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# Question based on Triangle Inequality $\displaystyle |x+y|\leq |x|+|y|$

If $x,y,z\in \mathbb{R}-\left\{0\right\}$. Then prove that $\displaystyle 1\leq \frac{|x+y|}{|x|+|y|}+\frac{| y+z|}{| y |+| z |}+\frac{| z+x|}{| z |+| x |}\leq 3$

My Try:: Using Triangle Inequality

$\displaystyle | x+y | \leq | x |+| y |\Rightarrow \frac{| x+y|}{| x |+| y |}\le 1$

Similarly $\displaystyle | y+z | \leq | y |+| z |\Rightarrow \frac{| y+z|}{| y |+| z |}\le 1$

Similarly $\displaystyle | z+x | \leq | z |+| x | \Rightarrow \frac{| z+x|}{| z |+| x |}\le 1$

$\displaystyle \frac{| x+y|}{| x |+| y |}+\frac{| y+z|}{| y |+| z |}+\frac{| z+x|}{| z |+| x |}\leq 3$

But I did not Understand How can I calculate Lower value of the Given expression.

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I think you forgot the $\le 1$ on each of your implications ;D –  DanZimm May 22 '13 at 4:29
The lower bound can be calculated differently. Note that for $x,y,z$ at least $2$ must be on the same side of $0$, so (taking $x,y$ as this pair) $|x+y|=|x|+|y|$. Does this help? –  user45150 May 22 '13 at 4:30
Thanks User 45150 would you like to explain it in more detail, thanks –  juantheron May 22 '13 at 4:41
It is not good to use \mid for absolute value, since it give incorrect spacing. TeX.SE Absolute Value Symbols. –  Martin Sleziak Jun 10 '13 at 10:35

Case 1: $x, y, z$ are all greater than $0$.

Then the expression simplifies to 3 as we can get rid of the absolute signs.

Case 2: Two of $x, y, z$ are greater than $0$.

Without loss of generality let $x, y$ be greater than $0$. Then the expression reduces to:

$$1 + \frac{|x+z|}{x+|z|} + \frac{|y+z|}{y+|z|}$$

The above expression reduces to $1$ when we set $x=y=-z$ and in every other situation the value of the expression is greater than $1$. Thus, $1$ is the minimum value that we have found so far.

Case 3: Only one of $x, y, z$ is greater than $0$.

Without loss of generality suppose that $z$ is greater than $0$. Then using the strategy outlined for case 4 we can flip this case to that of case 2 an show that the minimum is again 1.

Case 4: $x, y, z$ are all less than $0$.

Just as in case 1, the expression simplifies to 3. We can see this as follows:

Define X=-x, Y=-y and Z=-z. By definiton, $X, Y, Z$ are all greater than 0. Thus,

$$\frac{\mid x+y\mid}{\mid x \mid+\mid y \mid}+\frac{\mid y+z\mid}{\mid y \mid+\mid z \mid}+\frac{\mid z+x\mid}{\mid z \mid+\mid x \mid}$$

is equivalent to:

$$\frac{\mid -X+-Y\mid}{\mid -X \mid+\mid -Y \mid}+\frac{\mid -Y+-Z\mid}{\mid -Y \mid+\mid -Z \mid}+\frac{\mid -Z+-X\mid}{\mid -Z \mid+\mid -X \mid}$$

Factor out the '-1's and we are back to case 1.

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# 4-manifold

In mathematics, a 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different. There exist some topological 4-manifolds which admit no smooth structure and even if there exists a smooth structure it need not be unique (i.e. there are smooth 4-manifolds which are homeomorphic but not diffeomorphic).

4-manifolds are of importance in physics because, in General Relativity, spacetime is modeled as a pseudo-Riemannian 4-manifold.

## Topological 4-manifolds

The homotopy type of a simply connected compact 4-manifold only depends on the intersection form on the middle dimensional homology. A famous theorem of Michael Freedman (1982) implies that the homeomorphism type of the manifold only depends on this intersection form, and on a ${\displaystyle \mathbb {Z} /2\mathbb {Z} }$ invariant called the Kirby–Siebenmann invariant, and moreover that every combination of unimodular form and Kirby–Siebenmann invariant can arise, except that if the form is even then the Kirby–Siebenmann invariant must be the signature/8 (mod 2).

Examples:

• In the special case when the form is 0, this implies the 4-dimensional topological Poincaré conjecture.
• If the form is the E8 lattice, this gives a manifold called the E8 manifold, a manifold not homeomorphic to any simplicial complex.
• If the form is ${\displaystyle \mathbb {Z} }$, there are two manifolds depending on the Kirby–Siebenmann invariant: one is 2-dimensional complex projective space, and the other is a fake projective space, with the same homotopy type but not homeomorphic (and with no smooth structure).
• When the rank of the form is greater than about 28, the number of positive definite unimodular forms starts to increase extremely rapidly with the rank, so there are huge numbers of corresponding simply connected topological 4-manifolds (most of which seem to be of almost no interest).

Freedman's classification can be extended to some cases when the fundamental group is not too complicated; for example, when it is ${\displaystyle \mathbb {Z} }$, there is a classification similar to the one above using Hermitian forms over the group ring of ${\displaystyle \mathbb {Z} }$. If the fundamental group is too large (for example, a free group on 2 generators), then Freedman's techniques seem to fail and very little is known about such manifolds.

For any finitely presented group it is easy to construct a (smooth) compact 4-manifold with it as its fundamental group. As there is no algorithm to tell whether two finitely presented groups are isomorphic (even if one is known to be trivial) there is no algorithm to tell if two 4-manifolds have the same fundamental group. This is one reason why much of the work on 4-manifolds just considers the simply connected case: the general case of many problems is already known to be intractable.

## Smooth 4-manifolds

For manifolds of dimension at most 6, any piecewise linear (PL) structure can be smoothed in an essentially unique way,[1] so in particular the theory of 4 dimensional PL manifolds is much the same as the theory of 4 dimensional smooth manifolds.

A major open problem in the theory of smooth 4-manifolds is to classify the simply connected compact ones. As the topological ones are known, this breaks up into two parts:

1. Which topological manifolds are smoothable?
2. Classify the different smooth structures on a smoothable manifold.

There is an almost complete answer to the first problem of which simply connected compact 4-manifolds have smooth structures. First, the Kirby–Siebenmann class must vanish.

• If the intersection form is definite Donaldson's theorem (Donaldson 1983) gives a complete answer: there is a smooth structure if and only if the form is diagonalizable.
• If the form is indefinite and odd there is a smooth structure.
• If the form is indefinite and even we may as well assume that it is of nonpositive signature by changing orientations if necessary, in which case it is isomorphic to a sum of m copies of II1,1 and 2n copies of E8(−1) for some m and n. If m ≥ 3n (so that the dimension is at least 11/8 times the |signature|) then there is a smooth structure, given by taking a connected sum of n K3 surfaces and m  3n copies of S2×S2. If m ≤ 2n (so the dimension is at most 10/8 times the |signature|) then Furuta proved that no smooth structure exists (Furuta 2001). This leaves a small gap between 10/8 and 11/8 where the answer is mostly unknown. (The smallest case not covered above has n=2 and m=5, but this has also been ruled out, so the smallest lattice for which the answer is not currently known is the lattice II7,55 of rank 62 with n=3 and m=7.) The "11/8 conjecture" states that smooth structures do not exist if the dimension is less than 11/8 times the |signature|.

In contrast, very little is known about the second question of classifying the smooth structures on a smoothable 4-manifold; in fact, there is not a single smoothable 4-manifold where the answer is known. Donaldson showed that there are some simply connected compact 4-manifolds, such as Dolgachev surfaces, with a countably infinite number of different smooth structures. There are an uncountable number of different smooth structures on R4; see exotic R4. Fintushel and Stern showed how to use surgery to construct large numbers of different smooth structures (indexed by arbitrary integral polynomials) on many different manifolds, using Seiberg–Witten invariants to show that the smooth structures are different. Their results suggest that any classification of simply connected smooth 4-manifolds will be very complicated. There are currently no plausible conjectures about what this classification might look like. (Some early conjectures that all simply connected smooth 4-manifolds might be connected sums of algebraic surfaces, or symplectic manifolds, possibly with orientations reversed, have been disproved.)

