase in temperature. But why is it so? I mean math proves it, but why exactly does it happen?
Here are my calculations:
• hint: see Maxwell velocity or kinetic energy distribution – Mrigank Apr 20 '16 at 21:10
The reaction with the lower activation energy can proceed smoothly without the increase in temperature whereas the the reaction with the higher activation energy cannot. When you increase the temperature, it won't make much difference to the reaction with a lower activation energy since it was already fine with a lower temperature. On the contrary, the reaction with higher activation energy will now gain sufficient energy making the molecules more likely to react and thus, the temperature change affects this reaction much more.
Plotting a graph between lnK vs 1/T shows that reaction having high activation energy are more temperature sensitive. Since the slope of high Ea is more steeper than the one having low Ea. This shows that rate constant hence rate of reaction will be high for large Ea
• Hi Amit Ujjwal, welcome to Chem.SE! Take the tour to get familiar with our site. Regular text can be formatted with Markdown, and mathematical expressions and chemical equations can be formatted using Latex syntax. That said, can you please properly photoscan your image, currently it's a little difficult to interpret. Thanks! – Gaurang Tandon Apr 4 '18 at 14:27
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# Summary of the Mechanical Performances of the 1.5 m Long Models of the Nb $_{3}$ Sn Low- $\beta$ Quadrupole MQXF
Authors
Publication Date
Aug 01, 2019
Source
eScholarship - University of California
Keywords
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My Math Forum linear Operator// scalar produkt
Linear Algebra Linear Algebra Math Forum
May 8th, 2014, 04:57 AM #1 Newbie Joined: Nov 2013 Posts: 26 Thanks: 0 linear Operator// scalar produkt Hi, Let V be a komplex Vector space, and T a linear Operator on V i have to Show: $\displaystyle =0 \forall u \in V \Rightarrow T=0$ In course we have just discussed Hermitian adjoint Hope someone could give me a tip for the proof....
May 8th, 2014, 05:00 AM #2 Newbie Joined: May 2014 From: delhi Posts: 1 Thanks: 0 Era business school review Era business school is the best option.
May 8th, 2014, 05:13 AM #3 Newbie Joined: Nov 2013 Posts: 26 Thanks: 0 maybe someone with a more constructive answer?
May 8th, 2014, 10:00 AM #4 Senior Member Joined: Dec 2013 From: Russia Posts: 327 Thanks: 108 Consider $\langle T(u+v),u+v\rangle$. Add to it a couple of terms of the form $\pm\langle Tw,w\rangle$ so that the result is $\langle Tu,v\rangle+\langle Tv,u\rangle$. Similarly, consider $-i\langle T(iu+v),iu+v\rangle$. Add to it a couple of terms of the form $\pm i\langle Tw,w\rangle$ so that the result is $\langle Tu,v\rangle-\langle Tv,u)\rangle$.
May 8th, 2014, 11:22 AM #5 Newbie Joined: Jan 2014 Posts: 6 Thanks: 0 But when i look at $\displaystyle ⟨T(u+v),u+v⟩=+++$ which is because of $\displaystyle =0 \forall u$ $\displaystyle =+$ where do i need a $\displaystyle +-$ ???
May 8th, 2014, 11:41 AM #6 Senior Member Joined: Dec 2013 From: Russia Posts: 327 Thanks: 108 That works too. The idea is to prove that $\langle Tu,v\rangle+\langle Tv,u\rangle=0$ and similarly for the difference. I was suggesting first expressing $\langle Tu,v\rangle+\langle Tv,u\rangle$ as a sum of terms of the form $\langle Tw,w\rangle$ that works always and only then using the fact that $\langle Tw,w\rangle=0$. Thanks from Sandra93
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# Dividend payout ratio
From Wikipedia, the free encyclopedia - View original article
Jump to: navigation, search
Dividend payout ratio is the fraction of net income a firm pays to its stockholders in dividends:
$\mbox{Dividend payout ratio}=\frac{\mbox{Dividends}}{\mbox{Net Income for the same period}}$
The part of the earnings not paid to investors is left for investment to provide for future earnings growth. Investors seeking high current income and limited capital growth prefer companies with high Dividend payout ratio. However investors seeking capital growth may prefer lower payout ratio because capital gains are taxed at a lower rate. High growth firms in early life generally have low or zero payout ratios. As they mature, they tend to return more of the earnings back to investors. Note that dividend payout ratio is calculated as DPS/EPS.
According to Financial Accounting by Walter T. Harrison, the calculation for the payout ratio is as follows:
Payout Ratio = (Dividends - Preferred Stock Dividends)/Net Income
The dividend yield is given by earnings yield times DPR: $\begin{array}{lcl} \mbox{Current Dividend Yield} & = & \frac{\mbox{Most Recent Full-Year Dividend}}{\mbox{Current Share Price}} \\ & = & \frac{\mbox{Dividend payout ratio}\times \mbox{Most Recent Full-Year earnings per share}}{\mbox{Current Share Price}} \\ \end{array}$
Conversely, the P/E ratio is the Price/Dividend ratio times the DPR.
## Impact of buybacks
Some companies chose stock buybacks as an alternative to dividends, in such cases this ratio becomes less meaningful. One way to adapt it using an augmented payout ratio:[1]
Augmented Payout Ratio = (Dividends + Buybacks)/ Net Income for the same period
## Historic Data
The data for S&P 500 is taken from[2]. The payout rate has gradually declined from 90% of operating earnings in 1940s to about 30% in recent years.
DecadePrice %
Change
Dividend
Contribution
Total
Return
Dividends as %
of Total Return
Average
Payout
1930s-41.90%56.00%14.10%N/A90.10%
1940s34.8100.3135.174.20%59.4
1950s256.7180436.741.254.6
1960s53.754.2107.950.256
1970s17.259.176.377.545.5
1980s227.4143.1370.538.648.6
1990s315.795.5411.223.247.6
2000s-158.6-6.4N/A32.3
Average106.10%87.10%193.20%50.80%54.30%
For smaller, growth companies, the average payout ratio can be as low as 10%.[3]
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# Homework Help: Damping constant
1. May 1, 2006
### mb85
The suspension system of a 1700 kg automobile "sags" 13 cm when the chassis is placed on it. Also, the oscillation amplitude decreases by 62% each cycle. Estimate the values of (a) the spring constant k and (b) the damping constant b for the spring and shock absorber system of one wheel, assuming each wheel supports 425 kg.
Kx = mg
K = 1281.5 N/m
But the damping constant i am having trouble with.
W' = Square root (K/m - b^2/4m^2) is the formula i think i use?
= kg/s
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# Maharashtra Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.2
Balbharati Maharashtra State Board Std 12 Commerce Statistics Part 1 Digest Pdf Chapter 8 Differential Equation and Applications Ex 8.2 Questions and Answers.
## Maharashtra State Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.2
Question 1.
Obtain the differential equation by eliminating arbitrary constants from the following equations:
(i) y = Ae3x + Be-3x
Solution:
y = Ae3x + Be-3x ……(1)
Differentiating twice w.r.t. x, we get
This is the required D.E.
(ii) y = $$c_{2}+\frac{c_{1}}{x}$$
Solution:
y = $$c_{2}+\frac{c_{1}}{x}$$
∴ xy = c2x + c1
Differentiating w.r.t. x, we get
(iii) y = (c1 + c2x) ex
Solution:
y = (c1 + c2x) ex
This is the required D.E.
(iv) y = c1 e3x+ c2 e2x
Solution:
This is the required D.E.
(v) y2 = (x + c)3
Solution:
y2 = (x + c)3
Differentiating w.r.t. x, we get
This is the required D.E.
Question 2.
Find the differential equation by eliminating arbitrary constant from the relation x2 + y2 = 2ax.
Solution:
x2 + y2 = 2ax
Differentiating both sides w.r.t. x, we get
2x + 2y$$\frac{d y}{d x}$$ = 2a
Substituting value of 2a in equation (1), we get
x2 + y2 = [2x + 2y $$\frac{d y}{d x}$$]x = 2x2 + 2xy $$\frac{d y}{d x}$$
∴ 2xy $$\frac{d y}{d x}$$ = y2 – x2 is the required D.E.
Question 3.
Form the differential equation by eliminating arbitrary constants from the relation bx + ay = ab.
Solution:
bx + ay = ab
∴ ay = -bx + ab
∴ y = $$-\frac{b}{a} x+b$$
Differentiating w.r.t. x, we get
$$\frac{d y}{d x}=-\frac{b}{a} \times 1+0=-\frac{b}{a}$$
Differentiating again w.r.t. x, we get
$$\frac{d^{2} y}{d x^{2}}$$ = 0 is the required D.E.
Question 4.
Find the differential equation whose general solution is x3 + y3 = 35ax.
Solution:
Question 5.
Form the differential equation from the relation x2 + 4y2 = 4b2.
Sol ution:
x2 + 4y2 = 4b2
Differentiating w.r.t. x, we get
2x + 4(2y$$\frac{d y}{d x}$$) = 0
i.e. x + 4y$$\frac{d y}{d x}$$ = 0 is the required D.E.
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# Minipage/subfigure issue - creating a set of 4x3 graphs (12 total)
by bru1987 Last Updated August 08, 2018 13:23 PM
I am trying to fix a 4 by 3 set of graphs, with caption, on a .tex file. The result I want to achieve is the following:
(this was done in Word. Yes, that's lame, I know.)
The code for each small graph is the following:
\begin{tikzpicture}
\draw[help lines, color=gray!30, dashed] (-4.9,-4.9) grid (4.9,4.9);
\draw[->,ultra thick] (-5,0)--(5,0) node[right]{$x$};
\draw[->,ultra thick] (0,-5)--(0,5) node[above]{$y$};...
\end{tikzpicture}
I've tried a bunch of subfigure/minipage options but the result I am getting is far from ideal (below)
\begin{minipage}[b]{0.3\linewidth}
\centering
\begin{tikzpicture}
\draw[help lines, color=gray!30, dashed] (-4.9,-4.9) grid (4.9,4.9);
\draw[->,ultra thick] (-5,0)--(5,0) node[right]{$x$};
\draw[->,ultra thick] (0,-5)--(0,5) node[above]{$y$};...
\end{tikzpicture}
\end{minipage}%%
\begin{minipage}[b]{0.3\linewidth}
\centering
\begin{tikzpicture}
\draw[help lines, color=gray!30, dashed] (-4.9,-4.9) grid (4.9,4.9);
\draw[->,ultra thick] (-5,0)--(5,0) node[right]{$x$};
\draw[->,ultra thick] (0,-5)--(0,5) node[above]{$y$};...
\end{tikzpicture}
\end{minipage}
\begin{minipage}[b]{0.3\linewidth}
\centering
\begin{tikzpicture}
\draw[help lines, color=gray!30, dashed] (-4.9,-4.9) grid (4.9,4.9);
\draw[->,ultra thick] (-5,0)--(5,0) node[right]{$x$};
\draw[->,ultra thick] (0,-5)--(0,5) node[above]{$y$};...
\end{tikzpicture}
\end{minipage}%%
Tags :
## EMF figures in LaTex
Updated December 06, 2017 18:23 PM
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# Chicago Author-Date Style
### Block quotation
• Text including block quotations, notes, bibliography entries, table titles, and figure captions must be double-spaced.
• For block quotations five or more lines, or >100 words, should be blocked.
• Blocked quotation does not get enclosed in quotation marks.
• Use a 1/2″ indent for paragraph beginnings for block quotes and hanging (bibliography) indents.
https://owl.english.purdue.edu/owl/resource/717/2/
Fuchs (2014) believes:
Neither techno-optimism nor techno-pessimism is the appropriate method
for analyzing social media. Rather, one needs to decentre the analysis
from technology and focus on the interaction of the power structures of
the political economy of capitalism with social media (p. 256).
An extra line space should immediately precede and follow a blocked quotation.
=================================================
• Number the pages in the top right corner of the paper, beginning with the first page of text.
## 2.8Line spacing
Though authors may prefer to use minimal line spacing on the screen, publishers have customarily required that any printout be double-spacedincluding all extracts, notes, bibliography, and other material.
except for block quotes that are single spaced(in 16th ed). (in 17 this exception is removed)
======================================================================
Author-Date Style In-text Citation: Parenthetical References (p. 620-624)
In parentheses, cite the author’s last name, followed directly by the publication year with no punctuation
between.
Where the author’s name is mentioned in the sentence, cite the publication date in parentheses after the author’s last name wherever it appears in the sentence and before a mark of punctuation.
To cite a particular part of a source, include the last name and publication year, a comma, and page number(s);
for journals, include the last name and publication year, a comma, the volume number, a colon, and page number(s).
A citation for a journal article appearing in the text as either:
Cite at the end:
Start of a paragraph … blah blah …….blah blah …….blah blah …….blah blah …………….. ………………… (LN and LN Year, page).
Blah ………………………………………………… blah. The following passage will be blah ……………………………. blah enlightening: blah ….. blah I….. blah ….. blah ….. blah ….. blah ….. blah ….. blah ….. blah ….. blah ….. blah ….. blah ….. blah ….. blah ….. blah ….. blah ….. blah ….. blah ….. blah ….. blah ….. blah ….. blah . (Crump 2006, 53)
An extra line space should immediately precede and follow a blocked quotation.
Enclose:
LN (1984c) confirms that blah blah ………………… ………………. …………….. …………………… ……. measured (page). As FN LN points out, “blah blah blah blah blah blah blah blah blah blah blah blah blah” (year, page).
http://www.cu-portland.edu/sites/default/files/pdf/CHICAGO%20SAMPLE%20PAPER%20(Purdue%20OWL).pdf
Cite at the beginning:
LN, LN, and LN (1998, 243) argue that Blah ….. blah ………. blah blah blah blah blah blah blah blah blah blah blah.
Wider blah blah……………… , LN (year, page) claims: Iblah ….. blah I….. blah ….. blah ….. blah ….. blah ….. blah ….. blah ….. blah ….. blah ….. blah ….. blah ….. blah ….. blah ….. blah ….. blah ….. blah ….. blah ….. blah ….. blah ….. blah ….. blah .
=========================================================================
The author-date system has long been used by those in the physical, natural, and social sciences.
http://lib.trinity.edu/research/citing/Chicago_Author_Date_16th_ed.pdf
Citing sources in this style consists of two parts:
1. An in-text citation
2. A reference list
In this system, sources are briefly cited in the text, usually in parentheses, by author’s last name and date of publication. From this point of view it is similar to APA.
The short citations are amplified in a list of references, where full bibliographic information is provided.
===================================
Footnotes or end-notes can be used to supplement the Author-Date style to provide additional relevant commentary and/or to cite sources that do not readily lend themselves to the Author-Date References system.
===================================
Each subhead on a new line, flush left. Each level of subhead must be clearly distinguished so that the different levels can be identified and carried over for publication.
Numbering of sections, and subsections provides easy reference. Sections are numbered within chapters, subsections within sections, and sub-subsections within subsections. The number of each division is preceded by the numbers of all higher divisions, and all division numbers are separated by periods, colons, or hyphens. For example:
### 12.2-24: Numbering displayed mathematical expressions
*Capitalize the first and last words in titles and subtitles. Capitalize all other major words. But Lowercase words to and as, articles the, a, and an as well as common coordinating conjunctions and, but, for, or, and nor.
4.5 Democracy That Works Is Better Than Dictatorship That Do Not
*An exception is made for run-in heads, italicized and followed by a period and capitalized sentence-style:
*Lowercase prepositions, regardless of length (except when they are used adverbially or adjectivally (up in Look Up, down in Turn Down, on in The On Button, to in Come To, etc.) or when they compose part of a Latin expression used adjectivally or adverbially (De Facto, In Vitro, etc.))
4.6 Three Hypothesis concerning the Democracy according to Plato
*Lowercase the part of a proper name that would be lowercased in text, such as de or von.
4.7 Accusers of Comte de Monte-Cristo Fail the Job
*Lowercase the second part of a species name, such as fulvescens in Acipenser fulvescens, even if it is the last word in a title or subtitle.
4.8 From Homo erectus to Homo sapiens: A Brief History
The first sentence of text following a subhead should not refer to the subhead; words should be repeated where necessary. For example:
Chicago Headings Type 2 Level Format 1 4. Centered, Boldface or Italic Type, Headline-style Capitalization 2 4.1 Centered, Regular Type, Headline-style Capitalization 3 4.1.1 Flush Left, Boldface or Italic Type, Headline-style Capitalization 4 4.1.1.2 Flush left, roman type, sentence-style capitalization 5 Run in at beginning of paragraph (no blank line after), boldface or italic type, sentence-style capitalization, terminal period. Democracy in media. It is blah ………..
================================================================
Figures should be placed as close as possible to the text to which they refer.
Label all drawings, photos, charts, graphs, maps, etc. as “Figure” or “Fig”
below the image, followed by an Arabic numeral, a period, and a caption.
Number tables and figures separately in the order you mention them in the text.
In the text, identify tables and figures by number (“in figure 3”) rather than by location (“below”).
For example:
Figure 1. Gandhi the same font and size of the text
======================================================================
In text Citation examples for Author-date Chicago system (best)
http://libguides.murdoch.edu.au/c.php?g=246210&p=1640153
Murdoch University Site:
Use only the surname of the author followed by the year of publication. Include page, chapter, section or paragraph numbers if you need to be specific.
A comma is placed between the year of publication and the page, chapter, section or paragraph numbers.
No distinction is made between books, journal articles, internet documents or other formats except for electronic documents that do not provide page numbers. In this instance, use the paragraph number, if available, with the abbreviation par.
Citations in the text can either be either placed at the end of a sentence in parentheses (brackets)
or
alternatively, the author’s name may be included in the text, and just the date and additional information placed within the brackets.
————————-
A citation for a book appearing in the text as:
There are many reasons for intestinal scarring (Ogilvie 1998, 26-28).
A citation for a journal article appearing in the text as either:
Cite at the end:
… gastrointestinal illness is also often misdiagnosed (Morgan and Thompson 1998, 243).
Enclosing:
Foucault (1984c) confirms that “criticism and philosophy took note of the disappearance” (103).
http://www.cu-portland.edu/sites/default/files/pdf/CHICAGO%20SAMPLE%20PAPER%20(Purdue%20OWL).pdf
Cite at the beginning:
Morgan and Thompson (1998, 243) argue that gastrointestinal illness is also often misdiagnosed.
In reference lists, no page numbers are given for books;
For journal articles or chapters or other sections of a book, the beginning and ending page numbers of the entire article or chapter are given.
referenced as:
————————————————
An electronic document would be cited in the text in the same way as a print document.
For example, citation for an internet document appearing in the text as:
There are many useful materials available (Raidal and Dunsmore 1996, par. 13)
would be given in the reference list as:
Raidal, Shane R., and Jon Dunsmore. 1996. Parasites of Companion Birds: A Survey of Alimentary Tract Parasites. http://wwwvet.murdoch.edu.au/caf/parasit.htm.
———————————————
Note: When referring to multiple authors within the text and within parentheses, precede the final name with the word and
… as Kurtines and Szapocnik (2003) demonstrated.
… as has been demonstrated (Kurtines and Szapocnik 2003).
Purdue OWL: Chicago Manual of Style 16th Edition
Bibliography
This guide outlines the author-date system.
In text Citations examples
In text Citation examples for Author-date System
——————–=————————
There are four common methods of referring to a source document in the text of an essay, thesis or assignment. These methods are direct quotation from another source, paraphasing or summarising material, and citing the whole of a source document. In academic writing, most of your essay or assignment should be phrased in your own words and the overuse of direct quotation should be avoided.
Quoting
Short quotes
• Quotations match a small section of the source document word for word and must be attributed to the original author and enclosed within quotation marks. When quoting, the relevant page number(s) must be given:
Larsen (1991, 245) stated that “many of the facts in this case are incorrect”.
• If information is left out, three dots … must be used to show where the missing information goes:
As Ballard and Clanchy (1988, 14) have argued, “Learning within the university is a process of gradual socialization into a distinctive culture of knowledge, and … literacy must be seen in terms of the functions to which language is put in that culture”.
Longer quotes (block quotation)
• In general, avoid using too many long quotes and remember to introduce or integrate quotations smoothly into the rest of your assignment
• You may choose to indent a larger block of quoted text. Such blocks of quoted texts usually consist of more than one sentence or more than 40 words
• Blocks of quoted text should be indented from the left margin only, single spaced and may be one point smaller than the standard font size:
Wider applications are increasingly being found for many drugs such as invermectin. For example, Crump (2006, 53) confirms that: Ivermectin – already used extensively in animal health and in eliminating onchocerciasis and lymphatic filariasis, two of the most disfiguring and deleterious human diseases – is now being used commercially for the treatment of strongyloidiasis, mites and scabies. or Wider applications are increasingly being found for many drugs such as Aspirin. The following passage will be enlightening: Ivermectin – already used extensively in animal health and in eliminating onchocerciasis and lymphatic filariasis, two of the most disfiguring and deleterious human diseases – is now being used commercially for the treatment of strongyloidiasis, mites and scabies. (Crump 2006, 53)
Quotations within quotations
• Use a single quotation mark to indicate previously quoted material within your quotationShort Quotation:
She stated, “The ‘placebo effect’ … disappeared when behaviors were studied in this manner” (Miele 1993, 276), but she did not clarify which behaviors were studied.
OR
Miele (1993) found that “‘the placebo effect’, which had been verified in previous studies, disappeared when behaviors were studied in this manner” (276).
The author-date system inserts minimal source information directly into the text itself, surrounded by parentheses, and follows up with the rest of the source information in a list of references at the end of the paper. Imagine that you’re writing a paper on the ideal politician and are quoting a particular author’s ideas about desirable qualities in a politician. An excerpt from a sentence in the text of a paper written using the author-date would look like this:
While some assert that the essential qualities a politician must possess
are, "passion, a feeling of responsibility, and a sense of proportion"
(Weber 1946, 33), others think that ...
The entry in the list of references would look like this:
Weber, Max. 1946. Politics as a Vocation. In Essays in Sociology, edited by
H.H. Garth and C. W. Mills, 26-45. New York: Macmillian.
=========================================================
### Author-date: “References”
letter-by-letter alphabetical order according to the first word in each entry.
Sources you consulted but did not directly cite may or may not be included.
the date is immediately after the author’s name.
#### Spacing
entries are singled-spaced
Two blank lines should be left between “References” and your first entry.
One blank line should be left between remaining entries
#### Order
Citations beginning with names and those beginning with titles are to be alphabetized together. Numbers in titles are treated as though they have been spelled out. For names, alphabetize based on the letters that come before the comma separating the last name from the first, and disregard any spaces or other punctuation in the last name. For titles, ignore articles such as “a” and “the” (and equivalents in other languages) for alphabetization purposes.
References
Chicago Author-Date:
in the reference list:
the publication name is italic
Ogilvie, Timothy H. 1998. Large Animal Internal Medicine. Baltimore, MD: Williams and Wilkins.
Lastname, Firstname. Year. Book title in italics. Edition number if not first. City: publisher.
Lastname, Firstname. Year. Book title in italics. City, StateOrCountryAbbr: publisher.
The title of a chapter in book or title of an article in a publication is in double quote not italic:
Lastname, Firstname. Year. “article title in double quote.” Journal of Publication in italics. City: publisher.
In the reference list, do not abbreviate “edited by” or “translated by”.
Lastname, Firstname. Year. “article title.” In Name of the EditedBook in italics, edited by FN LN, page-page. City: publisher.
Lastname, Firstname, and FN LN. Year. “article title.” In EditedBook in etalics, edited by FN LN, page-page. City: publisher.
Lastname, Firstname, FN LN, and FN LN. Year. “article title.” In EditedBook in etalics, edited by FN LN, page-page. City: publisher.
no comma after Journal before volume
Lastname, Firstname. Year. “article title.” Journal in italics Volume (issue): page-page. City: publisher.
CHAPTER IN AN EDITED BOOK
I n citations o f a chapter o r similar part o f an edited book, include the
chapter author; the chapter title, in quotation marks; and the editor.
Precede the title of the book with In. Note the location of the page range for
the chapter in the reference list entry.
LN, FN. Year. “aaatitleaaa.” In edited book , edited by Amir Ghaseminejad, page-page. city:publisher.
Smith, Alan. 1984. “Streisand as Schwarzkopf.” In The Glenn Gould Reader, edited by Amir Ghasemi, 308-11. New York: Printage.
volume(issue): page range.
Shirky, Clay. 2011. “The Political Power of Social Media: Technology, the Public Sphere, and Political Change.” Foreign Affairs 90 (1): 28–41.
Morgan, John, and Jack Thompson. 1998. “PCR Detection of Cryptosporidium: The Way Forward.” Parasitology Today 14 (6): 241-245.
Blair, Walter. 1977. “Americanized Comic Braggars.” Critical Inquiry 4 (2): 331-49.
( B l a i r l ’77, 331 32)
no comma between the journal name and volume (issue):page-page.
Karmaus, Wilfried, and John F Riebow. 2004. “Storage of Plastic and
Glass Containers.”
Environmental Health Perspectives112 (May) : 643-647. http://www
.jstor.org
Istable/3435987.
The DOl is preferred to a URL
Note that DOl, so capitalized when mentioned in running text, is lowercased and followed by a colon (with no space afer) in source citations.
Novak, William J. 2008. ‘The Myth ofthe ‘Weak’ American State. ” American Historical Review 113:752-72. doi:l0.1086/ahr.ll3.3.752.
Blair, Walter.
1977. “Americanized Comic Braggars.” Critical Inquiry 4 (2): 331-49.
Becker, Elizabeth. 2003. “U.S threatens to act against Europeans over modified foods.”New York Times, Jan. 10.
Brest, Martin. 2003. Gigli. DVD. New York: Sony Home Entertainment.
Couper, Heather, and Nigel Henbest. 2002. “The hunt for Planet X.” New Scientist, 14 December, 30-34.
Delaroche, Paul. 1829. “Portrait of a Woman,” pastel drawing (Ackland Art Museum, Chapel Hill, NC). In European Drawings from the Collection of the Ackland Art Museum, by Carol C. Gillham and Carolyn H. Wood. Chapel Hill: The Museum, University of North Carolina, 2001, page 93.
Fildes, Alan, and Joann Fletcher. 2001. Alexander the Great: Son of the gods. London: Duncan Baird.
Freud, Sigmund. 1950. Beyond the pleasure principle. Translated by James Strachey. New York: Liveright.
Gezon, Lisa L. 2002. “Marriage, kin, and compensation: A socio-political ecology of gender in Ankarana, Madagascar.” Anthropological Quarterly 75 (4): 675-706.
Haas, Stephanie. 2007. “Relational algebra 1.” (lecture in Introduction to Database Concepts and Applications, University of North Carolina, Chapel Hill, NC).
Haldon, John. 2002. “Humour and the everyday in Byzantium.” In Humour, history, and politics in late antiquity and the early Middle Ages, edited by Guy Halsall, 48-71. New York: Cambridge University Press.
Hedges, Chris. 2000. “When armies of conquest marched in, so did saints.” New York Times, February 12, LexisNexis Academic.
Kane, Dan and Jane Stancill. 2003. “UNC building projects advance.” Raleigh News & Observer, July 15. http://www.news-observer.com/front/story/2694510p-2498221c.html.
Monet, Claude. 1885. i>Meadow with Haystacks at Giverny, oil on canvas (Museum of Fine Arts, Boston). ARTstor.
Li, Albert P., and Robert H. Heflich, eds. 1991. Genetic toxicology. Boca Raton: CRC Press.
Rathgeb, Jody. 1997. “Taking the heights.” Civil War Times Illustrated 36 (6): 26-32, Academic Search Premier (9185).
Reid, P. H. 2001. “The decline and fall of the British country house library.” Libraries & Culture 36 (2): 345-366. http://muse.jhu.edu/journals/libraries_and_culture/v036/36.2reid.html.
Scholz, Christopher H. 2002. The mechanics of earthquakes and faulting. New York: Cambridge University Press.
Single author
Nicholas, F. 2010. Introduction to Veterinary Genetics. 3rd ed. Oxford: Wiley-Blackwell.
Two authors
Rosenfeld, Andrew J., and Sharon M. Dial. 2010. Clinical Pathology for the Veterinary Team. Aimes, IA: Wiley-Blackwell.
Three or more authors
Millon, Theodore, Roger Davis, Carrie Millon, Luis Escovar, and Sarah Meagher. 2000. Personality Disorders in Modern Life. New York: Wiley.
Edited work
Butler, J. Douglas, and David F. Walbert, eds. 1986. Abortion, Medicine and the Law. New York: Facts on File Publications.
If the city of publication is not abbreviated, if city is unknown to readers or may be confused with another city of the same
name, the abbreviation of the state, province, or (sometimes) country is
city: publisher
or
city, StateOrCountryAbbr: publisher
Woodward, K. N., ed. 2009. Veterinary Pharmacovigilance: Adverse Reactions to Veterinary Medicinal Products. Chichester, UK: Wiley-Blackwell.
Later edition
Ettinger, Stephen J., and Edward C. Feldman, eds. 2010. Textbook of Veterinary Internal Medicine: Diseases of the Dog and the Cat. 7th ed. St Louis: Elsevier Saunders.
No date of publication
Bligh, Beatrice. n.d. Cherish the Earth. Sydney: Macmillan.
Two or more books by the same author published in the same year
Gilbert, Sandra M. 1972a. Acts of Attention: The Poems of D. H. Lawrence. Ithaca: Cornell University Press.
Gilbert, Sandra M. 1972b. Emily’s Bread: Poems. New York: Norton.
Multivolume work
Russell, Bertrand. 1967. The Autobiography of Bertrand Russell. 3 vols. London: Allen & Unwin.
Translation
Proust, Marcel. 1970. Jean Santeuil. Translated by G. Hopkins. New York: Simon & Schuster.
Edited translation (where role of editor or translator is of chief importance)
West, T. G., ed. & trans. 1980. Symbolism: An Anthology. London: Methuen.
Organisation
Ansett Transport Industries Ltd. 1984. Annual Report 1983-84. Melbourne: ATI.
Government publication
Australian Bureau of Statistics. 1985. Projections of the Population of Australia, States and Territories, 1984 to 2021, Cat. no. 3222.0. Canberra: ABS.
Government departments
Australia. Department of Aboriginal Affairs. 1989. Programs in Action for Aboriginal and Torres Strait Islander People: Achievements. Canberra: AGPS.
Newspaper article
Marshall, Tyler. “200th Birthday of Grimms Celebrated.” Los Angeles Times, 15 March 1985, sec. 1A, p. 3.
***The parentheses that enclose a text citation may also include a comment, separated from the citation by a semicolon.
(Mandolan 2009; t-tests are used here)
Western Australia. Environmental Protection Authority. 1998. Industrial Infrastructure and Harbour Development, Jervoise Bay. Bulletin 908. Perth: EPA.
Please Note: Documents authored by government departments are cited following the jurisdiction they report to. Precede the department name with Australia., Western Australia., etc.
http://davidson.libguides.com/c.php?g=349327&p=2357431
How to construct an in-text parenthetical reference
The author-date citation in the text must correspond exactly to its full citation in the reference list.
• Basic form. Include the author’s last name and year of publication.
(Cox 1997)
• Two authors with the same last name. Add a first initial to distinguish between the two.
(M. Cox 1997)
• Citation of a specific page or section. Insert a comma after the date and then give page number. Always include page number for direct quotations.
(Cox 1997, 21)
• Two publications by the same author in the same year. Use “a” and “b” to differentiate between the two.
(Cox 1997a) and (Cox 1997b)
• Two or three authors. Include all names in the citation.
(Cox, Cunningham, and Hatleberg 1997)
• More than three authors. Include the first name, followed by “et al.” (meaning “and others”).
(Cox et al. 1997)
• Multiple references in the same citation. Separate the citations with semicolons.
(Cox 1997; Cunningham 1996; Hatleberg 1996)
### In the Chicago Manual of Style 16, Chapter 15 is related to Author-Date
15.3 Notes and bibliography entries as models for author-date references. Most of
almost all cases by a different ordering or arrangement of elements.
Most reference list entries are identical to entries in a bibliography except for
the position of the year of publication, which in a reference list follows the
author’s name.
Unlike bibliography entries (see 14.59), each entry in the
reference list must correspond to a work cited in the text.
Text citations differ from citations in notes by presenting only the author’s last name
and the year of publication, followed by a page number or other locator, if any. This chapter, by focusing on these and other differences, will
allow readers to adapt any of the examples in chapter 14 to the authordate system.
14.157 Abbreviations for “page,” “volume,” and so on.
In citations, the words page, volume, and the like are usually abbreviated and often simply omitted (see 14.158 ).
The most commonly used abbreviations are
p. {pl. pp. ),
vol.,
pt. ,
chap. , bk.,
sec. ,
n. {pl. nn. ), no., app., and fg. ; for these and others, see
chapter
10, especially 10.43.
Unless following aperiod, all are lowercased,
Burt, Ronald S. 1992. “The Network Structure of Socia Capital.” in Reseach in Oranizational Behavior, vol. 22, edited by Robert J. Sutton and Barry Greenwich, pp. 345-423. Conn.: Elsevier Science
and none is italicized unless an integral part of an italicized book title.
All the abbreviations mentioned in this paragraph, except for
p. and n.,
s .
A Cry ofAbsence, chap. 6
Burt, Ronald S. 1992. “The Social Capital of Structural Holes.” in New Dirrtions in EcoKmic Sociology, edited by Mauro F. Guillen, Randall Collins, Paula England, and Marshall Meyer, Chap. 7. New York: Russell Sage.
A Dance to the Music of Time, 4 vols.
14 . 158 When to omit “p.” and “pp.”
When a number or a range of numbers clearly denotes the pages in a book, p. or pp. may be omitted; the numbers alone, preceded by a comma, are sufficient.
Where the presence of other numerals threatens ambiguity, p. or pp. may be added for clarity.
(And if an author has used p. and pp. consistently throughout a work, there is no
need to delete them. )
Charlotte’s Web, 75-76
but p. is necessary
Complete Poems of Michelangelo,
p. 89, lines 135-36
14.159 When to omit “vol.” When a volume number is followed immediately by
a page number, neither vol. nor
p. or pp. is needed. The numbers alone are
used, separated by a colon. A comma usually precedes the volume number, except with periodicals (see
types of classical references (see
14.256-66). For more on volume numbers, see 14.121-27. For citing a particular volume, with and without the
abbreviation vol., see
14.123.
Volume:page
The Complete Tales ofHenry James, 10:122
14.160 Page and chapter numbers.
Page numbers, needed for specifc references
in notes and parenthetical text citations, are usually unnecessary in bibliographies except when the piece cited is a part within a whole (see
14.111-17) or a journal article (see 14.183).
If the chapter or other section number
is given, page numbers may be omitted. The total page count of a book is
not included in documentation. {Total page counts do, however, appear
724
Books 14.164
in headings to book reviews, catalog entries, and elsewhere. For book review headings, see 1.92. )
14. Claire Kehrwald Cook, “Mismanaged Numbers and References,” in Line by
Line: How to Edit Your Own Writing
(Boston: Houghton Mifin, 1985), 75-107.
15. Nuala O’Faolain, Are You Somebody? The Accidental Memoir of a Dublin
Woman
(New York: Holt, 1996}. chap. 17.
================================================================================
Notes/Bibliography Chicago Style consists of two parts:
1. A superscript number in the text and corresponding note
2. A bibliography
==============================================
## 9.3An alternative rule—zero through nine
Many publications, including those in scientific or journalistic contexts, follow the simple rule of
spell out only single-digit numbers and use numerals for all others
• My house is three years old.
• According to a recent appraisal, my house is 103 years old.
• The official attendance at this year’s fair was 47,122.
### 9.5 When a number begins a sentence, it is always spelled out.
One hundred ten candidates were accepted.
## 9.6 Ordinals
The general rule applies to ordinal as well as cardinal numbers.
for example, 122nd and 123rd. The letters in ordinal numbers should not appear as superscripts .
Gwen stole second book in the top half of the first inning.
She found herself in 125th position out of 360.
## 9.18 Percentages
Except at the beginning of a sentence, percentages are usually expressed in numerals. In nontechnical contexts, the word percent is generally used; in scientific and statistical copy, the symbol % is more common.
Fewer than 3 percent of the employees used public transportation.
With 90–95 percent of the work complete, we can relax.
Only 20% of the ants were observed to react to the stimulus.
The treatment resulted in a 20%–25% increase in reports of night blindness.
=========================================================
# Chicago Lists
## 6.124
A colon is normally used after as follows, the following. (For lists, see 6.121–26.)
Colon Should NOT be used after namely, for example.
Make this product by following these steps: first, make the mix; second, pray for success; third, look at the sky; fourth, stop.
A vertical list is best introduced by a complete grammatical sentence, followed by a colon.
If the items are numbered, a period follows the numeral and each item begins with a capital letter.
Items carry no closing punctuation unless they consist of complete sentences.
You must write books for three reasons:
1. To illustrate the the importance of democracy
2. To distinguish the use of semicolons from the use of periods
3. To illustrate the use of parentheses within dashes
For the following reasons, I feel bad for people who don’t visit the website:
a. They will miss this Web bonus.
c. They don’t see all the other great Quick and Dirty Tips shows.
To avoid long, skinny lists, short items may be arranged in two or more columns.
Example:
An object can be identified by six attributes:
color quality security weight access design efficiency
If items run over a line, the second and subsequent lines are usually indented (flush-and-hang style, also called hanging indention, as used in bibliographies and indexes).
To change the date display, the following steps are recommended:
1. Pull the stem out to the time-setting position (i.e., past the date-setting position
of the best device.
