text:
A category is small if its objects form a set. But we know that there are things larger than sets. Famously, a set of all sets cannot be formed within the standard set theory (the Zermelo-Fraenkel theory, optionally augmented with the Axiom of Choice). So a category of all sets must be large. There are mathematical tricks like Grothendieck universes that can be used to define collections that go beyond sets. These tricks let us talk about large categories.

A category is locally small if morphisms between any two objects form a set. If they don’t form a set, we have to rethink a few definitions. In particular, what does it mean to compose morphisms if we can’t even pick them from a set? The solution is to bootstrap ourselves by replacing hom-sets, which are objects in Set, with objects from some other category V. The difference is that, in general, objects don’t have elements, so we are no longer allowed to talk about individual morphisms. We have to define all properties of an enriched category in terms of operations that can be performed on hom-objects as a whole. In order to do that, the category that provides hom-objects must have additional structure — it must be a monoidal category. If we call this monoidal category V, we can talk about a category C enriched over V.

Beside size reasons, we might be interested in generalizing hom-sets to something that has more structure than mere sets. For instance, a traditional category doesn’t have the notion of a distance between objects. Two objects are either connected by morphisms or not. All objects that are connected to a given object are its neighbors. Unlike in real life; in a category, a friend of a friend of a friend is as close to me as my bosom buddy. In a suitably enriched category, we can define distances between objects.

There is one more very practical reason to get some experience with enriched categories, and that’s because a very useful online source of categorical knowledge, the nLab, is written mostly in terms of enriched categories.

## Why Monoidal Category?

When constructing an enriched category we have to keep in mind that we should be able to recover the usual definitions when we replace the monoidal category with Set and hom-objects with hom-sets. The best way to accomplish this is to start with the usual definitions and keep reformulating them in a point-free manner — that is, without naming elements of sets.

Let’s start with the definition of composition. Normally, it takes a pair of morphisms, one from `C(b, c)` and one from `C(a, b)` and maps it to a morphism from `C(a, c)`. In other words it’s a mapping:

`C(b, c) × C(a, b) -> C(a, c)`

This is a function between sets — one of them being the cartesian product of two hom-sets. This formula can be easily generalized by replacing cartesian product with something more general. A categorical product would work, but we can go even further and use a completely general tensor product.

Next come the identity morphisms. Instead of picking individual elements from hom-sets, we can define them using functions from the singleton set 1:

`ja :: 1 -> C(a, a)`

Again, we could replace the singleton set with the terminal object, but we can go even further by replacing it with the unit `i` of the tensor product.

As you can see, objects taken from some monoidal category V are good candidates for hom-set replacement.

## Monoidal Category

We’ve talked about monoidal categories before, but it’s worth restating the definition. A monoidal category defines a tensor product that is a bifunctor:

`⊗ :: V × V -> V`

We want the tensor product to be associative, but it’s enough to satisfy associativity up to natural isomorphism. This isomorphism is called the associator. Its components are:

`αa b c :: (a ⊗ b) ⊗ c -> a ⊗ (b ⊗ c)`

It must be natural in all three arguments.