## Special phenomena in 4-dimensions

There are several fundamental theorems about manifolds that can be proved by low-dimensional methods in dimensions at most 3, and by completely different high-dimensional methods in dimension at least 5, but which are false in dimension 4. Here are some examples:

• In dimensions other than 4, the Kirby–Siebenmann invariant provides the obstruction to the existence of a PL structure; in other words a compact topological manifold has a PL structure if and only if its Kirby–Siebenmann invariant in H4(M,Z/2Z) vanishes. In dimension 3 and lower, every topological manifold admits an essentially unique PL structure. In dimension 4 there are many examples with vanishing Kirby–Siebenmann invariant but no PL structure.
• In any dimension other than 4, a compact topological manifold has only a finite number of essentially distinct PL or smooth structures. In dimension 4, compact manifolds can have a countable infinite number of non-diffeomorphic smooth structures.
• Four is the only dimension n for which Rn can have an exotic smooth structure. R4 has an uncountable number of exotic smooth structures; see exotic R4.
• The solution to the smooth Poincaré conjecture is known in all dimensions other than 4 (it is usually false in dimensions at least 7; see exotic sphere). The Poincaré conjecture for PL manifolds has been proved for all dimensions other than 4, but it is not known whether it is true in 4 dimensions (it is equivalent to the smooth Poincaré conjecture in 4 dimensions).
• The smooth h-cobordism theorem holds for cobordisms provided that neither the cobordism nor its boundary has dimension 4. It can fail if the boundary of the cobordism has dimension 4 (as shown by Donaldson).[2] If the cobordism has dimension 4, then it is unknown whether the h-cobordism theorem holds.
• A topological manifold of dimension not equal to 4 has a handlebody decomposition. Manifolds of dimension 4 have a handlebody decomposition if and only if they are smoothable.
• There are compact 4-dimensional topological manifolds that are not homeomorphic to any simplicial complex. In dimension at least 5 the existence of topological manifolds not homeomorphic to a simplicial complex was an open problem. Ciprian Manolescu showed that there are manifolds in each dimension greater than or equal to 5, that are not homeomorphic to a simplicial complex.[3]

## Failure of the Whitney trick in dimension 4

According to Frank Quinn, "Two n-dimensional submanifolds of a manifold of dimension 2n will usually intersect themselves and each other in isolated points. The "Whitney trick" uses an isotopy across an embedded 2-disk to simplify these intersections. Roughly speaking this reduces the study of n-dimensional embeddings to embeddings of 2-disks. But this not a reduction when the embedding is 4: the 2 disks themselves are middle-dimensional, so trying to embed them encounters exactly the same problems they are supposed to solve. This is the phenomenon that separates dimension 4 from others."[4]

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Rendering

This page provides an introduction to the way iink SDK manages the rendering. You will learn how to use the reference implementation provided by MyScript and properly plug things together.

Key concepts

With these concepts, you will be able to better understand how to use the reference rendering implementation or build your own (not documented yet).

Render target

A render target (implementing the IRenderTarget or the IRenderTarget2 interface) represents the platform “view” where the drawing operations will occur.

Canvas

A canvas object provides a platform implementation of the drawing commands called by iink SDK to render content. It is defined in the ICanvas or the ICanvas2 interface.

Renderer

A renderer is a component in charge of deciding how to render content of each layer, knowing which area of the model needs to be refreshed, as well as parameters such as zoom factor or view offset. It will issue rendering commands, through a canvas object that will do the actual drawing operations.

Version 1.4 introduced a new rendering capacity for the renderer based on the drawing of offscreen surfaces. As a shortcut, we call it the “offscreen rendering”. This increases the rendering speed and is even necessary for new features like math animation. So it is definitively our recommended choice.

To use this rendering, you must implement IRenderTarget2 and ICanvas2, or use the ones provided with the reference implementation, to handle the drawing requests of offscreen surfaces.

Only the UWP reference implementation implements these interfaces for you. For WPF, you should make your own implementation.

The iink SDK 1.5 renderer is still compatible with the legacy rendering mode that we name “direct rendering”. This legacy renderer relies on the IRenderTarget and a ICanvas interfaces.

Layer

For performance reasons, the renderer works on two different layers.

The reference rendering implementation implements these layers for you. However, you may sometimes need to interact with them.

The two layers are:

• A model layer, corresponding to everything in the model that was already processed by the engine (guidelines, strokes, images, typeset text, …),
• A capture layer, rendering the ink drawn on the screen but not yet processed by the engine.

Each layer can be refreshed independently from the other, so that it is not needed to redraw everything.

Reference implementation

To make it easy to build applications, MyScript provides in its examples repository a default rendering implementation.

It is released as a library, under a permissive license and can be reused as-is or modified should you feel the need. As an integrator, you just have to link against it and do a bit of plumbing:

Instanciate a EditorUserControl object in your application xaml. It is a ready-to-use implementation of a UserControl that holds an Editor and a Renderer, and that handles capture, rendering and some editing commands.

Back to the example

For the rendering part of the calculator, you can just reuse the reference implementation coming with the MyScript iink code examples.

Once the reference implementation has been integrated with your project, all you have to do is the corresponding plumbing:

<Page
x:Class="Company.Calculator"
...
xmlns:Uc="using:MyScript.IInk.UIReferenceImplementation.UserControls">

<Grid ...>
...
<Uc:EditorUserControl Grid.Row="1"  x:Name="Editor"></Uc:EditorUserControl>
</Grid>
</Page>

If you compile and run the application at this point, you will notice that the view currently shows nothing, as you have not added any content to the model yet. This is the goal of the next step of this guide.


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# Problem: What type of quantity (for example, length, volume, density) do the following units indicate.mL

###### FREE Expert Solution

We’re being asked to determine what type of quantity (for example, length, volume, density) does mL indicate.

Recall that the SI units for length volume and density are given as:

• length → m
• Volume  → L or m3
• density → kg/cm3

Also recall that metric Prefixes are “labels” that can be placed in front of base units. Whenever using the metric prefix chart, remember that “1” is associated with the metric prefix

This means that mL is L with a metric prefix m: (m)L, therefore:

###### Problem Details

What type of quantity (for example, length, volume, density) do the following units indicate.

mL


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TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as quantity (number theory), structure (algebra), space (geometry), and change (analysis). It has no generally accepted definition. Mathematicians seek and use patterns to formulate ...
, a Cauchy sequence (; ), named after
Augustin-Louis Cauchy Baron Augustin-Louis Cauchy (; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He was ...

, is a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ...

whose elements become arbitrarily close to each other as the sequence progresses.Lang, Serge (1993), Algebra (Third ed.), Reading, Mass.: Addison-Wesley Pub. Co., , More precisely, given any small positive distance, all but a finite number of elements of the sequence are less than that given distance from each other. It is not sufficient for each term to become arbitrarily close to the term. For instance, in the sequence of square roots of natural numbers: :$a_n=\sqrt n,$ the consecutive terms become arbitrarily close to each other: :$a_-a_n = \sqrt-\sqrt = \frac < \frac.$ However, with growing values of the index , the terms become arbitrarily large. So, for any index and distance , there exists an index big enough such that . (Actually, any suffices.) As a result, despite how far one goes, the remaining terms of the sequence never get close to , hence the sequence is not Cauchy. The utility of Cauchy sequences lies in the fact that in a
complete metric space In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in or, alternatively, if every Cauchy sequence in converges in . Intuitively, a space is complete if ...
(one where all such sequences are known to converge to a limit), the criterion for
convergence Convergence may refer to: Arts and media Literature *''Convergence'' (book series), edited by Ruth Nanda Anshen *"Convergence" (comics), two separate story lines published by DC Comics: **A four-part crossover storyline that united the four Weirdo ...
depends only on the terms of the sequence itself, as opposed to the definition of convergence, which uses the limit value as well as the terms. This is often exploited in
algorithm of an algorithm (Euclid's algorithm) for calculating the greatest common divisor (g.c.d.) of two numbers ''a'' and ''b'' in locations named A and B. The algorithm proceeds by successive subtractions in two loops: IF the test B ≥ A yields "yes" ...

s, both theoretical and applied, where an iterative process can be shown relatively easily to produce a Cauchy sequence, consisting of the iterates, thus fulfilling a logical condition, such as termination. Generalizations of Cauchy sequences in more abstract
uniform spaces In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and uniform ...
exist in the form of
Cauchy filter Baron Augustin-Louis Cauchy (; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He was ...
s and
Cauchy net In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a generalization of the notion of a sequence. In essence, a sequence is a function whose domain is the natural numbers. The codomain ...
s.

# In real numbers

A sequence :$x_1, x_2, x_3, \ldots$ of real numbers is called a Cauchy sequence if for every
positive Positive is a property of positivity and may refer to: Mathematics and science * Converging lens or positive lens, in optics * Plus sign, the sign "+" used to indicate a positive number * Positive (electricity), a polarity of electrical charge * ...
real number ''ε'', there is a positive
integer An integer (from the Latin ''integer'' meaning "whole") is colloquially defined as a number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , and  are not. The set of intege ...
''N'' such that for all
natural numbers In mathematics, the natural numbers are those used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, words colloquially used ...
''m'', ''n'' > ''N'' :$, x_m - x_n, < \varepsilon,$ where the vertical bars denote the
absolute value of the absolute value function for real numbers In mathematics, the absolute value or modulus of a real number , denoted , is the non-negative value of  without regard to its sign. Namely, if is positive, and if is negative (in whi ...

. In a similar way one can define Cauchy sequences of rational or complex numbers. Cauchy formulated such a condition by requiring $x_m - x_n$ to be
infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not exist in the standard real number system, but do exist in many other number systems, such a ...
for every pair of infinite ''m'', ''n''. For any real number ''r'', the sequence of truncated decimal expansions of ''r'' forms a Cauchy sequence. For example, when ''r'' = ''π'', this sequence is (3, 3.1, 3.14, 3.141, ...). The ''m''th and ''n''th terms differ by at most 101−''m'' when ''m'' < ''n'', and as ''m'' grows this becomes smaller than any fixed positive number ε.

## Modulus of Cauchy convergence

If $\left(x_1, x_2, x_3, ...\right)$ is a sequence in the set $X$, then a ''modulus of Cauchy convergence'' for the sequence is a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriented ...
$\alpha$ from the set of
natural number In mathematics, the natural numbers are those used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, words colloquially used ...
s to itself, such that $\forall k \forall m, n > \alpha\left(k\right), , x_m - x_n, < 1/k$. Any sequence with a modulus of Cauchy convergence is a Cauchy sequence. The existence of a modulus for a Cauchy sequence follows from the
well-ordering property In mathematics, a well-order (or well-ordering or well-order relation) on a set ''S'' is a total order on ''S'' with the property that every non-empty subset of ''S'' has a least element in this ordering. The set ''S'' together with the well-order ...
of the natural numbers (let $\alpha\left(k\right)$ be the smallest possible $N$ in the definition of Cauchy sequence, taking $r$ to be $1/k$). The existence of a modulus also follows from the principle of dependent choice, which is a weak form of the axiom of choice, and it also follows from an even weaker condition called AC00. ''Regular Cauchy sequences'' are sequences with a given modulus of Cauchy convergence (usually $\alpha\left(k\right) = k$ or $\alpha\left(k\right) = 2^k$). Any Cauchy sequence with a modulus of Cauchy convergence is equivalent to a regular Cauchy sequence; this can be proven without using any form of the axiom of choice. Moduli of Cauchy convergence are used by constructive mathematicians who do not wish to use any form of choice. Using a modulus of Cauchy convergence can simplify both definitions and theorems in constructive analysis. Regular Cauchy sequences were used by Errett Bishop in hi
Foundations of Constructive Analysis
and by Douglas Bridges in a non-constructive textbook ().