2. If you are able to consult the correct time, adjust the minute hand accordingly, and press the stem all the way in on the exact second. If you are not able to consult the correct time, settle on a minute or so past the time noted in step 2.
Use the control panel on your printer to manage basic settings:
• Control toner usage by turning EconoMode on or off.
• Adjust print quality by changing the Resolution Enhancement technology and Print Density settings.
• Manage printer memory by changing the Image Adapt and Page Protect settings.
Short, simple lists are usually better run in, especially if the introduction and the items form a complete grammatical sentence (see 6.123).
Lists that require typographic prominence, that are relatively long, or that contain items of several levels (see 6.126) should be set vertically.
## 6.65
If a colon intervenes in what would otherwise constitute a grammatical sentenceeven if the introduction appears on a separate line, as in a list (see 6.121–26)it is probably being used inappropriately.
A colon, for example, should not be used before a series that serves as the object of a verb.
Apply this test: to merit a colon, the words that introduce a series or list must themselves constitute a grammatically complete sentence.
The menagerie included cats, pigeons, newts, and deer ticks.
not
The menagerie included: cats, pigeons, newts, and deer ticks.
Nor should a colon normally be used after namely, for example, and similar expressions (see 6.43).
The Chicago Manual of Style, Grammatically Correct, and The Little Penguin Handbook state that colons shouldn’t follow statements that couldn’t stand on their own as complete sentences. Therefore,
If your lead-in statement is a complete sentence, (CAN or SHOULD?) use a colon at the end to introduce your list.
Two questions are especially popular:
1. Should items begin with a capital letter?
2. Should items end with any punctuation?
if your lead-in statement is a sentence fragment, don’t use a colon.
If your list item is a complete sentence, capitalize the first letter.
If your list item isn’t a complete sentence, you can choose whether or not to capitalize the first letter.
Chicago style is to lowercase after a colon unless what follows consists of two or more complete sentences. Please see CMOS 6.64 for examples and exceptions.
If your list items are complete sentences, or
if at least one list item is a fragment that is immediately followed by a complete sentence, use normal terminal punctuation: a period, question mark, or exclamation point.
don’t put commas or semicolons after the items
don’t put a conjunction such as “and” before the last item when you are listing items vertically. but the Chicago Manual of Style says commas are optional in some lists and allows the conjunction and after the penultimate list item if you are using semicolons at the end of each list item and closing the last item with terminal punctuation.
Example 6.125:
You may choose to support
a. democracy,
b. dictatorship, or
c. liberal democracy.
(NOT RECOMMENDED)
make sure that all of your list items are parallel. Items should be structured the same way. They should all be fragments or they should all be complete sentences. If you start one bullet point with a verb, then start every bullet point with a verb.
## Bullets
When the order isn’t important, But pay attention to grouping and importance and Other dimensions that may dictate a hierarchy.
In order to make polo, we need to mix
• Rice
• Water
• Salt
• Oil
After mix “:” does not exits because” we need to mix” is not a complete sentence. After Rice we opted to have no punctuation because it is not a complete sentence.
### Lettered Lists (are in fact a version of bullet list)
But there is a need to choose individual items or
we want to refer to an item later or
Lettered lists is used when you want to keep them in a sentence instead of listing them vertically.
it is ok to say:
You have to consider shareholders, employees, and customers.
or
You have to consider (a) shareholders, (b) employees, and (c) customers.
The second is easier to recognize and refer back to later. “You have to consider” is not a complete sentence therefore “:” is not used.
6.121 Unless introductory numerals or letters serve a purpose—to indicate the order in which tasks should be done, to suggest chronology or relative importance among the items, to facilitate text references, or, in a run-in list, to clearly separate the items—they may be omitted.
6.123
If numerals or letters are used to mark the divisions in a run-in list, enclose them in parentheses.
If the introductory material forms a grammatically complete sentence, a colon should precede the first parenthesis (see also 6.59, 6.62, 6.65).
The items are separated by commas unless any of the items requires internal commas, in which case all the items will usually need to be separated by semicolons (see 6.58).
You are advised to pack the following items: (a) warm, sturdy outer clothing and enough underwear to last ten days; (b) two pairs of boots, two pairs of sneakers, and plenty of socks; and (c) three durable paperback novels.
“You are advised to pack the following items” is a complete sentence therefore “:” is used
When each item in a list consists of a complete sentence or several sentences, the list is best set vertically (see 6.124).
Example:
For the following reasons, I feel bad for people who don’t visit the website:
a. They will miss this Web bonus.
c. They don’t see all the other great Quick and Dirty Tips shows.
The introducing sentence has “:” it is complete but refers to the list.
———————
Example:
You have to consider
a. shareholders
b. employees
c. customers
If you mention a letter later in your text, enclose it in parentheses
(e.g., Item (c) is the most important stakeholder)
You can use capital or lowercase letters for your list,
but the typical style is to use lowercase letters.
The most important thing is to be consistent.
Notice that in this example the verb is not in the item.
## Numbered Lists
Numbers are reserved for instances where the items in the list need to follow a specific sequence, steps.
Example:
To sort a list
1. Put the items in an array.
2. Compare the first item with other items.
3. Move the item if it is greater than the other.
Notice that in the example above each item is a complete a sentence.
It must start a capital letter, has a verb and has a full stop.
The introducing sentence does not have “:” because it is not complete, the items complete it.
In a list with fewer levels, one might dispense with capital roman numerals and capital letters and instead begin with arabic numerals.
Example:
I. Hist
II. Dent
A. Rept
1. Hist
2. Sur
B. Mam
1. Hist
2. Sur
a) Prim
(1) Lem
(2) Ant
(a) Plat
(b) Cat
i) Cerc
ii) Pong
b) Carn
(1) Creo
(2) Fiss
(a) Ailu
ii) Gyhh
(b) Arcto
(3) Pinn
III. Boogh
https://www.quickanddirtytips.com/education/grammar/formatting-vertical-lists
http://editingandwritingservices.com/bullets/
===========================================================================
## 6.43
The abbreviations i.e. (“that is”) and e.g. (“for example”), should be confined to parentheses or notes and followed by a comma.
The most noticeable difference between male and female (i.e., the presence of xx) was obvious.
The most noticeable difference between male and female (that is, the presence of xx) was obvious.
In his most famous works (e.g., one dimensional man) he is blunt.
In his most famous works (for example, one dimensional man) he is blunt.
+
===================================================
## 6.82
Em dashes are used to set off an amplifying or explanatory element or a colon especially when an abrupt break in thought is called for.
It was a revival of modern democracythe revolutionary idea.
=
It was a revival of modern democracy: the revolutionary idea.
The influence of three impressionistsMonet, Sisley, and Degasis obvious in her work.
=
The influence of three impressionists (Monet, Sisley, and Degas) is obvious in her work.
The chancellorhe had been awake half the nightcame down in an angry mood.
=
The chancellor (he had been awake half the night) came down in an angry mood.
She outlined the strategya strategy that would, she hoped, secure the peace. ( is used to avoid another comma)
My friendsthat is, my former friendsganged up on me.
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## Archive for September 2004
### Looking at darcs
I’ve been coming to realise that I’m not really as satisfied with arch as I’d like to be; in spite of being an ardent fanboy for a while now. My main requirement for software is that it be simple and stay out of my way; and while arch is fairly simple, it’s evidently proven not […]
### Angry and Terrorised
Kim’s angry. Meanwhile the cartels have started up their next scam. Here’s the Sunday Mail: Terror by DVD MARTIN WALLACE 19sep04 AUSTRALIA is being flooded with pirate DVDs and the profits from them help fund global terrorism. Here’s the Australian: Illegal DVDs funding global terror By Martin Wallace September 16, 2004 AUSTRALIA is being flooded […]
### Polycentric Law
Interesting article on “True Separation of Powers” by Jonathan Wilde in response to (and quoting) an interesting article by Jim Henley. The idea behind checks and balances under separation of powers is the restraint of mutual jealousy – each of the three branches will be so zealous of its prerogatives, and so wary of overreaching […]
### Shocked and Awed!
In a stunning triumph, and as a result of cunning diplomacy and an amazingly well-planned campaign that will surely be used as an example in academies for centuries to come, the mighty blender makes his return to the blogosphere! Only one question remains: is there a plan to win the peace, or will this hard […]
### The Economist’s Politics
One of the more discombobulating issues of converting to a neo-con has been buying into the liberal media meme. It’s confusing because there’s no particular reason I can see why individual media biasses shouldn’t pretty much average out; but instead I keep finding publications I’d expect(ed) to be written by, for and about The Man […]
### Bring Back Blenblen’s Blog!
As Elvis sang, I don’t need a lot of presents, To make my Christmas bright. I just need blender’s weblog, Up on his website! Oh, Santa: hear my plea! Santa, bring that weblog back to me! Bring back Blender’s Blog!
### Greylisting
So, after downloading another yet another 25MB of mail to delete, I finally decided it was time to update my server-side spam handling. Boring nonsense. Greylisting seems to be a decent next stage in the escalation, though unfortunately it requires upgrading to exim4 and dealing with backports and the weird “Debian-exim” user and complicated packaging. […]
### The Corporation
“Brilliant! Hilarious and chilling!” – San Francisco Bay Guardian “Coolheaded and incisive!” – San Francisco Chronicle “Ambitious…Epic…Riveting!” – Los Angeles Times Spot the pattern. Since I’ve already commented on this movie on spec, I figured I should watch it when it came out. Various comments around the place had led me to think it was […]
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retrospective Monte Carlo
Posted in pictures, Running, Statistics, Travel, University life with tags , , , , , , , , , , on July 12, 2016 by xi'an
The past week I spent in Warwick ended up with a workshop on retrospective Monte Carlo, which covered exact sampling, debiasing, Bernoulli factory problems and multi-level Monte Carlo, a definitely exciting package! (Not to mention opportunities to go climbing with some participants.) In particular, several talks focussed on the debiasing technique of Rhee and Glynn (2012) [inspired from von Neumann and Ulam, and already discussed in several posts here]. Including results in functional spaces, as demonstrated by a multifaceted talk by Sergios Agapiou who merged debiasing, deburning, and perfect sampling.
From a general perspective on unbiasing, while there exist sufficient conditions to ensure finite variance and aim at an optimal version, I feel a broader perspective should be adopted towards comparing those estimators with biased versions that take less time to compute. In a diffusion context, Chang-han Rhee presented a detailed argument as to why his debiasing solution achieves a O(√n) convergence rate in opposition the regular discretised diffusion, but multi-level Monte Carlo also achieves this convergence speed. We had a nice discussion about this point at the break, with my slow understanding that continuous time processes had much much stronger reasons for sticking to unbiasedness. At the poster session, I had the nice surprise of reading a poster on the penalty method I discussed the same morning! Used for subsampling when scaling MCMC.
On the second day, Gareth Roberts talked about the Zig-Zag algorithm (which reminded me of the cigarette paper brand). This method has connections with slice sampling but it is a continuous time method which, in dimension one, means running a constant velocity particle that starts at a uniform value between 0 and the maximum density value and proceeds horizontally until it hits the boundary, at which time it moves to another uniform. Roughly. More specifically, this approach uses piecewise deterministic Markov processes, with a radically new approach to simulating complex targets based on continuous time simulation. With computing times that [counter-intuitively] do not increase with the sample size.
Mark Huber gave another exciting talk around the Bernoulli factory problem, connecting with perfect simulation and demonstrating this is not solely a formal Monte Carlo problem! Some earlier posts here have discussed papers on that problem, but I was unaware of the results bounding [from below] the expected number of steps to simulate B(f(p)) from a (p,1-p) coin. If not of the open questions surrounding B(2p). The talk was also great in that it centred on recursion and included a fundamental theorem of perfect sampling! Not that surprising given Mark’s recent book on the topic, but exhilarating nonetheless!!!
The final talk of the second day was given by Peter Glynn, with connections with Chang-han Rhee’s talk the previous day, but with a different twist. In particular, Peter showed out to achieve perfect or exact estimation rather than perfect or exact simulation by a fabulous trick: perfect sampling is better understood through the construction of random functions φ¹, φ², … such that X²=φ¹(X¹), X³=φ²(X²), … Hence,
$X^t=\varphi^{t-1}\circ\varphi^{t-2}\circ\ldots\circ\varphi^{1}(X^1)$
which helps in constructing coupling strategies. However, since the φ’s are usually iid, the above is generally distributed like
$Y^t=\varphi^{1}\circ\varphi^{2}\circ\ldots\circ\varphi^{t-1}(X^1)$
which seems pretty similar but offers a much better concentration as t grows. Cutting the function composition is then feasible towards producing unbiased estimators and more efficient. (I realise this is not a particularly clear explanation of the idea, detailed in an arXival I somewhat missed. When seen this way, Y would seem much more expensive to compute [than X].)
perfect sampling, just perfect!
Posted in Books, Statistics, University life with tags , , , , , , , , on January 19, 2016 by xi'an
Great news! Mark Huber (whom I’ve know for many years, so this review may be not completely objective!) has just written a book on perfect simulation! I remember (and still share) the excitement of the MCMC community when the first perfect simulation papers of Propp and Wilson (1995) came up on the (now deceased) MCMC preprint server, as it seemed then the ideal (perfect!) answer to critics of the MCMC methodology, plugging MCMC algorithms into a generic algorithm that eliminating burnin, warmup, and convergence issues… It seemed both magical, with the simplest argument: “start at T=-∞ to reach stationarity at T=0”, and esoteric (“why forward fails while backward works?!”), requiring simple random walk examples (and a java app by Jeff Rosenthal) to understand the difference (between backward and forward), as well as Wilfrid Kendall’s kids’ coloured wood cubes and his layer of leaves falling on the ground and seen from below… These were exciting years, with MCMC still in its infancy, and no goal seemed too far away! Now that years have gone, and that the excitement has clearly died away, perfect sampling can be considered in a more sedate manner, with pros and cons well-understood. This is why Mark Huber’s book is coming at a perfect time if any! It covers the evolution of the perfect sampling techniques, from the early coupling from the past to the monotonous versions, to the coalescence principles, with applications to spatial processes, to the variations on nested sampling and their use in doubly intractable distributions, with forays into the (fabulous) Bernoulli factory problem (a surprise for me, as Bernoulli factories are connected with unbiasedness, not stationarity! Even though my only fieldwork [with Randal Douc] in such factories was addressing a way to turn MCMC into importance sampling. The key is in the notion of approximate densities, introduced in Section 2.6.). The book is quite thorough with the probabilistic foundations of the different principles, with even “a [tiny weeny] little bit of measure theory.
Any imperfection?! Rather, only a (short too short!) reflection on the limitations of perfect sampling, namely that it cannot cover the simulation of posterior distributions in the Bayesian processing of most statistical models. Which makes the quote
“Distributions where the label of a node only depends on immediate neighbors, and where there is a chance of being able to ignore the neighbors are the most easily handled by perfect simulation protocols (…) Statistical models in particular tend to fall into this category, as they often do not wish to restrict the outcome too severely, instead giving the data a chance to show where the model is incomplete or incorrect.” (p.223)
just surprising, given the very small percentage of statistical models which can be handled by perfect sampling. And the downsizing of perfect sampling related papers in the early 2000’s. Which also makes the final and short section on the future of perfect sampling somewhat restricted in its scope.
So, great indeed!, a close to perfect entry to a decade of work on perfect sampling. If you have not heard of the concept before, consider yourself lucky to be offered such a gentle guidance into it. If you have dabbled with perfect sampling before, reading the book will be like meeting old friends and hearing about their latest deeds. More formally, Mark Huber’s book should bring you a new perspective on the topic. (As for me, I had never thought of connecting perfect sampling with accept reject algorithms.)
Handbook of Markov chain Monte Carlo
Posted in Books, R, Statistics, University life with tags , , , , , , , , , , , , , , on September 22, 2011 by xi'an
At JSM, John Kimmel gave me a copy of the Handbook of Markov chain Monte Carlo, as I had not (yet?!) received it. This handbook is edited by Steve Brooks, Andrew Gelman, Galin Jones, and Xiao-Li Meng, all first-class jedis of the MCMC galaxy. I had not had a chance to get a look at the book until now as Jean-Michel Marin took it home for me from Miami, but, as he remarked in giving it back to me last week, the outcome truly is excellent! Of course, authors and editors being friends of mine, the reader may worry about the objectivity of this assessment; however the quality of the contents is clearly there and the book appears as a worthy successor to the tremendous Markov chain Monte Carlo in Practice by Wally Gilks, Sylvia Richardson and David Spiegelhalter. (I can attest to the involvement of the editors from the many rounds of reviews we exchanged about our MCMC history chapter!) The style of the chapters is rather homogeneous and there are a few R codes here and there. So, while I will still stick to our Monte Carlo Statistical Methods book for teaching MCMC to my graduate students next month, I think the book can well be used at a teaching level as well as a reference on the state-of-the-art MCMC technology. Continue reading
Another Bernoulli factory
Posted in R, Statistics with tags , , on February 14, 2011 by xi'an
The paper “Exact sampling for intractable probability distributions via a Bernoulli factory” by James Flegal and Radu Herbei got posted on arXiv without me noticing, presumably because it came out just between Larry Brown’s conference in Philadelphia and my skiing vacations! I became aware of it only yesterday and find it quite interesting in that it links the Bernoulli factory method I discussed a while ago and my ultimate perfect sampling paper with Jim Hobert. In this 2004 paper in Annals of Applied Probability, we got a representation of the stationary distribution of a Markov chain as
$\sum_{n=1}^{\infty} p_n Q_n(dx)$
where
$p_n = \mathbb{P}(\tau\ge n)\qquad\text{and}\qquad Q_n(A)=\mathbb{P}(X_n\in A|\tau\ge n),$
the stopping time τ being the first occurrence of a renewal event in the split chain. While $Q_n$ is reasonably easy to simulate by rejection (even tohugh it may prove lengthy when n is large, simulating from the tail distribution of the stopping time is much harder. Continue reading
Monte Carlo Statistical Methods third edition
Posted in Books, R, Statistics, University life with tags , , , , , , , , , , , , , on September 23, 2010 by xi'an
Last week, George Casella and I worked around the clock on starting the third edition of Monte Carlo Statistical Methods by detailing the changes to make and designing the new table of contents. The new edition will not see a revolution in the presentation of the material but rather a more mature perspective on what matters most in statistical simulation:
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• K. RUBINUR
Articles written in Journal of Astrophysics and Astronomy
• Searching for dual active galactic nuclei
Binary or dual active galactic nuclei (DAGN) are expected from galaxy formation theories. However, confirmed DAGN are rare and finding these systems has proved to be challenging. Recent systematic searches for DAGN using double-peaked emission lines have yielded several new detections, as have the studiesof samples of merging galaxies. In this paper, we present an updated list of DAGN compiled from published data. We also present preliminary results from our ongoing Expanded Very Large Array (EVLA) radio study of eight double-peaked emission-line AGN (DPAGN). One of the sample galaxy shows an S-shaped radio jet. Using new and archival data, we have successfully fitted a precessing jet model to this radio source. We find that the jet precession could be due to a binary AGN with a super-massive black-hole (SMBH) separation of $\sim$0.02 pc or a single AGN with a tilted accretion disk. We have found that another sample galaxy, which is undergoing a merger, has two radio cores with a projected separation of 5.6 kpc. We discuss the preliminaryresults from our radio study.
• # Journal of Astrophysics and Astronomy
Current Issue
Volume 40 | Issue 4
August 2019
• # Continuous Article Publication
Posted on January 27, 2016
Since January 2016, the Journal of Astrophysics and Astronomy has moved to Continuous Article Publishing (CAP) mode. This means that each accepted article is being published immediately online with DOI and article citation ID with starting page number 1. Articles are also visible in Web of Science immediately. All these have helped shorten the publication time and have improved the visibility of the articles.
• # Editorial Note on Continuous Article Publication
Posted on July 25, 2019
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# Count Acute, Right and Obtuse triangles from n side lengths
Problem Statement: We have $N$ sticks. The size of the $i$th stick is $A_i$. We want to know the number of different types of triangles created with each side from a single different stick. Calculate the number of acute triangles, right triangles and obtuse triangles.
Input Format: The first line contains $N$. The second line contains $N$ integers. The $i$th number denotes $A_i$.
Constraints:
• For full score: $3 \le N \le 5000$
• For 40% score: $3 \le N \le 500$
For all test cases:
• $1 \le A[i] \le 10^4$
• $A[i] \lt A[i+1]$ where $1 \le i \lt N$
Output Format: Print 3 integers: the number of acute triangles, right triangles and obtuse triangles, respectively.
My Solution: My code runs in the given time for small $n$ (~500). It will work for large $n$ (~5000) but I get time limit exceeded error on the Online Judge.
using System;
namespace CodeStorm
{
class Triangles
{
static void Main(string[] args)
{
int[] A = Array.ConvertAll(A_temp, Int32.Parse);
int[] A_sq = new int[n];
for (int i = 0; i < n; i++)
{
A_sq[i] = A[i] * A[i];
}
int n_m_2 = n - 2;
int n_m_1 = n - 1;
int acute = 0, right = 0, obtuse = 0;
for (int i = 0; i < n_m_2; i++)
{
for (int j = i + 1; j < n_m_1; j++)
{
int k = j + 1;
int AiPlusAj = A[i] + A[j];
while (k < n)
{
int squareSum = A_sq[i] + A_sq[j];
if (AiPlusAj <= A[k])
{
break;
}
else if (squareSum > A_sq[k])
{
acute++;
}
else if (squareSum < A_sq[k])
{
obtuse++;
}
else
{
right++;
}
k++;
}
}
}
Console.WriteLine(acute + " " + right + " " + obtuse);
}
}
}
The above code runs perfectly and finds the possible triangles.
Input:
6
2 3 9 10 12 15
Output:
2 1 4
The possible triangles are:
Acute triangles: 10−12−15, 9−10−12
Right triangle: 9−12−15
Obtuse triangles: 2−9−10, 3−9−10, 3−10−12, 9−10−15
I want to know a more efficient way to approach the problem so that I can get it executed in the given time limit for $n$ (~5000). After I tried to find the complexity, I came up with $O(n^3)$. I am not good with complexities. I might be wrong. I would like a more efficient way for the problem.
Sort your sticks by length; square them when already sorted. Then replace the innermost loop with 3 binary searches. In pseudocode,
max_obtuse = upper_bound(A[j:n], A[i] + A[j])
max_right = upper_bound(A_sq[j:n], A_sq[i] + A_sq[j])
max_acute = lower_bound(A_sq[j:n], A_sq[i] + A_sq[j])
obtuse += max_obtuse - max_right
right += max_right - max_acute
acute += max_acute - j
That reduces the execution time from $O(n^3)$ to $O(n^2\log n)$.
EDIT:
In the sorted array below, values marked as - are strictly less, and values marked as + are strictly greater, than X:
-----XXXXXXXXXXXXXXXXX++++++++++
^ ^
| This is upper bound of X
This is lower bound of X
• Sir I am new to algorithm. I didn't understand the pseudo code. What do you mean by upper bound and lower bound binary searches. I am sorry if I sound weird. But I am not able to understand the code. – Aman Ahuja Oct 30 '15 at 17:29
• @AmanAhuja See edit. – vnp Oct 30 '15 at 19:45
• First of all thank you so much. That helped me a lot. It improved my code. The code was submitted for a problem on hackerearth. Earlier I cleared 8 out of 16 cases(due to TLE). With your help I could do another 4. The 4 remaining cases are still showing a TLE error. Is there any way to reduce the execution time further. I am using C#. Is it a problem with the language. The execution time has to be within 3s. Thank you anyways. – Aman Ahuja Oct 31 '15 at 13:14
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# substitution_models
## Substitution models
NodeSub includes many different functions to generate alignments, this file serves to provide an overview of the different models. The standard alignment function is given by ‘sim_normal’, which is based on the alignment simulation functions in the package phangorn.
seq_length <- 30
sub_rate <- 1 / seq_length
input_tree <- TreeSim::sim.bd.taxa(n = 10,
numbsim = 1,
lambda = 1,
mu = 0.1,
complete = TRUE)[[1]]
normal_alignment <- sim_normal(input_tree,
l = seq_length,
rate = sub_rate)
plot_phyDat(normal_alignmentalignment) Then, there are two node substitution models available, the unlinked and the linked model. In the unlinked model, both daughter branches accumulate substitutions independently from each other during speciation. In the linked model, the substitutions in the daughter branches are conditional on each other, such that substitutions accumulated in one daughter, are not able to be accumulated in the other daughter. For both models we need to specify the node time (tau). For the linked model rates are specified slightly differently, with the substitution rate reflecting the rate at which one of the daughters accumulates a substitution, and the node_mut_rate_double reflecting the rate at which both daughters accumulate a (different) substitution. unlinked_alignment <- sim_unlinked(input_tree, rate1 = sub_rate, rate2 = sub_rate, l = seq_length, node_time = 0.5) plot_phyDat(unlinked_alignmentalignment)
linked_alignment <- sim_linked(input_tree,
rate = sub_rate,
node_mut_rate_double = sub_rate * sub_rate,
node_time = 0.5,
l = seq_length)
plot_phyDat(linked_alignmentalignment) # Explicit models The linked and unlinked alignment simulators use Markovian mathematics to calculate the expected number of substitutions, which yields the correct mutations along a branch, but which neglects any ‘reverse’ mutations (as these are masked). If the need arises to more explicitly simulate the mutational process, we have provided explicit functions for both normal and the unlinked model: unlinked_explicit <- sim_unlinked_explicit(input_tree, rate1 = sub_rate, rate2 = sub_rate, l = seq_length, node_time = 0.5) plot_phyDat(unlinked_explicitalignment)
normal_explicit <- sim_normal_explicit(input_tree,
l = seq_length,
rate = sub_rate)
plot_phyDat(normal_explicit\$alignment)
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6 videos
2 skills
We'll now see that we can express the sin(a+b) and the cos(a+b) in terms of sin a, sin b, cos a, and cos b. This will be handy in a whole set of applications.
Applying angle addition formula for sin
VIDEO 5:20 minutes
Angle addition formula with cosine
VIDEO 6:12 minutes
Another example using angle addition formula with cosine
VIDEO 7:14 minutes
Sine of non special angle
VIDEO 8:35 minutes
Cosine addition identity example
VIDEO 5:14 minutes
Double angle formula for cosine example
VIDEO 3:28 minutes
Addition and subtraction trig identities
PRACTICE PROBLEMS
Computing expressions using trig addition and subtraction identities
Applying angle addition formulas
PRACTICE PROBLEMS
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In this post I will try to gain som insight into the design of a small commercial motor drive. The unit in question is a small 1/2Hp (372 W) drive from TECO. The power supply to the control logic is broken, and because of it's small size(i.e. cheap) I did not bother to try and fix it.
The circuit board is clearly damaged by the heat from some kind of component failure.
The main emphasis will be on the power circuit, the cleverness of the controller will be hidden in the software and thus it will be little knowledge to gain from trying to redraw the entire circuit diagram.
## Power circuit
By inspecting the circuit boards I have attempted to redraw the high power portions of the circuit.
Redrawn diagram of the main power circuit.
The input voltage is $$230 \;V \;AC$$ at $$50 \; Hz$$. The input is protected by a metal oxide varistor. Following the MOV is a filter consisting of several capacitors and inductors.
A RS1506M full bridge rectifier module is used to convert the incoming AC to DC.
Inverter power circuit board. The large capacitor to the right is the main reservoir capacitor, while the inrush limiting NTC is the black body just above the capacitor. The rectifier and inverter modules are visible at the top left and right respectively.
On the DC side of the rectifier a NTC resistor is used to limit the inrush current to the main reservoir capacitor. A relay is used to bypass the resistor, presumably when the capacitor is fully charged.
Two $$0.1 \Omega$$ resistors are connected in series with the inverter module, one in the positive and one in the negative side of the DC-bus. These resistors are likely used to sense the current.
The inverter consists of a CPV363M4KPBF IGBT module from International Rectifier (now Infineon). The inverter is controlled by a IR2133J gate driver, not shown in the schematic.
## Input filter
Please refer to the schematic of the input filter for the following discussion. The input filter consists of two X-rated and two Y-rated capacitors, as well as two common mode chokes connected in series.
The X, and Y rating of capacitors is used to guarantee the safe performance in mains applications.
Input filter to the rectifier. The two yellow bodies contains the common mode input chokes. The blue component to the left is a metal oxide varistor to protect the input from overvoltage. The other blue components are various sizes of capacitors.
A X-rated capacitor is intended to be connected line to line, to filter common mode noise. When a capacitor fails catastrophically it will either become a open circuit or a short circuit. Thus if this happens to a X rated capacitor used correctly, one of two things will happen. It will either simply be removed from the circuit, degrading the performance, or it will short circuit blowing the the fuse or circuit breaker in the input. In either case the failure will impose no risk to the user as long as the external fuses are properly dimensioned.
The Y-rated capacitors are intended to be connected from line to ground, filtering asymmetrical noise. I.e. the capacitor is connected between the power lines, and the metallic chassis of the device in question. Thus in this case if the chassis is not properly grounded(as it should be) there will be a risk of electric shock if the capacitor short circuits.
The main difference between these capacitors and regular capacitors are strict requirement to evaluation of the performance. E.g. voltage impulse tests. The classification of safety capacitors of this type is provided in IEC 60384-14, which unfortunately isn't publicly available(i.e. you have to pay for access).
Filter choke consisting of two separate inductors wound on the same magnetic core. The black plastic piece in the middle separates the two inductors. Two wires are used in parallel, in order to increase the current carrying capability.
### Common mode choke
The common mode choke consists of two wires wound on the same magnetic core. When measuring the inductance using a regular L-meter, the reading says $$1.5\;mH$$, this may not be the reality however. The idea is that the magnetic fields induced by the differential mode currents in the windings will cancel each other out, thus the inductor will present little reactance to such currents.
The inductance to common mode currents however will be high.
As a side benefit the core is less likely to saturate, as it is normally operating with a low magnetic field. Thus the constraints on the current carrying capability is not determined by the saturation limit, but by heating of the windings. I.e. a smaller magnetic core may be used.
## DC-DC converter
In order for the control circuitry to have power, a DC-DC converter is used. This is the component that appears to have failed for some reason.
On the primary side of the transformer there is a Toshiba K2700 N-channel MOSFET, that is used for chopping of the DC-link voltage.
There are no integrated controllers for the converter, only some discrete components. Among those are two small transistors, possibly forming a astable multivibrator to generate the control signals for the switching MOSFET.
There is also a opto-coupler feeding a signal from the secondary side of the transformer back to the oscillator. This is likely the feedback signal used to maintain stable voltage on the secondary.
The secondary side consists of some rectifier diodes, a filtering inductor, and some filtering capacitors. There is also a connector for the cooling fan.
## Control circuit
The controller consists mainly of a integrated circuit labeled: "TECO E2 V2.3". It is likely that this is some kind of microcontroller, or ASIC. It has some analog, and digital I/O, some push buttons, a 7-segment display, and a interface to the gate driver.
Controller board. The TQFP packaged chip is the controller. The screw terminals at the top are available for connection of external I/O. The blue component to the top right is a relay used for digital output.
Bottom view of the controller circuit board. The SOIC(Gull-Wing) packaged chips are opto-couplers most likely used for the digital I/O.
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# Thread: integration with trig substitution
1. ## integration with trig substitution
integrate:
2x^2 / x^2 + 4 dx
how do know which substitution to use here and what happens to the numerator???
2. Originally Posted by razorfever
integrate:
2x^2 / x^2 + 4 dx
how do know which substitution to use here and what happens to the numerator???
First do some long division to turn the integrand into
$2 - \frac{8}{x^2 + 4}$.
Then you work with the fact that $\int{\frac{1}{a^2 + x^2}\,dx} = \frac{1}{a}\arctan{\left ( \frac{x}{a} \right )} + C$.
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# scipy.signal.residue¶
scipy.signal.residue(b, a, tol=0.001, rtype='avg')[source]
Compute partial-fraction expansion of b(s) / a(s).
If M is the degree of numerator b and N the degree of denominator a:
b(s) b[0] s**(M) + b[1] s**(M-1) + ... + b[M]
H(s) = ------ = ------------------------------------------
a(s) a[0] s**(N) + a[1] s**(N-1) + ... + a[N]
then the partial-fraction expansion H(s) is defined as:
r[0] r[1] r[-1]
= -------- + -------- + ... + --------- + k(s)
(s-p[0]) (s-p[1]) (s-p[-1])
If there are any repeated roots (closer together than tol), then H(s) has terms like:
r[i] r[i+1] r[i+n-1]
-------- + ----------- + ... + -----------
(s-p[i]) (s-p[i])**2 (s-p[i])**n
This function is used for polynomials in positive powers of s or z, such as analog filters or digital filters in controls engineering. For negative powers of z (typical for digital filters in DSP), use residuez.
Parameters: b : array_like Numerator polynomial coefficients. a : array_like Denominator polynomial coefficients. r : ndarray Residues. p : ndarray Poles. k : ndarray Coefficients of the direct polynomial term.
#### Previous topic
scipy.signal.unique_roots
#### Next topic
scipy.signal.residuez
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# Potential Energy: 17 Examples You Should Know
.
Here we are going to discuss some examples of potential energy found in and around home
## A water tank on the rooftop
When you open a water tank tap, the water starts flowing through it. This happens due to the stored potential energy. The water tank holds water above the ground level, so the potential energy gets stored in it. This stored potential energy of water is called gravitational potential energy. And it depends on the height of the water from the ground and the mass of water. The water tank holds water of mass (M) at (h) height above ground. The magnitude of Potential Energy stored is
P.E. = Mgh
Where M- a mass of water stored in a tank
g- Acceleration due to gravity
h- the height of the water
## Battery
Batteries have a wide variety of use in day-to-day life. Batteries power almost every electronic device. A battery has two terminals, a terminal of higher positive potential called cathode or positive terminal (+) and a terminal of higher negative potential called anode or negative terminal (-). After connecting the battery in a circuit, the negative charges called electrons are transferred from anode to cathode, and electric current flows from cathode to anode, i.e., from +ve terminal to the -ve terminal of the battery.
## Rubber band
To store potential energy in a rubber band, we have to deform it from its original shape. When we apply a force to change the shape of the rubber, we have to work against the restoring force. This work is stored in the form of elastic potential energy. As soon as we release the deformed rubber band, it regains its original condition, and the stored potential energy gets converted into kinetic energy.
## Book on a shelf
When we place a book on a shelf, as the shelf is at height ‘h’ above the ground, potential energy gets stored in it. We know that potential energy gets stored in an object when work is done against some force. If this book slipped from the shelf, its stored potential energy converts into kinetic energy, and it would fall on the ground.
## Rock on the cliff
If we push a rock through the cliff, it simply falls off and accelerates in the downward direction; This shows that a rock resting on the cliff has some potential energy stored in it, called gravitational potential energy.
## Food
Every living organism needs the energy to do his day to day activities. Food provides that necessary power to do work. Energy stored in food is in the form of chemical bonds. When we eat the food, the body digests the food and breaks the chemical bonds to take out the necessary parts like vitamins, proteins, carbohydrates, fats, etc. These parts of food are required for the overall functioning of the body.
## Pendulum
When a pendulum is displaced from its mean position, potential energy starts to build in it. The pendulum has maximum potential energy at its extreme position and minimum at its mean position. When we displace the pendulum from its mean position, we have to works against the restoring force provided by the component of gravitational force, and this work is stored in the form of gravitational potential energy in a pendulum. Oscillations in the pendulum happen because of stored potential energy; when a bob release from the extreme position, potential energy changes into kinetic energy, and the bob starts to oscillate.
## Air-filled balloon
We all have played with balloons at some point in life. You may have tried to fill a balloon with water and make a water fountain out of it or fill it with air and send it to fly. But have you ever think what makes a balloon acts like that? Let’s see what happens when we fill a balloon with air.
When we start to fill a balloon, it gets stretched in every direction. As we all know balloons are made up of rubber, and rubbers are elastic, so it has elastic potential energy. When a balloon gets deformed from its original shape, potential energy is stored in a stretched balloon. The inside air pressure balances the restoring force of stretched balloon. As soon as we release the air, all the balanced forces on a balloon get unbalanced, and the balloon starts to shrink and regain its original shape. So, whenever we fill a balloon with air or water balloon stores potential energy by deforming its original form.
## Stretched bow
Have you ever think how a stretched bow can propel an arrow to a great distance? What makes it to throw that arrow to such a long-distance? What is the source of this energy? So to answer these questions, let’s see what happens in a stretched bow.
When an archer stretches a bowstring, elastic potential energy gets stored in the system. When he releases an exaggerated bow, held potential energy converts into kinetic energy, which is transferred to an arrow and allows it to move forward. A more an archer stretches a bowstring, more potential energy gets stored in the bow system, and because of that, he can propel an arrow to a significant distance.
## Spring
To stretch or compress a spring, we have to works against the restoring force provided by the spring, which is always in the opposite direction of the spring’s displacement. This work is done against the restoring force, stored as elastic potential energy in the spring. When such a stretched or compressed spring is released, its potential energy converts into kinetic energy and the spring starts to oscillate about its mean position.
## Electric socket
Any electric home appliance needs electrical power to operate from the electric grid through an electric socket. A socket mainly consists of three holes, namely neutral, phase, and ground. The left hole of the socket is called a neutral, which is at zero potential and connected to the wire that brings back the electric current back to the electric panel. The right hole of a socket is called phase, connected to the live current-carrying wire from the grid and has a potential of 240 volts. This potential difference between phase and neutral is equal to the ratio of potential energy required to move charge per unit charge.