# In a metric space

Since the definition of a Cauchy sequence only involves metric concepts, it is straightforward to generalize it to any metric space ''X''. To do so, the absolute value is replaced by the distance ''d''(''x''''m'', ''x''''n'') (where ''d'' denotes a metric (mathematics), metric) between ''x''''m'' and ''x''''n''. Formally, given a metric space , a sequence : is Cauchy, if for every positive real number there is a positive
integer An integer (from the Latin ''integer'' meaning "whole") is colloquially defined as a number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , and  are not. The set of intege ...
such that for all positive integers , the distance :. Roughly speaking, the terms of the sequence are getting closer and closer together in a way that suggests that the sequence ought to have a limit of a sequence, limit in ''X''. Nonetheless, such a limit does not always exist within ''X'': the property of a space that every Cauchy sequence converges in the space is called ''completeness'', and is detailed below.

# Completeness

A metric space (''X'', ''d'') in which every Cauchy sequence converges to an element of ''X'' is called Complete metric space, complete.

## Examples

The real numbers are complete under the metric induced by the usual absolute value, and one of the standard Construction of the real numbers, constructions of the real numbers involves Cauchy sequences of rational numbers. In this construction, each equivalence class of Cauchy sequences of rational numbers with a certain tail behavior—that is, each class of sequences that get arbitrarily close to one another— is a real number. A rather different type of example is afforded by a metric space ''X'' which has the discrete space, discrete metric (where any two distinct points are at distance 1 from each other). Any Cauchy sequence of elements of ''X'' must be constant beyond some fixed point, and converges to the eventually repeating term.

## Non-example: rational numbers

The rational numbers Q are not complete (for the usual distance):
There are sequences of rationals that converge (in R) to irrational numbers; these are Cauchy sequences having no limit in Q. In fact, if a real number ''x'' is irrational, then the sequence (''x''''n''), whose ''n''-th term is the truncation to ''n'' decimal places of the decimal expansion of ''x'', gives a Cauchy sequence of rational numbers with irrational limit ''x''. Irrational numbers certainly exist in R, for example: * The sequence defined by $x_0=1, x_=\frac$ consists of rational numbers (1, 3/2, 17/12,...), which is clear from the definition; however it converges to the irrational number, irrational square root of two, see Methods of computing square roots#Babylonian method, Babylonian method of computing square root. * The sequence $x_n = F_n / F_$ of ratios of consecutive Fibonacci numbers which, if it converges at all, converges to a limit $\phi$ satisfying $\phi^2 = \phi+1$, and no rational number has this property. If one considers this as a sequence of real numbers, however, it converges to the real number $\varphi = \left(1+\sqrt5\right)/2$, the Golden ratio, which is irrational. * The values of the exponential, sine and cosine functions, exp(''x''), sin(''x''), cos(''x''), are known to be irrational for any rational value of ''x''≠0, but each can be defined as the limit of a rational Cauchy sequence, using, for instance, the Maclaurin series.

## Non-example: open interval

The open interval $X = \left(0, 2\right)$ in the set of real numbers with an ordinary distance in R is not a complete space: there is a sequence $x_n = 1/n$ in it, which is Cauchy (for arbitrarily small distance bound $d > 0$ all terms $x_n$ of $n > 1/d$ fit in the $\left(0, d\right)$ interval), however does not converge in $X$ — its 'limit', number $0$, does not belong to the space $X$.

## Other properties

* Every convergent sequence (with limit ''s'', say) is a Cauchy sequence, since, given any real number ''ε'' > 0, beyond some fixed point, every term of the sequence is within distance ''ε''/2 of ''s'', so any two terms of the sequence are within distance ''ε'' of each other. * In any metric space, a Cauchy sequence is bounded function, bounded (since for some ''N'', all terms of the sequence from the ''N''-th onwards are within distance 1 of each other, and if ''M'' is the largest distance between and any terms up to the ''N''-th, then no term of the sequence has distance greater than from ). * In any metric space, a Cauchy sequence which has a convergent subsequence with limit ''s'' is itself convergent (with the same limit), since, given any real number ''r'' > 0, beyond some fixed point in the original sequence, every term of the subsequence is within distance ''r''/2 of ''s'', and any two terms of the original sequence are within distance ''r''/2 of each other, so every term of the original sequence is within distance ''r'' of ''s''. These last two properties, together with the Bolzano–Weierstrass theorem, yield one standard proof of the completeness of the real numbers, closely related to both the Bolzano–Weierstrass theorem and the Heine–Borel theorem. Every Cauchy sequence of real numbers is bounded, hence by Bolzano–Weierstrass has a convergent subsequence, hence is itself convergent. This proof of the completeness of the real numbers implicitly makes use of the least upper bound axiom. The alternative approach, mentioned above, of the real numbers as the Completion (metric space), completion of the rational numbers, makes the completeness of the real numbers tautological. One of the standard illustrations of the advantage of being able to work with Cauchy sequences and make use of completeness is provided by consideration of the summation of an infinite series of real numbers (or, more generally, of elements of any complete normed linear space, or Banach space). Such a series $\sum_^ x_$ is considered to be convergent if and only if the sequence of partial sums $\left(s_\right)$ is convergent, where $s_ = \sum_^ x_$. It is a routine matter to determine whether the sequence of partial sums is Cauchy or not, since for positive integers ''p'' > ''q'', :$s_ - s_ = \sum_^ x_.$ If $f \colon M \rightarrow N$ is a uniformly continuous map between the metric spaces ''M'' and ''N'' and (''x''''n'') is a Cauchy sequence in ''M'', then $\left(f\left(x_n\right)\right)$ is a Cauchy sequence in ''N''. If $\left(x_n\right)$ and $\left(y_n\right)$ are two Cauchy sequences in the rational, real or complex numbers, then the sum $\left(x_n + y_n\right)$ and the product $\left(x_n y_n\right)$ are also Cauchy sequences.

# Generalizations

## In topological vector spaces

There is also a concept of Cauchy sequence for a topological vector space $X$: Pick a local base $B$ for $X$ about 0; then ($x_k$) is a Cauchy sequence if for each member $V\in B$, there is some number $N$ such that whenever $n,m > N, x_n - x_m$ is an element of $V$. If the topology of $X$ is compatible with a translation-invariant metric $d$, the two definitions agree.

## In topological groups

Since the topological vector space definition of Cauchy sequence requires only that there be a continuous "subtraction" operation, it can just as well be stated in the context of a topological group: A sequence $\left(x_k\right)$ in a topological group $G$ is a Cauchy sequence if for every open neighbourhood $U$ of the identity element, identity in $G$ there exists some number $N$ such that whenever $m,n>N$ it follows that $x_n x_m^ \in U$. As above, it is sufficient to check this for the neighbourhoods in any local base of the identity in $G$. As in the Complete metric space#Completion, construction of the completion of a metric space, one can furthermore define the binary relation on Cauchy sequences in $G$ that $\left(x_k\right)$ and $\left(y_k\right)$ are equivalent if for every open Neighbourhood (mathematics), neighbourhood $U$ of the identity in $G$ there exists some number $N$ such that whenever $m,n>N$ it follows that $x_n y_m^ \in U$. This relation is an equivalence relation: It is reflexive since the sequences are Cauchy sequences. It is symmetric since $y_n x_m^ = \left(x_m y_n^\right)^ \in U^$ which by continuity of the inverse is another open neighbourhood of the identity. It is Transitive relation, transitive since $x_n z_l^ = x_n y_m^ y_m z_l^ \in U\text{'} U\text{'}\text{'}$ where $U\text{'}$ and $U\text{'}\text{'}$ are open neighbourhoods of the identity such that $U\text{'}U\text{'}\text{'} \subseteq U$; such pairs exist by the continuity of the group operation.

## In groups

There is also a concept of Cauchy sequence in a group $G$: Let $H=\left(H_r\right)$ be a decreasing sequence of normal subgroups of $G$ of finite index of a subgroup, index. Then a sequence $\left(x_n\right)$ in $G$ is said to be Cauchy (w.r.t. $H$) if and only if for any $r$ there is $N$ such that $\forall m,n > N, x_n x_m^ \in H_r$. Technically, this is the same thing as a topological group Cauchy sequence for a particular choice of topology on $G$, namely that for which $H$ is a local base. The set $C$ of such Cauchy sequences forms a group (for the componentwise product), and the set $C_0$ of null sequences (s.th. $\forall r, \exists N, \forall n > N, x_n \in H_r$) is a normal subgroup of $C$. The factor group $C/C_0$ is called the completion of $G$ with respect to $H$. One can then show that this completion is isomorphic to the inverse limit of the sequence $\left(G/H_r\right)$. An example of this construction, familiar in number theory and algebraic geometry is the construction of the p-adic number, ''p''-adic completion of the integers with respect to a prime ''p''. In this case, ''G'' is the integers under addition, and ''H''''r'' is the additive subgroup consisting of integer multiples of ''p''''r''. If $H$ is a cofinal (mathematics), cofinal sequence (i.e., any normal subgroup of finite index contains some $H_r$), then this completion is Canonical form, canonical in the sense that it is isomorphic to the inverse limit of $\left(G/H\right)_H$, where $H$ varies over normal subgroups of finite index of a subgroup, index. For further details, see ch. I.10 in Serge Lang, Lang's "Algebra".

## In a hyperreal continuum

A real sequence $\langle u_n: n\in \mathbb \rangle$ has a natural hyperreal number, hyperreal extension, defined for hypernatural values ''H'' of the index ''n'' in addition to the usual natural ''n''. The sequence is Cauchy if and only if for every infinite ''H'' and ''K'', the values $u_H$ and $u_K$ are infinitely close, or adequality, adequal, i.e. :$\, \mathrm\left(u_H-u_K\right)= 0$ where "st" is the standard part function.

## Cauchy completion of categories

introduced a notion of Cauchy completion of a category (mathematics), category. Applied to Q (the category whose objects are rational numbers, and there is a morphism from ''x'' to ''y'' if and only if ''x'' ≤ ''y''), this Cauchy completion yields R (again interpreted as a category using its natural ordering).

*Modes of convergence (annotated index) *Dedekind cut

# References

* * * * * (for uses in constructive mathematics)


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Home > Percent Error > Percent Error If Accepted Value Is Zero

Percent Error If Accepted Value Is Zero

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# zbMATH — the first resource for mathematics

Lindelöf theorems for monotone Sobolev functions on uniform domains. (English) Zbl 1070.31001
Let $$(X,d)$$ be a metric space with a Borel measure $$\mu$$ with $$\mu(B(z,r)) \in (0, \infty)$$ for all $$z \in X, r >0,$$ where $$B(z,r)$$ stands for the ball centered at $$z \in X$$ and with radius $$r>0.$$ Assume further that there is a constant $$C_1\geq 1$$ such that $$\mu(2B) \leq C_1 \mu(B)$$ holds for all balls $$B$$ and that there are constants $$Q \geq 1, C_2 > 0,$$ with $\mu(B) /\mu(B_0) \geq C_2 (\text{diam}( B) / \text{diam} (B_0))^Q$ for all balls $$B, B_0$$ with $$B \subset B_0.$$ Let $$D \subset X,$$ consider functions $$u : D \to \mathbb R$$ for which there exists a nonnegative Borel measurable $$g \in L^p(D; \mu)$$, $$p>1$$, $$M >0$$, $$\lambda \in (0,1],$$ with $| u(x)-u(x')| \leq M r \left( {1 \over \mu(B(x,r))} \int_{B(x,r)} g(y)^p d \mu(y)\right)^{1/p}$ for $$x,x' \in B(z, \lambda r)$$ whenever $$B(x,r) \subset D$$ and $\int_D g(y)^p \delta_D(y)^\alpha \, d \mu(y) < \infty \,,$ $$\delta_D(z) = d(z, \partial D)$$. Call functions satisfying these conditions functions of class $${\mathcal U}(D)$$. The author proves the following result.
Theorem. Let $$D$$ be a uniform domain in $$X,$$ $$u \in {\mathcal U}(D)$$, and for $$p > Q +\alpha-1$$ set $E= \biggl\{ \xi \in \partial D : \lim \sup_{r \to 0} {r^{p-\alpha} \over \mu(B(\xi,r))} \int_{B(\xi,r) \cap D} g(y)^p \delta_D(y)^{\alpha} d \mu(y) > 0 \biggr\}.$ If $$\xi \in \partial D \setminus E$$ and there exists a curve $$\gamma$$ in $$D$$ tending to $$\xi$$ along which $$u$$ has a finite limit $$L$$, then $$u$$ has nontangential limit $$L$$ at $$\xi$$.

##### MSC:
 31B25 Boundary behavior of harmonic functions in higher dimensions 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
##### Keywords:
nontangential limits; uniform domains


text:
# Plotting with pandas and matplotlib¶

At this point we are familiar with some of the features of pandas and explored some very basic data visualizations at the end of Chapter 3. Now, we will wade into visualizing our data in more detail, starting by using the built-in plotting options available directly in pandas. Much like the case of pandas being built upon numpy, plotting in pandas takes advantage of plotting features from the matplotlib 1 plotting library. Plotting in pandas provides a basic framework for quickly visualizing our data, but as you’ll see we will need to also use features from matplotlib for more advanced formatting and to enhance our plots. In particular, we will use features from the the pyplot 2 module in matplotlib, which provides MATLAB-like 3 plotting. We will also briefly explore creating interactive plots using the hvplot 4 plotting library, which allows us to produce plots similar to those available in the bokeh plotting library 5 using plotting syntax very similar to that in pandas.

## Creating a basic x-y plot¶

The first step for creating a basic x-y plot is to import pandas and read in the data we want to plot from a file. We will be using a datetime index for our weather observation data as we learned in Chapter 3. In this case, however, we’ll include a few additional parameters in order to read the data with a datetime index. We will read in the data first, and then discuss what happened.

Let’s start by importing the libraries we will need (pandas and Matplotlib), and then read in the data.

import pandas as pd
import matplotlib.pyplot as plt

fp = "data/029740.txt"

fp,
delim_whitespace=True,
na_values=["*", "**", "***", "****", "*****", "******"],
usecols=["YR--MODAHRMN", "TEMP", "MAX", "MIN"],
parse_dates=["YR--MODAHRMN"],
index_col="YR--MODAHRMN",
)

So, let us now examing what is different here compared to files read in Chapter 3. There are two significant changes in the form of two new parameters: parse_dates and index_col.

• parse_dates takes a Python list of column name(s) for data file columns that contain date data and pandas will parse and convert data in these column(s) to the datetime data type. For many common date formats pandas will automatically recognize and convert the date data.

• index_col is used to state a column that should be used to index the data in the DataFrame. In this case, we end up with our date data as the DataFrame index. This is a very useful feature in pandas as we will see below.

Having read in the data file, we can now have a quick look at what we have using data.head().

data.head()

TEMP MAX MIN
YR--MODAHRMN
1952-01-01 00:00:00 36.0 NaN NaN
1952-01-01 06:00:00 37.0 NaN 34.0
1952-01-01 12:00:00 39.0 NaN NaN
1952-01-01 18:00:00 36.0 39.0 NaN
1952-01-02 00:00:00 36.0 NaN NaN

As mentioned above, you can now see that the index column for our DataFrame (the first column) contains date values related to each row in the DataFrame. With this we are already able to create our first plot using pandas. We will start by using the basic line plot in pandas to visualize at our temperature data.

ax = data.plot()

Figure 4.3. A basic plot of the example temperature data using Matplotlib.

Now, let’s break down what just happened. First, we first created the plot object using the plot() method of the data DataFrame. Without any parameters given, this makes the plot of all columns in the DataFrame as lines of different color on the y-axis with the index, time in this case, on the x-axis. Second, in case we want to be able to modify the plot or add anything to it after the basic plot has been created, we assign the plot object to the variable ax. Why don’t we check its data type below using the type() function.

type(ax)

matplotlib.axes._subplots.AxesSubplot

OK, so this is a data type we have not seen previously, but clearly it is part of matplotlib. In case you were skeptical before, we now have evidence that pandas is using matplotlib for generating plots.

### Selecting plot data based on date¶

Now we can make a few small changes to our plot to further explore plotting with pandas. We can begin by plotting only the observed temperatures in the data['TEMP'] column. In addition, we can restrict ourselves to observations from only the afternoon of October 1, 2019 (the last day in this dataset). We will do this by first creating a pandas series for only the desired data column and restricting the dateof interest. Once we have created the new pandas series we can plot the results.

# Create basic plot with subset of data
oct1_temps = data["TEMP"].loc[data.index >= "201910011200"]
ax = oct1_temps.plot()

Figure 4.4. A plot of the example temperature data for the afternoon of October 1, 2019.

So, what did we change this time? First, we selected only the 'TEMP' column from the data DataFrame by using data['TEMP'] instead of data. Second, we added a restriction to the date range using loc[] to select only rows where the index value data.index is greater than '201910011200'. In that case, the number in the string is in the format 'YYYYMMDDHHMM', where YYYY is the year, MM is the month, DD is the day, HH is the hour, and MM is the minute. This will result in temperatures only from noon onwards on October 1, 2019. Finally, by saving this selection to the variable oct1_temps we’re able to now use oct1_temps.plot() to plot only our selection. As you can see, we are able to easily control the values plotted in pandas, but we can do even better.

## Basic plot formatting¶

We can control the appearance of our plots, making them look nicer and provide more information by using a few additional plotting options available in pandas and/or matplotlib. Let’s start by changing the line format, adding some axis labels, and adding a title.

# Change line and symbol format, and add axis labels/title
ax = oct1_temps.plot(
style="ko--",
title="Helsinki-Vantaa temperatures",
xlabel="Date",
ylabel="Temperature [°F]",
)

Figure 4.5. A plot of the example temperature data with additional formatting, a title, and axis labels.

Now we see that our temperature data as a black dashed line with circles indicating the temperature values from the data file. This comes from the additional parameter style='ko--'. In this case, k tells the oct1_temps.plot() function to use black color for the lines and symbols, o tells it to show circles at the data points, and -- says to use a dashed line between points. You can use help(oct1_temps.plot) to find out more about formatting plots. We have also added a title using the title parameter, and axis labels using the xlabel and ylabel parameters 6.