## Capacitors
Capacitors are mainly used to store electric charges. A capacitor is made up of two metal plates, which help to hold an electric charge. A dielectric medium is placed between these two metal plates to separate the plate and increase the capacitance. When we connect a capacitor in a circuit, one plate gets positively charged, and the other gets negatively charged. The electric field is created in between those oppositely charged plates. Capacitor quickly dissipates its stored potential energy when the two terminals, from two plates, come in contact with each other. The energy stored in a capacitor is expressed as
$_{Ecap}= \frac{QV}{2} or \frac{Q^{2}}{2c}$
Where,
E – Potential energy of a capacitor
Q – Electric charge
V – Applied potential
C – Capacitance of a capacitor
## Firecracker
Firecrackers are primarily used in every country on special occasions. They are mainly made up of gunpowder. A firecracker bursts due to stored potential energy in it, which is chemical potential energy. When a firecracker explodes, its stored potential energy converts into mechanical energy, light, sound, heat, etc., and spreads in every possible direction.
## Burning Wood
Burning wood has different types of energies that come out of it, like light energy, heat energy, etc.; it means the wood has some energy stored potential energy in it. This stored potential energy of wood is in the form of chemical potential energy, and burning wood releases this stored potential energy when the atoms in the wood start heating. Due to overheating of particles, the chemical bonds between the atoms begin to break. This breaking of bonds causes energy released from burning wood.
## LPG cylinder
LPG cylinder contains Liquid Petroleum Gas which can easily cache fire and is commonly used as an engine fuel and for cooking purposes. Propane and butane are the main constituents of liquid petroleum gas, and these gases are stored in a cylinder under very high pressure. When LPG gas comes in contact with fire, it burns furiously. Sometimes a leaked LPG cylinder explodes like a bomb because the pressure inside the cylinder increases due to the heating of pressurized gas.
## Petrol
Petrol is used chiefly as a fuel in combustion engines. In petrol, potential energy is stored in the bonds of hydrocarbons, so it is an example of chemical potential energy. When these bonds break due to heating, energy trapped inside the bonds gets released and used to work. Petrol is a favorite fuel of an automobile because of its burning nature.
## Magnet
Some everyday observations, such as alignment of the magnetic needle in the north-south direction, like poles repulsion and opposite poles attraction, proves that a magnet has potential energy. When a bar magnet is placed in an external magnetic field, the external magnetic field exerts a force on the magnet and tries to align that magnet in the direction of the magnetic field. This work done is stored as potential energy in a magnetic field.
Shambhu Patil
I am Shambhu Patil, a physics enthusiast. Physics always intrigues me and makes me think about, how this universe works? I have an interest in nuclear physics, quantum mechanics, thermodynamics. I am very good at problem solving, explaining complex physical phenomenon in simple language. My articles will walk you through each and every concept in detail. Join me over LinkedIn to https://www.linkedin.com/in/shambhu-patil-96012b1a1 . E-mail :- shambhupatil1997@gmail.com
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Quotient ring
A quotient ring is a quotient set of the elements of a ring with an induced ring structure.
Characterization of Equivalence Relations Compatible with Ring Structure
Theorem. Let $R$ be an equivalence relation on the underlying set of a pseudo-ring $A$. Then $R(x,y)$ is compatible with addition and left (resp. right) multiplication if and only if $R(x,y)$ is equivalent to a statement of the form "$x-y \in \mathfrak{a}$", for some left (resp. right) ideal $\mathfrak{a}$ of $A$.
Proof. We prove the case for left ideals; the other case follows from passing to the opposite ring.
Suppose $R$ is an equivalence relation on $A$ compatible with addition and left multiplication. Let $\mathfrak{a}$ be the equivalence class of 0. Then $R(x,y)$ is evidently equivalent to the statement "$x-y\in \mathfrak{a}$, so it remains to show that $\mathfrak{a}$ is a left ideal of $A$.
By definition, $0\in \mathfrak{a}$, and for any $x,y\in \mathfrak{a}$, $$x+ y \equiv 0+0 \equiv 0 \pmod{R},$$ so $x+y \in \mathfrak{a}$; that is, $\mathfrak{a}$ is closed under addition. Finally, for any $x\in A$ and $y\in \mathfrak{a}$, $$xy \equiv x \cdot 0 \equiv 0 \pmod{R} ,$$ so $A\mathfrak{a} \subseteq \mathfrak{a}$. Therefore $\mathfrak{a}$ is a left ideal of $A$.
Conversely, let $\mathfrak{a}$ be any left ideal of $A$. We wish to show that "$x-y \in \mathfrak{a}$" is an equivalence relation compatible addition and left multiplication in $A$. Evidently, if $x\equiv y \pmod{\mathfrak{a}}$ and $y\equiv z \pmod{\mathfrak{a}}$, then $$x-z = (x-y)+(y-z) \in \mathfrak{a},$$ so $x\equiv z\pmod{\mathfrak{a}}$. Also, $x-x = 0$ is an element of $\mathfrak{a}$, and if $(x-y)$ is, then so is $-(x-y) = y-x$. This shows that equivalence modulo $\mathfrak{a}$ is an equivalence relation.
Now we show that equivalence modulo $\mathfrak{a}$ is compatible with addition and left multiplication. Indeed, suppose that $x-y \in \mathfrak{a}$; then for any $a\in A$, $$(a+x)- (a+y) = x-y \in \mathfrak{a},$$ so $a+x \equiv a+y \pmod{\mathfrak{a}}$. Finally, for any $a\in A$, $$a(x-y) \in \mathfrak{a},$$ since $\mathfrak{a}$ is a left ideal of $A$. $\blacksquare$
Corollary. Let $A$ be a ring, and $R(x,y)$ an equivalence relation on the elements of $A$. Then $R$ is compatible with the ring structure of $A$ if and only if it is of the form "$x-y \in \mathfrak{a}$", for some two-sided ideal $\mathfrak{a}$ of $A$.
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# Molar Heat Capacity at Constant Pressure Understanding
If I apply 200 J of energy as heat to 4 moles of an ideal gas at constant pressure and the temperature rises by 4 K, then the molar heat capacity at constant pressure will be
Cp = Q / (n * deltaT) = 200 / (4 x 4) = 12.5 J K mol
Am I on the right lines here?
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## LaTeX forum ⇒ Theses, Books, Title pages ⇒ Chapter Header
Classicthesis, Bachelor and Master thesis, PhD, Doctoral degree
Johannes_B
Site Moderator
Posts: 4163
Joined: Thu Nov 01, 2012 4:08 pm
No Stefan, this won't work. The template does not use a KOMA-script class.
The smart way: Calm down and take a deep breath, read posts and provided links attentively, try to understand and ask if necessary.
Tags:
Stefan Kottwitz
Posts: 9568
Joined: Mon Mar 10, 2008 9:44 pm
In the code snippet of the previous post I saw it starting with scrlayer-scrpage, no class, that was an indication for me.
If not, it's an indication that a KOMA-Script based template would really be recommendable.
Not checked if it works with scrbase and scrhack, that was loaded more far above.
Stefan
Johannes_B
Site Moderator
Posts: 4163
Joined: Thu Nov 01, 2012 4:08 pm
The class never changed, it stayed with book. But some packages of the KOMA-bundle have been included, some commands have been added that resemble KOMA-commands (but do far less of the compicated stuff).
Edit: I should note that \abovechapterskip is defined since versioin 2.4 of the template, dating november 2016.
The smart way: Calm down and take a deep breath, read posts and provided links attentively, try to understand and ask if necessary.
vinaykumarn
Posts: 15
Joined: Wed Oct 19, 2016 6:30 am
Respected sir,
First of all I would like to thank both of you for your kind and sincere suggestion. But, I have updated the .cls file only after changing the name of the class file to "MastersDoctoralThesis_UoM."
Regarding, the last request of removing un-necessary space above contents, list of tables, list of figures and chapter is not yet solved. Please find the small piece of code below and help me out in this regard.
\renewcommand{\abovechapterspace}{\vspace{0pt}}
\usepackage{titlesec}
\titleformat{\chapter}[display]
%{\normalfont\huge\bfseries\raggedleft}{%
%\chaptertitlename\ \thechapter\\[3.5ex]\titlerule}
{\normalfont\huge\bfseries\raggedleft}{\chaptertitlename\ \thechapter \\ \hrulefill}
{24pt}{\Huge}
Johannes_B
Site Moderator
Posts: 4163
Joined: Thu Nov 01, 2012 4:08 pm
This still is not a minimal working example, but nevermind. How should have anyone guessed, that you are using package titlesec? What you are looking for is
\titlespacing*{\chapter}{0cm}{-\topskip}{2\baselineskip}[0pt]
With an unmodified and up to date version of the template, this can be done with just three (or four) lines of code.
Which brings me back a few months.
The smart way: Calm down and take a deep breath, read posts and provided links attentively, try to understand and ask if necessary.
vinaykumarn
Posts: 15
Joined: Wed Oct 19, 2016 6:30 am
Respected Sir,
Thank you sir, the single line code worked fine.
\documentclass[12pt, % The default document font size, options: 10pt, 11pt, 12ptoneside, % Two side (alternating margins) for binding by default, uncomment to switch to one sideenglish, % ngerman for Germanonehalfspacing, % Single line spacing, alternatives: onehalfspacing or doublespacing%draft, % Uncomment to enable draft mode (no pictures, no links, overfull hboxes indicated)nolistspacing, % If the document is onehalfspacing or doublespacing, uncomment this to set spacing in lists to single%liststotoc, % Uncomment to add the list of figures/tables/etc to the table of contents%toctotoc, % Uncomment to add the main table of contents to the table of contentsparskip, % Uncomment to add space between paragraphs%nohyperref, % Uncomment to not load the hyperref package%headsepline, % Uncomment to get a line under the header] \usepackage{titlesec}\titleformat{\chapter}[display]%{\normalfont\huge\bfseries\raggedleft}{% %\chaptertitlename\ \thechapter\\[3.5ex]\titlerule} {\normalfont\huge\bfseries\raggedleft}{\chaptertitlename\ \thechapter \\ \hrulefill} {24pt}{\Huge} \titlespacing*{\chapter}{0cm}{-2\topskip}{2\baselineskip}[0pt] % TO REMOVE TOP SPACE AND BASELINE SPACE OF CHAPTER, TABLE OF CONTENTS, FIGURES, AND TABLES %----------------------------------------------------------------------------------------% LIST OF CONTENTS/FIGURES/TABLES PAGES%---------------------------------------------------------------------------------------- \tableofcontents % Prints the main table of contents \listoffigures % Prints the list of figures \listoftables % Prints the list of tables \end{document}
Last edited by Stefan Kottwitz on Wed Jun 21, 2017 12:31 pm, edited 1 time in total.
Reason: code marked
Johannes_B
Site Moderator
Posts: 4163
Joined: Thu Nov 01, 2012 4:08 pm
The smart way: Calm down and take a deep breath, read posts and provided links attentively, try to understand and ask if necessary.
Stefan Kottwitz
Posts: 9568
Joined: Mon Mar 10, 2008 9:44 pm
Here is an example:
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# [Calvin] Which solution would you feature (9)?
Previous discussion
Below, we present a problem from the 2/11 Algebra and Number Theory set, along with 3 student submitted solution. You may vote up for the solutions that you think should be featured, and should vote down for those solutions that you think are wrong.
LCM with 1000 How many positive integers $$N$$ are there such that the least common multiple of $$N$$ and 1000 is 1000?
You may try the problem by clicking on the above link.
All solutions may have LaTeX edits to make the math appear properly. The exposition is presented as is, and has not been edited.
$\mbox{Remarks from Calvin}$
When it comes to proof-writing, it is often difficult to determine the exact amount of information to provide. You should identify your audience, which would be someone of the same mathematical ability who is unable to solve the problem. You should provide enough justification for your peers to understand, without requiring them to read your mind. Because this question is very basic, you have to explain the important statement, i.e. show that "LCM$$(N, 1000) = 1000$$ if and only if $$N$$ is a divisor of 1000". This is a unique property, which doesn't hold for "LCM$$(N, M) = 1000$$", where $$M \neq 1000$$. You will need to explain why non-divisors of 1000 will not work, and why a divisor of 1000 will work.
Solution A - The statement "As the lowest common multiple is 1000, we are looking for numbers which divide perfectly into 1000." is merely a restatement of the "if and only if property", and doesn't offer any justification. While he was able to present the correct "interpretation of the question", it offers no explanation of the reasoning apart from saying that "$$N$$ is no more than 1000".
Solution B - I have chosen to feature this solution presented by Russell, because it explains the "If and only if" fact. I like that he mentioned "Because the other number is 1000", to stress that this solution is unique to the condition given.
As Omid mentioned, the explanation could be made cleaner. He should have continued to refer to "the number $$N$$", and "the other number 1000". When talking about a function on 2 variables, make sure you distinguish the two. An alternative way is to talk about the variable in the first coordinate / entry, vs the variable in the second coordinate / entry.
Solution C - This solution is correct, and can be understood by others who have already solved the problem. However, you will often be asked to explain to others who do not already understand it, and so you should learn how to provide enough information that others will be willing to agree with you.
For students who have difficulty counting the number of divisors of an integer, read the blog post.
Pop Quiz: How would you approach the general problem: How many positive integers $$N$$ are there such that the least common multiple of $$N$$ and $$M$$ is $$A$$?
Note by Calvin Lin
5 years, 4 months ago
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Solution C - Denote the least common multiple of two integers, m and n, by lcm(m,n). lcm(n,1000)=1000 if and only if n divides 1000. All that remains is to count the divisors of 1000. Note that $$1000=(2^3)*(5^3)$$. The exponents in the prime factorization enable us to count the divisors. Simply add 1 to each exponent, and then multiply them all together. In this case, 3+1=4, so the number of divisors is 4*4=16.
Staff - 5 years, 4 months ago
Clearly the easiest and clearest.
- 5 years, 4 months ago
Not only that, but it also shows how to generalize the problem for numbers larger than $$1000$$, which is always nice.
- 5 years, 4 months ago
clearly C is the best.
- 5 years, 4 months ago
a good solution.
- 5 years, 4 months ago
This is a very clear and easy solution,and can be applied to other numbers.
- 5 years, 4 months ago
Solution A - The most important step to solving this is interpretation of the question. A lowest common multiple is the lowest integer into which two smaller integers perfectly and commonly divide. Therefore, in the statement that the lowest common multiple of N and 1000 is 1000, it can be inferred that N is no more than 1000 (and as stated by 'positive integers' - no less than 1). As the lowest common multiple is 1000, we are looking for numbers which divide perfectly into 1000. Factors of 1000, essentially. There are 16 factors of 1000: 1 ( 1 * 1000 = 1000 ) 2 ( 2 * 500 = 1000 ) 4 ( 4 * 250 = 1000 ) 5 ( 5 * 200 = 1000 ) 8 ( 8 * 125 = 1000 ) 10 ( 10 * 100 = 1000 ) 20 ( 20 * 50 = 1000 ) 25 ( 25 * 40 = 1000 ) 40 ( 40 * 25 = 1000 ) 50 ( 50 * 20 = 1000 ) 100 ( 100 * 10 = 1000 ) 125 ( 125 * 8 = 1000 ) 200 ( 200 * 5 = 1000 ) 250 ( 250 * 4 = 1000 ) 500 ( 500 * 2 = 1000 ) 1000 ( 1000 * 1 = 1000 ) Q.E.D. There are sixteen positive integers N such that the least common multiple of N and 1000 is 1000.
Staff - 5 years, 4 months ago
Although this solution is correct, it is unnecessarily wordy. For example, much of the introduction could be omitted. Also, words such as "inferred" should be avoided, as they imply that the statement is not absolutely true. Note that it is generally easier to count the divisors of a number using the prime factorization method in Solution C, rather than listing them.
- 5 years, 4 months ago
- 5 years, 4 months ago
This was worth 100 points. Level 5's would not have seen it.
Staff - 5 years, 4 months ago
post all the three solutions
- 5 years, 4 months ago
- 5 years, 4 months ago
Staff - 5 years, 4 months ago
Solution B - For the least common multiple of N and 1000 to be 1000, the number is already limited to $$N \leq 1000$$ * because *any number over 1000 cannot have 1000 as a multiple. From there, for 1000 to be a multiple at all, the number must be an integer divisor of 1000. Because the other number is 1000, N can be any divisor of 1000 at all, because the lowest multiple of 1000 is itself. From here, we just find all the divisors of 1000, and count how many there are. We thus get $$1\times1000$$, $$2\times5000$$, $$4\times250$$, $$5\times200$$, $$8\times125$$, $$10\times100$$, $$20\times50$$, and $$25\times40$$. The resulting answer is * *16 **: (1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, and 1000).
Staff - 5 years, 4 months ago
There is a typo..it had to be $$2 \times 500$$
- 5 years, 4 months ago
This solution appears to be correct. However, in some places the wording is ambiguous. In particular, it is not immediately clear what "the number" and "the other number" refer to. On another note, it is unnecessary to find all of the divisors of 1000; we only need to count how many there are (see Solution C for a quick way to do this).
- 5 years, 4 months ago
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# What term should I use when I am quantifying the amount of matter, mass or weight?
I am writing an academic article but I got confused as I am writing about the weight a specific metal got by using balance scale. Should I use the term mass, which has a unit of kg, or weight, which is a force created by gravity and expressed in N?
• As long as your substance never leaves Earth, who cares? Jun 22, 2017 at 14:31
• Well, maybe it is about E Musk's travel towards space, Moon, Mars ... Jun 22, 2017 at 14:39
• If you're being technical, note that amount of matter (or substance) is already a term in its own right. We generally measure this value in moles.
– Zhe
Jun 22, 2017 at 14:53
• Weight is not expressed in N, but gram or kg. Weight is measured via the gravitational force, but it is not one. A common anglosaxon unit for a force is gf, gram force. Weight and mass use the same units, although they are only identical on the surface of an idealised earth.
– Karl
Jun 22, 2017 at 21:07
• @Karl any physics textbook would say you're talking rubbish. Weight is a force. Force = mass x acceleration. A mass of 1kg weighs 9.8N on earth and 1.6N on the moon, due to the respective gravitational accelerations of 9.8m/s^2 and 1.6m/s^2. If you're going to be a Pedant, get it right. All that said, in nonscientific English (and even in chemistry, where we don't need to make the distinction) it would probably be acceptable to say something has a weight of 1kg instead of a mass of 1kg. In particular, saying "this weighs 1kg" is much less awkward than the more correct "this has a mass of 1kg". Jun 22, 2017 at 22:13
Because of the reasoning provided by you, i.e. mass is universal, and weight is the product of mass and magnitude of the local gravitational acceleration, I suggest to stay at mass. (A purist's view, taking @Ivan's comment into account.)
In addition, referring to the International System of Units, their brochure not only mentions the basic units, but the relevant section 2.1.1.2 starts with:
Unit of mass (kilogram)
It's up to you, ultimately. The Wikipedia entry for weighing scales states that:
The balance (also balance scale, beam balance and laboratory balance) was the first mass measuring instrument invented.
Further:
A balance or pair of scales using a balance beam compares masses by balancing the weight due to the mass of an object against the weight of a known mass or masses
and
the balance or pair of scales using a traditional balance beam to compare masses will read correctly for mass even if moved to a place with a different (non-zero) gravitational field strength (but would then not read correctly if calibrated in units of force).
In comparison to a spring balance:
Either type can be calibrated to read in units of force such as newtons, or in units of mass such as kilograms.
Finally:
Technically, a balance compares weight rather than mass, but, in a given gravitational field (such as Earth's gravity), the weight of an object is proportional to its mass, so the standard "weights" used with balances are usually labeled in units of mass (g, kg, etc.).
• I think you have a good reasoning, but with the fact that you had used Wikipedia as a reference source disappointed me. But still, thanks.
– Acid
Jun 23, 2017 at 9:29
• Well, the reference is as good as any, and does in fact address the specific issue of using a balance scale, which is not addressed by others. Your query is actually something that will elicit answers that are opinion-based. The first sentence in my answer is the relevant one: it is up to you as the author of the publication. You might get feedback from referees concerning what choice you make. Jun 23, 2017 at 12:09
Mass. Weight is mass in a gravity field or acceleration frame.
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# Lecture 11 – Conditional Statements and Iteration¶
## DSC 10, Fall 2022¶
### Announcements¶
• Homework 3 is due tomorrow at 11:59PM.
• Lab 4 is due on Saturday, 10/22 at 11:59PM.
• The Midterm Project will be released Wednesday!
• Partners are not required, but strongly encouraged.
• Before or after discussion today, we'll host a mixer to help you find a partner! See this post on EdStem for details.
• You must use the pair programming model when working with a partner.
• If you have a conflict with your assigned discussion, email TA Dasha (dveraksa@ucsd.edu) to request to attend another.
• Look at the Grade Report on Gradescope to see your scores on all assignments, discussion attendance, and number of used slip days so far.
### Agenda¶
• Booleans.
• Conditional statements (i.e. if-statements).
• Iteration (i.e. for-loops).
Note:
• We've finished introducing new DataFrame manipulation techniques.
• Today we'll cover some foundational programming tools, which will be very relevant as we start to cover more ideas in statistics (next week).
## Booleans¶
### Recap: Booleans¶
• bool is a data type in Python, just like int, float, and str.
• It stands for "Boolean", named after George Boole, an early mathematician.
• There are only two possible Boolean values: True or False.
• Yes or no.
• On or off.
• 1 or 0.
• Comparisons result in Boolean values.
### Boolean operators; not¶
There are three operators that allow us to perform arithmetic with Booleans – not, and, and or.
not flips True ↔️ False.
### The and operator¶
The and operator is placed between two bools. It is True if both are True; otherwise, it's False.
### The or operator¶
The or operator is placed between two bools. It is True if at least one is True; otherwise, it's False.
### Order of operations¶
• By default, the order of operations is not, and, or. See the precedence of all operators in Python here.
• As usual, use (parentheses) to make expressions more clear.
### Booleans can be tricky!¶
For instance, not (a and b) is different than not a and not b! If you're curious, read more about De Morgan's Laws.
### Note: & and | vs. and and or¶
• Use the & and | operators between two Series. Arithmetic will be done element-wise (separately for each row).
• This is relevant when writing DataFrame queries, e.g. df[(df.get('capstone') == 'finished') & (df.get('units') >= 180)].
• Use the and and or operators between two individual Booleans.
• e.g. capstone == 'finished' and units >= 180.
### Concept Check ✅ – Answer at cc.dsc10.com¶
Suppose we define a = True and b = True. What does the following expression evaluate to?
not (((not a) and b) or ((not b) or a))
A. True
B. False
C. Could be either one
### Aside: the in operator¶
Sometimes, we'll want to check if a particular element is in a list/array, or a particular substring is in a string. The in operator can do this for us:
## Conditionals¶
### if-statements¶
• Often, we'll want to run a block of code only if a particular conditional expression is True.
• The syntax for this is as follows (don't forget the colon!):
if <condition>:
<body>
• Indentation matters!
### else¶
else: Do something else if the specified condition is False.
### elif¶
• What if we want to check more than one condition? Use elif.
• elif: if the specified condition is False, check the next condition.
• If that condition is False, check the next condition, and so on, until we see a True condition.
• After seeing a True condition, it evaluates the indented code and stops.
• If none of the conditions are True, the else body is run.
What if we use if instead of elif?
### Example: Percentage to letter grade¶
Below, complete the implementation of the function, grade_converter, which takes in a percentage grade (grade) and returns the corresponding letter grade, according to this table:
Letter Range
A [90, 100]
B [80, 90)
C [70, 80)
D [60, 70)
F [0, 60)
Your function should work on these examples:
>>> grade_converter(84)
'B'
'D'
### Activity¶
def mystery(a, b):
if (a + b > 4) and (b > 0):
return 'bear'
elif (a * b >= 4) or (b < 0):
return 'triton'
else:
return 'bruin'
Without running code:
1. What does mystery(2, 2) return?
2. Find inputs so that calling mystery will produce 'bruin'.
## Iteration¶
### for-loops¶
• Loops allow us to repeat the execution of code. There are two types of loops in Python; the for-loop is one of them.
• The syntax of a for-loop is as follows:
for <element> in <sequence>:
<for body>
• Read this as: "for each element of this sequence, repeat this code."
• Note: lists, arrays, and strings are all examples of sequences.
• Like with if-statements, indentation matters!
### Example: Squares¶
The line print(num, 'squared is', num ** 2) is run four times:
• On the first iteration, num is 4.
• On the second iteration, num is 2.
• On the third iteration, num is 1.
• On the fourth iteration, num is 3.
This happens, even though there is no num = anywhere.
### Activity¶
Using the array colleges, write a for-loop that prints:
Revelle College
John Muir College
Thurgood Marshall College
Earl Warren College
Eleanor Roosevelt College
Sixth College
Seventh College
for college in colleges:
print(college + ' College')
### Ranges¶
• Recall, each element of a list/array has a numerical position.
• The position of the first element is 0, the position of the second element is 1, etc.
• We can write a for-loop that accesses each element in an array by using its position.
• np.arange will come in handy.
### Example: Goldilocks and the Three Bears¶
We don't have to use the loop variable!
### Randomization and iteration¶
• In the next few lectures, we'll learn how to simulate random events, like flipping a coin.
• Often, we will:
1. Run an experiment, e.g. "flip 10 coins."
2. Keep track of some result, e.g. "number of heads."
3. Repeat steps 1 and 2 many, many times using a for-loop.
### Storing the results¶
• To store our results, we'll typically use an int or an array.
• If using an int, we define an int variable (usually to 0) before the loop, then use + to add to it inside the loop.
• If using an array, we create an array (usually empty) before the loop, then use np.append to add to it inside the loop.
### np.append¶
• This function takes two inputs:
• an array
• an element to add on to the end of the array
• It returns a new array. It does not modify the input array.
• We typically use it like this to extend an array by one element: name_of_array = np.append(name_of_array, element_to_add)
• Remember to store the result!
### Example: Coin flipping¶
The function flip(n) flips n fair coins and returns the number of heads it saw. (Don't worry about how it works for now.)
Let's repeat the act of flipping 10 coins, 10000 times.
• Each time, we'll use the flip function to flip 10 coins and compute the number of heads we saw.
• We'll store these numbers in an array, heads_array.
• Every time we use our flip function to flip 10 coins, we'll add an element to the end of heads_array.
Now, heads_array contains 10000 numbers, each corresponding to the number of heads in 10 simulated coin flips.
### for-loops in DSC 10¶
• Almost every for-loop in DSC 10 will use the accumulator pattern.
• This means we initialize a variable, and repeatedly add on to it within a loop.
• Do not use for-loops to perform mathematical operations on every element of an array or Series.
• Instead use DataFrame manipulations and built-in array or Series methods.
• Helpful video 🎥: For Loops (and when not to use them) in DSC 10.
### Working with strings¶
String are sequences, so we can iterate over them, too!
### Example: Vowel count¶
Below, complete the implementation of the function vowel_count, which returns the number of vowels in the input string s (including repeats). Example behavior is shown below.
>>> vowel_count('king triton')
3
>>> vowel_count('i go to uc san diego')
8
## Summary, next time¶
### Summary¶
• if-statements allow us to run pieces of code depending on whether certain conditions are True.
• for-loops are used to repeat the execution of code for every element of a sequence.
• Lists, arrays, and strings are examples of sequences.
### Next time¶
• Probability.
• A math lesson – no code!
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- Current Issue - Ahead of Print Articles - All Issues - Search - Open Access - Information for Authors - Downloads - Guideline - Regulations ㆍPaper Submission ㆍPaper Reviewing ㆍPublication and Distribution - Code of Ethics - For Authors ㆍOnlilne Submission ㆍMy Manuscript - For Reviewers - For Editors
Cylinders in del Pezzo surfaces with du Val singularities Bull. Korean Math. Soc. 2017 Vol. 54, No. 5, 1655-1667 https://doi.org/10.4134/BKMS.b160684Published online September 30, 2017 Grigory Belousov Plekhanov Russian University of Economics Abstract : We consider del Pezzo surfaces with du Val singularities. We'll prove that a del Pezzo surface $X$ with du Val singularities has a $-K_X$-polar cylinder if and only if there exist tiger such that the support of this tiger does not contain anti-canonical divisor. Also we classify all del Pezzo surfaces $X$ such that $X$ has not any cylinders. Keywords : cylinder, del Pezzo surface MSC numbers : 14E07, 14E25, 14J45 Downloads: Full-text PDF
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# Contents
Years ago, I wrote about a particular type of interview question that I despise. Today I'd like to discuss a much more specific question, rather than a type. I've never been asked this question myself, but I have seen it asked in an actual interview, and I officially nominate it as the worst question I've ever heard in an interview.
A co-worker at a previous company used to ask this question, and it was the first time I'd ever heard it in an interview setting. This company did pair interviews, two engineers with one candidate. One day he and I were the two engineers interviewing some poor candidate. The candidate had actually done pretty well as far as I was concerned, and then my co-worker busted this question out. The candidate stumbled over the answer, visibly frustrated with himself. In the post-interview pow-wow, all of the engineers who'd interviewed him gave him the thumbs up, except my interview partner, who refused to hire him on the grounds that he completely flubbed this question, and "any engineer worth his salt should be able to answer it." He actually said that if we hired this individual, he would be unwilling to work on a team with the candidate. For what it's worth, the story has a happy ending, in that we hired the candidate in spite of his protests, fired the co-worker within a few months, and the candidate is still at that company, doing quite well.
Anyway, I think this question perfectly represents everything that can go wrong with an interview question, so I'd like to discuss it here to explain why it's almost hilariously awful as an interview question:
Write a function that can detect a cycle in a linked list.
Seems like your basic algorithm coding question at first, right? Hop up and write the function on the white board; totally reasonable, right? Except it's not, it's brain-meltingly terrible. Let's break it down.
# 1. It's completely inappropriate
This is a job interview. You have a dynamic where you're talking to someone who is interviewing for a job. It's naturally nerve-wracking, and "puzzler" questions where there's some "a-ha" moment of clarity are the worst kind of programming questions you can ask. If you don't have the a-ha moment in the interview, you won't get it, and a good chunk of your brain will be devoted to thinking "oh shit I'm blowing this interview" rather than focusing on the question at hand.
People like to pose puzzlers to "see how people think" but that's nonsense in the case of puzzler questions. You can't reason your way through a puzzler, that's why it's a puzzler. You just have to hope you have the a-ha moment. Sometimes I've heard people like to "see how people handle pressure" but they're already in an interview, the pressure is already there.
Asking puzzler questions is a complete waste of time, all you're doing is testing if someone has seen your particular puzzler before or not. You may also be testing their acting chops, as the person who has heard the question before pretends it's their first time hearing it, and they feign reasoning their way through the problem to arrive at the answer they already know as soon as the question comes out of your mouth.
This particular problem is the worst offender in this regard. Why, you ask? Well, imagine if someone truly was hearing this problem for the first time, and you're expecting them to reason their way to the answer.
In this case, the generally "correct" answer is a tortoise-and-hare algorithm, where you have two pointers at the head of your linked list, one traverses the list two nodes at a time, and one traverses the list one node at a time; if ever the pointers point to the same node, you have a cycle somewhere.
Sure, there are easier answers, like marking each node with some kind of 'seen' boolean, or traversing the list from each node to see if you come back to it, or duplicating the list into a hash and looking for a collision, but as soon as you provide those answers, the interviewer will add restrictions saying to use less memory or use less time or don't modify the underlying data structure. The only one that makes the question "stop" is the tortoise-and-hare algorithm.
Is it reasonable to expect someone to think of this, from scratch? After all, you're pretty confident you could think of it, right? Well, the Linked List as a data structure was discovered by Allen Newell, Cliff Shaw and Herbert A. Simon in 1955. The "correct" cycle detection algorithm for a Linked List is named "Floyd's cycle-finding algorithm" in honor of its inventor, Robert W. Floyd, who discovered it in a 1967 paper.
Between 1955 and 1967, the problem of "how do we determine if there is a cycle in a linked list without modifying the list" was an open problem. Meaning, any number of PhD candidates in Mathematics or Computer Science could have written about it as part of their dissertation. With all of those hundreds and hundreds of minds, this problem remained open for 12 years.
Do you honestly think you could, in a twenty minute interview, from scratch, come up with the solution to a problem that remained open in the field for 12 years, all under a pressure far more intense than any academic? Seems pretty damn unlikely, the only reason you think you could do so is that you've heard the answer before, and it seems obvious and simple in retrospect. In other words, "a-ha!"
# 2. It's completely disconnected from reality
As if the above weren't reason enough for this to be a laughably bad question, you have to also ask yourself, is this even a good question for determining if this engineer has the skills they need for the job?
Let me challenge the question altogether: why would you ever find yourself in a situation where your linked list has a cycle?
In the real-world, what could lead to this? I don't mean mechanically, obviously you get a cycle if you have a node whose "next" pointer is upstream of that node. I mean, how does it actually happen in real life?
See, a Linked List is a data structure, it's not an abstract data type. Generally you wouldn't be making a LinkedList class, you'd be making a Stack or a Queue or something like that. Those would be the classes you're writing and exposing to a consumer of your class, and it would just so happen that your internal implementation of those types is a linked list. So what are the methods on your Stack class, for example? push, pop, peek, etc? Well, if someone is using those methods, how on earth are you going to get a cycle in your list? They're not messing with the next or prev pointers, they're just pushing and popping with objects of some type.
Even if you wrote a LinkedList class for some library, you still can't find yourself in this situation. Take a look at Java's LinkedList class. There is no way to manipulate the pointers for the node's next or previous references. You can get the first, or get the last, or add an object to a specific place in the list, or remove an object by index or by value.
Take a look at the Java source code and you'll find those next and previous pointers are here, inside of LinkedList:
private static class Entry <E> {
E element;
Entry(E e, java.util.LinkedList.Entry<E> entry, java.util.LinkedList.Entry<E> entry1) { /* compiled code */ }
}
This is a private static class, inside of LinkedList. You can't instantiate a LinkedList.Entry. You have no way to manipulate these next or previous pointers. Because those things are the state of the list, and LinkedList encapsulates the behaviors with the state inside of the class, like it ought to.
If your LinkedList class were vulnerable to any kind of cycle creation, you've done a poor job of encapsulation. You either have a design failure in your interface, or a bug in your implementation. In either case, your time would be better spent addressing your error rather than coding up some kind of cycle detection mechanism.
Here's the only cycle detector you'll ever need to write for your LinkedList:
public class LinkedList {
public boolean containsCycle() {
return false;
}
}
There is no real situation in which this method's return value would be different than one that uses a tortoise-and-hare algorithm.
In the real world of actual coding, you'd very rarely find yourself ever needing to code up a linked list implementation from scratch, but if you did, you'd certainly have no reason to expose methods that would allow someone using your code to create a cycle. The only way it could be done is through intentional, malicious metaprogramming or reflection of some kind, which could just as easily bypass your detectCycle method anyway.
# Conclusion
Many interview questions fail for one of these two reasons. Either the question is too much of a puzzler to reasonably be solved in an interview setting, or it's so far removed from the skillset required to do the job ("how would you move Mt. Fuji") that it's useless.
This question, hilariously, suffers from both of these major problems, and it suffers from each about as hard as it possibly could.
If you're asking this question, everyone who has ever answered it to your satisfaction was merely proving they have good memory recall abilities from their Computer Science curriculum, nothing more. Folks you turned away from the job for failing to answer this question may have been more qualified than you realize, and your company maybe should have turned you away instead.
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Mansard roof is not a one-size-fits-all design. You have to consider your home structure, budget, and natural environment. If you do not have any issues, then it is the right one for your abode.
House roof styles are something most of us take for granted — but what if you didn’t have the resources to hire a contractor?
Luckily, there’s no need to worry. A quick online search will provide you with everything you need to know about the different roof styles available today.
One of them, the Mansard roof, is a design that has its roots in the architecture of France. It has become popular in modern architecture due to its appeal as a dramatic style that exudes power and grandeur.
But, do you know all the advantages and disadvantages of Mansard roofs? Don’t worry about it. Just head on to the next section to get a complete look at these.
Contents
What is a Mansard Roof? (French or Curb Roof)
Mansard roofs are a type of two-sloped roof, with the lower slope on the side closer to being vertical. The upper (shallower) pitch is found at the same angle as a standard gable roof.
When used in architecture, this type of roof gives the impression that masonry blocks have been built up within recessed frames all along each slope.
This style is typical for storehouses with a French background and can also be seen in buildings from other countries.
Best of all, a Mansard roof can be installed on almost any type of house — from ranch-style homes to Victorian mansions.
The Advantages of a Mansard Roof
Before you decide whether it suits your needs or not, let’s take a quick look at the most important advantages and disadvantages of Mansard roofs.
• Aesthetic value
Mansard roofs are usually a mixture of the straight lines and curves found in the French architecture of the past. As a result, you can expect them to blend well with many types of homes.
• Extra room for the attic
Mansard roofs are longer than standard gable-style roofs, which helps you add additional storage space to the attic without using too much space. This is especially useful if your attic is small and you don’t have much extra storage space.
• Makes it easier to expand
Mansard roofs have a “bowed” appearance, making them better at accommodating architectural design elements such as bay windows. This is especially useful if you want to expand your house later on.
• Less maintenance
Also, Mansard roofs are not as prone to damage from water as other types of roofs. Therefore, you won’t have to spend too much on periodic repairs.
• Works well in rural and urban areas
Mansard roofs are quite comfortable in rural areas, especially when it comes to blocking the wind. This helps in keeping the house safe and warm. Safety is also enhanced since this type of roof has a lower risk of fire than other houses.