## Formatting and annotating the figure¶

In addition to labeling the plot axes and adding a title, there are several other common plot attributes one might like to utilize. We briefly present some of the most common plot features here and then demonstrate how they work in a modified example plot below.

Changing the figure size

While the default plot sizes we’re working with are fine, it is often helpful to be able to control the figure size. Fortunately, there is an easy way to change the figure size in pandas and matplotlib. In order to define the figure size, we simply include the figsize parameter with a tuple (set of values in normal parentheses) that lists the width and height of the figure (in inches!).

figsize=(12, 6)

to the ax.plot() function will increase the figure size to be 12 inches wide and 6 inches tall.

Note that it is also possible to change the default figure size for all figures in a Jupyter Notebook by importing the pyplot module from matplotlib (i.e., import matplotlib.pyplot as plt)and then defining the default figure size using plt.rcParams['figure.figsize'] = [12, 6]. In this case the figure size should be given as a Python list.

It is also possible to add text that can be displayed on a plot using ax.text(). For example,

ax.plot(...)
x, y = "201910011800", 42
ax.text(x, y, "This is my text.")

would add the text “This is my text.” aligned to the left starting from the date and time 6:00 pm on October 1, 2019 (x="201910011800") and at temperature y=42. Notice that the x coordinate is passed as a text following the formatting in the original data.

Changing the axis ranges

In some cases you may want to plot only a subset of the data you are working with. You can modify the range of values that are plotted by definiing the axis ranges. Changing the plot axes can be done using the xlim and ylim parameters of the plot() function, where xmin is the minimum bound of the x-axis, xmax is the maximum bound, and the same goes for the y-axis with ymin and ymax. For example, adding

ax.plot(
...
xlim=["201910011500", "201910012200"],
ylim=[38, 44]
)

to the parameter list for ax.plot() would change the x-axis range to 3:00 pm to 10:00 pm on October 1, 2019 and the y-axis range to 38 to 44 degrees.

It is also possible to use the datetime library to make it somewhat easier to format ranges when defining axis limits. The benefit is that it can be more clear to define dates in the datetime format as opposed to a long text string. As an example, we could define the same date ranges as above using the format below. Note that we must first import the datetime library before using it.

from datetime import datetime

# Use datetime to define axis limits
xmin = datetime(2019, 10, 1, 15)
xmax = datetime(2019, 10, 1, 22)

...
ax.plot(
...
xlim=[xmin, xmax]
)

In this case, 3:00 pm on October 1, 2019 is defined by datetime(2019, 10, 1, 15) using the datetime library. In this format we are able to specify a set of comma-separated values for the year, month, day, hour, minute, seconds, and microseconds to define a date. If only year, month, and day are given, datetime assumes the time on that day is midnight at the start of the day.

Defining axis ranges this way is handy becaues it will adjust the range of values shown on the plot, but not alter the source data in any way. This can be quite nice when exploring your datasets.

The final example of a common plot feature we can add is a legend. The legend allows you to provide some additional information about the lines, points, or other features of a plot, and adding a legend is quite simple. To do so, we need to add two things: a label parameter in the plot function that lists the text that should be displayed in the legend, and a call to the legend() function to display the plot legend. For example, adding the following to the plot will add and display a legend.

ax.plot(
...
label="Observed temperature"
)
ax.legend()

This would add the label “Observed temperature” to the legend for the line formatted using the ax.plot() function. The legend can be displayed by adding ax.legend() to the code.

Note that by default the legend will automatically be positioned in the top right corner of a plot, or in a location where it minimizes interference with other plot elements. The location of the legend can be controlled using the loc parameter in the call to the legend() function. The matplotlib documentation 7 contains more information about how to control the location of the plot legend.

The modified example plot

Based on the additions above (changing the figure size, adding text to the plot, etc.), the modified plot can be generated using the code below.

from datetime import datetime

# Use xlim and ylim to define the axis ranges
xmin, xmax = datetime(2019, 10, 1, 15), datetime(2019, 10, 1, 22)
ymin, ymax = 38, 44

ax = oct1_temps.plot(
style="ko--",
title="Helsinki-Vantaa temperatures",
xlabel="Date",
ylabel="Temperature [°F]",
xlim=[xmin, xmax],
ylim=[ymin, ymax],
label="Observed temperature",
figsize=(12, 6),
)

x, y = "201910011800", 42
ax.text(x, y, "This is my text.")

# Display plot legend
ax.legend()

<matplotlib.legend.Legend at 0x16e64b520>

Figure 4.6. A plot of the example temperature data further modified with additional formatting and reduced axis ranges.

As you can see, we now have a more informative plot with only a few changes in the plotting code.

## Dealing with datetime axes¶

One issue we will encounter both with placing text on the plot and changing the axis ranges is the datetime index for our DataFrame. In order to do either thing, we need to define x-values using a datetime object. The easiest way to do this is to use the pandas pd.to_datetime() function, which converts a character string date to a datetime object. For example, we can convert 13:00 on October 1, 2019 from the character string '201910011300' to a datetime equivalent by typing

pd.to_datetime("201910011300")

Timestamp('2019-10-01 13:00:00')

With this datetime issue in mind, let’s now create a modified version of the plot above. In this plot we can

1. Limit our time range to 12:00 to 15:00 on October 1, 2019

2. Only look at temperatures between 40-46° Fahrenheit

3. Add text to note the coldest part of the early afternoon

# Define dates
start_time = pd.to_datetime("201910011200")
end_time = pd.to_datetime("201910011500")
cold_time = pd.to_datetime("201910011205")

ax = oct1_temps.plot(
style="ko--",
title="Helsinki-Vantaa temperatures",
xlabel="Date",
ylabel="Temperature [°F]",
xlim=[start_time, end_time],
ylim=[40.0, 46.0],
label="Observed temperature",
figsize=(12, 6),
)

ax.text(cold_time, 42.0, "<- Coldest temperature in early afternoon")
ax.legend()

<matplotlib.legend.Legend at 0x16fc52fd0>

Figure 4.7. A plot of the example temperature data further modified with additional formatting and reduced axis ranges.

### Question 4.1¶

Create a line plot similar to the examples above with the following attributes:

• Temperature data from 18:00-24:00 on October 1, 2019

• A dotted red line connecting the observations (do not show the data points)

• A title that reads “Evening temperatures at Helsinki-Vantaa on October 1, 2019”

• A text label indicating the warmest temperature in the evening

# Use this cell to enter your solution.

# Solution

# Define dates
start_time = pd.to_datetime("201910011800")
end_time = pd.to_datetime("201910020000")
warm_time = pd.to_datetime("201910012055")

ax = oct1_temps.plot(
style="k--",
title="Helsinki-Vantaa temperatures",
xlabel="Date",
ylabel="Temperature [°F]",
xlim=[start_time, end_time],
ylim=[35.0, 44.0],
label="Observed temperature",
figsize=(12, 6),
)

ax.text(warm_time, 43.0, "Warmest temperature in the evening ->")
ax.legend(loc=4)

<matplotlib.legend.Legend at 0x16e3511c0>

Figure 4.8. The solution plot for Question 4.1.

## Bar plots in pandas¶

In addition to line plots, there are many other options for plotting in pandas. Bar plots are one option, which can be used quite similarly to line plots with the addition of the kind=bar parameter. Note that it is easiest to plot our selected time range for a bar plot by selecting the dates in our data series first, rather than adjusting the plot limits. Pandas sees bar plot data as categorical, so the date range is more difficult to define for x-axis limits. For the y-axis, we can still define its range using the ylim=[ymin, ymax] parameter. Similarly, text placement on a bar plot is more difficult, and most easily done using the index value of the bar where the text should be placed.

oct1_afternoon = oct1_temps.loc[oct1_temps.index <= "201910011500"]

# Plot temperature data as a bar plot
ax = oct1_afternoon.plot(
kind="bar",
title="Helsinki-Vantaa temperatures",
xlabel="Date",
ylabel="Temperature [°F]",
ylim=[40, 46],
label="Observed temperature",
figsize=(12, 6),
)

ax.text(0, 42.1, "Coldest \ntemp \nv")
ax.legend()

<matplotlib.legend.Legend at 0x16e33a8e0>

Figure 4.9. A bar plot using pandas with the example temperature data.

You can find more about how to format bar charts on the pandas documentation website 8.

## Saving your plots as image files¶

Saving plots created using pandas can be done in several ways. The recommendation for use outside of Jupyter notebooks is to use Matplotlib’s plt.savefig() function. When using plt.savefig(), you simply give a list of commands to generate a plot and include plt.savefig() with some parameters as the last command in the Python cell. The file name is required, and the image format will be determined based on the listed file extension.

Matplotlib plots can be saved in a number of useful file formats, including PNG, PDF, and EPS. PNG is a nice format for raster images, and EPS is probably easiest to use for vector graphics. Let’s check out an example of how to save our lovely bar plot.

oct1_afternoon = oct1_temps.loc[oct1_temps.index <= "201910011500"]

ax = oct1_afternoon.plot(
kind="bar",
title="Helsinki-Vantaa temperatures",
xlabel="Date",
ylabel="Temperature [°F]",
ylim=[40, 46],
label="Observed temperature",
figsize=(12, 6),
)

ax.text(0, 42.1, "Coldest \ntemp \nv")
ax.legend()

# Write figure to PNG file
plt.savefig("bar-plot.png")

The commands above would both plot the figure and save a copy in a raster image titled bar-plot.png. If working in the JupyterLab environment, you would see the new plot file in the file browser on the left side of the notebook window.

Just to demonstrate once again, we can save another version of the plot at a higher resolution with a minor change to the plot commands above.

oct1_afternoon = oct1_temps.loc[oct1_temps.index <= "201910011500"]

ax = oct1_afternoon.plot(
kind="bar",
title="Helsinki-Vantaa temperatures",
xlabel="Date",
ylabel="Temperature [°F]",
ylim=[40, 46],
label="Observed temperature",
figsize=(12, 6),
)

ax.text(0, 42.1, "Coldest \ntemp \nv")
ax.legend()

# Write figure to PDF file with 600 dpi resolution
plt.savefig("bar-plot-hi-res.pdf", dpi=600)

This would save the output plot as a PDF file with a resolution of 600 dots per inch.

## Interactive plotting, a teaser¶

We have seen in the previous parts of this chapter that there are many ways to configure and visualize data using Python. Up to this point, however, we have focussed on static visualizations in Jupyter notebooks or those saved as images. As you might imagine, there are numerous Python options to also create interactive visualizations that allow those viewing them to dive deeper into the data and explore. Interactive visualizations are increasingly popular online, and here we provide a brief example of how to create an interactive visualization using the data we have already plotted above. In this example, we will be using the hvPlot library 4, which we will return to using later in Part 2 of this book.

hvPlot is a high-level plotting package that can accept several different common Python data types and plot them using using the bokeh visualization library 5. The great thing with hvPlot is that it takes almost no effort to move from plotting in pandas to creating interactive plots. We can explore this in the following example.

Let us start by importing the pandas submodule of hvPlot.

import hvplot.pandas


text:
# Draw Vectors in Spherical Shell and Then Animate

I want to draw three vectors inside a spherical shell. One of the vectors will be moving, the others will be stationary. However, the animation is the next step. For now, I would like to get the vectors drawn.

The three vectors need to go from the origin to the surface of the sphere. I am using an external program to create a list of angles that change with Temperature and these vectors represent an external magnetic field and easy axis. Those two angles are stationary, one perfectly vertical (along the z-axis) another at phi = pi/4. The third will change, but, can be stationary drawing. Obviously r is constant since they go to the surface.

I tried Show[Graphics3D[Sphere[{0, 0, 0}]], Boxed -> False], but I would prefer something more transparent with axes at the origin not outside the sphere.

Something like this with two more vectors and 3D:

• Please provide more details. As it is, it's unclear what you really want. Vectors pointing from the origin to the surface of the sphere? Vectors tangent to the sphere? Random vectors or particular ones? Anyway, look at Graphics3D and cousins. – march Jan 28 '16 at 21:18
• all three vectors from origin to the surface of the sphere – Joseph Jan 28 '16 at 21:25
• Please edit your post with the new details (and include more details about how they should be oriented, etc.). Also make sure to read the documentation on Graphics3D and Arrow in particular. There should be plenty of examples in the documentation of how to do this. – march Jan 28 '16 at 21:27
• You may be interested in my answer here as an example of how to do it. – Jens Jan 28 '16 at 22:09

Manipulate[
Graphics3D[{{Opacity[0.5], Sphere[]},
{EdgeForm[{Dashed, Red}],
FaceForm[None], Cylinder[{{0, 0, -.001}, {0, 0, .001}}]},
{Dashed, Line[{{0, 0, 0}, {1, 0, 0}}]},
{Dashed, Line[{{0, 0, 0}, {0, 1, 0}}]},
{Dashed, Line[{{0, 0, 0}, {0, 0, 1}}]},
Arrow[{{1, 0, 0}, {1.3, 0, 0}}],
Arrow[{{0, 1, 0}, {0, 1.3, 0}}],
Arrow[{{0, 0, 1}, {0, 0, 1.3}}],
Text[Style["x", 18], {1.4, 0, 0}],
Text[Style["y", 18], {0, 1.4, 0}],
Text[Style["z", 18], {0, 0, 1.4}],
Text[Style["r", 18, Red], .5 {Cos[θ] Sin[φ], Sin[θ] Sin[φ], Cos[φ]}],
{Red, Arrow[{{0, 0, 0}, {Cos[θ] Sin[φ],
Sin[θ] Sin[φ], Cos[φ]}}]}},
Boxed -> False],
{θ, 0, 2 π},
{{φ, .3}, 0, π}]

• This is the perfect starting point! Thank you! I will work with this and get it setup the way I need before asking anything else. – Joseph Jan 28 '16 at 22:37


text:
# Math Help - Simplifying trig functions

1. ## Simplifying trig functions

I'm sure that I'm missing an identity here, but how do I get from:

-4cos(t)cos(2t)-2sin(t)sin(2t)

to

-4cos3(t)

Thanks for any help.

2. ## Re: Simplifying trig functions

Originally Posted by MajMinor
I'm sure that I'm missing an identity here, but how do I get from:

-4cos(t)cos(2t)-2sin(t)sin(2t)

to

-4cos3(t)

Thanks for any help.
Use the product to sum identities to get

$-4\left( \frac{\cos(3t)+\cos(t)}{2} \right)-2\left( \frac{\cos(t)-\cos(3t)}{2} \right) = -\cos(3t)-3\cos(t)$

by the triple angle forumla you get the desired result

$-4\cos^3(t)$

3. ## Re: Simplifying trig functions

Thanks for the help.


text: 08:30 to 09:00 Registration 09:10 to 09:50 E Vanden-Eijnden (Courant Institute of Mathematical Sciences)Dimension reduction, coarse-graining and data assimilation in high-dimensional dynamical systems: modeling and computational issues Modern computing technologies, such as massively parallel simulation, special-purpose high-performance computers, and high-performance GPUs permit to simulate complex high-dimensional dynamical systems and generate time-series in amounts too large to be grasped by traditional "look and see" analyses. This calls for robust and automated methods to extract the essential structural and dynamical properties from these data in a manner that is little or not depending on human subjectivity. To this end, a decade of work has led to the development of analysis techniques which rely on the partitioning of the conformation space into discrete substates and reduce the dynamics to transitions between these states. A particular successful class of methods of this type are Markov state models (MSMs), in which the transitions between the states in the partition are assumed to be memoryless jumps. The accuracy of these models crucially depends on the choice of these states. In this talk, I will discuss systematic strategies that permit to identify these states and quantify the error of the resulting approximation. These methods will be illustrated on examples arising from molecular dynamics simulations of biomolecules. INI 1 09:55 to 10:35 G Csanyi (University of Cambridge)Gaussian process regression in molecular modelling The talk will summarise our recent work in using Gaussian process regression (a form of Bayesian non-parametric inference) to model moderate to high dimensional functions in chemistry. Applications range from constructing interatomic potentials (i.e. parametrisations of the Born-Oppenheimer potential energy surface) starting from total energy electronic structure calculations, to free energy surface reconstruction based on umbrella sampling trajectories. In every case studied so far, a careful treatment of data representation and hyperparameters leads to huge increases in computational efficiency and model fidelity. INI 1 10:40 to 11:25 Coffee/Tea Break 11:30 to 12:10 D Crommelin (Centrum voor Wiskunde en Informatica (CWI))Modeling of unresolved scales with data-inferred stochastic processes I will discuss a data-driven stochastic approach to modeling unresolved scales, in which feedback from micro-scale processes is represented by a network of Markov processes. The Markov processes are conditioned on macro-scale model variables, and their properties are inferred from pre-computed high-resolution (micro-scale resolving) simulations. These processes are designed to emulate, in a statistical sense, the feedback observed in the high-resolution simulations, thereby providing a statistical-dynamical coupling between micro- and macro-scale models. This work is primarily aimed at applications in atmosphere-ocean science (stochastic parameterizations of atmospheric convection and of mesoscale oceanic eddies). INI 1 12:15 to 12:55 R Miller (Oregon State University)Implicit particle methods for high dimensional highly nonlinear systems The implicit particle filter is one of a number of recently-proposed particle filtering schemes in which the trajectory of each particle is informed by observations within each assimilation cycle. In the case of observations defined by a linear function of the state vector, taken every time step of the numerical model, the implicit particle filter is equivalent to the optimal importance filter, i.e., at each step, any given particle is drawn from the density of the system conditioned jointly upon the observation and the state of the particle at the previous time. The optimal importance filter was implemented for a shallow water model with O(10^4) state variables, and performed well with nominal demands on computing resources, but it exhibited some characteristics of the degeneracy some authors have predicted. We note the similarity of our scheme to other recently-devised schemes, and propose a potential solution in the form of a fixed-lag smoother. INI 1 13:00 to 14:25 Sandwich Lunch in INI 14:30 to 15:10 L Colwell (University of Cambridge)Using evolutionary sequence variation to make inferences about protein structure and function The explosive growth in the number of protein sequences gives rise to the possibility of using natural variation in sequences of homologous proteins to find residues that control different protein phenotypes. Because in many cases phenotypic changes are controlled by a group of residues, the mutations that separate one phenotype from another will be correlated. We show that correlations between amino acid mutations at different sites in a protein can be used to predict, de novo, tertiary protein structure of both globular and transmembrane proteins from large sequence alignments. In addition, residues that determine the specificity of protein interactions can be identified from inter-protein residue pairs that co-vary. Those amino acids whose mutation patterns are most highly constrained by evolution are found to often involve known functional sites of proteins, such as enzyme active sites, and ligand binding sites. These findings raise questions about the relationship between protein structure and function, and the evolutionary constraints that this relationship imposes on different proteins. Our maximum entropy based analysis identifies a global probability with a minimal set of amino acid pair interactions that reproduce all the observed pairwise correlations in the data. The resulting probability model for the sequence of the protein of interest raises the possibility that we may be able to identify amino acids that control different protein phenotypes, and hence re-programme existing proteins. INI 1 15:15 to 15:55 D Sanz-Alonso (University of Warwick)Filtering partially observed chaotic deterministic dynamical systems Many physical systems can be successfully modelled by a deterministic dynamical system for which, however, the initial conditions may contain uncertainty. In the presence of chaos this can lead to undesirable growth of uncertainty over time. However, when noisy observations of the system are present these may be used to compensate for the uncertainty in the initial state. This scenario is naturally modelled by viewing the initial state as given by a probability distribution, and to then condition this probability distribution on the noisy observations, thereby reducing uncertainty. Filtering refers to the situation where the conditional distribution on the system state is updated sequentially, at the time of each observation. In this talk we investigate the asymptotic behaviour of this filtering distribution for large time. We focus on a class of dissipative systems that includes the Lorenz '63 and '96 models, and the Navier-Stokes equations on a 2D torus. We first study the behaviour of a variant on the 3DVAR filter, creating a unified analysis which subsumes the existing work in [1,2] which, itself, builds on [3]. The optimality property of the true filtering distribution is then used, when combined with this modified 3DVAR analysis, to provide general conditions on the observation of our wide class of chaotic dissipative systems which ensure that the filtering distributions concentrate around the true state of the underlying system in the long-time asymptotic regime. [1] C.E.A. Brett, K.F. Lam, K.J.H. Law, D.S. McCormick, M.R. Scott and A.M. Stuart, Accuracy and stability of filters for dissipative PDEs.'' Physica D 245(2013). [2] K.J.H. Law, A. Shukla and A.M. Stuart, Analysis of the 3DVAR Filter for the Partially Observed Lorenz '63 Model.'' Discrete and Continuous Dynamical Systems A, 34(2014). [3] K. Hayden, E. Olsen and E.S. Titi, Discrete data assimilation in the Lorenz and 2D Navier-Stokes equations.'' Physica D 240(2011). INI 1 16:00 to 16:45 Coffee/Tea Break 16:50 to 17:30 J Lu (Duke University)Transition path processes Understanding rare events like transitions of chemical system from reactant to product states is a challenging problem due to the time scale separation. In this talk, we will discuss some recent progress in mathematical theory of transition paths. In particular, we identify and characterize the stochastic process corresponds to transition paths. The study of transition path process helps to understand the transition mechanism and provides a framework to design and analyze numerical approaches for rare event sampling and simulation. INI 1 17:30 to 18:30 Wine Reception