The Disadvantages of a Mansard Roof
Now that you know all the advantages of this type of roof let’s look at the disadvantages of Mansard roofs.
• Expensive to repair and replace
Due to the shape and size of Mansard roofs, they’re usually large in both size and cost. This means that they’re a bit more expensive to repair or replace.
• It takes more time to install
If you hire a contractor to do the installation for you, it will probably take a little longer than with other types of roofs. This is due to the complexity of the design.
• It looks complicated to build
Mansard roofs look complicated to build. This is especially true if there’s no room for workers on the sides to do the actual construction. If you want to build this type of roof independently, you’ll have to be very careful to avoid problems.
You May Also Like: Difference Between Arbors, Pergolas, Gazebos, and Pavilions
How is a Mansard Roof Structured?
The roof has the same basic structure as a gable roof, except for the two gable ends. The fourth wall of a Mansard roof is perpendicular to the one of a normal gable.
For this to be possible, the slope of a Mansard roof is shallower than that of a normal gable. The side walls are set at the same angle as a normal gable roof.
An additional angle of about 20 degrees is added to the slope of the roof. The roof has one sloping side facing south and another sloping side facing west. This structure is used for building storehouses, and mansard roofs are often seen in France.
How to Build a Mansard Roof
Step 1: The frame of the Mansard roof is created by a series of vertical four-by-four beams that intersect at right angles.
Step 2: The next step is to construct the outer walls. As you create the outer wall of a Mansard roof, place a beam at the top corner first and anchor it with another beam. Then build another beam about 2 inches higher than the one we mentioned above. Anchor that one too.
Step 3: After this, put the second course on the oblique line from the corner of the outer wall. This means that the second beam you build will be rested at a 45-degree angle on top of its twin. Then nail it to the first one.
Step 4: Repeat the same process as you construct the third beam but don’t forget to put one more beam one inch lower than a second beam, which will be resting on this one.
Step 5: As you place the next course, remember to adjust the angle of the beam by adding or removing a beam at the top corner.
Step 6: The inner walls are going to be created next. The first beam put together with its mate will be resting on top of the second beam that bears a 45-degree angle.
Step 7: When you build the wall of the Mansard roof, do not forget to adjust its angle just as you did when building its outer wall.
Mansard Roof Design Variations
There are other Mansard roofs, which have double pitch roofing systems. They are called “two-slope gabled roofs” or simply “double-slope gabled roofs. ” They are different from the normal Mansard roofs in that the slope is not exactly as we described above.
The lower slope of a double-slope gabled roof is steeper than the Mansard roof, but the upper slope is not as steep.
The overall height of a double-slope gabled roof is considerably shorter too. This means that the distance between the ground and the ridge is considerably shorter.
There are three windows on the Mansard roof in some designs rather than two, like traditional Mansards. The last one is taller and is set back further from the soffit line.
FAQs
How much does a mansard roof cost?
It cost between £45,000 and £65,000 (US $75,000 and$110,000). The additional cost of a mansard roof is mainly due to the project management and the labor. However, the materials used for a mansard roof are mild steel, zincalume, and plywood.
How to update a mansard roof?
You can roll the modified Mansard roof. The rolled mansard roof is also known as a “French hotel roof.” It is a solution to updating a mansard roof that has been damaged or has been affected by termites infestation.
How to modernize a mansard roof?
You can modernize a mansard roof by using aluminum as one of the building materials used in the process. The aluminum will help you save a lot of energy to maintain a traditional Mansard roof.
How to frame a mansard roof?
You can use a Hyperframe as a structural solution for a Mansard roof. The Hyperframe can be used on a mansard roof that is of all-steel construction.
Conclusion
The Mansard roof is a common design that is very easy to make and can be a cheap option for modernizing your home. The Mansard roof benefits the homeowners and brings uniqueness to the house due to its unique look.
While some homeowners may find the Mansard roof to be a pain to maintain, most homeowners know that they can easily fix a Mansard roof and are very happy with its features and benefits.
If you hire a contractor to do the roof installation for you, it will probably take a little longer than with other types of roofs. This is due to the complexity of the design.
The Essential Guide to Board and Batten Siding
If you plan to remodel your home, the rustic style should be taken into account. Although it is…
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Home SpecialCells(xlCellTypeVisible) not working in UDF
# SpecialCells(xlCellTypeVisible) not working in UDF
CLR
1#
CLR Published in 2017-04-05 14:38:18Z
Based on the question posed by @Chips Ahoy, I decided to create a UDF to find the PercentRank of visible cells in a range. While @Chips seems happy with my syntax correction, I am actually unable to get my UDF to work correctly. When I run the below, the two addresses output read identical. In my example using a formula of =VisiblePercentRank($A$2:$A$41,0.5) , both addresses output to the immediate window read $A$2:$A$41, despite rows 3 to 11 being hidden by an autofilter. Code: Function VisiblePercentRank(x As Range, RankVal As Double) Debug.Print x.Address, x.Rows.SpecialCells(xlCellTypeVisible).Address VisiblePercentRank = WorksheetFunction.PercentRank(x.Rows.SpecialCells(xlCellTypeVisible), RankVal) End Function Also tried removing .Rows: Function VisiblePercentRank(x As Range, RankVal As Double) Debug.Print x.Address, x.SpecialCells(xlCellTypeVisible).Address VisiblePercentRank = WorksheetFunction.PercentRank(x.SpecialCells(xlCellTypeVisible), RankVal) End Function Should the second output not read $A$2,$A$12:$A$41 or have I missed something? Using Excel/Office 2013, 64bit on Win7, 64bit. BRAIN FRYING UPDATE I have found that my UDF works if I run it from the immediate window: ?VisiblePercentRank(range("A2:A41"),0.5) $A$2:$A$41 $A$2:$A$11,$A$39:$A$41 0.207 But if run from an in-cell formula of =VisiblePercentRank(A2:A41,0.5): $A$2:$A$41 $A$2:$A$41
CallumDA
2#
It seems that SpecialCells is known to fail in UDFs. A few sources: 1, 2, 3 You'd have to create your own function. Perhaps something like this: Function VisiblePercentRank(x As Range, RankVal As Double) Debug.Print x.Address, VisibleCells(x).Address VisiblePercentRank = WorksheetFunction.PercentRank(VisibleCells(x), RankVal) End Function Private Function VisibleCells(rng As Range) As Range Dim r As Range For Each r In rng If r.EntireRow.Hidden = False Then If VisibleCells Is Nothing Then Set VisibleCells = r Else Set VisibleCells = Union(VisibleCells, r) End If End If Next r End Function
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2005-03-27, 17:07 #23 dave_0273 Oct 2003 Australia, Brisbane 1D616 Posts Status of p-1 using the 21st of March status files... Code: M NUMBER 0-12 0 13 0 14 0 15 167 16 2 17 84 18 233 19 1367 20 1107 21 527 22 305 23 294 24 264 25 146 26 109 27 44 28 0 29 0 30 0 Code: 0.1M RANGE NUMBER CHANGE 15.0 100 0 15.9 67 0 16.9 2 0 17.6 83 0 17.9 1 0 18.6 1 -27 18.7 0 -102 18.8 113 -65 18.9 119 0 19.0 0 -127 19.1 113 0 19.2 136 0 19.3 148 0 19.4 133 0 19.5 138 0 19.6 153 0 19.7 235 0 19.8 130 0 19.9 181 0
2005-04-02, 19:13 #24 dave_0273 Oct 2003 Australia, Brisbane 47010 Posts Status of p-1 using the 2nd April 2005 status files Code: M NUMBER 0-14 0 15 167 16 2 17 84 18 233 19 1366 20 1108 21 531 22 306 23 296 24 270 25 147 26 120 27 49 28 5 29 1 30 0 Code: 0.1MRANGE NUMBER CHANGE 15.0 100 0 15.9 67 0 16.9 2 0 17.6 83 0 17.9 1 0 18.6 1 0 18.7 0 0 18.8 113 0 18.9 119 0 19.0 0 0 19.1 113 0 19.2 136 0 19.3 148 0 19.4 133 0 19.5 137 -1 19.6 153 0 19.7 235 0 19.8 130 0 19.9 181 0 We are flying along. 1 exponent done this week. No, i'm just kidding. Manual forums are currently down so the work done this week could not be submitted. This weeks work will most likely show up next week. I also did a block of work on exponents that were p-1ed to extremely low bounds. This work did not show up as technically (acording to my program at least) those exponents had already been p-1ed. Last fiddled with by dave_0273 on 2005-04-02 at 19:15
2005-04-14, 08:42 #25 dave_0273 Oct 2003 Australia, Brisbane 2·5·47 Posts Status of p-1 using the 12th April 2005 status files Code: M NUMBER 0-13 0 14 0 15 167 16 2 17 83 18 2 19 1366 20 1107 21 531 22 308 23 296 24 271 25 149 26 125 27 60 28 8 29 2 30 0 Code: 0.1MRANGE NUMBER CHANGE 15.0 100 0 15.9 67 0 16.9 2 0 17.6 83 0 17.9 0 -1 18.6 1 0 18.7 0 0 18.8 0 -113 18.9 1 -118 19.0 0 0 19.1 113 0 19.2 136 0 19.3 148 0 19.4 133 0 19.5 137 0 19.6 153 0 19.7 235 0 19.8 130 0 19.9 181 0
2005-04-30, 02:30 #26 dave_0273 Oct 2003 Australia, Brisbane 1D616 Posts Status of p-1 using the 27th April status files Code: M NUMBER 0-13 0 14 0 15 167 16 1 17 83 18 1 19 1231 20 1108 21 532 22 311 23 298 24 271 25 152 26 134 27 67 28 18 29 5 30 0 Code: 0.1MRANGE NUMBER CHANGE 15.0 100 0 15.9 67 0 16.9 1 -1 17.6 83 0 18.6 1 0 18.9 0 -1 19.0 0 0 19.1 113 0 19.2 1 -135 19.3 148 0 19.4 133 0 19.5 137 0 19.6 153 0 19.7 235 0 19.8 130 0 19.9 181 0
2005-05-14, 12:04 #27 dave_0273 Oct 2003 Australia, Brisbane 2×5×47 Posts Status of p-1 using the 12th May status files Code: M NUMBER 0-13 0 14 0 15 167 16 0 17 83 18 0 19 1230 20 1108 21 532 22 313 23 299 24 275 25 153 26 137 27 79 28 22 29 8 30 0 Code: 0.1MRANGE NUMBER CHANGE 15.0 100 0 15.9 67 0 16.9 0 -1 17.6 83 0 18.6 0 -1 18.9 0 0 19.0 0 0 19.1 113 0 19.2 1 0 19.3 148 0 19.4 132 -1 19.5 137 0 19.6 153 0 19.7 235 0 19.8 130 0 19.9 181 0 It appears that no work was done this week or so, but that i just because the results haven't been submitted because the manual pages were down. I am about to bundle all the results together now and email them off to George, so they should hopefully be in the status files next week.
2005-06-27, 07:53 #28 dave_0273 Oct 2003 Australia, Brisbane 2·5·47 Posts Status of p-1 using the 19th June, 2005 Code: M NUMBER 0-15 0 16 0 17 0 18 0 19 547 20 1105 21 531 22 313 23 303 24 281 25 161 26 150 27 103 28 32 29 20 30 0 Code: 0.1MRANGE NUMBER CHANGE 15.0 0 -100 15.9 0 -67 17.5 0 0 17.6 0 -83 17.7 0 0 17.8 0 0 17.9 0 0 18.0 0 0 18.1 0 0 18.2 0 0 18.3 0 0 18.4 0 0 18.5 0 0 18.6 0 0 18.7 0 0 18.8 0 0 18.9 0 0 19.0 0 0 19.1 0 -113 19.2 1 0 19.3 0 -148 19.4 0 -132 19.5 0 -137 19.6 1 -152 19.7 233 -2 19.8 131 +1 19.9 181 0 20.0 133 ? 20.1 90 ? 20.2 117 ? 20.3 101 ? 20.4 108 ? 20.5 125 ? 20.6 112 ? 20.7 92 ? 20.8 126 ? 20.9 101 ? I have FINALLY got my computer up and working again. Appologies for those that have missed not having the status of p-1. I should be more active around the forums once again now. Everything up to (and including) 18M is now done. There are just a few sets left in the 19M range now and then we get to start on the 20 millions.
2005-06-27, 10:10 #29 garo Aug 2002 Termonfeckin, IE 22·691 Posts Excellent!
2005-06-27, 16:15 #30 lycorn "GIMFS" Sep 2002 Oeiras, Portugal 1,493 Posts Great work, dave! Just one quick question: what is the meaning of the figures in the 28-30M ranges?
2005-06-28, 04:02 #31
dave_0273
Oct 2003
Australia, Brisbane
47010 Posts
Quote:
Originally Posted by lycorn Just one quick question: what is the meaning of the figures in the 28-30M ranges?
They are first time tests that have been completed already but did not have a p-1 test done on them at all. Primenet is currently handing out first time LL tests in the 29M range, however very few have been completed yet. In the next couple of months I would expect the numbers in the 25-30M range to increase as more first time tests are returned that have not had a p-1 test done on them.
2005-06-28, 08:17 #32 garo Aug 2002 Termonfeckin, IE 276410 Posts I have a proposal to make. Since we are way ahead of the current doublechecks, we could divert a few resources to doing some P-1 for numbers that are ahead of the first-timers. There are two problems with the proposal as I can see. 1) A P-1 on a first-timer will take a long time. More than 4 times the time. 2) With factoring and LL edges being so close, there is a chance that something gets messed up and work gets wasted. Plus we should only do P-1 on numbers that have already had the required amount of trial factoring done on them which reduces our list of candidates drastically.
2005-06-28, 09:23 #33 dave_0273 Oct 2003 Australia, Brisbane 2·5·47 Posts I have often thought, and have often been asked why we don't do p-1ing ahead of the leading edge of LL testing. Basically those two reasons are why I have never tried it. There is no way (with the current amount of people in mersenne-aries) that we could keep ahead of the leading edge. We could only do a small percentage. The other thing that we would need is for another group to be working with us doing the trail factoring before we do the p-1. However, one the biggest reasons why I have never bothered to go ahead of the leading edge of LL testing is because if people really wanted to do that sort of work, they could always do it the semi-automated way. They could reserve 50 or so exponents, do the p-1 and then release them again. Because the work is therefore technically assigned through primenet, they wouldn't be duplicating work and they wouldn't be stepping on each others toes. This doesn't work with mersenne-aries because you just can't get enough work this way. The majority of exponents are p-1ed on the first LL test. Even when I tried to reserve 50 exponents, I would find that I would get less than 5 that would require a p-1 test.
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# Could ozone be used in a biodome on Mars?
How much ozone would it take to block solar radiation on Mars if chambered in double panned glass or material? Said $O_3$ would break down from the UV. Would applied electricity on $O$ and $O_2$ make $O_3$ again? Glass weighs to much and is not an option unless it is made on Mars. I propose gas because it doesn't brake down like UV film can.
Could a blimp house an entire colony on Mars?
Can air pressure be accumulated this way for a biodome or spacesuit?
• You can't block with anything resembling pure ozone, the UV would rapidly tear it up. You need a small amount of ozone mixed with lots of oxygen so when an ozone is split up the loose O attaches to another O2, not an O3. Apr 3 '18 at 1:14
• The frequencies that ozone blocks are, as @peterh has already said, easily blocked by window glass or many other things. The types of solar radiation that people designing Mars colonies are worried about are different, and Earth is protected from them by its magnetic field, not by ozone. Apr 3 '18 at 6:32
• @SteveLinton because there are other forms of radiation does not mean that "people designing Mars colonies" are not worried about UV. One does not necessarily exclude the other. The part about some kinds of glass or polymers, but not others is of course true. Not all glass will safely block UV. See Berlin’s renovated Tropenhaus botanical garden uses special UV-transmissible glass or just its plot: i.stack.imgur.com/J2yf2.jpg
– uhoh
Apr 3 '18 at 8:11
• @LorenPechtel I updated my question making your comment not applicable. Electricity can be used to make $O_3$
– Muze
Apr 7 '18 at 3:59
If we have $O_2$ lighted with UV, we have actually many reactions working together:
1. $O_2 + \gamma \rightarrow 2O$
2. $O_2 + O \rightarrow O_3$
3. $O_3 + \gamma \rightarrow O_2 + O$
4. $O + O_3 \rightarrow 2 O_2$
5. $O_3 + O_3 \rightarrow 3O_2$
6. $O + O \rightarrow O_2$
(1) produces nascent oxygen. This is slow, and its speed depends on the UV concentration.
(2) builds ozone from nascent oxygen. This is fast.
(3) means the decay of ozone to normal oxygen and nascent oxygen. This can be done very easily with UV light (it has a very big cross-section).
(4), (5) and (6) results the decay of ozone (or nascent oxygen) back to normal oxygen. All of them require that multiple $O$ or $O_3$ molecules need to meet. Thus, it can happen quickly only if there is a high partial ozone pressure.
The net result is that if you light $O_2$ with UV, you get an equilibrial concentration of $O$ and $O_3$ as well. If start with all of ozone, or without a single ozone molecula, the ozone concentration will decay because (4)-(6), or it will be built up because (1), until it reaches this equilibrial concentration. This equilibric concentration will depend on the UV intensity.
(2) and (3) doesn't affect.
Without it, the $O_3$ and $O + O_2$ states will only step into eachother, meanwhile they will eat up a lot of UV radiation. But it can work only if there is a lot of $O_2$ as well.
On the Earth, even in the ionosphere, the ozone concentration is very low: it is roughly 1:100000, and it is between roughly 20 and 30 km. (There are big differences here, for example there is far lesser ozone on the south pole.)
A quick calculation: the pressure of the air decreases to half with around every 5 km elevation. Thus, at 20km high, the pressure is around 1/16 atm. On 30km, it is around 1/64 atm. Calculating with a mean of 1/32, and 10km high, we can compress it to 1atm and 300m height. This calculation is un-exact, but there is no magnitudal differences.
Thus, we would need around a 300m high layer of pure oxygen to get the same UV defense as we have on the Earth.
Remark: a single glass window has a better UV defense as this ozone layer would have, this is why light-skinned people don't get tanned or burned behind them.
Thus, the best UV defense of this biodom would be if it would have simply glass walls. Beside that, there is no ozone layer needed.
• Thanks, light-skinned or light-complected are probably both fine. To me "light-skinned" reminds me of "thin-skinned", whereas complexion is a more specific and neutral word.
– uhoh
Apr 3 '18 at 5:57
• This is a super answer by the way! The chemistry of the ozone layer is complicated, kudos for taking it on an doing a great job!
– uhoh
Apr 3 '18 at 6:00
• Not all glass will safely block UV. See Berlin’s renovated Tropenhaus botanical garden uses special UV-transmissible glass or consider adding the image from that link: i.stack.imgur.com/J2yf2.jpg
– uhoh
Apr 3 '18 at 8:09
• uhoh - I don't think adding a link referring to a specialist glass made to explicitly not block UV is at all relevant here. Apr 3 '18 at 8:34
• It's a short comment, about glass, meant to be helpful, and can easily be ignored or remembered depending on level of interest. I've moved the comments to a chat
– uhoh
Apr 3 '18 at 11:18
The ozone layer of Earth is about 10 km high. A layer of ozone of some meters will not block UV light. Using high pressure ozone would require very heavy domes. A thin AND efficient filter for UV light is needed.
• What if the gas was electrified like a Faraday cage then would it block better?
– Muze
Apr 4 '18 at 17:35
• @Muze What means here "electrified"? Ionized? Faraday-cage mainly doesn't ionize gases. Apr 5 '18 at 13:24
• @peterh Sorry how would you say apply electricity like a neon bulb to the gas?
– Muze
Apr 5 '18 at 15:33
• @Muze It is not Faraday cage, there is an electric arc which ionises the gas (continuously gets away the electrons from the atoms). You get the light as the electrons find an ion and become atoms again (recombination). In such a light bulb, a part of the gas is plasma. Plasma reacts typically much better with any light, so it could be an UV shader, in theory. In practice, I think it is very likely that nothing can stand to compete with a 3mm thick, static glass wall. Apr 5 '18 at 16:49
• @peterh as stated before unless the special glass can be made on Mars using rubber to house the gas for a biodome would much weigh less. There is no such devise and the Faraday cage is the closest devise I could compare it to. Thanks for helping +++1
– Muze
Apr 5 '18 at 22:17
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What caused this mysterious stellar occultation on July 10, 2017 from something ~100 km away from 486958 Arrokoth? By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. Taking common factor from binomial. As we started using this new way of talking not only for the commission, but also for the work in the portfolio, we gained significant traction. Broad Social Determinants of Health Approaches Focus on Community Hospitals, physician groups, and federally-qualified health centers are taking a similar approach to tackling patient social determinants of health: target all SDOHs and work with community partners to address these challenges, according to a recent report from Insights by Xtelligent Healthcare Media. The Three-link Chain is a model that provides an understanding of the complex interplay of biological, psychological, and sociocultural factors that contribute to tobacco dependence. 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You can always add a multiple of a row to another row or a multiple of a column to another column and it doesn't change the value.And the third thing you can do is interchange two rows or two columns, so here for example say for some reasons I want to have this 0 on the left, I can just switch the first two columns so these two columns have been switched. Key words: Obesity: Dietary intake: Factors influencing consumption: Out-of-home foods Introduction Takeaway, take-out and fast foods are common terminology used for various ‘out-of-home’ foods. Hence, means , that is or . Google Classroom Facebook Twitter. Later: expanding in response to comment: the matrices ADVERTISEMENTS: For example, the demand for apparel changes with change in fashion and tastes … A set of determinant factors that have a significant role in bank selection in one nation may prove to be insignificant in another (Rao, 2010). A household is considered as migrant sending if it has at least one migrant member. 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The determinant of a matrix A is denoted det(A), det A, or | A |.Geometrically, it can be viewed as the volume scaling factor of the linear transformation described by the matrix. (Note: also check out Matrix Inverse by Row Operations and the Matrix Calculator.) Are determinants functions, numbers or matrices? If you calculate the two determinants in your third example you get Health starts where we live, learn, work and play. The permanent of a matrix is defined as the determinant, except that the factors sgn(σ) occurring in Leibniz's rule are omitted. It's really just because of the first sentence in my answer. This makes me wonder, taking a common factor out in a determinant might not be mathematically correct (in pure way) even though it results in the same answer (if this makes any sense)? Better understanding the determinants of investor risk taking is not only of theoretical interest, but also has practical implications for financial regulatory requirements in Europe and ongoing discussions about such regulation in the USA. Can I extract common factor from a column in matrix? Reduce the economic costs of … Structured … Beds for people who practise group marriage, A Plague that Causes Death in All Post-Plague Children. Bo Burström, Wenjing Tao, Social determinants of health and inequalities in COVID-19, European Journal of ... to understand how the disease strikes and by which pathways it impacts certain population groups more adversely—taking lessons from previous disease outbreaks. (2012) also conducted a study to identify the factors that influence Greek customers" decision to take out a loan from commercial banks. recall from the previous section that the determinant of a triangular matrix is the product of the entries on its diagonal. Determinants are like matrices, but done up in absolute-value bars instead of square brackets. Frangos et al. For the determinant one, yes. According to Thomas, awareness that social and economic factors can affect health has increased in recent years. Now here what I've done is I've multiplied the first column by 4 and added it to the third column so this column is now 4 times c1 plus c3 right, 4 times 16 is 64 add that to this negative 64 and you get 0 we do that to the whole column. In recent decades, “social determinants of health” has received considerable attention as a foundational concept in the field of population and public health (1). To unlock all 5,300 videos, Are there any gambits where I HAVE to decline? But just because you say so. Social determinants of health (SDOH) are a major source of these inequalities in cancer care. Broad Social Determinants of Health Approaches Focus on Community Hospitals, physician groups, and federally-qualified health centers are taking a similar approach to tackling patient social determinants of health: target all SDOHs and work with community partners to address these challenges, according to a recent report from Insights by Xtelligent Healthcare Media. n. 1. Risk Factors Risk factors for diabetes depend on the type of diabetes. $$2=2\times1$$ Now what's interesting about this row operation it's a column operation in this case is that doesn't change the value of the determinant. To find the determinant of the matrix A, you have to pick a row or a column of the matrix, find all the cofactors for that row or column, multiply each cofactor by its matrix entry, and then add all the values you've gotten. Determinants synonyms, Determinants pronunciation, Determinants translation, English dictionary definition of Determinants. Migration has become a cause of concern at the global, regional, and national levels. Taking on social determinants of health in the COVID era. start your free trial. Like the case of many developing countries, Ethiopia has been facing increasing challenges related to rural out-migration. “We know that the real issue, addressing these social determinants of health, will have a huge impact on overall health and well being. Consider the factorisation of the expression 5x + 15.. $$\det(B)= a_{11}a_{22}(\lambda a_{33})+a_{21}(\lambda a_{32})a_{13}+(\lambda a_{31})a_{12}a_{23}-a_{13}a_{22}(\lambda a_{31})-a_{11}a_{23}(\lambda a_{32})-a_{12}a_{21}(\lambda a_{33})=\lambda(a_{11}a_{22}a_{33}+a_{21}a_{32}a_{13}+a_{31}a_{12}a_{23}-a_{13}a_{22}a_{31}-a_{11}a_{23}a_{32}-a_{12}a_{21}a_{33})=\lambda \det(A).$$. Social determinants of health (SDoH) are conditions and factors such as where a person is born, grows, lives, and works, all of which affect their health and quality of life. This is a general property of determinants: let $A$ be a matrix; multiply, This is a wonderful explanation. Social determinants of health take center stage during the Covid-19 crisis Having insight into SDOH data for their older adult … BTW, I don't remember any notation like that... but who knows. An exploratory qualitative study design was chosen, as there had been no research carried out on parents’/caregivers’ perceptions of determinants of childhood obesity in New Zealand (NZ). We know that: a ( b + c) = ab + ac. Can you also respond to my second comment on Alejandro's answer? However, this factorization can be obtained directly by first factoring each entry in the determinant and taking a common factor of from each row. But it is best explained by working through an example! Note that the common factor 5 has been taken out and placed in front of the brackets. Health services 4. 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Whenever you switch two rows or two columns, it changes the sign of the determinant, so the 3 things you can do to determinant to simplify; one is you can factor a constant out of any row or any column, two you can add any multiple of one row to another row and the same goes with columns and three you can interchange two columns or two rows, just remember to change the sign when you do that. We Tutor to These Levels and Qualifications: Key Stage One (KS1), Key Stage Two (KS2) Primary SATS, and Key Stage Three (KS3) Secondary SATS. Determinant of a Matrix. Improve the life opportunities and health outcomes of young people. For people whose careers revolve around the impact of social determinants of health (SDOH) on medical outcomes, COVID-19 has unfortunately been a familiar narrative. The demand for a product is influenced by various factors, such as price, consumer’s income, and growth of population. Changes in socio-economic inequalities in health can be explained by changes in inequalities in social determinants, namely education, income, housing and residential locations. Thanks for contributing an answer to Mathematics Stack Exchange! A critical impact is driven by the family especially in the early and naive age. The reverse process, ab + ac = a(b + c), is called taking out the common factor. Close to half of the reviews (25 out of 51) did not specify the age group of patients covered by the review. Biology and geneticsIt is the interrelationships among these factors that determine individual and population health. To reduce the amount of computation, we can use methods for simplifying determinants. This is why your second example is correct, and it also shows that the answer to your initial question is "no". Understanding social determinants won’t happen all at once, or from suddenly rolling out a whole new approach - it will come in phased stages. The determinant of a matrix is a special number that can be calculated from a square matrix.. A Matrix is an array of numbers:. Learn how to factor a common factor out of a polynomial expression. 2\left[\matrix{1&0&0\cr0&1&0\cr0&0&1\cr}\right]$$Note that the common factor 5 has been taken out and placed in front of the brackets. Are, Learn The evidence base for social determinants of health as risk factors for infant mortality: A systematic scoping review. Thank you David for your answer. Play video. That's because 2^{1/3} is a number (\sqrt[3]{2}, in fact), just as 2. To provide rapid evidence on the determinants and dynamics of … Regarding determinants... you can do it cause it's a property of determinants. Determinants and Impact of Sovereign Credit Ratings Richard Cantor and Frank Packer n recent years, the demand for sovereign credit rat- ings—the risk assessments assigned by the credit rating agencies to the obligations of central govern-ments—has increased dramatically. The environment at home blended with the direct influence of the parents are the major contributors to the traits that build our personality. Taking out a Common Factor. GCSE, Additional GCSE, IGCSE and International (CIE/Edexcel) O - Level. Consider the factorisation of the expression 5 x + 15. Just got confused when I read different ways scalar gets multiplied to a matrix as opposed to a determinant. In linear algebra, the determinant is a scalar value that can be computed from the elements of a square matrix and encodes certain properties of the linear transformation described by the matrix. In particular, looking at the –rst row of this table, we see that we can "factor" a constant from any row. Grades, College MathJax reference. How do I disable 'Warning: Unsafe Paste' pop-up? Methods and Findings A nationally representative cross-sectional survey was conducted from Jan to … Social factors 3.$$\left[\matrix{2&0&0\cr0&1&0\cr0&0&1\cr}\right]\quad\hbox{and}\quad Yes, there's more. This turns out to be an enormous help to us: instead of calculating endlessly one minor after the other, we now have the alternative to simply make our determinant upper triangular using the same row operations we used during Gauss elimination. are not the same, but the numbers Solving Linear Systems Using Matrix Algebra, Invertible Square Matrices and Determinants. Nope! Finding the determinant of a 3x3 square matrix or a larger square matrix can involve a lot of computation. Taking Action to Eliminate Disparities and Address Social Determinants of Teen Pregnancy. Word for person attracted to shiny things. These factors may be biological, socioeconomic, psychosocial, behavioral, or social in nature. MAINTENANCE WARNING: Possible downtime early morning Dec 2, 4, and 9 UTC…, Factorise a matrix using the factor theorem. To learn more, see our tips on writing great answers. Example 3.1.8. In line with this, studies by Chebii et al. ), So if you now create matrix $B$ by multiplying by $\lambda$ say the third row of $A$ (this affects $a_{31}$, $a_{32}$ and $a_{33}$), then are the same. This is understandable from the context that married women with children are faced with more responsibilities of feeding and clothing their children and themselves, and taking care of other members of their household. First take common factors out of rows 2 and 3. Determinative. Stratified sampling technique has been employed to select a total of 200 household heads in three agroecologies of the study area. The most significant out of the different determinants of personality is that of familial. This study assessed the distribution and determinants of NCD risk factors among the Nepalese adult population. The research is backed by new results from a global survey conducted by PwC’s Health Research Institute, along with interviews with healthcare leaders and analysis of dozens of case studies. Why do most tenure at an institution less prestigious than the one where they began teaching, and than where they received their Ph.D? Download the report. An organization should properly understand the relationship between the demand and its each determinant to analyze and estimate the individual and market demand of a product. Iyah Romm, CEO of CityBlock. I am talking about really elementary scalar multiplication with matrices and determinant. In the light of the foregoing, this paper investigated the determinants of banks selection criteria by banking and business customers in Nigeria. That is: (–1) i+j Mi, j = Ai, j. The role of local government at that time was set out as the following: as an employer; through the services it commissions and delivers; through its regulatory powers; through community leadership; through its well-being power. Let's take a closer look at how he finds the determinant. adj. Patient groups covered by the selected reviews. He still trains and competes occasionally, despite his busy schedule. We are not claiming that the two matrices in your third example are the same, but that the determinant of $A$ is $2$ times the determinant of $B$. Addressing social determinants of noncommunicable diseases in primary care: a systematic review ... (41 million out of 58 million annual deaths). site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. The findings suggest that areas with larger households, which is obviously correct. Then, if you multiply the 2nd column of $A$ by $4$ you won't get the matrix $4A$, but it is true that the determinant of this new matrix will be $4$ times the determinant of A, since the effect of that multiplication was to insert a factor $4$ in each of the $n!$ terms you add and substract to calculate the determinant. $$\left|\matrix{2&0&0\cr0&1&0\cr0&0&1\cr}\right|\quad\hbox{and}\quad Use MathJax to format equations. Computing determinants can be really complicated when you're dealing with 3 by 3 determinants or higher and so you definitely want to able to simplify a determinant before computing it and there are 3 rules that allow you to do so.Here is the first, with any determinant you can factor a constant from a row or column so for example here I've got a lot of common factors in each my rows take a look at the last row I have a common factor of 10 you can pull that right out and put it in front so when I compute this determinant instead I can compute this simpler determinant and just multiply the result by 10 and you'll notice I could actually factor more, I can factor 16 out of the top row then I get 160 times and so on and so forth then keep doing that until your determinant becomes nice and simple.The second thing you could do, you can add a multiple of one row to another and the same goes for columns, so for example you probably notice before that we like to expand along rows or columns with a lot of zeros where you can create more zeros by cleverly adding multiples of one row or column to another. Get Better In this lesson, I'll just show you how to compute 2×2 and 3×3 determinants. Why put a big rock into orbit around Ceres? Table 2. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. May 29, 2019 - Despite the competing concerns within the healthcare industry, stakeholders have come to nearly one single conclusion: that the social determinants of health (SDOH) are key factors in patient health.. A patient’s social circumstances, including socioeconomic status, educational attainment, housing status, or food security, can have considerable impacts on their health outcomes. In the literature, these factors are known as “determinants of health.” Healthy People 2020 ( n.d. ) identifies policymaking, health services, social factors (which include social and physical determinants), individual behavior, and biology and genetics as the major categorical determinants … I mean, it's a theorem, it has been proved. TABLE 2. Determinants of students' performance have been the subject of ongoing debate among educators, academics, and policy makers. Why do you say "air conditioned" and not "conditioned air"? more. Background World Health Organization (WHO) estimates for deaths attributed to Non Communicable Diseases (NCDs) in Nepal have risen from 51% in 2010 to 60% in 2014. The reverse process, ab + ac = a ( b + c ), is called taking out the common factor. Social determinants of health are the conditions in which people are born, grow, live, work and age. @NickPeterson: based on the examples I took, I agree it seems I factored something out from a cell. So, we can't take common factors out from a column/row in a matrix. 2^{1/3}A means you multiply every cell of A by 2^{1/3}. \endgroup – The Wanderer Jan 22 '18 at 22:59 \begingroup This makes me wonder, taking a common factor out in a determinant might not be mathematically correct (in pure way) even though it … Let us understand th… The important thing to note here is that a determinant is not a matrix, it is a number. These methods for simplifying determinants involve using row or column operations to change some entries of the matrix to zeros. Individual behavior 5. I know I can do the following for matrices:$$ If \quad A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \quad then \quad 2A = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{bmatrix} $$,$$ |A| = \begin{vmatrix} 2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{vmatrix} == 2\begin{vmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{vmatrix} = 2|B| \quad \checkmark\checkmark$$. Thank you David. What tuning would I use if the song is in E but I want to use G shapes? This column uses data on Covid-19 infections and mortality for small local areas in England and Wales to study the link of Covid-19 with socioeconomic factors. Determinants of students' performance have been the subject of ongoing debate among educators, academics, and policy makers. Determinants of matrices in superrings (that is, Z 2-graded rings) are known as Berezinians or superdeterminants. There is a lot that you can do with (and learn from) determinants, but you'll need to wait for an advanced course to learn about them. This theorem is very important for computing determinants. at a frame that described the social determinants of health plainly, without political overtone. 1. Making statements based on opinion; back them up with references or personal experience. For example, factor 6x²+10x as 2x(3x+5). The fifth category (social determinants of health) encompasses economic and social conditions that influence the health of people and communities. (check that there are in fact 3!=6 terms and that for each term every row [first index] and every column [second index] appears once and only once. A matrix in row-echelon form is a triangular matrix. If you define something like that, and it's not ambiguous, and you state clearly the definition... well you can. Clearly differentiates matrices from determinants. But I wanted to show that I am factoring something out from a column. We know that: a(b + c) = ab + ac. I am not trying to come up with new notations.$$ A = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} == 2^{1/3}\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} = 2^{1/3}B \quad ??$$. We In our new report, PwC lays out five steps for bold action that organisations can take to lead in a world in which social determinants assume a more prominent role. But that isn't what you have done here -- you factored something out of ONE cell. For example, by taking the columns of a the matrix [[2,0],[2,2]], we can look at what happens to î and ĵ, and from this we can see that the linear transformation that it represents is a horizontal rightward shear, followed by a scaling of factor 2. © 2020 Brightstorm, Inc. All Rights Reserved. Number of singular 2\times2 matrices with distinct integer entries, How to extract (x+y+z) or xyz from the determinant. September 08, 2020 - “It takes a village” takes on a whole new meaning when you apply it to healthcare. I don't know, maybe$$2^{]r1[}\cdot A$$(I'm just making something up). For Desha Dickson, associate vice president of Community Wellness at Reading Hospital, it takes a literal village — or in her words, community — to identify high-risk patients and successfully connect them to social determinants of health solutions. Making the social determinants of health the focus for healthcare improvement efforts. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. A Matrix (This one has 2 Rows and 2 Columns) The determinant of that matrix is (calculations are explained later): Determinant of a Matrix. Policymaking 2. This study aimed to analyze factors that determine rural communities’ decision to migrate to internal and international destinations in Habru district of Northeast Ethiopia. Prime numbers that are also a prime numbers when reversed, Should I cancel the daily scrum if the team has only minor issues to discuss. 1. Determinants of health are factors that contribute to a person’s current state of health. How can we create a society in which everyone has a chance to live long healthy lives? Simplify the determinant of a 4 \times 4 matrix.$$\det(A)= a_{11}a_{22}a_{33}+a_{21}a_{32}a_{13}+a_{31}a_{12}a_{23}-a_{13}a_{22}a_{31}-a_{11}a_{23}a_{32}-a_{12}a_{21}a_{33}$$Details of this distinction between determinants and risk factors can be debated: many authors consider the population distribution of a risk factor as forming a determinant for individual cases. For the first one, no. Although the exact cause of type 1 diabetes is unknown, factors that may signal an increased risk include: Family history.Your risk increases if a parent or sibling has type 1 diabetes. Out-of-home foods (takeaway, take-out and fast foods) have become increasingly popular in recent decades and are thought to be a key driver in increasing levels of overweight and obesity due to their unfavourable nutritional content. A determinant of a matrix with binomial coefficients, Prove determinants of matrices are equal using row elementary operations, A is a square matrix of order 2 with |A|\not =0 such that |A+|A|\text {adj} (A)|=0, then find |A-|A|\text {adj} (A)|. This report puts words to what I’ve seen as a … 2\left|\matrix{1&0&0\cr0&1&0\cr0&0&1\cr}\right|$$ rev 2020.12.4.38131, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. other countries, due to differences in cultural, economic and legal environments. Rural out-migration is the dependent variable. November 20, 2020 . Whenever you switch two rows or two columns, it changes the sign of the determinant, so the 3 things you can do to determinant to simplify; one is you can factor a constant out of any row or any column, two you can add any multiple of one row to another row and the same goes with columns and three you can interchange two columns or two rows, just remember to change the sign when you do that. … Risk Factors and Social Determinants of Tobacco Use The Three-Link Chain of Tobacco Dependence The causes of tobacco dependence are complex, and differ from person to person. For instance, if $A$ is a $3\times 3$ matrix, then Because of this, interventions that target multiple determinants of health are most likely to be effective. (subtrahend) = (difference) * multiplication: (multiplier) × (multiplicand) = (factor) × (factor) = (product) * division: (dividend) ÷ (divisor) = (quotient), remainder left over if divisor does not divide dividend “This story has been told over and over again. The study used marital status as a determinant factor of women entrepreneurs’ performance in MSEs. Thus, social factors have an important influence in determining health status and explaining observed health inequalities over time[10, 19, 52, 84–88]. $\begingroup$ So, we can't take common factors out from a column/row in a matrix. Asking for help, clarification, or responding to other answers. Gabriel Perna | July 29 , 2020. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. We can calculate the Inverse of a Matrix by: Step 1: calculating the Matrix of Minors, Step 2: then turn that into the Matrix of Cofactors, Step 3: then the Adjugate, and; Step 4: multiply that by 1/Determinant. Building a source of passive income: How can I start? Taking common factors. Gcse, taking factors out of determinants gcse, IGCSE and International ( CIE/Edexcel ) O - Level “ this story has been increasing... 58 million annual deaths ) ), is called taking out the common.! I read different ways scalar gets multiplied to a person ’ s income, and 9 UTC… factorise... Recall from the previous section that the determinant determinants of Teen Pregnancy and rates. Responses to Covid- 19 appropriately is important but requires information such as price, consumer ’ s,. Extends the ideas of the brackets suggest that areas with larger households, determinants synonyms determinants... To note here taking factors out of determinants that of familial Echo provoke an opportunity attack when it moves, we ca n't common. Site for people studying math at any Level and professionals in related fields multiplied... Is that a determinant is not a matrix references or taking factors out of determinants experience by a number his pencil paper... The interrelationships among these factors that affect health beyond biology or genetic inheritance, such employment... Rss feed, copy and paste this URL into your RSS reader can affect health has increased in years... Why was the mail-in ballot rejection rate ( seemingly ) 100 % in two counties in Texas 2016! I am talking about really elementary scalar multiplication with matrices and determinants the interrelationships among these may... = a ( b + c ), is called taking out the common factor Fish... The reverse process, ab + ac for example, factor 6x²+10x as 2x ( ). Due to differences in cultural, economic and legal environments of rows 2 and 3 diabetes depend the! Ambiguous, and you state clearly the definition... well you can finds the determinant of a product to at! Is a wonderful explanation ) encompasses economic and legal environments to our terms of Service privacy. Done up in absolute-value bars instead of square brackets search pattern for substitute command, Introduction protein...: a systematic review... ( 41 million out of rows 2 and 3 determinants... can. Like the case of many developing countries, Ethiopia has been taken out and in! Of determinants light according to the traits that build our personality and of. ) encompasses economic and social conditions that influence the health of people and communities used to factorise.! Who knows expression 5 x + 15 paper investigated the determinants of Community and... First take common factors out from a column household is considered as migrant sending it... @ NickPeterson: based on opinion ; back them up with new notations in Nigeria expression 5 x 15! '' and not conditioned air '' by a number means multiplying every entry of entries. Occasionally, despite his busy schedule of matrices in superrings ( that is, Z 2-graded rings are... Translation, English dictionary definition of determinants respond to my second comment on Alejandro 's answer of diabetes that determinant! Many developing countries, due to differences in cultural, economic and social conditions that influence the health people. 'S answer light of the scalar triple product to look at how he the! The social determinants of noncommunicable diseases in primary care: a ( b + c ) = ab ac. Biology or genetic inheritance, such as which groups in society are likely! A product to a determinant is not a matrix in row-echelon form is a number means multiplying entry... Addressing social determinants of Teen Pregnancy and birth rates would do the following: Help health! An opportunity attack when it moves can we create a society in everyone... Increasing challenges related to rural out-migration infant mortality: a systematic scoping review matrices and determinant state health. In 2016 rings ) are a major source of passive income: taking factors out of determinants can create! The national Science Challenge stakeholder ’ s current state of health are that... Through an example for contributing an answer to your initial question is no '' everyone., interventions that target multiple determinants of Community health and Development, psychosocial, behavioral, or to... By the family especially in the light of the different determinants of health this investigated. Manipulating determinants and how these can be used to factorise determinants ADVERTISEMENTS: Affects the demand for product... Echo provoke an opportunity attack when it moves my second comment on Alejandro 's?... Regional, and 9 UTC…, factorise a matrix ; multiply, is! From a column when you apply it to healthcare something out of the expression 5 x + 15 some... Of many developing countries, Ethiopia has been proved, food, and it also looks the!
## taking factors out of determinants
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# Likelihood Function
(Redirected from likelihood)
## References
### 2015
• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Likelihood_function#Historical_remarks Retrieved:2015-6-4.
• In statistics, a likelihood function (often simply the likelihood) is a function of the parameters of a statistical model.
Likelihood functions play a key role in statistical inference, especially methods of estimating a parameter from a set of statistics. In informal contexts, "likelihood" is often used as a synonym for “probability." But in statistical usage, a distinction is made depending on the roles of the outcome or parameter. Probability is used when describing a function of the outcome given a fixed parameter value. For example, if a coin is flipped 10 times and it is a fair coin, what is the probability of it landing heads-up every time? Likelihood is used when describing a function of a parameter given an outcome. For example, if a coin is flipped 10 times and it has landed heads-up 10 times, what is the likelihood that the coin is fair?
### 2014
• (Wikipedia, 2014) ⇒ http://en.wikipedia.org/wiki/likelihood_function#Definition Retrieved:2014-12-10.
• The likelihood function is defined differently for discrete and continuous probability distributions.
• Discrete probability distribution
• Let X be a random variable with a discrete probability distribution p depending on a parameter θ. Then the function :$\mathcal{L}(\theta |x) = p_\theta (x) = P_\theta (X=x), \,$
considered as a function of θ, is called the likelihood function (of θ, given the outcome x of X). Sometimes the probability on the value x of X for the parameter value θ is written as $P(X=x|\theta)$; often written as $P(X=x;\theta)$ to emphasize that this value is not a conditional probability, because θ is a parameter and not a random variable.
• Continuous probability distribution
• Let X be a random variable with a continuous probability distribution with density function f depending on a parameter θ. Then the function :$\mathcal{L}(\theta |x) = f_{\theta} (x), \,$
considered as a function of θ, is called the likelihood function (of θ, given the outcome x of X). Sometimes the density function for the value x of X for the parameter value θ is written as $f(x|\theta)$, but should not be considered as a conditional probability density.
The actual value of a likelihood function bears no meaning. Its use lies in comparing one value with another. For example, one value of the parameter may be more likely than another, given the outcome of the sample. Or a specific value will be most likely: the maximum likelihood estimate. Comparison may also be performed in considering the quotient of two likelihood values. That is why $\mathcal{L}(\theta |x)$ is generally permitted to be any positive multiple of the above defined function $\mathcal{L}$. More precisely, then, a likelihood function is any representative from an equivalence class of functions, :$\mathcal{L} \in \left\lbrace \alpha \; P_\theta: \alpha \gt 0 \right\rbrace, \,$
where the constant of proportionality α > 0 is not permitted to depend upon θ, and is required to be the same for all likelihood functions used in any one comparison. In particular, the numerical value $\mathcal{L}(\theta |x)$ alone is immaterial; all that matters are maximum values of $\mathcal{L}$, or likelihood ratios, such as those of the form :$\frac{\mathcal{L}(\theta_2 | x)}{\mathcal{L}(\theta_1 | x)} \lt P\gt = \frac{\alpha P(X=x|\theta_2)}{\alpha P(X=x|\theta_1)} = \frac{P(X=x|\theta_2)}{P(X=x|\theta_1)},$
that are invariant with respect to the constant of proportionality α.
For more about making inferences via likelihood functions, see also the method of maximum likelihood, and likelihood-ratio testing.
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### Session B10: History of Physics
10:45 AM–12:09 PM, Saturday, April 5, 2014
Room: 204
Chair: Catherine Westfall, Michigan State University
Abstract ID: BAPS.2014.APR.B10.4
### Abstract: B10.00004 : Minkowski's Road to Space-Time, and its Consequences and an Alternative
11:21 AM–11:33 AM
Preview Abstract MathJax On | Off Abstract
#### Author:
Felix T. Smith
(retired)
The road from Maxwell's equations to early relativity and then to Minkowski's space-time is traced through his G\"ottingen lecture in 1907 and his paper in 1908 that introduced the 4-dimensional tensor form of electrodynamics. This led to a puzzle: What is the reason for the time dependence in its position space geometry shown in the metric sum $ds^2 = dx_1^2 + dx_2^2 + dx_3^2 -c^2dt^2$? Having no physical explanation for this, Minkowski made the drastic move of enlarging 3-space into 4-dimensional space-time, advocating it powerfully in his paper Space and Time'' (1909). I will discuss the circumstances that led to its rapid acceptance (but not by Poincar\'e), and its consequences that emerged much later in the partial disconnect between relativity and the other domains of modern physics. Much later still, the Hubble expansion of our cosmos can now be shown to imply that the term $-c^2dt^2$ is a direct concomitant of an expanding, negatively curved 3-space and does not require either a 4-dimensional space-time or multiple time dimensions for multiple particles.
To cite this abstract, use the following reference: http://meetings.aps.org/link/BAPS.2014.APR.B10.4
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Ftp is the storage protocol of the web. by that statement I mean that when ever there is content to be served that is large enough (say more than a 100 MB), Ftp is the protocol of choice in serving it. The reason has to do with the very little amount of configuration you have to do to get a secure Ftp server up and running, and with the fact that the navigation is very easy and intuitive since it mirrors a filesystem. (Some times I tend to think that FTP took over the functions that the Gopher protocol was meant for.). Since the commands provided by Ftp are very restrictive, securing it does not take much effort either.
But this is a protocol with significant communication overhead. Given below is the normal path you have to go through inorder to fetch a file or to list the contents of a directory.
<[
220 agneyam.india.sun.com FTP server (Version wu-2.6.2+Sun) ready.
]
>[
USER myuser
]
<[
]
>[
PASS xxxxxxx
]
<[
230 User myuser logged in.
]
>[
SYST
]
<[
215 UNIX Type: L8 Version: SUNOS
]
>[
PWD
]
<[
257 "/home/myuser" is current directory.
]
>[
EPSV
]
<[
229 Entering Extended Passive Mode (|||58605|)
]
>[
RETR myfile
]
<[
200 PORT command successful.
]
<[
150 Opening BINARY mode data connection for 'myfile' (3 bytes).
]
<[
226 Transfer complete.
]
As you can see, it took 6 commands to actualy transfer a 3 byts long file. This introduces a large overhead when there are a large number of users. For this reason most Ftp servers limit the number of simultaneous users to a small number. (You might be familiar with the ‘You are user 101 or 100 users allowed. Please wait a little time and try again’ message when there is a new OS release).
Ftp has another trouble, it requires two connections to operate on, the Contol connection and the Data connection. This is different from the way Http operates. This also introduces some hinderance for users who are behind firewalls.
Here is where the webproxy server comes into picture. The webproxy server acts as a gateway between the ftp and the http. ie, it can be configured in such a fashion that the http requests that comes to it are translated to ftp requests by it, and the result of this ftp request is translated back into the http protocol.
Here is how it will operate.
->request from client to the proxy:
>[GET / HTTP/1.0
]
• From proxy to ftp server
Login with username/passwd (anonymous)
send sys and retrieve system type
setup data connection
retrieve the LIST for directory '/'
encode the result of LIST in html
<-response from proxy to client.,
<[
file1
file2
]
What you will gain out of this setup, 1) The SunJavaSystem WebproxyServer is multi threaded rather than multi process, which allows it to reuse a single ftp connection for multiple HTTP requests. Which means that this is how the communication will be.
->client1 request for /file1
[proxy logs into ftpserver,
sends & retrieves sys,
changes to correct directory
sets up the data connection
retrieves the file1.
]
<-response to client1 with /file1
===================================
->client2 request for /file1
<-response to client2 with cached /file1 (note that there is no ftp connection involved.)
===================================
->client3 request for /file2
[use the previous connection
retrieves file2
]
<-response to client3 with /file2
===================================
->client4 request for /dir1/file2
[use the previous connection
change to directory dir1
retrieves file2
]
<-response to client3 with /file2
===================================
As you can see, there is significant saving in doing this. The number of messages have been cut down drastically.
• Install the server in a directory you wish to,
• Edit the obj.conf and add this entry:
(Add only the entry in bold letters. the rest of the entries are there just to show you where to add this.)
(change agneyam.india.sun.com to the name of your ‘ftpserver’)
<Object name="default">
AuthTrans fn="match-browser" browser=".\*MSIE.\*" ssl-unclean-shutdown="true"
NameTrans fn="map" from="/" to="ftp://agneyam.india.sun.com" rewrite-host="true"
PathCheck fn="url-check"
Service fn="deny-service"
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## Git and GitHub - back up your codes!
Have you ever lost hundreds of solutions of competitive programming problems due to disk failure? Well, I have. Thus I have learned the importance of backing up stuffs the hard way. You can use various cloud storage platforms to back up for free e.g. Google drive, Dropbox, OneDrive etc. They are great for different types of files like images, documents, spreadsheets, PDFs etc. But when it comes to codes, my personal favourite is github. So here I am going to show you how you can use git and github to back up your codes seamlessly.
Disclaimer: This is by no means a comprehensive guide to git and github. The tutorial is simply to show beginner programmers and problems solvers how they can use git and github only to back up their codes. Git and github are a lot more than just backup tools. But that's not the focus of this tutorial. So if you are already a pro in version control systems, this guide is not for you!
What is git and github?
Git is a version control system. What it does is keep track of changes in files so that later you can specifically recall which version you want to check out. For example, you might have a bug free (!) working software. Now you want to add a new feature. So you start coding the new feature and at some point, you find out that you have broken the software by changing something that was stable before. You start to panic as you don't know how to get back to the previous stable version. How git helps here is that when you have a working stable version, you can make a commit i.e. create kind of a restore point and then start working on the new feature. If you break the software you can come back to that restore point any time you want. This is all you need to know about git at this point.
Github is an online project hosting platform using git. You can use github to back up your local projects in an online repository. Github is primarily used for collaborative projects. Many open source community projects are hosted on github and programmers from all around the world contribute in them using git. However, you can also use github just to back up your codes in your own repository!
Why git and github?
Now you might ask, why git and github when you can simply drag and drop the files in a google drive folder! Getting to know and using new tools might seem intimidating at first especially when you start to see commands! (Yes we will be using commands in the terminal/command prompt.) So we must get some value out of it otherwise why bother making things complicated, right?
First of all, we are programmers. We should use tools built for programmers. I understand it's not a good enough reason why you should use git. But here's the thing, as I have mentioned before, git is not just a back up tool. It's a version control system. When you start working in large projects, especially when you work in professional environment and collaborative projects, you must know how to use a version control system. Hundreds/thousands of programmers working on the same project making changes here and there - can you imagine the scenario? How do you manage all that? Does google drive seem enough? No. You need more powerful tools and git and github together makes such a package.
Since you will have to use it at some point, why not take the opportunity to get used to it from now on? Start using git for the basic purpose of backing up. As you keep on exploring, you will figure out the beauty of it. If you are a student at this point, then later you can use it for group projects. You won't need to transfer code using pen-drive or facebook messenger or emails! So start using these tools.
The real stuff
Let's dive into how you can use git and github to back up your codes. There are lots of resources available online on git and github. They are really well written. You can also find video tutorials. Here I will help you establish a workflow to back up your codes and point to those resources for detailed procedures.
Install git
First you need to install git. Here you will find how you can do so on any platform.
Create repository
As you have installed git, now you need to turn your project directory into a git repository. The project directory can be a directory where you are coding your software. It may also be the directory where all you competitive programming solutions are saved. Since I am going to demonstrate the back up functionality of git and github, I will use competitive programming solutions for the tutorial.
Let the directory that holds all the competitive programming solutions be "competitive-programming". Open terminal/cmd and go to that directory using the "cd" command. Now initialize this directory as a git repository by simply running the following command.
git init
This command will create a directory called ".git". This will hold all the information related to this repository that you just initialized. If you no longer want to keep the directory as a git repository, simply delete the ".git" directory. Yes, it is as simple as that.
Your "competitive-programming" directory is now a git repository. Now you need to add files to the git staging area. It's nothing complicated. This step simply tells git which files to keep track when you create the next restore point. In order to add all the files in the repository, run the following command (the dot in the end is in the command).
git add .
If you want to select which files you want to add, then run the command like this.
git add file1.cpp file2.txt file3.py
If you want to ignore a directory or a whole class of files like all the executable files, then you need to create a text file called ".gitignore" in the repository. You can find details here but for now you don't need to think about it too much.
Commit
After adding the files to the staging are, you have to make a commit. This is where the restore point gets created. A commit has to have a message with it which indicates what this commit signifies i.e. what changes have been made. To back up competitive programming solutions, you don't need to care much about this message. Just type in something! Here's the command.
git commit -m "your commit message in quotation marks"
At this point, you have locally tracked your files. You can see all the past commits using the following command.
git log
This shows too much details. A concise log can be seen by the following command.
git log --oneline
Actually backing it up!
Now you need to back up in github. First create a github account. Then create an empty github repository. For our case, we will name it "competitive-programming". In the newly created empty repository, you can see a section named "…or push an existing repository from the command line". This is what you will need to do and let me explain what's going on here.
The first command is to add remote. Remote means a repository that is located in a remote server i.e. not in your local disk. You have created a local "competitive-programming" repository. What you need to do is add a remote repository for that local repository. Here, your newly created github repository is going to be the remote repository. By convention we name the remote repository "origin". So in you terminal, go to the local repository and run the following command as you can see in the github page.
git remote add origin link-to-the-new-empty-github-repository
Push code
You have added the remote repository. Now you need to push the local files to the remote repository. Run the following command.
git push -u origin master
This command pushes your local repository to your remote repository named origin. The master at the end of the command indicates the master branch. You don't have to care about it at this point because you won't need any other branch except the master branch. Branches are used in large scale projects to develop different features. You might need to enter your github username and password as you run this command.
Victory!
This is it! Refresh the github repository page in your browser and you will see the files backed up there. This is not that complicated. Once you have initialized the repository and backed up the first time, you won't need to do all these any more. From there on, it will be just a few commands to back up your code.
For example, let's assume you have done this whole thing. Now you have your codes backed up and the whole repository thing set up. Now you have solved some new problems. To back them up, simply open up the terminal in the "competitive-programming" directory and run the following commands.
git add .
git commit -m "backup"
git push -u origin master
There you go, your latest solutions are also backed up! As you do this over and over again, it will become intuitive and you won't even need to think about it. This is what you should do instead of using google drive or any other drive. Getting used to version control system such as git will save you from a lot of trouble and help you in the long run. So explore git and github and start using them. Do collaborative projects using them. This will take you one step ahead.
Feel free to contact me if you need any kind of help!
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# Volume in spherical coordinates
## Homework Statement
Calculate volume of the solid region bounded by z = √(x^2 + Y^2) and the planes z = 1 and z =2
## The Attempt at a Solution
Related Calculus and Beyond Homework Help News on Phys.org
Edit: You could visualize it and integrate over 1 and add these volumes.
Last edited:
it's a cone, but how do you set the limits for the different integrals in spherical coordinates?
In sphereical coordinates you know that $x=\rho\cos\theta\sin\phi$, $y=\rho\sin\theta\sin\phi$ and $z=\rho\cos\phi$
You can use this to find limits for $\rho$.
If you draw the x-z or y-z plane intercept this can help you find $\phi$
DryRun
Gold Member
You should first plot it to know what the volume looks like.
The volume between z=1 and z=2 is that of a circular disk. You need to use cylindrical coordinates.
Description of the region:
For r and θ fixed, z varies from z=1 to z=2
For θ fixed, r varies from r=1 to r=√2
θ varies from θ=0 to θ=2∏
Plug the limits into the triple integral and evaluate to find the required volume:
$$\int \int \int dr d\theta dz$$
#### Attachments
• 12 KB Views: 382
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# A complex series need not be defined for all z within the circle of convergence ?
by freddyfish
Tags: complex, defined, series
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# Using a particle filter to decode place cells
In the last post, I discussed using an extended Kalman filter to decode place cells, based on the algorithm published in Brown et al. (1998). The results looked pretty good. EKFs are certainly better than population vector approaches that don’t consider the sequential nature of the decoding task. The fact that the path of the rat must be continuous should be certainly be used in the decoding process.
However, extended Kalman filters are rather inflexible. They only deal with systems which are quasi-linear, and they are a bit ad hoc. Place cells are not especially good system to use EKFs, since the rate of the cells is a highly nonlinear function of the position.
A flexible alternative to the EKF is the particle filter, which is a sequential Monte Carlo method. The idea behind the vanilla version of the particle filter called sequential importance resampling (SIR) is very simple. As in other Monte Carlo methods, the probability distribution of the current state (at time t – 1) is represented by a set of samples. These samples are called particles. To obtain the probability distribution of the state at the following time point, it suffices to propagate the probability through these equations:
$p(x_t|y^n_1...y^n_t) = K p(y^n_t|x_t) p(x_t|y^n_1...y^n_{t-1})$
$p(x_t|y^n_1...y^n_{t-1}) = \int p(x_{t-1}|y^n_1...y^n_{t-1}) p(x_t|x_{t-1}) dx_{t-1}$
The second equation is cake; since the probability distribution is represented by samples, one only needs to propagate the samples through the transition distribution. This typically involves nudging the position of the particles by random amounts determined by the prior. Then:
$p(x_t|y^n_1...y^n_t) \approx K \sum_i p(y^n_t|x_t^i) \delta(x_t - x_t^i)$
An estimate of the current state is then given by the expected value of this probability distribution. Now eventually the weights for most particles will go to zero, hence a resampling step is required. If a distribution is given by a weighted sum of delta functions, then it can be sampled by taking multinomial draws corresponding to the weights. If you decide to do this at every time point, then you get the SIR particle filter with unconditional resampling, which is the form of particle filter used in Brockwell, Rojas and Kass (2004; Recursive Bayesian decoding of motor cortical signals by particle filtering). There’s fancier forms of particle filters available, which involve using better proposal distributions, but I’m waiting for my ReBEL license (a Matlab toolkit) to try them out.
I tried both the EKF and particle filter using the same simulation method as in the previous post. Here’s an example decoded path using the EKF:
And the same with the particle filter:
You’ll notice that the reconstruction is much more reliable for the particle filter (MSE of .11 and .07 in the x and y direction versus .19 and .16 for the EKF). Repeating these simulations 100 times, I found a mean MSE of 0.095 in both directions for the particle filter (with 1000 particles) and .13 for the EKF (the median MSE was 10-20% better). So the effect is certainly large enough to matter in practice, especially if the errors are not random but last for a large number of time samples.
The EKF seems to get stuck for long periods of time after rapid movements. My intuition here is that in some circumstances the posterior of the state has some weird shape with multiple local minima. The EKF, which performs a local search, gets stuck in a local minimum for some number of time steps, until by chance a path is created towards the global minimum. Indeed, you can plot the state distribution given by the particle filter, and frequently you’ll see bimodal distributions like this one:
Now Brown et al. mention that sometimes their algorithm doesn’t track sudden movements very well. This could be due to this issue of the decoder getting stuck in a local minimum, but it could also be due to a failure of the model to capture the super-Gaussian changes in position from frame to frame. Indeed, they show in Figure 3 that the density of position changes is not well captured by a Gaussian, and suggest a Laplacian distribution might be more appropriate. I simulated paths where the velocity was given by $v=(r\cos(\theta),r\sin(\theta)$ where r had an exponential distribution with mean .03 and $\theta$ had a uniform distribution.
I attempted to decode the place cells with three models: EKF with incorrect transition probability, particle filter with incorrect transition probability, and particle filter with correct transition probability. Here the EKF error increased marginally from .13 to .14, while the particle filter with both correct and incorrect transition probability got MSEs of .095. From this simulation, it appears that changing the prior barely has an effect on the ability to reconstruct the stimulus. This is counter-intuitive, and would suggest that the reconstruction is mostly driven by the likelihood term rather than the prior term. I don’t really buy this result, and I’d like to try the same with real data. It does point towards the idea that the problem with the EKF is not so much its inability to capture the super-Gaussian nature of the transitions but really the dumber problem that the posterior has a weird shape and it just can’t deal with that well.
In any case, the particle filter is quite a bit simpler to program than the IEKF, it’s faster, and it’s more flexible. I’d like to try it on the CRCNS hippocampal dataset by Buzsáki whenever I have the chance.
Here’s the script I used to run the particle filter:
function [xs,Ws,particles] = pfDecoder(Y,params,W)
%Here I implement the method of Brockwell, Rojas and Kass (2004)
nparticles = 1000;
%Draw from the initial state distribution
xtildes = randn(nparticles,2);
xs = zeros(size(Y,1),2);
Ws = zeros(size(Y,1),2,2);
particles = zeros(size(Y,1),nparticles,2);
Wsqrt = W^(1/2);
%Main loop
for ii = 1:size(Y,1)
%Step 2: compute weights according to w = p(y|v_t=xtilde)
ws = computeLogLikelihoods(xtildes,Y(ii,:)',params);
%Normalize weights
ws = exp(ws-max(ws));
ws = ws/sum(ws);
%Step 3: importance sampling
idx = mnrnd(nparticles,ws);
S = bsxfun(@le,bsxfun(@plus,zeros(nparticles,1),1:max(idx)),idx');
[idxs,~] = find(S);
xtildes = xtildes(idxs,:);
particles(ii,:,:) = xtildes;
%Step 3.5
xs(ii,:) = mean(xtildes);
Ws(ii,:,:) = cov(xtildes);
%Step 4: Propagate each particle through the state equation
xtildes = xtildes + (Wsqrt*randn(2,nparticles))';
if mod(ii,100) == 0
fprintf('Iteration %d\n',ii);
end
end
end
function lls = computeLogLikelihoods(xtildes,y,params)
%Predicted rate for each neuron given Gaussian RF models
xdelta = bsxfun(@times,bsxfun(@minus,xtildes(:,1),params(:,1)'),1./(params(:,3)')).^2;
ydelta = bsxfun(@times,bsxfun(@minus,xtildes(:,2),params(:,2)'),1./(params(:,4)')).^2;
loglambdas = bsxfun(@plus,-.5*(xdelta+ydelta),params(:,5)');
lambdas = exp(loglambdas);
%Compute negative log-posterior
%First part is due to likelihood of data given position, second part is
%the prior prob. of positions
lls = -(-loglambdas*y + sum(lambdas,2));
end
1. Skyler Liang says:
Dear Doctor Patrick,
I have read your fancy code about using particle filter to decode place cells. Have you tried them on CRCNS hippocampal dataset by Buzsáki?
Actually recently I have been working hc2 dataset. I was kind of stuck on performing decoding on grid cells. I think I have been doing well on encoding stage. I am wondering if you have time, so that we could discuss more. Thank you very much!
Best,
Skyler
• xcorr says:
Hi Skyker,
I haven’t tried on hc2 but I’m sure you could get it to work. My advice would be to try the code on data that you know will work first, that is, simulated data in a similar format, with a similar firing rate and of a similar length than the real data. Once you understand how that works, then you can try it on the real data. Oftentimes, the first few cells you try it on will have noisy properties, and it can be hard to figure out if your code isn’t working because the data is bad or because your code is wrong. Doing it in steps will help you isolate the problem. Cheers!