text:
# Reviewer Checklist¶

• The code must be readable without extra effort from your part. The code should be easily readable (for infos e.g. here)

• Include references to the used algorithms in the docstring

• If the algorithm is new, please include a description in the docstring, or be sure to include a reference as soon as you publish the work

• Variable names should be chosen to be clear. Avoid item, element, var, list, data etc… A good variable name makes it immediately clear what it contains.

• Avoid as much as possible hard-coded indices for list (no x = l[0], y = l[1]). Rather, use tuple unpacking (see here). Note that tuple unpacking can also be used to update variables. For example, the Fibonacci sequence next number pair can be written as n1, n2 = n2, n1+n2.

• Do not use mutable (lists, dictionaries, …) as default values for functions and methods. Do not write: def function(default=[]):

but use def function(default=None):      if default is None: default=[]

• Use pythonic loops, list comprehensions

• Make sure the unit tests are testing all the lines of the code. Do not only check for working cases, but also the most common wrong use cases.

• Check the docstrings (Do they follow the Numpydoc conventions, is everything clearly explained, are the default values given and is it clear why they are set to this value)

• Keep the code simple. Avoid using complex Python functionalities whose use is opaque to non-expert developers unless necessary. For example, the @staticmethod decorator should only be used if really necessary. Another example, for counting the dictionary colors = ['red', 'green', 'red', 'blue', 'green', 'red'], version:

d = {}
for color in colors:
d[color] = d.get(color, 0) + 1

is perfectly fine, no need to complicate it to a maybe more pythonic version d = collections.defaultdict(int)  for color in colors:      d[color] += 1

• Did the code writer perform a static code analysis? Does the code respect Pep8 (see also the pylint config file)?

• Did the code writer perform a profiling and checked that there are no obviously inefficient (computation time-wise and memory-wise) parts in the code?


text:
Public Group

# SDL KeyRepeat

This topic is 4936 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

## Recommended Posts

Hi, I have the following piece of code /************************************************************************** while (SDL_PollEvent(&event)) { if (event.type == SDL_QUIT) { done = 1; } if (event.type == SDL_KEYDOWN) { if (event.key.keysym.sym == SDLK_ESCAPE) { done = 1; } if(event.key.keysym.sym == SDLK_p) { musicPaused=!musicPaused; } keys = SDL_GetKeyState(NULL); if(keys[SDLK_PAGEDOWN]) { theEngine->Sound->music("setVolume", 0, theEngine->Sound->music("getVolume", 0, 0)-8); } if(keys[SDLK_PAGEUP]) { theEngine->Sound->music("setVolume", 0, theEngine->Sound->music("getVolume", 0, 0)+8); } } } \****************************************************************************/ Near the bottom of the code if PAGEDOWN or PAGEUP is pressed it calls a function to change the level of the volume. My problem is I want to be able to hold PAGEDOWN or PAGEUP to continually change the volume. I find that if I press and hold the button down it only makes once change to the level. I looked at the SDL documentation and found SDL_EnableKeyRepeat() I added it to the code just above if(keys[SDLK_PAGEDOWN]), but this didn't make any difference. Any suggestions?

##### Share on other sites
I think your loop is kind of off.

Take a look at my comments -

while (SDL_PollEvent(&event)){   if (event.type == SDL_QUIT)    {       done = 1;    }   if (event.type == SDL_KEYDOWN)   {      // Good, check to see if escape was preseed      if (event.key.keysym.sym == SDLK_ESCAPE)       {          done = 1;       }      // Good, check to see if p was preseed      if(event.key.keysym.sym == SDLK_p)      {         musicPaused=!musicPaused;      }    }   }       keys = SDL_GetKeyState(NULL);      // Let's try the previous method         //if(keys[SDLK_PAGEDOWN])       if(event.key.keysym.sym == SDLK_PAGEDOWN)      {         theEngine->Sound->music("setVolume", 0,    theEngine->Sound->music("getVolume", 0, 0)-8);      }         if(keys[SDLK_PAGEUP])      {         theEngine->Sound->music("setVolume", 0,    theEngine->Sound->music("getVolume", 0, 0)+8);      }  }

Now see the changes I made? Your method that you were using looked like it was working fine, so try using that first. If that does notwork, you will need to do this.

while (SDL_PollEvent(&event)){   if (event.type == SDL_QUIT)    {       done = 1;    }   if (event.type == SDL_KEYDOWN)   {      // Good, check to see if escape was preseed      if (event.key.keysym.sym == SDLK_ESCAPE)       {          done = 1;       }      // Good, check to see if p was preseed      if(event.key.keysym.sym == SDLK_p)      {         musicPaused=!musicPaused;      }      // wtf mate?!?      //keys = SDL_GetKeyState(NULL);      // Let's try the previous method         //if(keys[SDLK_PAGEDOWN])       if(event.key.keysym.sym == SDLK_PAGEDOWN)      {         theEngine->Sound->music("setVolume", 0,    theEngine->Sound->music("getVolume", 0, 0)-8);      }   //      if(keys[SDLK_PAGEUP])       // Let's try the previous method      if(event.key.keysym.sym == SDLK_PAGEUP)      {         theEngine->Sound->music("setVolume", 0,    theEngine->Sound->music("getVolume", 0, 0)+8);      }   }   }

That main idea is to let everything update event-wise, then get the state of the keys and process that. I think one of those methods should do what you are trying to accompish. Just note that now, for every key press, you will be getting at least 12-15 events, so one press of PAGEUP will make the volume go up at least to 96! You will have to compensate for this.

As a side note, you should look into a more efficient main loop. I will give you a hint to look up SDL_PeepEvent. Hope this helps.

- Drew

##### Share on other sites
Yes you were very right.
I moved the code, down to where I process the rest of the input, such as moving the character about the screen. This is controlled by setting a frame rate, and checking SDL_GetTicks.

This provides a nice smooth volume up and down.

Cheers

I'll take a look at SDL_PeepEvent too, thanks!

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text:
Mapping out the landscape of a proof

 composite of src1, src2

Kenneth and Laurel Appel, our friend Andrew Appel’s father and sister, passed away in the past two months. Kenneth Appel partnered with Wolfgang Haken and some helpers to prove the Four-Color Map Theorem in 1976. One of those helpers was Laurel, then 13. Laurel became an adjunct associate professor of biology at Wesleyan University, where her husband Michael Weir is a professor, and directed the Ronald McNair Research Scholars program at Wesleyan. Her father was chair of mathematics at the University of New Hampshire, where he hired Yitang Zhang and is remembered for mentoring many junior faculty.

Today we remember both of them, and celebrate the nature of their work on the four-color proof.

The proof was the first truly famous one to employ computer calculations. We do not want to revisit here a long-term debate on the acceptance of computer-assisted proofs. Instead we wish to talk about whether the setting of a proof one is trying to construct helps the process converge toward finding one, or has a “slippery slope” sure to magnify any gap. The recognition of a helpful setting by Appel and Haken was an important human step before starting up the computer, and prefigured that Laurel’s role afterwards would be a happy one.

I (Ken) have known Andrew Appel as a Princeton ’81 classmate since 1977, and Dick has known him as a longtime colleague at Princeton. Our deepest condolences go to the entire family.