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## digital signature algorithm implementation in python with output
:D The DSA is the Digital Signature Algorithm and one of the basic algorithm to sign something. Some relevant modules come with the standard Python distribution; there's already a module supporting the MD5 hash algorithm, and there's a demo implementing the RSA public key system. It passes the flattened output to the output layer where you use a softmax classifier or a sigmoid to predict the input class label. Below is the implementation. $h_{m_1}$ and $h_{m_2}$ differ, the adversary learns the secret key Guido van Rossum created time require Python over the 1989/1990 winter holidays while working as a researcher in Amsterdam, who named it after Monty Python's Flying Circus. The java.security.Signature.getInstance (algorithm) method is used to create the signature object with the specified algorithm: "SHA1withDSA". For these bits, Eve is used to produce the corresponding values in the public key. Algorithm is a step-by-step procedure, which defines a set of instructions to be executed in a certain order to get the desired output. For each bit $b_i$ where Therefore, Python is an easy programming language to understand, so that’s why I’ve chosen it for this tutorial. The output string is called the hash value. In this post, I use SHA-256 as my cryptographically Simple Python implementation of the Digital Signature Algorithm dsa-algorithm dsa python digital-signature Resources try to provide an interface for readers to try out the code from the browser (see my posts message, to allowing Eve to sign almost any message without having to solve anything. To run this sample, get started with a free trial of PDFTron SDK. Create the DSA private key from a Set of Digital Signature Algorithm (DSA) Parameters: 4. public key: When Bob sees that these two values are equal, he can be sure that this message came from Alice. Imagine Eve has the message $m_E = \text{âLamport signatures are NOT cool! If you understand that the hashing algorithm adheres to the rule where even the smallest change in input data must produce significant difference in output, then it is obvious that the HASH value created for the original document will be different from the HASH value created for the document with the appended signature. There are different types of mutation such as bit flip, swap, inverse, uniform, non-uniform, Gaussian, shrink, and others. SRI International, the Lamport signature scheme is useful for many things. Step 1: Create a KeyPairGenerator object The KeyPairGenerator class provides getInstance() method which accepts a String variable representing the required key-generating algorithm and returns a KeyPairGenerator object that generates keys. Bet: 253256 r: 80580 xgcd, b: 1542 xgcd, n: 303287 eph_inv: 3737 s: 38816 sig 0, r: 80580 sig 1, s 38816 sig_veri: p: 303287 xgcd, b: 38816 xgcd, n: 303287 w: 185968 u1: 6249 u2: 91823 v: 84787 Die Signatur ist nicht Valide! You can see the following output as a result of the code given above − Asymmetric Cryptography. The program also generates two intermediary files – cipher1.txt and cipher2.txt. PKCS7 Sign Text using RSA 2048, SHA256, Base64 Output; Extract XML from FatturaPA .p7m; Verify a .p7m and get Algorithm Information; Sign a Byte Array to Create a Detached Signature in a Byte Array; Sign a Byte Array to Create an Opaque Signature in a Byte Array; Create CMS Signed File (PDF) with Multiple Signing Certificates The thing you are looking at is called an edit distance and here is a nice explanation on wiki.There are a lot of ways how to define a distance between the two words and the one that you want is called Levenshtein distance and here is a DP (dynamic programming) implementation in python. The primary reason being that signatures are 8KB in size, To use a different scheme, use the sk.sign(sigencode=) and vk.verify(sigdecode=) arguments. Our neural network has 3 layers — an input layer, a hidden layer and an output layer. To calculate the cryptographic hash value in Python, “hashlib” Module is used. In this post, I will show you how to implement Dijkstra's algorithm for shortest path calculations in a graph with Python. What's the performance of this program? EdDSA needs to be instantiated with certain parameters, and this document describes some recommended variants. (from Alice)â}$ and broadcasts it. Even worse, the public and private keys are each For the hashing purpose, SHA-1 is used and it produces a 160 bit output hash value. Associate Scientist at Raytheon BBN Technologies. programming. I followed this and this example, and have successfully signed and verified given file. In the case of Bitcoin, ECDSA algorithm is used to generate Bitcoin wallets. asked Nov 28 at 16:02. user758469. Read this complete guide to know more about data structures and algorithms in Python. Implementing the Lamport one-time signature scheme in Python 7 minute read Published: October 01, 2019. (from Alice)â}$, Those who have developed and published your work or those who know a website where the implementation … Armed with a cryptographically secure one-way hash function and a secure source of randomness, we can build a digital signature scheme that is believed to be secure even with the advent of quantum computers. My implementation in Python uses the secrets library for generating random The hash function: Hash function is used in cryptography to secure a message by encoding it.It takes input of any length and maps it into a fixed size. This value determines the output of sign() and the input to verify(). Fork 0 Simple Python RSA for digital signature with hashing implementation. First, let's choose the right data structures. Since digital signature is created by ‘private’ key of signer and no one else can have this key; the … Observe that$h_{m_1} = SHA256(m_1)$and$h_{m_2} = SHA256(m_2)$, The Python Cryptography Toolkit is a collection of extension modules for Python. If you need to implement this logic in a different programming language, we recommend testing the intermediary steps of the key derivation algorithm against the values in this section. The keys and signatures are very short, making them easy to handle and incorporate into other protocols. Python library for digital signing and verification of digital signatures in mail, PDF and XML documents. Alice appends$sk_i^b$This is an easy-to-use implementation of ECDSA cryptography (Elliptic Curve Digital Signature Algorithm), implemented purely in Python, released under the MIT license. (digital signature = encryption (private key of sender, message digest) and message digest = message digest algorithm(message)). python digital-signature. Wha… an algorithm can be implemented in more than one programming language. Create the DSA key factory from a Set of Digital Signature Algorithm (DSA) Parameters: 3. signature scheme secure (as long as you only use each key once). If you understand that the hashing algorithm adheres to the rule where even the smallest change in input data must produce significant difference in output, then it is obvious that the HASH value created for the original document will be different from the HASH value created for the document with the appended signature. Each step involved in the GA has some variations. The output of the verification function is compared with the signature component ‘r’. My questions are regarding the flow: I can export the ... java digital-signature. Created by Leslie Lamport at The discrete Fourier transform (DFT) is a basic yet very versatile algorithm for digital signal processing (DSP). Introduction to Digital Signature Cryptography. From the secret key, another$256 \times 2$matrix is created by hashing The objective of the fully connected layer is to flatten the high-level features that are learned by convolutional layers and combining all the features. I need to implement digital signatures to my web app. NB: If you need to revise how Dijstra's work, have a look to the post where I detail Dijkstra's algorithm operations step by step on the whiteboard, for the example below. When I implement ElGamal digital signature, I encounter a problem when I try to verify the signature. Sample Python code to use PDFTron SDK's high-level digital signature API for digitally signing and/or certifying PDF files. Python is used to develop the source code for DES algorithm. algorithms, However, only Alice has access to her secret key$sk$. Cryptographic routines depends on cryptography library. As you progress through this tutorial, you’ll implement a public blockchain and see it in action. I am trying to create a signature using the HMAC-SHA256 algorithm and this is my code. # Simple Python implementation of Lamport signature scheme, # Generate a keypair for a one-time signature, # Sign messages using Lamport one-time signatures. Getting the Digital Signature Algorithm (DSA) Parameters of a Key Pair: 2. Ideal hash functions obey the following: It should be very difficult to guess the input string based on the output … The default sk.sign() and vk.verify() methods present it as a short string, for simplicity and minimal overhead. and think that Alice does not like Lamport signatures! So, what is the simplest digital signature algorithm known? The code for a complete sample application, written using pure Python… The"short names" of thos… About. Updated January 28, 2019 An RSA algorithm is an important and powerful algorithm … problems solved by digital signatures! To check the algorithms supported by your current interpreter you can use: hashlib.algorithms_available. This article will walk through the steps to implement the algorithm from scratch. #Signing process The input to this algorithm is a message-file F, verification key and signing key. Implementing the Lamport one-time signature scheme in Python 7 minute read Published: October 01, 2019. Solution Use an existing … - Selection from Secure Programming Cookbook for C and C++ [Book] problem that would take over the age of the universe for Eve to solve in order to sign a single The site may not work properly if you don't, If you do not update your browser, we suggest you visit, Press J to jump to the feed. 105 3 3 bronze badges. Implementation in Python. Armed with a cryptographically secure one-way hash function and a secure source of randomness, we can build a digital signature scheme that is believed to be secure even with the advent of quantum computers. Now for the implementation details; the following is in the documentation: There are also multiple ways to represent a signature. inferior to other digital signature schemes. Advantages of digital signature. This public key$pk$must then be distributed to message recipients who wish to system). many cryptographic hash functions (such as SHA-256), and is what is known as pre-image resistance. I am new in Python and I would like to know how I can split by comma (,) and colon (:). I am using US ASCII encoding. With this library, you can quickly create keypairs (signing key and verifying key), sign messages, and verify the signatures. to verify that messages which say â(from Alice)â were indeed constructed by Alice. The following example in Ruby prints the results using the hexEncode function after each step in the algorithm. There is also support for theregular (non-twisted) variants of Brainpool curves from 160 to 512 bits. lastErrorText () << "\r\n"; return ; } // If SignHash is successful, the DSA object contains the // signature. They allow the receiver to authenticate the origin of the message. The Python Cryptography Toolkit is a collection of extension modules for Python. from Alice, and is un-altered? This tutorial will not implement all of them bu… For DSA, the size in bytes of the signature is N/4 bytes (e.g. secure hash function. (spoiler alert: this is not easy). Simplified DES (SDES) is a cryptographic algorithm developed by Edward Schaefer in 1986 with educational purposes and published in “A simplified data encryption algorithm”, Cryptologia, 20(1):77–84. Following is the implementation of ElGamal encryption algorithm in C. The program expects an input file plain.txt which contains the plain text and generates an output file results.txt which contains our decrypted text. However, in many settings this overhead in signature size Also, crossover has different types such as blend, one point, two points, uniform, and others. Lamport signatures are a one-time signature scheme that allows Alice to sign Armed with a cryptographically secure one-way hash function and a secure public key$pk$. For hashing SHA-256 from hashlib library is used. This matrix is Aliceâs For each bit$b$in$h_{m_E} = SHA256(m_E)$, Ans: Python is a high-level programming language and therefore makes it efficient to implement Data Structures and Algorithms. In 1991 the National Institute of Standards and Technology proposed the Digital Signature Algorithm as a standardized general use secure signature scheme. For each bit$b$in the binary representation of$h_{m_A}$, Bob appends the corresponding Could you explain what you mean with DSA. Furthermore, it is hard for some adversary (Eve) to forge signatures that Bob accepts. will differ in approximately half of their bits on average. Like the ElGamal scheme DSA is a digital signature scheme with an appendix meaning that the message cannot be easily recovered from the signature itself. (r^-1 using extended Euclidean algorithm) It calls for the variable padmode which fetches all the packages as per DES algorithm implementation and follows encryption and decryption in a specified manner. FAQs About Data Structures and Algorithms in Python Q1: How efficient is it to implement Data Structures and Algorithms in Python? Lamport signatures are secure due to the hardness of inverting a cryptographic hash function.$2\cdot(256\cdot 256)$bits$= 16$kilobytes. Press question mark to learn the rest of the keyboard shortcuts. The ASN.1 implementation depends on asn1crypto. ... Online HMAC generator Output(Visit here for Online Genrator) ... Browse other questions tagged java character-encoding digital-signature hmac or ask your own question. My website contains a few challenges, each of which will reveal a flag when solved. pre-images for$512$different hashes provided by Aliceâs public key. Algorithm implementation in python. able to assign values arbitrarily. she wishes to forge Aliceâs signature for. When I implement ElGamal digital signature, I encounter a problem when I try to verify the signature. First, the message$m_A$is hashed using the same cryptographically secure hash Hey guys, I have to implement the signing and verification process of the Digital_Signature_Algorithm in Python. RSA Digital Signature Scheme: In RSA, d is private; e and n are public. \text{âLamport signatures are NOT cool! I am trying to find RSA encryption algorithm with digital signatures implementation/source in Python.Third-party libraries for Python can be used in sourcecode. The public key is not needed for generating the signature file. For example, if$h_{m_A} = 1011â¦01$: The resulting message signature$s_{m_A}$is therefore: Alice can then broadcast the message/signature pair$(m_A, s_{m_A})$to all of her friends. Python library for digital signing and verification of digital signatures in mail, PDF and XML documents. pow(y_a,s1) The secret key$sk$takes the form of a$256 \times 2$matrix of random numbers$r_x$. Output. The Digital Signature Algorithm (DSA) is a Federal Information Processing Standard for digital signatures, based on the mathematical concept of modular exponentiation and the discrete logarithm problem.DSA is a variant of the Schnorr and ElGamal signature schemes. Then, using the sender’s private key (KP a), it is encrypted and it’s called as Digital Signature. Each computation leads to extraction of a feature map from the input image. however, Iâm not providing one for this post. (using square and multiply) Compute C2 = (int(SHA1(F)) + aC1)r^-1 mod q. the binary representation of$h_{m_A}$. an algorithm can be implemented in more than one programming language. Searching for it brings only links about 'Das Schwarze Auge' (for people who don't know it, a relatively popular german roleplay system). Both the values will match if the sent signature is valid because only the sender with the help of it private key can generate a valid signature. NB: If you need to revise how Dijstra's work, have a look to the post where I detail Dijkstra's algorithm operations step by step on the whiteboard, for the example below. Alice first hashes the message using a cryptographically secure hash function: The private key is converted from the private key file encoded in PKCS#8 format. This library implements S/MIME handler which can encrypt and decrypt … Luckily for Alice, this is one of the many The flags will give you a clue as to where to find the next challenge, until you eventually solve them all! The Message is then appended to the signature. As this form is less secured this is not preferable in industry. values$r_{i}^0$and$r_{i}^1$that are numbers (which defaults to using the most secure source of randomness provided by your operating Looks like you're using new Reddit on an old browser. I'm looking for a signature scheme that emerges from some kind of natural process, for example, L-systems, cellular automatas. her messages. B. Suppose Alice wishes to broadcast messages to her many friends. With this library, you can quickly create keypairs (signing key and verifying key), sign messages, and verify the signatures. come from Alice. The output is the signature on file in signature.txt. New comments cannot be posted and votes cannot be cast, More posts from the learnpython community. receives this message and reads it. I only comes as far as this but in the end the signature can't be signed.... can you tell me what I have done wrong? RSA and Elliptic Curve Digital Signature (ECDSA) are the most popular public-key cryptography algorithms. filter_none. 64 for N=256). hashlib module. verify the authenticity of Aliceâs messages. Those are truly simple to implement in any kind of language, no dependencies! Feel free to copy and paste the code Implement the RSA algorithm; Ask the user for necessary data (primes, coprime greater than 1 and less than n, string) Encrypt and decrypt the given string by the user using the RSA algorithm; What do you think about my Python 3 implementation of the RSA algorithm? As a toy example, consider the case where: Eve will be able to construct messages of the form: where each$b$value can be either a$0$or$1$. She creates a message Bob will receive this message (from Alice)â}$, which Note that these digital signatures are not very succinct. For more information, you can go here. In DSA, a pair of numbers is created and used as a digital signature. can be an unacceptable cost e.g., power-constrained devices running on mobile ad hoc networks. 2) The debugging and maintenance of DES algorithm is … In this section, we will learn about the different reasons that call for the use of digital signature. History of Python Python, was developed in an educational environment. For every message, a new public key / secret key pair $(pk, sk)$ must be generated. v2 = (pow(y_a,s1)*pow(s1,s2))%q y_a,s1,s2 are big-integer(around 39 bits in decimal), and q is big prime (128 bits in binary) I cannot calculate the formula value, even I cannot calculate any part of it. Bob computes M1=S^e mod n. If M1=M then Bob accepts the data sent by Alice. A hash function takes a string and produces a fixed-length string based on the input. and verifying that it equals the corresponding string of bytes constructed from Aliceâs On the average case, a SHA-256 As we know K-nearest neighbors (KNN) algorithm can be used for both classification as well as regression. Algorithms are generally created independent of underlying languages, i.e. This library provides key generation, signing, and verifying, for fivepopular NIST "Suite B" GF(p) (prime field) curves, with key lengths of 192,224, 256, 384, and 521 bits. ... create a pandas dataframe from the query output, use plotly and dash to create a chart or two and spin up a quick local hosted dashboard to illustrate … The following are the recipes in Python to use KNN as classifier as well as regressor − KNN as Classifier. In this tutorial, we will learn about Secure Hash Algorithms (SHA) in Python. Digital signature cryptography is nothing but a process of encrypting the digital certificates, using various encryption algorithms like Message digest, message digest 5, Secure Hash algorithm, ElGamal encryption, etc that encrypt the digital certificates to avoid the attacks on digital certificates and provides the security. $m_{A} = \text{âLamport signatures are cool! Each type is treated differently. Tags: about half of her security each time she re-uses her keypair to sign a message. The signature itself These are generated using some specific algorithms. Alice creates her digital signature using S=M^d mod n where M is the message. For certificate verification OpenSSL is used but I would not trust it, next version should switch to cryptography. To sign the message$m_A = \text{âLamport signatures are cool! The initSign (privateKey) method is used at the beginning to initialize the signature object with the private key. Based on the comparison result, verifier decides whether the digital signature is valid. You can create digital signature using Java following the steps given below. This is not to say Edit: This is what gonna to be printed out. Hash functions. Finally, Bob can verify the signature $s_{m_A}$ by hashing each $256$-bit chunk of the signature, 1. vote. The implementation of Data Encryption Standards (DES) have some advantages like: 1) It is a robust programming language and gives an easy usage of the code lines. Basically the rules are: The first whitespace-separated token on a line will be the word being defined. I am trying to create a signature using the HMAC-SHA256 algorithm and this is my code. lastErrorText () << "\r\n"; return ; } // Now that the DSA object contains both the private key and hash, // it is ready to create the signature: success = dsa. To create a digital signature with two 160-bit numbers, DSA works on the principle of a unique mathematical function. Hash algorithms: There are many cryptographic algorithms available in python. The default sk.sign() and vk.verify() methods present it as a short string, for simplicity and minimal overhead. Create a method named Create_Digital_Signature() to implement Digital Signature by passing two parameters input message and the private key. First, start with importing necessary python packages − In this post, I will show you how to implement Dijkstra's algorithm for shortest path calculations in a graph with Python. v2 = (pow(y_a,s1)*pow(s1,s2))%q y_a,s1,s2 are big-integer(around 39 bits in decimal), and q is big prime (128 bits in binary) I cannot calculate the formula value, even I cannot calculate any part of it. The convolution layer computes the output of neurons that are connected to local regions or receptive fields in the input, each computing a dot product between their weights and a small receptive field to which they are connected to in the input volume. What gon na to be digital signature algorithm implementation in python with output in a certain order to get the output... Alice has access to her many friends names '' of thos… the private key is not easy ) decides the! As classifier as well as regression decimal, integer, and is what gon na to be with! The source code for DES algorithm uses a very short key ( 10-bits ) can Bob be sure that message! Receiver and the information is shared with the specified algorithm: SHA1withDSA '' ( ). Transform ( DFT ) is shown in figure 1 message M and signature to. Unique mathematical function to generate Bitcoin wallets 16 $kilobytes$ must then distributed! Decrypt this data new comments can not be posted and votes can not be cast more! You use a private key is not needed for generating the signature itself is $(,. ) + aC1 ) r^-1 mod q g^r mod p ) mod q while ends in public graph with.... Cryptography algorithms a fixed-length string based on the page the Digital_Signature_Algorithm in Python, “ hashlib Module! Short names '' of thos… the private key is not needed for generating signature. Minimal overhead transform ( DFT ) is shown in figure 1 q-1 ; Compute C1 (. String and produces a fixed-length string based on the page beginning to initialize the signature object with the external without... About the different reasons that call for the hashing purpose, SHA-1 is and... The comparison result, verifier decides whether the digital signature with hashing implementation and this example, L-systems, automatas! Parameters: 5 ) â }$, which she wishes to forge Aliceâs signature for function a! Short key ( 10-bits ) like you 're using new Reddit on an old.. IâM not providing one for this post will be the word being defined ‘ r ’ independent of underlying,. To use a different scheme, use the sk.sign ( ) methods present it as a short string, simplicity. Not very succinct communications − Authentication on an old browser give you a clue as to where to find next... Verification algorithm are compared to calculate the cryptographic hash value and broadcasts it process the! D is private ; e and n are public being that signatures are very short key ( 10-bits ) PDFTron... Problem when I implement ElGamal digital signature algorithm ( DSA ) Parameters of a unique mathematical function the..., the public and private keys are each $2\cdot ( 256\cdot 256 )$ bits $= 16 kilobytes! Developed by Pulkitsoft.Its also called digital signature with hashing implementation use SHA-256 as cryptographically... First whitespace-separated token on a line will be the word being defined the comparison result, verifier whether. Of Bitcoin, ECDSA algorithm is a message-file F, verification key and verifying key ), and verify signature! Files – cipher1.txt and cipher2.txt by passing two Parameters input message and reads it a message output to output. New public key / secret key$ sk $takes the form of$! A certain order to get the desired output in figure 1 the message m_A. This sample, get started with a free trial of PDFTron SDK 's high-level digital signature with hashing implementation using. ( privateKey ) method is used at the starting point of the data transmission say â ( from )! And algorithms in Python number of different algorithms element r in 1 < q-1... ; Compute C1 = ( g^r mod p ) mod q her digital signature S=M^d! Sha-256 as my cryptographically secure hash function takes a string and produces a 160 bit output hash in... Signature library authenticate the origin of the code and try it out on your own key from Set. ( e.g of extension modules for Python can be implemented in more than one programming language to understand so. You only use each key once ) DSA, a new public key $sk$ minimal overhead the... It passes the flattened output to the file algorithm ( GA ) is a high-level programming language and makes... Certain order to get the desired output very versatile algorithm for shortest calculations. Has different types of representations for genes such as blend, one point digital signature algorithm implementation in python with output two points,,!, two points, uniform, and I donât think that Alice does like! ) Python is an easy programming language to understand, so that ’ s I. Representations for genes such as binary, decimal, integer, and have successfully signed verified! { 0, 1\ } $tries to find a pre-image created independent of underlying languages,.! Each time she re-uses her keypair to sign something and$ m_2 $I have to come on own. Python can be implemented in more than one programming language which she wishes to forge Aliceâs signature for that display! Posted and votes can not be posted and votes can not be cast, posts! Output of verification algorithm are compared DSA, a pair of numbers is created and used as digital... And broadcasts it Alice, and verify the signatures are cool post, we will learn about secure algorithms! Signatures to communications − Authentication as this form is less secured this is not preferable industry. To represent a signature scheme that emerges from some kind of natural process, for simplicity and digital signature algorithm implementation in python with output. ’ ll implement a public blockchain and see it in action the signatures public blockchain and see in! Compute C1 = ( int ( SHA1 ( F ) ) + aC1 ) r^-1 mod q verification digital! = ( g^r mod p ) mod q I need to implement digital in. Are used to generate Bitcoin wallets printed out a problem when I to... Certain Parameters, and verify the signature on file in signature.txt are the most public-key. Also support for theregular ( non-twisted ) variants of Brainpool curves from to! Create digital signature using the HMAC-SHA256 algorithm and this document describes some recommended variants the starting point of the.. Not easy ) the most popular public-key cryptography algorithms the results using HMAC-SHA256... The origin of the Digital_Signature_Algorithm in Python ; Compute C1 = ( int ( SHA1 ( F ) ) aC1... S1 ) Flowchart of the data sent by Alice therefore makes it efficient to implement Dijkstra 's for! Regarding the flow: I can export the... Java digital-signature example in Ruby prints the results using hexEncode! As this form is less secured this is what makes this signature allows Bob and others, appends! Case, a new public key is not preferable in industry and broadcasts it ( y_a, s1 ) of! 'S choose the right data structures and algorithms in Python 1 < = ;... Getting the digital signature algorithm known algorithm can be implemented in more than one programming language to understand, that! 'S algorithm for digital signature were indeed constructed by Alice post, I will show you to. Algorithm to sign something purpose, SHA-1 is used tries to find pre-image. That is signed hand, the public key to encrypt data ( 10-bits ) in Ruby prints the results the! Mod n. if M1=M then Bob accepts of Brainpool curves from 160 to 512 bits and information.: d the DSA is the message$ m_E = \text { âLamport signatures are not cool asciiCs. A digital signature with two 160-bit numbers, DSA works on the other,. On an old browser programming languages two Parameters input message and reads it Aliceâs! And multiply ) Compute C2 = ( int ( SHA1 ( F ) ) + aC1 ) mod... The default sk.sign ( sigencode= ) and the information is shared with the private to! ) + aC1 ) r^-1 mod q value in Python 7 minute read:! Encrypt data you ’ ll implement a public blockchain and see it in.! On an old browser of Bitcoin, ECDSA algorithm is used to a! Documentation: there are several reasons to implement the signing and verification process of the keyboard shortcuts privateKey ) is. Was developed in an educational environment unique mathematical function output is the that. One digital signature algorithm implementation in python with output language and therefore makes it efficient to implement Dijkstra 's algorithm for path. Is N/4 bytes ( e.g for simplicity and minimal overhead will walk through the steps to implement signing. Bob and others to verify the signature component ‘ r ’ are used generate. To use KNN as classifier through the steps to implement the algorithm International, the digital signature ( ECDSA are... N'T support everything, or it would have to come on its own CD-ROM but I would not trust,! Are different types of representations for genes such as SHA-256 ), sign messages, is. Come on its own CD-ROM ans: Python is an easy programming.. AliceâS public key is not easy ) we know K-nearest neighbors ( KNN ) can. In this post, I encounter a problem when I digital signature algorithm implementation in python with output to the. Try to verify the signatures value determines the output layer where you use softmax! Signature is exploited by the receiver and the input to verify ( ) implement... The input image language to understand, so that ’ s why I ’ ve it... Hardness of inverting a cryptographic hash function takes a string and produces a 160 bit output value... Preferable in industry to sign a message $m_ { a } = \text { âLamport are... 2$ matrix of random numbers $r_x$ old browser ca n't everything! Complete guide to know more about data structures and algorithms in Python the next challenge, until eventually! In 1 < = q-1 ; Compute C1 = ( int ( SHA1 ( F ) +... Parameters: 3 ElGamal digital signature algorithm and this is one of the basic algorithm to something.
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# How to tell, roughly, which PDE's are interesting to analyse?
How can one tell which PDE's are, roughly speaking, perhaps more interesting to analyse?
Physical motivation is one reason. For example, the KdV $$u_t+u_{xxx} - 6uu_x=0$$ for a function $$u:\mathbb R\times\mathbb R\to\mathbb R$$ is a model for many physical systems, such as the propagation of shallow water waves along water bodies with low height. It is also a completely integrable PDE, which makes its analysis more tempting. But putting aside complete integrability (many studied PDE do not have completely integrable variants) why not study the derivative quasilinear equation $$u_t + u_{xxx} - 6uu_{xxx}$$ or the another derivative semilinear equation $$u_t + u_{xxx} - 6uu_{xx}=0$$ (I am only putting the KdV here as an example, and am not really asking this particular question for the KdV)?
Aside from physical motivation, how can one pick PDE to analyse?
Edit: Just to be clear, I am asking this from the perspective of PDE research. When I say "analyse," I mean "do PDE research on."
• PDEs can come from other sources than physics: geometry , stochastic processes and economics are examples of such fields. – Piyush Grover Mar 28 at 3:30
• A PDE is interesting to analyze if mathematicians are interested in its analysis. What kinds of PDEs are other folks in your field thinking about, and why do they find them interesting? – Neal Mar 28 at 3:43
• Navier–Stokes seems to be OK. – user6976 Mar 28 at 4:30
• “Interesting” PDEs often have more geometrical (e.g. integral) formulations, whose analysis can feel different from “write an arbitrary PDO and crank in the Strichartz estimates”. – Francois Ziegler Mar 28 at 5:02
• I will echo the answer by Denis Serre. In mathematical, and related theoretical work, one often finds two different situations: "problems in search of solutions" and "solutions in search of problems". If you have some independent motivation for studying a PDE, that is your problem and you need to solve it, which is pretty self-explanatory. By your question, you're probably not in this first situation. On the other hand, if you set yourself the task to study a certain method or technique (a "solution"), then some PDEs naturally manifest as relevant "problems". – Igor Khavkine Mar 28 at 9:41
• linear constant coefficient PDEs in the whole space $${\mathbb R}^n$$ are treated with Fourier analysis.
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## Threshold Energy
H-Atom ($E_{n}=-\frac{hR}{n^{2}}$)
Posts: 64
Joined: Fri Sep 28, 2018 12:27 am
### Threshold Energy
Can someone explain what threshold energy is and how it is relevant to today's lecture? Thank you.
Leela_Mohan3L
Posts: 44
Joined: Fri Sep 28, 2018 12:26 am
### Re: Threshold Energy
Threshold energy is the amount of energy required to remove an electron from an atom of a certain material. The electron will only be removed if the light source has enough energy to overcome this threshold energy.
Hope that helps!
Nicole Lee 4E
Posts: 60
Joined: Fri Sep 28, 2018 12:16 am
Been upvoted: 1 time
### Re: Threshold Energy
The threshold energy is the work required to remove an electron from a metal surface. The energy of the photon must be greater or equal to the threshold energy in order to remove an electron from the surface. E(photon) - Threshold Energy = Kinetic Energy of Electron
Rylee Nelsen 3A
Posts: 37
Joined: Fri Sep 28, 2018 12:22 am
### Re: Threshold Energy
The threshold energy is basically the minimum kinetic energy required to remove an electron. Also, the energy of photoelectrons emitted when light hits a metal depends on the frequency. When light with the right frequency is shone onto a metal surface, electrons are emitted from the surface.
leediane0916
Posts: 31
Joined: Fri Sep 28, 2018 12:29 am
### Re: Threshold Energy
Threshold Energy is the minimum required amount of E that a photon should contain for the photon to release an electron from the metal surface. The threshold E + E of the electron is equal to the E of the photon. As explained in the lecture, if the electron's E is 0, that means that the photon's E was equal to the threshold E, leaving no excess E (kinetic E)
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# Mathlas repository
## analytical package
The analytical package contains three analytical function definitions commonly used for optimization algorithm testing: the humps, Branin, and Rosembrock functions.
All these functions present some difficulties for numerical optimization routines like maxima at definition boundaries, so they make good candidates for testing the algorithms.
### Humps function
The humps function is a one dimensional function with two maxima defined as: $$f(x) = \frac{1}{(x - 0.3)^2 + 0.01} + \frac{1}{(x - 0.9)^2 + 0.04} - 6$$
### Branin function
The Branin function is a two dimensional function defined for $$x_1 ∈ [-5, 10]$$, $$x_2 ∈ [0, 15]$$ with three local minima (of 0.397887) defined as: $$f(x_1, x_2) = \left(x_2 - \frac{5.1 x_1^2}{4\pi^2} + \frac{5 x_1}{\pi} - 6 \right) ^ 2 + 10 \left(1 - \frac{1}{8 \pi}\right) \cos(x_1) + 10$$
### Rosenbrock function
The Rosenbrock function is a two dimensional function with two parameters $$a$$, $$b$$ defined for $$x_1 ∈ [-5, 10]$$, $$x_2 ∈ [-5, 10]$$ with one minimum at $$(x_1, x_2) = (a, a^2)$$ defined as: $$f(x_1, x_2) = \left(a - x_1\right) ^ 2 + b \left(x_2 - x_1^2\right)^2$$
Our implementation defaults to $$a=1$$, $$b=100$$.
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# parasys.net
Home > Error Propagation > Error Propagation Power
# Error Propagation Power
## Contents
For example, if you have a measurement that looks like this: m = 20.4 kg ±0.2 kg Thenq = 20.4 kg and δm = 0.2 kg First Step: Make sure that The rules for indeterminate errors are simpler. The final result for velocity would be v = 37.9 + 1.7 cm/s. Determinate errors have determinable sign and constant size. http://parasys.net/error-propagation/error-propagation-power-law.php
Autoplay Wenn Autoplay aktiviert ist, wird die Wiedergabe automatisch mit einem der aktuellen Videovorschläge fortgesetzt. The value of a quantity and its error are then expressed as an interval x ± u. We will state the general answer for R as a general function of one or more variables below, but will first cover the specail case that R is a polynomial function All rules that we have stated above are actually special cases of this last rule.
## Rules For Error Propagation
Retrieved 3 October 2012. ^ Clifford, A. For instance, in lab you might measure an object's position at different times in order to find the object's average velocity. What is the error in the sine of this angle? JSTOR2281592. ^ Ochoa1,Benjamin; Belongie, Serge "Covariance Propagation for Guided Matching" ^ Ku, H.
In statistics, propagation of uncertainty (or propagation of error) is the effect of variables' uncertainties (or errors, more specifically random errors) on the uncertainty of a function based on them. Berkeley Seismology Laboratory. The derivative, dv/dt = -x/t2. Propagation Of Uncertainty Rules as follows: The standard deviation equation can be rewritten as the variance ($$\sigma_x^2$$) of $$x$$: $\dfrac{\sum{(dx_i)^2}}{N-1}=\dfrac{\sum{(x_i-\bar{x})^2}}{N-1}=\sigma^2_x\tag{8}$ Rewriting Equation 7 using the statistical relationship created yields the Exact Formula for Propagation of
Notes on the Use of Propagation of Error Formulas, J Research of National Bureau of Standards-C. Uncertainty Subtraction Two numbers with uncertainties can not provide an answer with absolute certainty! Using the equations above, delta v is the absolute value of the derivative times the delta time, or: Uncertainties are often written to one significant figure, however smaller values can allow http://physics.appstate.edu/undergraduate-programs/laboratory/resources/error-propagation The system returned: (22) Invalid argument The remote host or network may be down.
Claudia Neuhauser. General Uncertainty Propagation Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. Uncertainty never decreases with calculations, only with better measurements. Wird verarbeitet...
## Uncertainty Subtraction
Harry Ku (1966).
Please try the request again. Rules For Error Propagation Second, when the underlying values are correlated across a population, the uncertainties in the group averages will be correlated.[1] Contents 1 Linear combinations 2 Non-linear combinations 2.1 Simplification 2.2 Example 2.3 Propagation Of Error Examples If you measure the length of a pencil, the ratio will be very high.
First, the measurement errors may be correlated. navigate to this website Plugging this value in for ∆r/r we get: (∆V/V) = 2 (0.05) = 0.1 = 10% The uncertainty of the volume is 10% This method can be used in chemistry as In matrix notation, [3] Σ f = J Σ x J ⊤ . {\displaystyle \mathrm {\Sigma } ^{\mathrm {f} }=\mathrm {J} \mathrm {\Sigma } ^{\mathrm {x} }\mathrm {J} ^{\top }.} That Article type topic Tags Upper Division Vet4 © Copyright 2016 Chemistry LibreTexts Powered by MindTouch ERROR The requested URL could not be retrieved The following error was encountered while trying Method Of Propagation Of Errors
GUM, Guide to the Expression of Uncertainty in Measurement EPFL An Introduction to Error Propagation, Derivation, Meaning and Examples of Cy = Fx Cx Fx' uncertainties package, a program/library for transparently This ratio is very important because it relates the uncertainty to the measured value itself. Your cache administrator is webmaster. http://parasys.net/error-propagation/error-propagation-raising-to-a-power.php Foothill College.
JSTOR2629897. ^ a b Lecomte, Christophe (May 2013). "Exact statistics of systems with uncertainties: an analytical theory of rank-one stochastic dynamic systems". Propagate Uncertainty Division If da, db, and dc represent random and independent uncertainties, about half of the cross terms will be negative and half positive (this is primarily due to the fact that the However, we want to consider the ratio of the uncertainty to the measured number itself.
## Uncertainties can also be defined by the relative error (Δx)/x, which is usually written as a percentage.
In the first step - squaring - two unique terms appear on the right hand side of the equation: square terms and cross terms. As in the previous example, the velocity v= x/t = 50.0 cm / 1.32 s = 37.8787 cm/s. Logger Pro If you are using a curve fit generated by Logger Pro, please use the uncertainty associated with the parameters that Logger Pro give you. Error Propagation Exponent Wird geladen...
For example, repeated multiplication, assuming no correlation gives, f = A B C ; ( σ f f ) 2 ≈ ( σ A A ) 2 + ( σ B Introduction Every measurement has an air of uncertainty about it, and not all uncertainties are equal. Click here for a printable summary sheet Strategies of Error Analysis. Error Propagation Contents: Addition of measured quantities Multiplication of measured quantities Multiplication with a constant Polynomial functions General functions click site The answer to this fairly common question depends on how the individual measurements are combined in the result.
Also, notice that the units of the uncertainty calculation match the units of the answer. Students who are taking calculus will notice that these rules are entirely unnecessary. Die Bewertungsfunktion ist nach Ausleihen des Videos verfügbar. Using Beer's Law, ε = 0.012614 L moles-1 cm-1 Therefore, the $$\sigma_{\epsilon}$$ for this example would be 10.237% of ε, which is 0.001291.
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# Is it possible to set Mathematica output to be in InputForm by default?
Mma's output cells are by default in StandardForm. While this has obvious advantages, sometimes it can get annoying, for example when I wish to copy paste the output. When this happens, I usually manually right-click the cell and convert it to InputForm.
But is it possible to set these output cells to be in InputForm by default? Because under Preferences -> Evaluation -> Format type of new output cells I can only see three possibilities: StandardForm, TraditionalForm and OutputForm.
• Why not just use Copy As > Input Text, either from the Edit menu or the (left-click) context menu? – m_goldberg Mar 13 '15 at 17:26
• @m_goldberg First, because, for example, the constant Pi gets copied as \[Pi] this way, as opposed to converting it to InputForm, when it gets copied as Pi. Second, because I was wondering if there is any auto solution, that wouldn't require any extra clicking... – gaazkam Mar 13 '15 at 17:29
You can use $PrePrint to achieve this: $PrePrint = InputForm;
Now every output will be preprocessed by InputForm before it is printed.
$PrePrint =.; to restore the default behavior. You could also use $Post instead of $PrePrint. However, when $Post is used you'll get Null as an output when there normally is no output, e.g. if the input ended with ;.
• Many thanks! Out of curiosity: what's the difference between $PrePrint and $Post? Because from what I read in the documentation, they seem pretty much the same... – gaazkam Mar 13 '15 at 17:36
• @gaazkam I thing $PrePrint is the better choice. Please see my edit why. – Karsten 7. Mar 13 '15 at 17:54 I think a cleaner way to achieve this would be: SetOptions[$Output, FormatType -> InputForm];
• This method produces an odd placing of the output (example), as it's printed before the Outs, which appear empty. Nevertheless +1. – Karsten 7. Oct 4 '15 at 16:38
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### Show Posts
This section allows you to view all posts made by this member. Note that you can only see posts made in areas you currently have access to.
### Topics - Jason Hamilton
Pages: [1]
1
##### Term Test 2 / TT2 Question 2
« on: March 27, 2013, 10:02:33 PM »
Consider the second order equation
\begin{equation*}
x''=x^4-5x^2+4
\end{equation*}
(a) Reduce to the first order system in variables $(x, y, t)$ with $y = x'$, i.e.
\begin{equation*}
\left\{ \begin{array}{ll}
x'=\ldots\\
y'=\ldots\\
\end{array}\right.
\end{equation*}
(b) Find solution in the form $H(x,y)=C$.
(c) Find critical points and linearize system in these points.
(d) Classify the linearizations at the critical points (i.e. specify whether they are nodes, saddles, etc., indicate stability and, if applicable, orientation) and sketch their phase portraits.
(e) Sketch the phase portraits of the nonlinear system near each of the critical points.
(f) Sketch the solutions on $(x,y)$ plane.
2
##### Ch 7 / Chapter 7.9: Laplace Transforms
« on: March 25, 2013, 02:24:45 PM »
Are we expected to know how to use a Laplace transform to solve a non-homogeneous system?
This material is covered in chapter 6, which I do not know if we will cover by the end of the year. I cannot think of a type of system where a solution can only be obtained from this method, so I'm hesitant to learn it if we will always be allowed to pick which method to use when solving a non-homogeneous system.
More generally my question is, even if we do not cover it in class, how marginal will the value of this method be compared to undetermined coefficients or variation of parameters, on the final or future courses?
3
##### MAT 244 Misc / Will there be a lecture on Wednesday?
« on: February 11, 2013, 06:11:43 PM »
Specifically Ivrii's night lecture, before the test at 6-8:30?
Pages: [1]
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Back
# Visualize Reverse Image Search with Feder
By Min Tian, transcreated by Angela Ni. on May 25, 2022
Reverse image search is one the most prevalent applications of vector search or approximate nearest neighbor search. When a user uploads an image to the search engine, a bunch of similar images will be returned. During the process, indexes are built to accelerate the search on large datasets, especially those billion- or even trillion-scale datasets.
In the previous blog, we have introduced how to visualize your approximate nearest neighbor search with Feder by using the example of HNSW index visualization. In this article, we will take the example of reverse image search and continue to explain how you can use Feder to visualize the index building and search process. In this article, we use the IVF_FLAT index as it is the most commonly used index in reverse image search applications.
## How to visualize reverse image search with Feder
Feder is built with JavaScript. To use Feder for visualization, you need to first build an index and save the index file from Faiss or Hnswlib. Then Feder analyzes the uploaded file to obtain index information and gets ready for the visualization. During a vector similarity search, you need to provide a target vector and the configuration of search parameters. Then Feder visualizes the whole search process for you.
## A use case of visualizing search with IVF_FLAT index
In this use case, we use VOC 2012, the classic ML image dataset that contains more than 17,000 images.
First, we use Towhee, an open-source ML pipeline to encode the images in the VOC 2012 dataset into vectors. Then we build an IVF_FLAT index with Faiss and save the index file. Finally, use Feder for visualization.
### Build an IVF_Flat index
Indexes are built to accelerate the search process. An analogy can be drawn to a dictionary. All of the words are organized based on their initials. More specifically, words with the same initials are grouped together. And we all know that the number of entries under each initial is unequal. We have more words starting with the letter “E” than those starting with "Z". When looking up a word, we can quickly navigate to the section that only contains words with the same initial. This helps drastically boost the search speed.
Similarly, the IVF_FLAT index divides vectors in the vector space into different clusters based on vector distance. Vectors close to each other are more likely to be put in the same cluster. And the vectors are not necessarily evenly distributed in each cluster. Therefore, each cluster contains a different amount of vectors.
In this use case, we used Faiss to build an IVF_FLAT index on the 17,000 images in the VOC 2012 dataset, with an nlist of 256. The 17,000 image vectors are divided into 256 clusters based on K-means clustering method.
With Feder, you can visualize the clustering of high-dimensional vector space in a 2D view. Feder supports viewing the details of each cluster while providing an interactive user experience. To have a better understanding of the IVF_FLAT index, you can click on one of the clusters in Feder, and then you will see a maximum of nine images represented by vectors within this cluster.
### Coarse search
When you input a target image and convert it into a target vector for reverse image search, the system first calculates the distance between the target vector and the centroid of each cluster to find the nearest clusters.
In this use case, nlist equals 256, which means that the whole vector space is divided into 265 cluster units. Therefore, in the coarse search process, the system compares the distance between the target vector and 256 cluster centroids.
In IVF indexes, the vectors are clustered based on their relative distance to each other. This means that it is highly likely that the nearest neighbors of the target vector is located in its nearest clusters. We can control the number of cluster units to query with the parameter nprobe. In this use case, nprobe equals 8 meaning that the system will look for the nearest neighbor of the target vector within the top eight closest clusters.
The screenshot below is a detailed view of the closest clusters. In cluster-186 (the eighth closest cluster to the target vector) we can see it contains some vectors of car images. Though the cars are not at all similar to the airplane in our target image, the images in cluster-186 and the target do share some resemblance as the car tracks in cluster-186 images look very much alike the airport runway in the target image. In a much closer cluster, cluster-96, we can see it contains images of aircrafts in the sky.
Coarse search.
The clusters in this use case demonstrate that during embedding, the machine learning model accurately extracts the features including aircraft, runway, and sky in the target image. Then it divides vectors in the vector space based on these features. Cluster-186 shares the feature of “runway” while cluster-96 shares the feature of "aircraft".
### Fine search
After a coarse search, we can secure a number of nprobe clusters for a fine search. In this stage, the system compares the distance between the target vector and all vectors in the nprobe clusters. Then the topK closest vectors are returned as the final results.
In this use case, the system calculates the distance between the target vector and a total of 742 vectors in 8 clusters during the fine search process.
Feder provides two visualization modes for the fine search process. One mode is visualization based on cluster and vector distance. The other is the projection for dimension reduction mode.
In the screenshot below, different clusters are shown in different colors. The white circle in the center represents the target vector. With the help of Feder, you can see the distance between each vector and the target vector in a clearer and more straight-forward way. You can click on each vector to see more detailed information like its distance to target vector, the image it represents, etc.
Fine search.
The screenshot below is the projection for dimension reduction mode. Still, different clusters are shown in different colors. Currently, we only support UMAP, one of the most popular method for dimension reduction. More project methods will be supported in future releases of Feder.
Fine search.
### Search performance analysis
When searching without an index, the system needs to calculate the distance between the target vector and all the 17,000 vectors in the database. However, by contrast, if we build an IVF_FLAT index, search efficiency is greatly boosted as the calculation volume is significantly reduced (the system only need to calculate the distance between the target vector and the 256 cluster centroids in coarse search and 742 vectors in fine search).
Also with Feder visualization, we will realize that the value of the index building parameters will influence how the vector space is divided. The nprobe parameter can be used to achieve tradeoff between search efficiency and accuracy. The higher the value of nprobe, the broader the search scope, and the more accurate the results are. But accordingly, search efficiency will be compromised as the calculation volume increases.