## Laurel’s Role

When Dick covered the Four-Color Theorem four years ago, his main point was Appel and Haken’s prior mathematical confidence in the structure of the proof, which went back to 1972. Now we note that this confidence was actually enhanced, not dented, during the discovery of errors after the initial computer runs. Many were in transcribing the computer output into readable form, but some involved the connection between the initial “discharging” phase which generated sets of coloring configurations, and the inputs to a computer program written by John Koch to show the configurations are reducible. As recounted by Donald Mackenzie in his book Mechanizing Proof: Computing, Risk, and Trust, members of both families took part in checking, and:

[P]articularly important was the contribution of Appel’s daughter, Laurel. This collective checking effort was crucial in persuading Haken that their proof was sound enough to “go public.”

Mackenzie’s book then picks up the story by quoting Haken:

She…found, I think, 800 mistakes, which is not many on 700 pages. And 650 she could correct herself. There was just…a wrong number, or so… [O]ver the 4th of July weekend, Ken … re-computed those 50 configurations which really had been missing … and only 12 of them did not work. And then I worked on those for one day, and I found that they could be replaced … by something like 20 others, and only 2 of them did not work, and then again [it took] a day… So I said, ‘Then, here somebody has worked one month full-time and found 800 mistakes. And then we have needed only five days to repair all that. This looks so stable—it is this incredible stability.’

While noting that the published paper in 1976 had not shaken out all such glitches, Andrew described the landscape this way in private correspondence to me:

[T]he basic proof method was very robust: there was a huge “forest” of “geographically good” reducible configurations, and any direction you shoot an arrow you’ll hit a geographically good configuration… If a configuration in the unavoidable set was found to be geographically not-good or even irreducible, the unavoidability algorithm was easily adjusted to replace it with three or four good configurations. That’s what was meant by robustness. So even if there happened to be a minor error and a truly irreducible configuration snuck in, it would be fixable.

## A Useful Contrast

What Appel and Haken did first was find a particularly good implementation of the idea of converting a universal statement into an existential one. The universal statement was:

For all planar graphs ${G}$, ${G}$ is 4-colorable.

The existential statement is:

There exists a finite set ${S}$ of graph configurations that are collectively unavoidable, each of which is reducible.

The definition of “unavoidable” means that every (large enough) ${G}$ includes a member of ${S}$, which sounds universal, but they gave a finite criterion for verifying it. Thus their goal was always well in sight.

By contrast, the Twin Prime and Goldbach conjectures are known only as universal statements, as is the Riemann Hypothesis. If they are false, there is a finite counterexample. If you had high confidence in these conjectures being false, you could search for a counterexample.

For both Twin Prime and Goldbach, there are also “heuristic arguments” that connote high likelihood of truth. Namely, suppose every (odd) natural number ${n}$ could randomly declare “I am prime” with probability ${1/\pi(n)}$, where ${\pi(n)}$ is the density of primes in the region of ${n}$, or a good enough estimate of this density. Then as ${n_0}$ increases, the probability of there failing to be a number ${p}$ between ${n_0}$ and ${2n_0}$ such that ${p}$ and ${p+2}$ both declared “I am prime,” or of there being an even number in that range that is not the sum of two self-declared primes, tends rapidly to zero. (Jordan Ellenberg’s article in Slate on Zhang’s proof includes the former heuristic.) However, these arguments say nothing about the likelihood of finding a proof.

The brilliance of Appel and Haken was setting up the problem so that a randomized argument would be about finding a proof. Indeed they were able to tune the argument to gauge the amount of computation time needed to find one. And to make checking the proof literally child’s play. The 1989 book by Appel and Haken rounded up the few errors that remained, while Dick’s Tech colleague Robin Thomas and three others found a smaller ${S}$ with easier verification, and Georges Gonthier automated the entire process of generating and checking the proof.

## Open Problems

We have blogged before about the process of “making a heuristic into a theorem,” and about a case of a 300-page proof that foundered because one inequality sign was wrong. Can we recognize general ways to move toward proof landscapes that are “robust” rather than “fragile”?

Again Dick and I were saddened by the news. We note a New Hampshire State House honor for Kenneth earlier this month, and a memorial fund for Laurel at Wesleyan.

May 28, 2013 6:23 pm

Here’s what David Ruelle said in 1988:

“Perhaps in a few decades shall we see what nonhuman mathematics looks like. I’m not predicting the imminent arrival of little green men from outer space, but simply the invasion of mathematics by computers. Since the human brain is a sort of natural computer, I see no reason why the artificial variety couldn’t perform better in the specialized task of doing mathematical research. My guess is that, within 50 or 100 years – or it might be 150 – computers will successfully compete with the human brain in doing mathematics, and that their mathematical style will be rather different from ours. Fairly long computational verifications – numerical or combinatorial – won’t bother them at all, and this should lead not just to different sorts of proofs, but more importantly to different sorts of theorems being proved.”

He’s a specialist in statistical mechanics and chaos theory. Today, many logicians are making similar predictions.

May 31, 2013 7:44 pm

OK- but who is going drive that computer-doing-mathematics?

2. May 30, 2013 12:00 am

re serge & the ruelle quote– could you plz cite that? it was indeed prescient. there have been major advances, but breakthroughs are still rare. alas we have cars that can drive themselves [via google] but there seems some basic/fundamental principle of ATP (auto thm proving) that is eluding us. it does seem to have some close relationship to AI.
much more on appel & the subj of automated thm proving at this blog post, “adventures and commotions in automated theorem proving”. there is an NYT obituary. also gowers is recently looking into and blogging on ATP in a collaboration with a TCSist and recently posted a remarkable mathematical ATP related Turing test which a significant number of human judges failed! and there is much interest expressed via populist measurement of stackexchange and mathoverflow posts in ATP, all cited on the blog, lots of further leads there.

May 30, 2013 8:20 am

Regarding the role of computers in mathematical research, here’s my brief account of the situation.

1) Henri Poincaré discovered Chaos Theory by proving the chaotic behavior of the Solar System and by foreseeing that of the weather. Unfortunately, he was unable to see the fractal “strange” attractors travelled across by the chaotic trajectories. Only with suitable computing power could Edward Lorenz, Jacques Laskar and others – incuding David Ruelle – highlight their existence and non-integer dimension. Of course, the subsequent theorems of Chaos Theory had to be proved manually.

2) Fancis Guthrie was the first to have the beautiful insight of the 4-color theorem, but he and his successors were unable to achieve a valid proof. Only with suitable computing power were Kenneth Appel and Wolfgang Haken able to do so. Later, Georges Gonthier restated their proof so as to make it look like a “normal” theorem of mathematics – though provable only with the help of a proof assistant.

3) Daniel Gorenstein and his team were able to state and prove manually the theorems about the classification of finite simple groups. Recently, Georges Gonthier and his team used a proof assistant to restate their theorems in a proof assistant – so as to increase our confidence in their validity.

Thus, until now at most a suitable cooperation between minds and machines has been required in order to make some progress. However, I’m of the opinion that, in fifty years from now, students will find it difficult to figure out how it was once possible to do mathematics without a computer.

4. May 31, 2013 9:45 pm

On Giving A Ruelle To Abduction …


text: Title: Arithmetic fundamental lemma in the minuscule case Speaker: Yihang Zhu (Harvard University) Time: 2017-6-15, 11:00-12:00 Place: 610 Abstract: The arithmetic fundamental lemma (AFL), formulated by Wei Zhang, is a key local ingredient in the relative trace formula approach to the arithmetic Gan-Gross-Prasad conjecture, which generalizes the famous Gross-Zagier formula. The AFL compares intersection numbers on Rapoport-Zink spaces with derivatives of orbital integrals. We compute the Rapoport-Zink space side in the AFL, under a minuscule assumption. In the unitary case our work recovers the proof of Rapoport-Terstiege-Zhang of the AFL; in the orthogonal case our computation is new, suggesting that a relative trace formula approach should also be possible. This is joint work with Chao Li. Attachment:


text:
# Re-statement of Fundamental theorem of calculus

The Fundamental theorem of calculus states:

$f:[a,b]\rightarrow\mathbb{R}$ is integrable and $F:[a,b]\rightarrow\mathbb{R}$ is such that $\forall x\in\left(a,b\right)$ $F'\left(x\right)=f\left(x\right)$, then $\int_{a}^{b}f=F\left(b\right)-F\left(a\right)$.

Can we rewrite it without using $f$ explicitly. For example,

$F:[a,b]\rightarrow\mathbb{R}$ is differentiable on $[a,b]$, then $\int_{a}^{b}F'=F\left(b\right)-F\left(a\right)$.

If yes, then why mathematician introduced $f$?

If no, could you provide a counterexample that would illustrates the shortcomings of the re-statement.

• The shortcoming of the re-statement is that it is not the statement we are used to. If you ask me, wriring the $f$ explicitly helps me to understand the theorem. I see no reason why your rewriting is any better than the original. – 5xum Jun 15 '15 at 8:42
• If $F'$ isn't continuous or more general a regulated function, taking the Riemann integral may not be defined. – abcdef Jun 15 '15 at 8:46
• @abcdef, could you provide an example, please? – ji borrob Jun 15 '15 at 8:47
• math.stackexchange.com/questions/293076/… – abcdef Jun 15 '15 at 8:58
• Or another example: en.wikipedia.org/wiki/Volterra's_function – abcdef Jun 15 '15 at 8:59


text:
N = M = 0 結束程式。

## Sample Input

3 3
5 5 5
1 1 5
2 1 5
3 1 5
3 5
10 20 5
2 16 20
1 6 10
3 1 5
1 1 5
2 1 15
0 0

## Sample Output

Case 1: 0

Case 2: 10