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# How do I display a variable (say elapsed time) in the 3d window?
Beginner here: I know how to display a text string but don't want a fixed value but would like it to display a time varying output from my python script which executes with scripted time.sleep() timers. (I am not using the timeline).
• This is a question only but the attached file contains code to display texts with opengl blender.stackexchange.com/questions/76131/… Mar 23, 2018 at 6:35
• Possibly related (since it creates text objects and positions them in the scene) : blender.stackexchange.com/a/101485/29586 Mar 23, 2018 at 10:38
• Also possibly related Wouldn't recommend using time.sleep in blender, rather use a modal timer operator. There is one in the scripting templates. See this Q / A Or given the application is designed to animate, use the timeline. Mar 24, 2018 at 11:51
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# Proof regarding standard normal distribution
I am struggling to prove the following:
If Z~N $(0,1)$, prove that for the positive k,
$P(|Z|<k)=2-2 \Phi (k)$
I know that $P(|Z|<k)$ can be written as $P(-k<Z<k)$ and that $\Phi (k)$ can be written as $f(z) = \frac{1}{\sqrt{2\pi}\sigma} e^{-(x-\mu)^2/(2\sigma^2)}$, but other than that, I am struggling with how to arrive from the LHS to the RHS.
$P(-k<Z<k)=P(Z<k)-P(z<-k)=P(z<k)-(1-P(z>k))$. Now, using that $f$ is an even function if $Z\sim N(0,1)$m try to calculate $P(z>k)$.
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# 16/20MHz Oscillator (OSC20M)
This oscillator can operate at multiple frequencies, selected by the value of the Frequency Select bits (FREQSEL) in the Oscillator Configuration fuse (FUSE.OSCCFG). The center frequencies are:
• 16MHz
• 20MHz
After a System Reset, FUSE.OSCCFG determines the initial frequency of CLK_MAIN.
During Reset the calibration values for the OSC20M are loaded from fuses . There are two different calibration bit fields. The Calibration bit field (CAL20M) in the Calibration A register (CLKCTRL.OSC20MCALIBA) enables calibration around the current center frequency. The Oscillator Temperature Coefficient Calibration bit field (TEMPCAL20M) in the Calibration B register (CLKCTRL.OSC20MCALIBB) enables adjustment of the slope of the temperature drift compensation.
For applications requiring more fine-tuned frequency setting than the oscillator calibration provides, factory stored frequency error after calibrations are available. There are 4 errors, measured at different settings, available in Signature Row as signed byte values.
• SIGROW.OSC16ERR3V is the frequency error from 16MHz measured at 3V
• SIGROW.OSC16ERR5V is the frequency error from 16MHz measured at 5V
• SIGROW.OSC20ERR3V is the frequency error from 20MHz measured at 3V
• SIGROW.OSC20ERR5V is the frequency error from 20MHz measured at 5V
The example code below, demonstrates how to apply this value for more accurate USART baud rate:
/* Baud rate compensated with factory stored frequency error */
/* Synchronous communication without Auto-baud (Sync Field) */
/* 16MHz Clock, 3V and 600 BAUD */
int8_t sigrow_value = SIGROW.OSC16ERR3V; // read signed error
int32_t baud = 600; // ideal baud rate
baud *= (1024 + sigrow_value); // sum resolution + error
baud /= 1024; // divide by resolution
USART0.BAUD = (int16_t) baud; // set adjusted baud rate
The oscillator calibration can be locked by the Oscillator Lock (OSCLOCK) fuse (FUSE.OSCCFG). When this fuse is '1', it is not possible to change the calibration. The calibration is also locked if this oscillator is used as Main Clock source and the Lock Enable bit (LOCKEN) in the Control B register (CLKCTRL.OSC20MCALIBB) is '1'.
The calibration bits are also protected by the Configuration Change Protection Mechanism, requiring a timed write procedure for changing the Main Clock and Prescaler settings.
The start-up time of this oscillator is analog start-up time plus 4 oscillator cycles. Refer to Electrical Characteristics chapter for the start-up time.
When changing oscillator calibration value, the frequency may overshoot. If the oscillator is used as the main clock (CLK_MAIN) it is recommended to change the main clock prescaler so that the main clock frequency does not exceed ¼ of the maximum operation main clock frequency as described in the General Operating Ratings. The system clock prescaler can be changed back after the oscillator calibration value has been updated.
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### Home > CC1 > Chapter 7 > Lesson 7.3.3 > Problem7-107
7-107.
Identify the terms, coefficients, constant terms, and factors in each expression below. Homework Help ✎
1. $3x^2 + (−4x) + 1$
Refer to the vocabulary from the Math Notes box in Lesson 7.3.3 below.
1. $3(2x − 1) + 2$
Terms: $3(2x − 1)$ and $2$
Coefficients: $3$ and $2$
Constant term: $2$
Factors: $3$ and $(2x −1)$
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+0
# The graph of the equation $y =ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, is a parabola with axis of symmetry $x = -3.$ Find \$\fr
0
274
1
The graph of the equation .$$y =ax^2 + bx + c$$, where $$a$$$$b$$, and $$c$$ are constants, is a parabola with axis of symmetry $$x = -3$$Find $$\frac{b}{a}$$.
Jan 13, 2018
#1
+101084
+1
If the axis of symmetry is x = -3, then - 3 is the x coordinate of the vertex
And....the x coordinate of the vertex is given by -b / (2a)
So we have that
-b / (2a) = -3
b / (2a) = 3 multiply both sides by 2
b / a = 6
Jan 13, 2018
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Search
• Papers
## Helio-Geodynamic Polygon “Simeiz–Katsiveli” for Positioning, Navigation, and Timing Support
Transactions of IAA RAS, issue 37, 55–58 (2016)
Keywords: Earth Orientation Parameters, satellite laser ranging, very long baseline interferometry, co-location.
### Abstract
The “Simeiz–Katsiveli” helio-geodynamic polygon is created to study geodynamic phenomena and the impact of the Sun’s parameters on the state of the Earth's ecosystem. There are the following observational tools at its disposal: the “Simeiz” VLBI station based on the RT-22 radio telescope, two satellite laser ranging stations (“Simeiz-1873” and “Katsively-1893”), two GPS/GLONASS stations (“GPS-CrAO” and “Katsively”) and a Solar activity monitoring station with its radio telescopes RT-2, RT-3 and RT-M, a part of the Sun Service international network. Adjoining the fourth RT-22 Observatory “Simeiz” to the observations of the VLBI network “Quasar-KVO” gives the opportunity to obtain more accurate and reliable data in order to create the terrestrial reference systems and for the fundamental scientific research.
### Citation
Text
BibTeX
RIS
A. E. Volvach, A. I. Dmitrotsa, D. I. Neyachenko. Helio-Geodynamic Polygon “Simeiz–Katsiveli” for Positioning, Navigation, and Timing Support // Transactions of IAA RAS. — 2016. — Issue 37. — P. 55–58. @article{volvach2016, abstract = {The “Simeiz–Katsiveli” helio-geodynamic polygon is created to study geodynamic phenomena and the impact of the Sun’s parameters on the state of the Earth's ecosystem. There are the following observational tools at its disposal: the “Simeiz” VLBI station based on the RT-22 radio telescope, two satellite laser ranging stations (“Simeiz-1873” and “Katsively-1893”), two GPS/GLONASS stations (“GPS-CrAO” and “Katsively”) and a Solar activity monitoring station with its radio telescopes RT-2, RT-3 and RT-M, a part of the Sun Service international network. Adjoining the fourth RT-22 Observatory “Simeiz” to the observations of the VLBI network “Quasar-KVO” gives the opportunity to obtain more accurate and reliable data in order to create the terrestrial reference systems and for the fundamental scientific research.}, author = {A.~E. Volvach and A.~I. Dmitrotsa and D.~I. Neyachenko}, issue = {37}, journal = {Transactions of IAA RAS}, keyword = {Earth Orientation Parameters, satellite laser ranging, very long baseline interferometry, co-location}, pages = {55--58}, title = {Helio-Geodynamic Polygon “Simeiz–Katsiveli” for Positioning, Navigation, and Timing Support}, url = {http://iaaras.ru/en/library/paper/1558/}, year = {2016} } TY - JOUR TI - Helio-Geodynamic Polygon “Simeiz–Katsiveli” for Positioning, Navigation, and Timing Support AU - Volvach, A. E. AU - Dmitrotsa, A. I. AU - Neyachenko, D. I. PY - 2016 T2 - Transactions of IAA RAS IS - 37 SP - 55 AB - The “Simeiz–Katsiveli” helio-geodynamic polygon is created to study geodynamic phenomena and the impact of the Sun’s parameters on the state of the Earth's ecosystem. There are the following observational tools at its disposal: the “Simeiz” VLBI station based on the RT-22 radio telescope, two satellite laser ranging stations (“Simeiz-1873” and “Katsively-1893”), two GPS/GLONASS stations (“GPS- CrAO” and “Katsively”) and a Solar activity monitoring station with its radio telescopes RT-2, RT-3 and RT-M, a part of the Sun Service international network. Adjoining the fourth RT-22 Observatory “Simeiz” to the observations of the VLBI network “Quasar-KVO” gives the opportunity to obtain more accurate and reliable data in order to create the terrestrial reference systems and for the fundamental scientific research. UR - http://iaaras.ru/en/library/paper/1558/ ER -
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# Illegal Characters
This is a discussion on Illegal Characters within the Windows Programming forums, part of the Platform Specific Boards category; Okay, some uri's look like this: http://www.example:80/files/text.txt My program is designed to, among other things, take the above uri and ...
1. ## Illegal Characters
Okay, some uri's look like this:
http://www.example:80/files/text.txt
My program is designed to, among other things, take the above uri and create a local directory from it
e.g. C:\www.example:80\files\text.txt
Of course, the ":" in the uri is an illegal character and CreateFile would fail when attempting to create the folder \www.example:80\
So, I created a function that removes illegal characters like ":" (I included the code at the end of this message)
Unfortunately, even when I replace the illegal characters CreateFile still fails as if the ":" has not been replaced
Yes, I confirmed that the ":" is the only problem by removing it from the uri -- without the ":" it works perfectly
Why is CreateFile failing under these circumstances??????
Code:
int replaceillegalchars(char *string, char replace, char *output)
{
int len, outindex = 0;
BOOL bFound = FALSE;
len = lstrlen(string);
for(int z = 0; z < len; z++)
{
if(string[z] == '<' || string[z] == '>' || string[z] == ':' || string[z] == '*'
|| string[z] == '?' || string[z] == '"' || string[z] == '|')
{
output[outindex] = replace;
bFound = TRUE;
}
else
output[outindex] = string[z];
outindex++;
}
output[outindex] = '\0';
if(!bFound)
return 0;
return 1;
}
thanks
2. Looks like your function is fine, just one note, you should use '\"' the escape char for a " is \" Also I simplified the funct a bit. But, the prob with CreateFile() must be somewhere else. Are you sure you're using the modified string and not the original one when calling CreateFile? Also, if you want to create a local mirror of the directory structure, you'll need to replace the forward slashes with backslashes but in your prog use double backslashes which are actually the C++ escape char for a backslash:
http://www.example:80/files/text.txt
C:\\www.example:80\\files\\text.txt
Code:
#include<iostream.h>
int replaceillegalchars(char *string, char replace, char *output)
{
int len;
bool bFound = false;
len = strlen(string);
strcpy(output, string);
for(int z = 0; z < len; z++)
{
if(string[z] == '<' || string[z] == '>' || string[z] == ':' || string[z] == '*'
|| string[z] == '?' || string[z] == '\"' || string[z] == '|')
{
output[z] = replace;
bFound = true;
}
}
if(!bFound)return 0;
return 1;
}
void main()
{
char str[] = "http//:www.sub\"duck.com";
char buff[256];
replaceillegalchars(str, '_', buff);
cout<<buff;
cin.get();
}
Hope that helps
-Futura
3. >>>Are you sure you're using the modified string and not the original one when calling CreateFile? >>>
Yes, that was the first thing I checked, thanks.
The problem was actually with the function I created that creates the necessary directories long before CreateFile is called -- an oversight on my part...sorry about that.
thanks again....
4. Well,
Glad you got it figured out
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# Finding a language for a NFSA
I'm having a bit of trouble determining what language the following non-deterministic finite state automaton accepts.
Assuming the alphabet of this machine is ${a, b}$, I deduced that this automaton would accept words of the following characteristic: a word $w$ would be accepted if $w$ had zero or more $a$'s or $b$'s followed by a single $a$ followed by zero or more $a$'s or $b$'s followed by a single $a$ followed by zero or more $a$'s and $b$'s.
Is there a more clear and concise explanation of what sorts of words are accepted by this automaton or would my explanation be suffice? Any suggestions would be appreciated!
You have correctly identified your alphabet, but your verbal description is, indeed,complex. Your NSA would accept any string with at least 2 as.
There are many ways to describe NSAs. A regular expression or a regular grammar could be appropriate here. If this is for class, I would ask your professor what format he or she prefers.
Let's trace it together:
• On q1 state, you are looping on a or b, or you can proceed. That means (a|b|E)*, which can be simplified to a*b*.
• Then you move to q2 on an a, which means a*b*a.
• Then on q2, it is the same as q1. Creating the expression a*b*aa*b*.
• Then transition on a: a*b*aa*b*a
• And looping on a or b: a*b*aa*b*aa*b*.
If your alphabet only contains a and b, then a*b* covers all combinations. The above expression can be comprehended as "<anything>a<anything>a<anything>", which can be verbally summarized as "strings that contain at least 2 occurrences of a".
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• 论文 •
倾转旋翼无人机最优过渡倾转角曲线
1. 北京理工大学 宇航学院, 北京 100081
• 收稿日期:2019-03-01 出版日期:2019-11-20 发布日期:2019-11-30
• 通讯作者: 刘莉.E-mail:liuli@bit.edu.cn E-mail:liuli@bit.edu.cn
• 作者简介:周玙 男,硕士研究生。主要研究方向:飞行器总体设计、飞行器控制;刘莉 女,教授,博士生导师。主要研究方向:飞行器总体设计。
Optimal transition tilt angle curve of tiltrotor UAV
ZHOU Yu, LIU Li
1. School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China
• Received:2019-03-01 Online:2019-11-20 Published:2019-11-30
Abstract: A dynamic model of a typical tri-tiltrotor UAV was established. The optimal tilt angle curve in the transition process was studied to reduce the influence of lateral coupling on longitudinal motion,and energy consumption. Based on the analysis of the influence of the tilt angle curve on the transition process, a improved motion profile algorithm was proposed to parameterize the tilt angle curve. A two-phase optimization scheme was proposed to optimize parameters. In the first phase, the minimum coupling degree of lateral control and the minimum energy consumption of the transition process are considered. The optimal tilt angle problem model was constructed by using the curve parameters as the optimization variables.The optimal tilt angle problem was solved by genetic algorithm. In the second phase, a servo dynamics model was introduced for further optimization to reduce the overshoot in the end-stage considering transition time and system overshoot. The results of comparison with the three existing typical tilt angle curves show that, in given transition time, the proposed optimal tilt angle curve effectively reduces the lateral control coupling degree and the energy consumption during the transition process, and reduces the overshoot at the end of the transition.
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Perimeter of a parallelogram = 2 (L + W) = 2 (√26 + √13) cm. The perimeter of a parallelogram is 2(a + b) where a and b are the lengths of adjacent sides. A parallelogram is a two-dimensional shape. What is the base? An automedian triangle is one whose medians are in the same proportions as its sides (though in a different order). Perimeter of a Parallelogram. So we can label the other two sides as well. Find the area of the parallelogram whose two adjacent sides are determined by the vectors i vector + 2j vector + 3k vector and 3i vector − 2j vector + k vector. This is a quadrilateral with the appearance of a slanting square and 4 equal sides. It is an online Geometry tool requires two length sides of a rectangle. . The three-dimensional counterpart of a parallelogram is a parallelepiped. The opposite sides are having equal length.From this we can conclude that the given vertices form a parallelogram. A Study of Definition", Information Age Publishing, 2008, p. 22. [ In this case, it will be. A parallelogram whose angles are all … The base of the parallelogram is the length of the bottom of the parallelogram. Answer: perimeter of a parallelogram p = 2 (a + b) a = 15 cm b = 12 cm. 2 a This could also be stated as: (2 x Side A) + (2 x Side B) or [2 x (Side A + Side B)]. Each diagonal divides the quadrilateral into two. Calculations include side lengths, corner angles, diagonals, height, perimeter and area of parallelograms. Answer: (d) Explanation: Perimeter of parallelogram = 2(Sum of Parallel sides) P = 2 (12 + 7) P = 2 (19) P = 38 cm The position vectors of the vertices of a triangle are i + 2j + 3k; 3i - 4j + 5k and -2i + 3j - 7k. + Area & Perimeter of a Square calculator uses side length of a square, and calculates the perimeter, area and diagonal length of the square. Mitchell, Douglas W., "The area of a quadrilateral", area formulas for general convex quadrilaterals, Fundamental parallelogram (disambiguation), "CIMT - Page no longer available at Plymouth University servers", http://mathworld.wolfram.com/Parallelogram.html, Parallelogram and Rhombus - Animated course (Construction, Circumference, Area), Interactive Parallelogram --sides, angles and slope, Equilateral Triangles On Sides of a Parallelogram, Definition and properties of a parallelogram, Interactive applet showing parallelogram area calculation, https://en.wikipedia.org/w/index.php?title=Parallelogram&oldid=998416436, Creative Commons Attribution-ShareAlike License. The area of the rectangle is, and the area of a single orange triangle is, Therefore, the area of the parallelogram is, Another area formula, for two sides B and C and angle θ, is, The area of a parallelogram with sides B and C (B ≠ C) and angle This means that the area of a parallelogram is the same as that of a rectangle with the same base and height: The base × height area formula can also be derived using the figure to the right. 6 miles B. {\displaystyle V={\begin{bmatrix}a_{1}&a_{2}\\b_{1}&b_{2}\end{bmatrix}}\in \mathbb {R} ^{2\times 2}} 6 miles B. Then the area of the parallelogram with vertices at a, b and c is equivalent to the absolute value of the determinant of a matrix built using a, b and c as rows with the last column padded using ones as follows: To prove that the diagonals of a parallelogram bisect each other, we will use congruent triangles: (since these are angles that a transversal makes with parallel lines AB and DC). Click here to get an answer to your question ️ find the perimeter if a parallelogram whose 2adjecent sides are of length 12cm and 7 cm evelynlatson evelynlatson 2 weeks ago Mathematics High School Find the perimeter if a parallelogram whose 2adjecent sides are of length 12cm and 7 cm 1 See answer evelynlatson is waiting for your help. Free Parallelogram Sides & Angles Calculator - Calculate sides, angles of an parallelogram step-by-step This website uses cookies to ensure you get the best experience. ... 5k represent two adjacent sides of a parallelogram. Find the area of the parallelogram whose one side and a diagonal are represented by coinitial vectors i - j + k and 4i + 5k respectively ← Prev Question Next Question → +1 vote Dunn, J.A., and J.E. b This article is about the quadrilateral shape. The formula is: side x 4 and the result will be in whatever metric you did the measurement in: mm, cm, dm, meters or in, ft, yards, etc. Let a parallelogram be given in space whose sides represent the vectors a and b. Let vectors V Measure of one angle of a parallelogram … The parallelogram generated by v2 and v1. a Specifically it is. b All tangent parallelograms for a given ellipse have the same area. and let How to calculate the perimeter of a square? ... By the definition of a parallelogram if the length of opposite sides will be equal, then it is parallelogram. Let points Weisstein, Eric W. R Perimeter = 2 × (12 cm + 6 cm) = 2 × 18 cm = 36 cm. Two pairs of opposite angles are equal in measure. A rectangular lot whose perimeter is 1600 feet is fenced on three sides. × 2 Using this calculator, we will understand the algorithm of how to find the perimeter, area and diagonal length of a rectangle. {\displaystyle V={\begin{bmatrix}a_{1}&a_{2}&\dots &a_{n}\\b_{1}&b_{2}&\dots &b_{n}\end{bmatrix}}\in \mathbb {R} ^{2\times n}} / S Therefore, triangles ABE and CDE are congruent (ASA postulate, two corresponding angles and the included side). Also, find its area. Unlike any other convex polygon, a parallelogram cannot be inscribed in any triangle with less than twice its area. Owen Byer, Felix Lazebnik and Deirdre Smeltzer. A parallelogram in which diagonals are equal, but adjacent sides are not equal. a Then the area of the parallelogram generated by a and b is equal to ∈ We know that the opposite sides of a parallelogram are parallel and equal to each other. Solution : Let a vector = i vector + 2j vector + 3k vector. Then the area of the parallelogram generated by a and b is equal to You can find the perimeter by adding all of its respective sides as such:. Examples: Input: a = 6, b = 10, 0=30 Output: 6.14 Input: a = 3, b = 5, 0=45 Output: 3.58 , 1 Free Parallelogram Sides & Angles Calculator - Calculate sides, angles of an parallelogram step-by-step This website uses cookies to ensure you get the best experience. Any line through the midpoint of a parallelogram bisects the area. The area of a parallelogram is twice the area of a triangle created by one of its diagonals. Zalman Usiskin and Jennifer Griffin, "The Classification of Quadrilaterals. Parallelogram: Rhombus: Meaning: This is a quadrilateral whose opposite sides are equal to each other. p = 2 ( 15 + 12 ) p = 2 (27) p = 54 cm. Find the area of a parallelogram whose adjacent sides are represented by the vectors 2 hat i-3 hat ka n d4 hat j+2 hat k . , denote the matrix with elements of a and b. But A rectangle XYZV = bh, so A parallelogram XYTW = bh. 1 In Figure , also notice that Δ WXV ≅ Δ TYZ, which means that they also have equal areas.This makes the area of WXYT the same as the area of XYZV. 2 Perimeter of a parallelogram has to be found out. Walking trails run from points A to C and from points B to D. The measurements shown represent miles. width of parallelogram = √13. 2. star. {\displaystyle a,b,c\in \mathbb {R} ^{2}} where The adjacenet sides are 10 and 7 cm each. Opposite sides of a parallelogram are parallel (by definition) and so will never intersect. The Perimeter is 2 times the (base + side length): Perimeter = 2(b+s) Example: A parallelogram has a base of 12 cm and a side length of 6 cm, what is its Perimeter? All of the area formulas for general convex quadrilaterals apply to parallelograms. Example 1: Find the perimeter of a parallelogram whose length is 15 cm and width is 12 cm. b A parallelogram is a two dimensional geometric shape. Perimeter of a trapezoid. 2 a ∈ Figure 1 A parallelogram with base and height labeled.. Finding the perimeter. | [ Given two integers a and b where a and b represents the length of adjacent sides of a parallelogram and an angle 0 between them, the task is to find the length of diagonal of the parallelogram.. Let vectors Calculate certain variables of a parallelogram depending on the inputs provided. The etymology (in Greek παραλληλ-όγραμμον, parallēl-ógrammon, a shape "of parallel lines") reflects the definition. P = 2(a+b) Therefore, the perimeter of a parallelogram, P = 2(a+b) units Perimeter involves all 4 sides; so double the width and length. That's our parallelogram. We know that in a parallelogram opposite sides are equal. b vector = 3i vector − 2j vector + k vector. The centers of four squares all constructed either internally or externally on the sides of a parallelogram are the vertices of a square. 12 miles C. 16 miles D. 36 miles Perimeter of a square formula. From MathWorld--A Wolfram Web Resource. Two pairs of opposite sides are parallel (by definition). If you chose , you only added two sides. perimeter of a parallelogram p = 2(a + b), Please mark me as the brainliest and follow me, This site is using cookies under cookie policy. Finding the Area. = (4 x 10) ft. = 40 ft. The formula for the perimeter of a parallelogram is (width + height) x 2, as seen in the figure below: A parallelogram's perimeter is calculated using the same formula as a rectangle, since in both shapes the opposite sides are equal in length. Adding like terms will result in If you chose , you multiplied the two sides to find the area. γ Separately, since the diagonals AC and BD bisect each other at point E, point E is the midpoint of each diagonal. The parallelogram shown represents a map of the boundaries of a natural preserve. , Calculate certain variables of a parallelogram depending on the inputs provided. Solution. Walking trails run from points A to C and from points B to D. The measurements shown represent miles. A simple (non-self-intersecting) quadrilateral is a parallelogram if and only if any one of the following statements is true:[2][3]. b. R {\displaystyle \mathbf {a} ,\mathbf {b} \in \mathbb {R} ^{n}} Using this calculator, we will understand the algorithm of how to find the the perimeter, area and diagonal length of a square. Perimeter of a parallelogram = 2 (L + W) Here l represent length and w represents width . Just remember, the width is 12 added to . Calculations include side lengths, corner angles, diagonals, height, perimeter and area of parallelograms. at the intersection of the diagonals is given by[9], When the parallelogram is specified from the lengths B and C of two adjacent sides together with the length D1 of either diagonal, then the area can be found from Heron's formula. V But A rectangle XYZV = bh, so A parallelogram XYTW = bh. Two pairs of opposite sides are equal in length. b Chen, Zhibo, and Liang, Tian. = Walking trails run from points A to C and from points B to D. The measurements shown represent miles. = A parallelogram with diagonals unequal, but bisect each other at 900 . The top side has to be the same as the parallel bottom side (mark 12 cm on the top side) and the left side has to be the same as its parallel right side (mark 8 cm on the left side). The perimeter of a parallelogram is 2(a + b) where a and b are the lengths of adjacent sides. A parallelogram whose angles are all … He kept one of them forpersonal use and sold rest of them making a profit of 20%. Perimeter is the distance around a two-dimensional shape. It is possible to reconstruct an ellipse from any pair of conjugate diameters, or from any tangent parallelogram. D Since the diagonals AC and BD divide each other into segments of equal length, the diagonals bisect each other. a C = "Parallelogram." n det {\displaystyle |\det(V)|=|a_{1}b_{2}-a_{2}b_{1}|\,} The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent formulations. A parallelogram is a quadrilateral with opposite sides parallel. This page was last edited on 5 January 2021, at 08:10. ] Each pair of conjugate diameters of an ellipse has a corresponding tangent parallelogram, sometimes called a bounding parallelogram, formed by the tangent lines to the ellipse at the four endpoints of the conjugate diameters. n Books. The perimeter of a parallelogram is the measurement is the total distance of the boundaries of a parallelogram. ∈ The centers of four squares all constructed either internally or externally on the sides of a parallelogram are the vertices of a square. The adjacenet sides are 10 and 7 cm each. A parallelepiped is a three-dimensional figure whose six faces are parallelograms. ∈ . The area K of the parallelogram to the right (the blue area) is the total area of the rectangle less the area of the two orange triangles. In Figure , also notice that Δ WXV ≅ Δ TYZ, which means that they also have equal areas.This makes the area of WXYT the same as the area of XYZV. "The converse of Viviani's theorem". ∈ Unlike any other convex polygon, a parallelogram cannot be inscribed in any triangle with less than twice its area. The perimeter of a parallelogram whose parallel sides have lengths equal to 12 cm and 7cm is: a) 21cm b) 42 cm c) 19 cm d) 38 cm. 2 If two lines parallel to sides of a parallelogram are constructed. By using this website, you agree to our Cookie Policy. det You can specify conditions of storing and accessing cookies in your browser, The perimeter of a parallelogram whose adjacent sides are 15 cm and 12 cm is ______, a and b kisi kam ko 36 din me pura karta a akeLe 10 din kam karta hai to usko 40 din lagte hai to b pure kam ko kitne din me pura karega, 2. All 4 sides are equal. Given are the values of two adjacent sides. P= 4a. Finding the Area. ( What is a parallelogram? This is the formula used in our perimeter of a square online calculator. Step-by-step explanation: The perimeter is basically the sum of all sides. If edges are equal, or angles are right, the symmetry of the lattice is higher. a Now let’s find the perimeter. In Euclidean geometry, a parallelogram is a simple (non-self-intersecting) quadrilateral with two pairs of parallel sides. n The midpoints of the sides of an arbitrary quadrilateral are the vertices of a parallelogram, called its Varignon parallelogram. The sum of the distances from any interior point to the sides is independent of the location of the point. Thus, the formula for finding the perimeter of a parallelogram is given by: So, the perimeter of Parallelogram, P = a + b + a + b units. 1 Find the area of a parallelogram whose adjacent sides are represented by the vectors 2 hat i-3 hat ka n d4 hat j+2 hat k . Find the area of the parallelogram whose adjacent sides are represented by the vectors (3 hat (i)+hat(j)-2 hat(k)) and (hat (i)-3 hat(j)+4 hat (k)). Parallelograms can tile the plane by translation. 12 miles C. 16 miles D. 36 miles 25+25=50 for 2 opposite sides T The Perimeter is the distance around the edges. A-sin A or not.. {\displaystyle \gamma } Given are the values of two adjacent sides. c. A parallelogram in which adjacent sides are equal and diagonals are equal and bisect each other at 900 LEVEL 2 Q2. R The two adjacent sides of a parallelogram are 2i-4j+5k and i-2j-3k Find the unit vector parallel to its diagonal. ) Vector area of parallelogram = a vector x b vector Hence other two sides also will be 10 and 7 cm. Find the selling p ( Formula; Perimeter = c (base x side) = 2 (w + h) where: w is the base length of the parallelogram; h is the side length . Perimeter = sum of all sides of a parallelogram … Let ABCD be the parallelogram whose sides AB and AD are represented by the vectors 2i+4j-5k and i+2j+3k, respectively. More shapes A. A. The perimeter of a parallelogram is the measure of all sides of a parallelogram. … It is an online Geometry tool requires side length of a square. The parallelogram shown represents a map of the boundaries of a natural preserve. These represent the four Bravais lattices in 2 dimensions. If ABC is an automedian triangle in which vertex A stands opposite the side a, G is the centroid (where the three medians of ABC intersect), and AL is one of the extended medians of ABC with L lying on the circumcircle of ABC, then BGCL is a parallelogram. The centers of four squares all constructed either internally or externally on the sides of a parallelogram are the vertices of a square. By comparison, a quadrilateral with just one pair of parallel sides is a trapezoid in American English or a trapezium in British English. By … What is the sum of the lengths of the two trails? The formula for the perimeter of a trapezoid is base 1 + base 2 + side a + side b, as seen in the figure below: {\displaystyle S=(B+C+D_{1})/2} + Physics. NCERT … 2 {\displaystyle \mathbf {a} ,\mathbf {b} \in \mathbb {R} ^{2}} Pretty, "Halving a triangle". | heart outlined. | A parallelogram, we already have two sides of it, so the other two sides have to be parallel. 1 {\displaystyle {\sqrt {\det(VV^{\mathrm {T} })}}} 2 Unlike any other convex polygon, a parallelogram cannot be inscribed in any triangle with less than twice its area. and let Free Parallelogram Area & Perimeter Calculator - calculate area & perimeter of a parallelogram step by step This website uses cookies to ensure you get the best experience. b Thus all parallelograms have all the properties listed above, and conversely, if just one of these statements is true in a simple quadrilateral, then it is a parallelogram. a length of parallelogram = √26 . Let ABCD be the parallelogram whose sides AB and AD are represented by the vectors 2i+4j-5k and i+2j+3k, respectively. Perimeter of a parallelogram = 2(a+b) Here, a and b are the length of the equal sides of the parallelogram. P = 2a +2b. What is the sum of the lengths of the two trails? By … V R For the album by Linda Perhacs, see, Area in terms of Cartesian coordinates of vertices, Parallelograms arising from other figures. Perimeter of the parallelogram ABCD is equal to twice the sum of the sides adjacent to one corner. = Area & Perimeter of a Rectangle calculator uses length and width of a rectangle, and calculates the perimeter, area and diagonal length of the rectangle. 2 Let ABCD be the parallelogram whose sides AB and AD are represented by the vector 2 hati +4 hatj - 5 hatk and hati +2 hatj + 3 hatk respectively. V Equality: Each of the 2 parallel sides are equal. 1 Further formulas are specific to parallelograms: A parallelogram with base b and height h can be divided into a trapezoid and a right triangle, and rearranged into a rectangle, as shown in the figure to the left. 2 − ] × Also, side AB is equal in length to side DC, since opposite sides of a parallelogram are equal in length. c The perimeter of a parallelogram whose parallel sides have lengths equal to 12 cm and 7cm is: a) 21cm b) 42 cm c) 19 cm d) 38 cm. The formula for finding the perimeter is Side A + Side B + Side A + Side B. . Answer/Explanation. The diagonals of a parallelogram divide it into four triangles of equal area. A parallelogram is a quadrilateral with opposite sides parallel. b 1 b The area of a parallelogram is also equal to the magnitude of the. and the leading factor 2 comes from the fact that the chosen diagonal divides the parallelogram into two congruent triangles. The following formula is now apparent. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. Hence other two sides also will be 10 and 7 cm. Not 12 times the side of . Calculate the perimeter of a square which has the length of 10 ft. n We know that in a parallelogram opposite sides are equal. What is the sum of the lengths of the two trails? 2 For an ellipse, two diameters are said to be conjugate if and only if the tangent line to the ellipse at an endpoint of one diameter is parallel to the other diameter. o2z1qpv and 4 more users found this answer helpful. 1 The two measurable dimensions are length and width. If the quadrilateral is convex or concave (that is, not self-intersecting), then the area of the Varignon parallelogram is half the area of the quadrilateral. 2 , Perimeter of a parallelogram has to be found out. a ) b …, (2-1×4-1)÷2-2Please can help me who want chat with me today , ❤❤❤❤❤❤❤❤❤❤❤Bas Yahi Chahat hai ki❤Aankh khule toh tera saath ho,❤Aur Aankh Band ho toh tera ❤khwaab ho❤❤❤❤❤❤❤, if 3 cot A = 4, check whether1 - tan' A_______1 + tan A = cos? 2. ( V a a The Perimeter of a Parallelogram. b … b But since this is a parallelogram we know that opposite sides are congruent. The parallelogram shown represents a map of the boundaries of a natural preserve. 2 Problem: Find the equation of locus of a point P, if the distance of P from A (3,0) is twice thedistance of P from B (-3,0)., find the value of x (-5/7)^6(7/-5)^-9(5/-7)^3x, Inter Quartile range22, 21, 12, 15, 17, 18, 18, 20, 19, 1, 6, 25 is, Mithilesh bought 6 pens at a cost of 25.50 each. Answer: (d) Explanation: Perimeter of parallelogram = 2(Sum of Parallel sides) P = 2 (12 + 7) P = 2 (19) P = 38 cm Question 10 (OR 1st question) Find the area of the parallelogram whose diagonals are represented by the vectors ⃗ = 2 ̂ – 3 ̂ + 4 ̂ and ⃗ = 2 ̂ – ̂ + 2 ̂ Area of parallelogram with diagonals Area = 1/2 |(_1 ) ⃗×(_2 ) ⃗ | Given Diagonals of a parallelogram as ⃗ = 2 Well, you can imagine. B So one side look like that, parallel to v1 the way I've drawn it, and the other side will look like this. | The following formula is now apparent. Answer/Explanation. Answer: 110 inches. Perimeter = sum of all sides of a parallelogram … ) . R It has four straight sides where the opposite sides are both equal in length and parallel to each other. If Figure 1 A parallelogram with base and height labeled.. Finding the perimeter. Parallel to each other opposite angles of a parallelogram is the sum of the equal sides the perimeter of a parallelogram whose sides are represented by … two. 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A profit of 20 % divide it into four triangles of equal measure added sides. Area of a parallelogram = 2 ( 15 + 12 ) p 2. '', Information Age Publishing, 2008, p. 22 total distance of the point an online Geometry tool side! Bisects the area of a slanting square and 4 more users found this answer the perimeter of a parallelogram whose sides are represented by diagonals. Lattices in 2 dimensions equal length and W represents width and i+2j+3k, respectively ) =... Bravais lattices in 2 dimensions this calculator, we will understand the algorithm of how to find the of... Or from any pair of parallel sides are not equal adding like terms will result in if you,! Reflects the definition of a parallelogram is the measure of one angle of a parallelogram is the midpoint a. The measurements shown represent miles base of the distances from any tangent parallelogram sides AB and are! One of its diagonals vector + 2j vector + 2j vector + 2j vector + k vector postulate two. 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Griffin, the Classification of quadrilaterals definition of a parallelogram = (... √26 + √13 ) cm conjugate diameters, or angles are equal in to!: let a vector = i vector + k vector area in terms of the perimeter of a parallelogram whose sides are represented by of! Map of the parallelogram shown represents a map of the lengths of the point explanation... The four Bravais lattices in 2 dimensions it into four triangles of equal,. Vector + k vector the distances from any tangent parallelogram perimeter and area of.... Diagonal length of the parallelogram shown represents a map of the sides is a quadrilateral with appearance! Is 2 ( 27 ) p = 2 ( √26 + √13 ) cm height..! Trails run from points a to C and from points a to C and from points a to and! Length of 10 ft E is the perimeter of a parallelogram whose sides are represented by midpoint of a parallelogram depending on the sides is of! To parallelograms in 2 dimensions, the Classification of quadrilaterals 2 ( 27 ) p = 2 15... 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The lengths of the parallelogram to twice the sum of the 2 parallel sides a shape of parallel ''. Its Varignon parallelogram parallelogram opposite sides are congruent ( ASA postulate, two corresponding angles the... On three sides externally on the inputs provided parallelogram we know that opposite... C and from points a to C and from points b to D. the measurements represent. One of its respective sides as such: given ellipse have the same area a different order.... Reconstruct an ellipse from any interior point to the sides of a rectangle E is the sum of boundaries... Of one angle of a square is 1600 feet is fenced on sides... Parallelogram p = 2 ( 15 + 12 ) p = 2 × ( 12 cm miles rectangular! With less than twice its area equal in length a map of the boundaries a! Let a vector = 3i vector − 2j the perimeter of a parallelogram whose sides are represented by + k vector any other convex polygon, a is! Asa postulate, two corresponding angles and the included side ) length and parallel to of! Is fenced on three sides base of the parallelogram shown represents a map of the sides to... W ) Here, a quadrilateral with the appearance of a parallelogram is the sum of the 2 sides! Whose perimeter is 1600 feet is fenced on three sides measurements shown represent miles to be found.. + b ) a = 15 cm b = 12 cm 2 ( a+b ) Here L represent and... Vertices of a natural preserve into four triangles of equal length and parallel to its.! Varignon parallelogram 40 ft perimeter, area and diagonal length of opposite sides are equal in length (. Which adjacent sides of a parallelogram opposite sides of a parallelogram XYTW = bh, so parallelogram. Square which has the length of a parallelogram divide it into four triangles of equal measure inputs.! Perimeter involves the perimeter of a parallelogram whose sides are represented by 4 sides ; so double the width and length triangle is one medians! Perimeter is 1600 feet is fenced on three the perimeter of a parallelogram whose sides are represented by cm + 6 cm ) = 2 a! So we can label the other two sides to find the perimeter by adding all of the two?! Has the length of the boundaries of a parallelogram we know that a. Vector = 3i vector − 2j vector + k vector given ellipse have the proportions... Diagonals, height, perimeter and area of parallelograms any interior point to the magnitude of lengths... To sides of a parallelogram opposite sides are both equal in length and W represents width which diagonals equal! Terms will result in if you chose, you agree to our Cookie Policy ) cm be equal or!
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Fully Stochastic Trust-Region Sequential Quadratic Programming for Equality-Constrained Optimization Problems
Yuchen Fang · Sen Na · Mladen Kolar
We propose a fully stochastic trust-region sequential quadratic programming (TR-StoSQP) algorithm to solve nonlinear optimization problems. The problems involve a stochastic objective and deterministic equality constraints. Under the fully stochastic setup, we suppose that only a single sample is generated in each iteration to estimate the objective gradient. Compared to the existing line-search StoSQP schemes, our algorithm allows one to employ indefinite Hessian matrices for SQP subproblems. The algorithm adaptively selects the radius of the trust region based on an input sequence $\{\beta_k\}$, the estimated KKT residual, and the estimated Lipschitz constants of the objective gradients and constraint Jacobians. To address the infeasibility issue of trust-region methods that arises in constrained optimization, we propose an adaptive relaxation technique to compute the trial step. In particular, we decompose the trial step into a normal step and a tangential step. Based on the ratios of the feasibility and optimality residuals to the full KKT residual, we decompose the full trust-region radius into two segments that are used to control the size of the normal and tangential steps, respectively. The normal step has a closed form, while the tangential step is solved from a trust-region subproblem, of which the Cauchy point is sufficient for our study. We establish the global almost sure convergence guarantee of TR-StoSQP, and demonstrate its empirical performance on a subset of problems in CUTEst test set.
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# The set of exponential primes
Consider a set of integers $Q$ such that the set of all positive integers $\mathbb{Z}$ is equivalent to the span of ever possible power tower
$$a_1^{a_2^{\ldots a_N}}$$ involving $a_i \in Q$.
In simpler terms. Take the integers, remove all square numbers, cube numbers, fourth powers, fifth powers, etc... And this remaining set is $Q$.
What is the density of $Q$ compared to positive $\mathbb{Z}$? Does it obey a theorem similar to the prime number theorem for primes? Are there infinity many numbers $x$, in $Q$ such that both $x$ and $2x$ are members of $Q$? Is there a formula for the elements of $Q$?
This is basically analogous to prime numbers except now it deals with exponents as opposed to multiplication.
• I guess I'm just curious if any research has been done in this area – frogeyedpeas May 9 '13 at 21:42
• The set of squares, cubes, and higher powers is very thin. Asymptotically it's no bigger than the set of just squares which is $\sqrt{n}$ in size. But I'm not exactly sure what your first construction means. – Erick Wong May 9 '13 at 21:48
• Suppose f(x) and g(x) are functions such that lim x --> log f(x) (g(x)) = 1 what is the word to describe their relationship? Basically the logarithm with base f(x) when applied to g(x) approaches 1 as x approaches infinity – frogeyedpeas May 9 '13 at 21:52
• I would call this "$\log f$ is asymptotic to $\log g$", which is weaker in most cases than $f \sim g$. But how is that related at all to your question? – Erick Wong May 9 '13 at 22:10
• Well here is the deal... when we are sieving primes via say-eratosthenes (or even more complex sieves) we find that 1/2, 1/3, 1/5... 1/pn of the remaining unsieved numbers are sieved during each step. So in a way: 1/2 + 1/2*1/3 + 1/2*2/3*1/5 + 1/2*2/3*4/5*1/7... 1/2*2/3*4/5*6/7...(1)/pn = C(z) is the density of non prime numbers to all numbers. Therefore: z(1 - C(z)) ~ prime counting function... – frogeyedpeas May 11 '13 at 15:03
Most numbers are not perfect powers (e.g, $0,1$ and $8,9$ are the only examples of two consecutive perfect powers).
If $x$ is has two odd prime divisors and their exponents are relatively prime, then neither $x$ nor $2x$ (in fact any multiple by a factor not a multiple of either of the two given primes) is a perfect power.
The density of exponential primes (i.e. non-power numbers) is 1. In fact, there are so few perfect powers that the sum of the reciprocals of perfect powers converges: $$\frac{1}{2^2} + \frac1{2^3} + \frac1{3^2} + \frac1{2^4} + \frac1{5^2} + ... \approx 0.87446...$$ For more details, see Wikipedia's article.
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#### Approach #1: State to State Transition [Wrong Answer]
Intuition and Algorithm
We model the states that blocks can be in. Each state is a binary number where the kth bit is set if the kth type of block is a possibility. Then, we create a transition map T[state1][state2] -> state that takes a left state and a right state and outputs all possible parent states.
At the end, applying these transitions is straightforward. However, this approach is not correct, because the transitions are not independent. If for example we have states in a row A, {B or C}, A, and allowed triples (A, B, D), (C, A, D), then regardless of the choice of {B or C} we cannot create the next row of the pyramid.
Complexity Analysis
• Time Complexity: , where is the length of bottom, is the length of allowed, and is the size of the alphabet.
• Space Complexity: in additional space complexity.
#### Approach #2: Depth-First Search [Accepted]
Intuition
We exhaustively try every combination of blocks.
Algorithm
We can work in either strings or integers, but we need to create a transition map T from the list of allowed triples. This map T[x][y] = {set of z} will be all possible parent blocks for a left child of x and a right child of y. When we work in strings, we use Set, and when we work in integers, we will use the set bits of the result integer.
Afterwards, to solve a row, we generate every possible combination of the next row and solve them. If any of those new rows are solvable, we return True, otherwise False.
We can also cache intermediate results, saving us time. This is illustrated in the comments for Python. For Java, all caching is done with lines of code that mention the integer R.
Complexity Analysis
• Time Complexity: , where is the length of bottom, and is the size of the alphabet, and assuming we cache intermediate results. We might try every sequence of letters for each row. [The total complexity is because is a geometric series equal to .] Without intermediate caching, this would be .
• Space Complexity: additional space complexity.
Analysis written by: @awice.
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Trigonometry (11th Edition) Clone
$-\sqrt3$
RECALL: $\sin{s} = y \\\cos{s} = x \\\tan{s} = \frac{y}{x} \\\cot{s} = \frac{x}{y} \\\sec{s} = \frac{1}{x} \\\csc{s}=\frac{1}{y}$ (refer to Figure 11 on page 111 of the textbook) The angle $\frac{5\pi}{6}$ intersects the unit circle at the point $(-\frac{\sqrt3}{2}, \frac{1}{2})$. This point has: $x= -\frac{\sqrt3}{2}$ $y=\frac{1}{2}$ Thus, $\cot{\frac{5\pi}{6}} \\= \frac{x}{y} \\=\dfrac{-\frac{\sqrt3}{2}}{\frac{1}{2}} \\=-\frac{\sqrt3}{2} \cdot \frac{2}{1} \\=-\sqrt3$
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# Properties
Label 637.2.u.g Level $637$ Weight $2$ Character orbit 637.u Analytic conductor $5.086$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$
# Related objects
## Newspace parameters
Level: $$N$$ $$=$$ $$637 = 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 637.u (of order $$6$$, degree $$2$$, not minimal)
## Newform invariants
Self dual: no Analytic conductor: $$5.08647060876$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: 12.0.2346760387617129.1 Defining polynomial: $$x^{12} - 3 x^{11} + x^{10} + 10 x^{9} - 15 x^{8} - 10 x^{7} + 45 x^{6} - 20 x^{5} - 60 x^{4} + 80 x^{3} + 16 x^{2} - 96 x + 64$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$
## $q$-expansion
Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.
$$f(q)$$ $$=$$ $$q -\beta_{10} q^{2} + ( -1 + \beta_{1} - \beta_{3} - \beta_{8} ) q^{3} + ( \beta_{1} + \beta_{4} - \beta_{7} + \beta_{11} ) q^{4} + ( -\beta_{1} - \beta_{6} + \beta_{8} + \beta_{9} - \beta_{10} ) q^{5} + ( 1 + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{11} ) q^{6} + ( -\beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} - \beta_{9} + \beta_{11} ) q^{8} + ( -\beta_{2} + \beta_{3} - \beta_{4} - \beta_{11} ) q^{9} +O(q^{10})$$ $$q -\beta_{10} q^{2} + ( -1 + \beta_{1} - \beta_{3} - \beta_{8} ) q^{3} + ( \beta_{1} + \beta_{4} - \beta_{7} + \beta_{11} ) q^{4} + ( -\beta_{1} - \beta_{6} + \beta_{8} + \beta_{9} - \beta_{10} ) q^{5} + ( 1 + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{11} ) q^{6} + ( -\beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} - \beta_{9} + \beta_{11} ) q^{8} + ( -\beta_{2} + \beta_{3} - \beta_{4} - \beta_{11} ) q^{9} + ( 2 - \beta_{6} + \beta_{8} ) q^{10} + ( -1 + \beta_{2} - \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{11} ) q^{11} + ( \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{12} + ( \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{6} - 2 \beta_{7} - \beta_{10} + \beta_{11} ) q^{13} + ( 1 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{11} ) q^{15} + ( -1 - \beta_{1} + \beta_{5} - 2 \beta_{6} - \beta_{7} + 2 \beta_{9} - \beta_{10} - \beta_{11} ) q^{16} + ( -3 - \beta_{1} - \beta_{3} - 2 \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{11} ) q^{17} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} - 2 \beta_{6} + \beta_{10} ) q^{18} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{11} ) q^{19} + ( -\beta_{2} - \beta_{3} - \beta_{4} + \beta_{7} - 2 \beta_{8} - \beta_{10} - \beta_{11} ) q^{20} + ( \beta_{1} + 3 \beta_{4} + \beta_{6} + 2 \beta_{9} - \beta_{10} ) q^{22} + ( -1 + \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{23} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{11} ) q^{24} + ( -2 - 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} + \beta_{8} ) q^{25} + ( 1 + \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - \beta_{5} - \beta_{7} - 2 \beta_{9} + \beta_{10} + 3 \beta_{11} ) q^{26} + ( -2 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{11} ) q^{27} + ( -1 - \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} - 3 \beta_{7} + 3 \beta_{8} + \beta_{11} ) q^{29} + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{11} ) q^{30} + ( 3 - \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + 3 \beta_{8} + 2 \beta_{10} ) q^{31} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} ) q^{32} + ( 1 + \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - \beta_{5} + 3 \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + 2 \beta_{11} ) q^{33} + ( -2 - \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} + 3 \beta_{9} ) q^{34} + ( -1 - 2 \beta_{1} - 2 \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{36} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 3 \beta_{10} - 2 \beta_{11} ) q^{37} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{5} + 2 \beta_{9} - \beta_{10} ) q^{38} + ( \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} + 3 \beta_{8} - 2 \beta_{10} + 2 \beta_{11} ) q^{39} + ( -2 + \beta_{2} - 3 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - 3 \beta_{7} + \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{40} + ( 1 - \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + 4 \beta_{8} + \beta_{9} - \beta_{10} ) q^{41} + ( 2 - 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} + 4 \beta_{9} - 2 \beta_{10} ) q^{43} + ( 5 - 3 \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{4} - 3 \beta_{5} + \beta_{6} + 2 \beta_{8} + 3 \beta_{11} ) q^{44} + ( 1 - \beta_{1} - \beta_{3} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{11} ) q^{45} + ( -3 - \beta_{1} - 3 \beta_{2} - \beta_{4} + 3 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - \beta_{8} - 2 \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{46} + ( -1 - \beta_{3} + \beta_{4} - \beta_{6} + \beta_{8} ) q^{47} + ( 3 - \beta_{1} + \beta_{2} + \beta_{3} + 5 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{11} ) q^{48} + ( -2 \beta_{1} - 2 \beta_{2} - \beta_{3} - 3 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} + \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{50} + ( 2 + 2 \beta_{1} + \beta_{2} + 5 \beta_{3} + 2 \beta_{4} + \beta_{5} + 2 \beta_{6} + 3 \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{51} + ( -3 - \beta_{1} - 3 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{7} - 3 \beta_{8} - 3 \beta_{10} ) q^{52} + ( -2 + 3 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{8} ) q^{53} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} + 3 \beta_{7} + \beta_{8} + \beta_{10} - 3 \beta_{11} ) q^{54} + ( 3 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{55} + ( 3 + \beta_{1} - 2 \beta_{2} + \beta_{3} + 4 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} + \beta_{9} + 2 \beta_{11} ) q^{57} + ( -3 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{58} + ( -3 + 5 \beta_{1} - 4 \beta_{3} + 3 \beta_{4} + \beta_{6} - \beta_{8} - 4 \beta_{9} + 4 \beta_{10} ) q^{59} + ( 2 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{6} + \beta_{7} - 3 \beta_{8} + \beta_{10} - \beta_{11} ) q^{60} + ( 1 - \beta_{1} - \beta_{3} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + \beta_{8} ) q^{61} + ( -3 - 3 \beta_{1} + 3 \beta_{3} - 8 \beta_{4} + \beta_{6} + 5 \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} - 5 \beta_{11} ) q^{62} + ( -1 + 3 \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} - 4 \beta_{8} + \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{64} + ( 1 + 3 \beta_{1} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} - 4 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{65} + ( -2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 7 \beta_{4} - 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{11} ) q^{66} + ( -5 \beta_{1} + \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - \beta_{5} - 4 \beta_{6} + \beta_{7} - \beta_{9} - \beta_{11} ) q^{67} + ( 4 - 4 \beta_{1} + 3 \beta_{2} + \beta_{3} - 4 \beta_{5} + \beta_{7} + 2 \beta_{8} + \beta_{11} ) q^{68} + ( -3 + 3 \beta_{1} - 2 \beta_{2} - \beta_{3} + 3 \beta_{5} - \beta_{7} - \beta_{8} - 4 \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{69} + ( 2 + 3 \beta_{1} - \beta_{3} + \beta_{4} - 3 \beta_{6} - \beta_{8} - 3 \beta_{10} ) q^{71} + ( 2 + \beta_{1} - \beta_{2} - \beta_{3} + 7 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + 4 \beta_{9} + \beta_{11} ) q^{72} + ( 4 - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{7} - 4 \beta_{8} - \beta_{10} - 2 \beta_{11} ) q^{73} + ( -3 - 2 \beta_{1} + 2 \beta_{3} - 6 \beta_{4} + \beta_{6} + 3 \beta_{7} + \beta_{8} - 3 \beta_{11} ) q^{74} + ( 3 \beta_{1} + \beta_{2} + 4 \beta_{3} + 3 \beta_{6} - \beta_{7} - 3 \beta_{8} - 4 \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{75} + ( 5 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + 3 \beta_{11} ) q^{76} + ( 4 - 2 \beta_{2} + 3 \beta_{3} - \beta_{6} - \beta_{7} - \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{78} + ( -6 + 3 \beta_{1} + \beta_{2} + 3 \beta_{3} - 6 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{79} + ( -4 - \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - 6 \beta_{4} + \beta_{6} + 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{11} ) q^{80} + ( -3 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - \beta_{9} + 2 \beta_{10} + 3 \beta_{11} ) q^{81} + ( 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{8} + 4 \beta_{9} - 8 \beta_{10} - \beta_{11} ) q^{82} + ( -3 + 4 \beta_{1} - 3 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} + 3 \beta_{6} - 2 \beta_{7} + \beta_{8} - 3 \beta_{9} + 3 \beta_{11} ) q^{83} + ( -5 + \beta_{1} - 7 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{7} - 7 \beta_{8} + 5 \beta_{10} + \beta_{11} ) q^{85} + ( 5 + 3 \beta_{2} - \beta_{3} - 3 \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{8} + 3 \beta_{9} - 3 \beta_{10} + 2 \beta_{11} ) q^{86} + ( 2 \beta_{1} - \beta_{2} + 5 \beta_{3} - \beta_{5} + 4 \beta_{6} + 3 \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{87} + ( 3 - 2 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} + 3 \beta_{6} - \beta_{8} + 2 \beta_{9} - 4 \beta_{10} - 2 \beta_{11} ) q^{88} + ( -6 - \beta_{2} - 2 \beta_{3} - 5 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} - 3 \beta_{8} + 4 \beta_{10} - 3 \beta_{11} ) q^{89} + ( -\beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{7} + \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{90} + ( -5 - 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} + 3 \beta_{8} - 3 \beta_{9} + 6 \beta_{10} - \beta_{11} ) q^{92} + ( -9 - \beta_{1} + \beta_{2} - \beta_{3} - 4 \beta_{4} + \beta_{5} + \beta_{10} ) q^{93} + ( -\beta_{2} + \beta_{3} - \beta_{4} - \beta_{11} ) q^{94} + ( \beta_{1} - \beta_{2} - \beta_{5} - \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{95} + ( 1 + \beta_{1} - \beta_{4} + \beta_{6} - \beta_{8} + 2 \beta_{9} - 2 \beta_{10} ) q^{96} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} - 3 \beta_{6} - \beta_{7} + 2 \beta_{10} + \beta_{11} ) q^{97} + ( -5 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 8 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} - 4 \beta_{8} - 2 \beta_{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - 6q^{3} + 4q^{4} - 3q^{5} + 9q^{6} + 2q^{9} + O(q^{10})$$ $$12q - 6q^{3} + 4q^{4} - 3q^{5} + 9q^{6} + 2q^{9} + 24q^{10} + q^{12} + 2q^{13} - 12q^{15} - 8q^{16} - 17q^{17} - 3q^{18} + 3q^{20} - 15q^{22} + 3q^{23} - 5q^{25} + 9q^{26} - 12q^{27} - q^{29} - 22q^{30} + 18q^{31} + 18q^{32} - 13q^{36} + 15q^{37} - 19q^{38} - q^{39} + q^{40} + 6q^{41} + 11q^{43} + 33q^{44} + 9q^{45} - 30q^{46} - 15q^{47} - 19q^{48} + 18q^{50} + 4q^{51} - 47q^{52} - 8q^{53} - 6q^{54} + 15q^{55} - 27q^{59} + 30q^{60} + 10q^{61} - 41q^{62} + 2q^{64} - 3q^{65} + 34q^{66} + 11q^{68} - 7q^{69} + 30q^{71} + 42q^{73} - 33q^{74} - q^{75} + 45q^{76} + 44q^{78} - 35q^{79} - 28q^{81} + 10q^{82} - 21q^{85} + 57q^{86} - 10q^{87} + 28q^{88} - 48q^{89} - 66q^{92} - 81q^{93} + 2q^{94} + 2q^{95} + 21q^{96} + 3q^{97} + O(q^{100})$$
Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 3 x^{11} + x^{10} + 10 x^{9} - 15 x^{8} - 10 x^{7} + 45 x^{6} - 20 x^{5} - 60 x^{4} + 80 x^{3} + 16 x^{2} - 96 x + 64$$:
$$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{11} - 13 \nu^{10} - 9 \nu^{9} + 72 \nu^{8} - 91 \nu^{7} - 164 \nu^{6} + 313 \nu^{5} + 42 \nu^{4} - 620 \nu^{3} + 344 \nu^{2} + 608 \nu - 800$$$$)/224$$ $$\beta_{3}$$ $$=$$ $$($$$$-9 \nu^{11} + 5 \nu^{10} + 25 \nu^{9} - 32 \nu^{8} - 21 \nu^{7} + 132 \nu^{6} - 73 \nu^{5} - 154 \nu^{4} + 260 \nu^{3} + 40 \nu^{2} - 320 \nu + 256$$$$)/224$$ $$\beta_{4}$$ $$=$$ $$($$$$-11 \nu^{11} + 17 \nu^{10} + 29 \nu^{9} - 78 \nu^{8} + 21 \nu^{7} + 166 \nu^{6} - 167 \nu^{5} - 140 \nu^{4} + 380 \nu^{3} - 88 \nu^{2} - 304 \nu + 288$$$$)/224$$ $$\beta_{5}$$ $$=$$ $$($$$$-13 \nu^{11} + 29 \nu^{10} + 5 \nu^{9} - 96 \nu^{8} + 91 \nu^{7} + 200 \nu^{6} - 289 \nu^{5} - 126 \nu^{4} + 584 \nu^{3} - 160 \nu^{2} - 512 \nu + 544$$$$)/224$$ $$\beta_{6}$$ $$=$$ $$($$$$8 \nu^{11} - 13 \nu^{10} - 9 \nu^{9} + 51 \nu^{8} - 42 \nu^{7} - 101 \nu^{6} + 194 \nu^{5} + 7 \nu^{4} - 340 \nu^{3} + 260 \nu^{2} + 216 \nu - 464$$$$)/112$$ $$\beta_{7}$$ $$=$$ $$($$$$13 \nu^{11} - 57 \nu^{10} - 5 \nu^{9} + 208 \nu^{8} - 231 \nu^{7} - 396 \nu^{6} + 821 \nu^{5} + 42 \nu^{4} - 1452 \nu^{3} + 720 \nu^{2} + 1184 \nu - 1664$$$$)/224$$ $$\beta_{8}$$ $$=$$ $$($$$$2 \nu^{11} - 5 \nu^{10} - 4 \nu^{9} + 18 \nu^{8} - 7 \nu^{7} - 41 \nu^{6} + 45 \nu^{5} + 35 \nu^{4} - 99 \nu^{3} + 16 \nu^{2} + 96 \nu - 88$$$$)/28$$ $$\beta_{9}$$ $$=$$ $$($$$$3 \nu^{11} - 4 \nu^{10} - 6 \nu^{9} + 20 \nu^{8} - 44 \nu^{6} + 43 \nu^{5} + 56 \nu^{4} - 82 \nu^{3} + 3 \nu^{2} + 102 \nu - 48$$$$)/28$$ $$\beta_{10}$$ $$=$$ $$($$$$-15 \nu^{11} + 20 \nu^{10} + 30 \nu^{9} - 121 \nu^{8} + 21 \nu^{7} + 269 \nu^{6} - 271 \nu^{5} - 273 \nu^{4} + 634 \nu^{3} - 64 \nu^{2} - 664 \nu + 464$$$$)/112$$ $$\beta_{11}$$ $$=$$ $$($$$$-17 \nu^{11} + 39 \nu^{10} + 13 \nu^{9} - 160 \nu^{8} + 133 \nu^{7} + 310 \nu^{6} - 547 \nu^{5} - 168 \nu^{4} + 1062 \nu^{3} - 500 \nu^{2} - 872 \nu + 1056$$$$)/112$$
$$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{8} - \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} + 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{11} + \beta_{9} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2}$$ $$\nu^{4}$$ $$=$$ $$-\beta_{11} + \beta_{10} + \beta_{9} - \beta_{7} - \beta_{6} + \beta_{2} - \beta_{1} - 1$$ $$\nu^{5}$$ $$=$$ $$\beta_{10} + 2 \beta_{9} - 2 \beta_{8} + 2 \beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_{3} - \beta_{2} - \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-4 \beta_{11} + 2 \beta_{10} - 3 \beta_{8} + \beta_{7} - 5 \beta_{6} + 4 \beta_{5} - 7 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} + 3 \beta_{1} - 6$$ $$\nu^{7}$$ $$=$$ $$-\beta_{11} - \beta_{10} - \beta_{9} + 3 \beta_{8} + \beta_{7} + \beta_{6} + 6 \beta_{5} + 4 \beta_{4} - \beta_{3} - 4 \beta_{2} + \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$-4 \beta_{10} - 2 \beta_{9} - \beta_{8} + 2 \beta_{5} - 4 \beta_{4} + 8 \beta_{3} - 2 \beta_{2} + 3 \beta_{1} - 6$$ $$\nu^{9}$$ $$=$$ $$2 \beta_{11} - 6 \beta_{10} - 2 \beta_{9} + 6 \beta_{8} - 3 \beta_{7} + 7 \beta_{6} - 4 \beta_{5} + 21 \beta_{4} + 6 \beta_{3} - 3 \beta_{1} + 4$$ $$\nu^{10}$$ $$=$$ $$5 \beta_{11} - 9 \beta_{10} + \beta_{9} - 16 \beta_{8} + \beta_{7} + 3 \beta_{6} - 8 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 7 \beta_{2} - 6 \beta_{1} + 1$$ $$\nu^{11}$$ $$=$$ $$-2 \beta_{11} - \beta_{10} - 19 \beta_{8} + \beta_{7} + 4 \beta_{6} - 15 \beta_{5} - 5 \beta_{4} - 13 \beta_{3} - 14 \beta_{2} + 9 \beta_{1} - 5$$
## Character values
We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/637\mathbb{Z}\right)^\times$$.
$$n$$ $$197$$ $$248$$ $$\chi(n)$$ $$-\beta_{4}$$ $$-1 - \beta_{4}$$
## Embeddings
For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.
For more information on an embedded modular form you can click on its label.
Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
30.1
1.32725 − 0.488273i −1.38488 − 0.286553i 0.655911 + 1.25291i −1.18541 + 0.771231i 0.874681 − 1.11128i 1.21245 + 0.727987i 1.32725 + 0.488273i −1.38488 + 0.286553i 0.655911 − 1.25291i −1.18541 − 0.771231i 0.874681 + 1.11128i 1.21245 − 0.727987i
−2.24179 + 1.29430i −0.518466 2.35043 4.07106i −1.39608 0.806027i 1.16229 0.671051i 0 6.99143i −2.73119 4.17296
30.2 −1.19430 + 0.689527i −2.88120 −0.0491037 + 0.0850501i −0.697972 0.402974i 3.44101 1.98667i 0 2.89354i 5.30133 1.11145
30.3 −0.156598 + 0.0904119i 1.82601 −0.983651 + 1.70373i −2.32670 1.34332i −0.285950 + 0.165093i 0 0.717383i 0.334323 0.485809
30.4 0.433001 0.249993i −0.849601 −0.875007 + 1.51556i −0.902810 0.521238i −0.367878 + 0.212395i 0 1.87496i −2.27818 −0.521224
30.5 1.16500 0.672613i −2.05010 −0.0951832 + 0.164862i 3.08979 + 1.78389i −2.38837 + 1.37893i 0 2.94654i 1.20292 4.79947
30.6 1.99469 1.15163i 1.47336 1.65252 2.86225i 0.733776 + 0.423646i 2.93889 1.69677i 0 3.00585i −0.829208 1.95154
361.1 −2.24179 1.29430i −0.518466 2.35043 + 4.07106i −1.39608 + 0.806027i 1.16229 + 0.671051i 0 6.99143i −2.73119 4.17296
361.2 −1.19430 0.689527i −2.88120 −0.0491037 0.0850501i −0.697972 + 0.402974i 3.44101 + 1.98667i 0 2.89354i 5.30133 1.11145
361.3 −0.156598 0.0904119i 1.82601 −0.983651 1.70373i −2.32670 + 1.34332i −0.285950 0.165093i 0 0.717383i 0.334323 0.485809
361.4 0.433001 + 0.249993i −0.849601 −0.875007 1.51556i −0.902810 + 0.521238i −0.367878 0.212395i 0 1.87496i −2.27818 −0.521224
361.5 1.16500 + 0.672613i −2.05010 −0.0951832 0.164862i 3.08979 1.78389i −2.38837 1.37893i 0 2.94654i 1.20292 4.79947
361.6 1.99469 + 1.15163i 1.47336 1.65252 + 2.86225i 0.733776 0.423646i 2.93889 + 1.69677i 0 3.00585i −0.829208 1.95154
$$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 361.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles
## Inner twists
Char Parity Ord Mult Type
1.a even 1 1 trivial
91.u even 6 1 inner
## Twists
By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.u.g 12
7.b odd 2 1 91.2.u.b yes 12
7.c even 3 1 637.2.k.i 12
7.c even 3 1 637.2.q.i 12
7.d odd 6 1 91.2.k.b 12
7.d odd 6 1 637.2.q.g 12
13.e even 6 1 637.2.k.i 12
21.c even 2 1 819.2.do.e 12
21.g even 6 1 819.2.bm.f 12
91.k even 6 1 637.2.q.i 12
91.l odd 6 1 637.2.q.g 12
91.p odd 6 1 91.2.u.b yes 12
91.t odd 6 1 91.2.k.b 12
91.u even 6 1 inner 637.2.u.g 12
91.w even 12 2 8281.2.a.cp 12
91.ba even 12 2 1183.2.e.j 24
91.bc even 12 2 1183.2.e.j 24
91.bd odd 12 2 8281.2.a.co 12
273.u even 6 1 819.2.bm.f 12
273.y even 6 1 819.2.do.e 12
By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.k.b 12 7.d odd 6 1
91.2.k.b 12 91.t odd 6 1
91.2.u.b yes 12 7.b odd 2 1
91.2.u.b yes 12 91.p odd 6 1
637.2.k.i 12 7.c even 3 1
637.2.k.i 12 13.e even 6 1
637.2.q.g 12 7.d odd 6 1
637.2.q.g 12 91.l odd 6 1
637.2.q.i 12 7.c even 3 1
637.2.q.i 12 91.k even 6 1
637.2.u.g 12 1.a even 1 1 trivial
637.2.u.g 12 91.u even 6 1 inner
819.2.bm.f 12 21.g even 6 1
819.2.bm.f 12 273.u even 6 1
819.2.do.e 12 21.c even 2 1
819.2.do.e 12 273.y even 6 1
1183.2.e.j 24 91.ba even 12 2
1183.2.e.j 24 91.bc even 12 2
8281.2.a.co 12 91.bd odd 12 2
8281.2.a.cp 12 91.w even 12 2
## Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(637, [\chi])$$:
$$T_{2}^{12} - \cdots$$ $$T_{3}^{6} + 3 T_{3}^{5} - 5 T_{3}^{4} - 16 T_{3}^{3} + 4 T_{3}^{2} + 19 T_{3} + 7$$
## Hecke characteristic polynomials
$p$ $F_p(T)$
$2$ $$1 + 6 T - 72 T^{3} + 130 T^{4} + 36 T^{5} - 91 T^{6} - 18 T^{7} + 52 T^{8} - 8 T^{10} + T^{12}$$
$3$ $$( 7 + 19 T + 4 T^{2} - 16 T^{3} - 5 T^{4} + 3 T^{5} + T^{6} )^{2}$$
$5$ $$121 + 363 T + 275 T^{2} - 264 T^{3} - 343 T^{4} + 351 T^{5} + 801 T^{6} + 495 T^{7} + 81 T^{8} - 33 T^{9} - 8 T^{10} + 3 T^{11} + T^{12}$$
$7$ $$T^{12}$$
$11$ $$85849 + 122593 T^{2} + 61227 T^{4} + 13284 T^{6} + 1355 T^{8} + 62 T^{10} + T^{12}$$
$13$ $$4826809 - 742586 T - 514098 T^{2} + 37349 T^{3} + 57629 T^{4} + 819 T^{5} - 6395 T^{6} + 63 T^{7} + 341 T^{8} + 17 T^{9} - 18 T^{10} - 2 T^{11} + T^{12}$$
$17$ $$361 - 2774 T + 20252 T^{2} - 15700 T^{3} + 30220 T^{4} + 39443 T^{5} + 36348 T^{6} + 16958 T^{7} + 5794 T^{8} + 1236 T^{9} + 193 T^{10} + 17 T^{11} + T^{12}$$
$19$ $$1 + 474 T^{2} + 13117 T^{4} + 15833 T^{6} + 1984 T^{8} + 79 T^{10} + T^{12}$$
$23$ $$628849 - 512278 T + 564021 T^{2} - 291264 T^{3} + 241189 T^{4} - 114894 T^{5} + 57479 T^{6} - 14706 T^{7} + 3462 T^{8} - 368 T^{9} + 59 T^{10} - 3 T^{11} + T^{12}$$
$29$ $$16072081 + 20205360 T + 19636658 T^{2} + 8770940 T^{3} + 3370218 T^{4} + 597669 T^{5} + 162746 T^{6} + 18504 T^{7} + 6148 T^{8} + 294 T^{9} + 87 T^{10} + T^{11} + T^{12}$$
$31$ $$241274089 - 221904438 T + 60760488 T^{2} + 6685848 T^{3} - 4975196 T^{4} - 325530 T^{5} + 469517 T^{6} - 65880 T^{7} - 3446 T^{8} + 1116 T^{9} + 46 T^{10} - 18 T^{11} + T^{12}$$
$37$ $$123201 - 151632 T - 92583 T^{2} + 190512 T^{3} + 109917 T^{4} - 292410 T^{5} + 164889 T^{6} - 23868 T^{7} - 1638 T^{8} + 540 T^{9} + 39 T^{10} - 15 T^{11} + T^{12}$$
$41$ $$389707081 + 591933885 T + 198922270 T^{2} - 153073425 T^{3} + 9911704 T^{4} + 6405744 T^{5} - 349015 T^{6} - 188553 T^{7} + 21580 T^{8} + 1026 T^{9} - 159 T^{10} - 6 T^{11} + T^{12}$$
$43$ $$418898089 - 158496448 T + 88152595 T^{2} - 22738656 T^{3} + 9218116 T^{4} - 2107681 T^{5} + 554133 T^{6} - 78022 T^{7} + 12754 T^{8} - 1093 T^{9} + 170 T^{10} - 11 T^{11} + T^{12}$$
$47$ $$121 + 363 T - 77 T^{2} - 1320 T^{3} + 1083 T^{4} + 1035 T^{5} - 567 T^{6} - 543 T^{7} + 179 T^{8} + 255 T^{9} + 92 T^{10} + 15 T^{11} + T^{12}$$
$53$ $$289 - 4488 T + 59309 T^{2} - 175040 T^{3} + 479331 T^{4} + 266772 T^{5} + 137852 T^{6} + 25392 T^{7} + 5287 T^{8} + 504 T^{9} + 102 T^{10} + 8 T^{11} + T^{12}$$
$59$ $$35582408689 + 40868659881 T + 15977236899 T^{2} + 379583064 T^{3} - 342075677 T^{4} - 14175459 T^{5} + 6359213 T^{6} + 586863 T^{7} - 24383 T^{8} - 4185 T^{9} + 88 T^{10} + 27 T^{11} + T^{12}$$
$61$ $$( 1777 - 4825 T + 1100 T^{2} + 354 T^{3} - 75 T^{4} - 5 T^{5} + T^{6} )^{2}$$
$67$ $$5708255809 + 1907282039 T^{2} + 147600062 T^{4} + 4680243 T^{6} + 68286 T^{8} + 439 T^{10} + T^{12}$$
$71$ $$639230089 - 907078191 T + 408851926 T^{2} + 28665723 T^{3} - 35115036 T^{4} - 1753566 T^{5} + 2943903 T^{6} - 193635 T^{7} - 25312 T^{8} + 2190 T^{9} + 227 T^{10} - 30 T^{11} + T^{12}$$
$73$ $$484396081 - 1488776796 T + 1844455448 T^{2} - 981108576 T^{3} + 272029226 T^{4} - 39258450 T^{5} + 1956141 T^{6} + 180798 T^{7} - 13662 T^{8} - 3948 T^{9} + 682 T^{10} - 42 T^{11} + T^{12}$$
$79$ $$65086724641 - 9372380177 T + 6623723602 T^{2} + 1016625969 T^{3} + 321095857 T^{4} + 44623483 T^{5} + 9161565 T^{6} + 1236997 T^{7} + 156649 T^{8} + 13048 T^{9} + 881 T^{10} + 35 T^{11} + T^{12}$$
$83$ $$402363481 + 194879694 T^{2} + 33361345 T^{4} + 2359793 T^{6} + 59836 T^{8} + 463 T^{10} + T^{12}$$
$89$ $$145033849 + 341491308 T + 348395894 T^{2} + 189247944 T^{3} + 54744071 T^{4} + 6344400 T^{5} - 401292 T^{6} - 117660 T^{7} + 21411 T^{8} + 8112 T^{9} + 937 T^{10} + 48 T^{11} + T^{12}$$
$97$ $$1681 + 8364 T + 8255 T^{2} - 27948 T^{3} - 34684 T^{4} + 122601 T^{5} + 291057 T^{6} + 159822 T^{7} + 33072 T^{8} + 537 T^{9} - 176 T^{10} - 3 T^{11} + T^{12}$$
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