A monoidal category must also define a special unit object `i` that serves as the unit of the tensor product; again, up to natural isomorphism. The two isomorphisms are called, respectively, the left and the right unitor, and their components are:

```λa :: i ⊗ a -> a
ρa :: a ⊗ i -> a```

The associator and the unitors must satisfy coherence conditions:

A monoidal category is called symmetric if there is a natural isomorphism with components:

`γa b :: a ⊗ b -> b ⊗ a`

whose “square is one”:

`γb a ∘ γa b = ida⊗b`

and which is consistent with the monoidal structure.

An interesting thing about monoidal categories is that you may be able to define the internal hom (the function object) as the right adjoint to the tensor product. You may recall that the standard definition of the function object, or the exponential, was through the right adjoint to the categorical product. A category in which such an object existed for any pair of objects was called cartesian closed. Here is the adjunction that defines the internal hom in a monoidal category:

`V(a ⊗ b, c) ~ V(a, [b, c])`

Following G. M. Kelly, I’m using the notation `[b, c]` for the internal hom. The counit of this adjunction is the natural transformation whose components are called evaluation morphisms:

`εa b :: ([a, b] ⊗ a) -> b`

Notice that, if the tensor product is not symmetric, we may define another internal hom, denoted by `[[a, c]]`, using the following adjunction:

`V(a ⊗ b, c) ~ V(b, [[a, c]])`

A monoidal category in which both are defined is called biclosed. An example of a category that is not biclosed is the category of endofunctors in Set, with functor composition serving as tensor product. That’s the category we used to define monads.

## Enriched Category

A category C enriched over a monoidal category V replaces hom-sets with hom-objects. To every pair of objects `a` and `b` in C we associate an object `C(a, b)` in V. We use the same notation for hom-objects as we used for hom-sets, with the understanding that they don’t contain morphisms. On the other hand, V is a regular (non-enriched) category with hom-sets and morphisms. So we are not entirely rid of sets — we just swept them under the rug.

Since we cannot talk about individual morphisms in C, composition of morphisms is replaced by a family of morphisms in V:

`∘ :: C(b, c) ⊗ C(a, b) -> C(a, c)`

Similarly, identity morphisms are replaced by a family of morphisms in V:

`ja :: i -> C(a, a)`

where `i` is the tensor unit in V.

Associativity of composition is defined in terms of the associator in V:

Unit laws are likewise expressed in terms of unitors:

## Preorders

A preorder is defined as a thin category, one in which every hom-set is either empty or a singleton. We interpret a non-empty set `C(a, b)` as the proof that `a` is less than or equal to `b`. Such a category can be interpreted as enriched over a very simple monoidal category that contains just two objects, 0 and 1 (sometimes called False and True). Besides the mandatory identity morphisms, this category has a single morphism going from 0 to 1, let’s call it `0->1`. A simple monoidal structure can be established in it, with the tensor product modeling the simple arithmetic of 0 and 1 (i.e., the only non-zero product is `1⊗1`). The identity object in this category is 1. This is a strict monoidal category, that is, the associator and the unitors are identity morphisms.

Since in a preorder the-hom set is either empty or a singleton, we can easily replace it with a hom-object from our tiny category. The enriched preorder C has a hom-object `C(a, b)` for any pair of objects `a` and `b`. If `a` is less than or equal to `b`, this object is 1; otherwise it’s 0.

Let’s have a look at composition. The tensor product of any two objects is 0, unless both of them are 1, in which case it’s 1. If it’s 0, then we have two options for the composition morphism: it could be either `id0` or `0->1`. But if it’s 1, then the only option is `id1`. Translating this back to relations, this says that if `a <= b` and `b <= c` then `a <= c`, which is exactly the transitivity law we need.

What about the identity? It’s a morphism from 1 to `C(a, a)`. There is only one morphism going from 1, and that’s the identity `id1`, so `C(a, a)` must be 1. It means that `a <= a`, which is the reflexivity law for a preorder. So both transitivity and reflexivity are automatically enforced, if we implement a preorder as an enriched category.

## Metric Spaces

An interesting example is due to William Lawvere. He noticed that metric spaces can be defined using enriched categories. A metric space defines a distance between any two objects. This distance is a non-negative real number. It’s convenient to include inifinity as a possible value. If the distance is infinite, there is no way of getting from the starting object to the target object.

There are some obvious properties that have to be satisfied by distances. One of them is that the distance from an object to itself must be zero. The other is the triangle inequality: the direct distance is no larger than the sum of distances with intermediate stops. We don’t require the distance to be symmetric, which might seem weird at first but, as Lawvere explained, you can imagine that in one direction you’re walking uphill, while in the other you’re going downhill. In any case, symmetry may be imposed later as an additional constraint.

So how can a metric space be cast into a categorical language? We have to construct a category in which hom-objects are distances. Mind you, distances are not morphisms but hom-objects. How can a hom-object be a number? Only if we can construct a monoidal category V in which these numbers are objects. Non-negative real numbers (plus infinity) form a total order, so they can be treated as a thin category. A morphism between two such numbers `x` and `y` exists if and only if `x >= y` (note: this is the opposite direction to the one traditionally used in the definition of a preorder). The monoidal structure is given by addition, with zero serving as the unit object. In other words, the tensor product of two numbers is their sum.

A metric space is a category enriched over such monoidal category. A hom-object `C(a, b)` from object `a` to `b` is a non-negative (possibly infinite) number that we will call the distance from `a` to `b`. Let’s see what we get for identity and composition in such a category.

By our definitions, a morphism from the tensorial unit, which is the number zero, to a hom-object `C(a, a)` is the relation:

`0 >= C(a, a)`

Since `C(a, a)` is a non-negative number, this condition tells us that the distance from `a` to `a` is always zero. Check!

Now let’s talk about composition. We start with the tensor product of two abutting hom-objects, `C(b, c)⊗C(a, b)`. We have defined the tensor product as the sum of the two distances. Composition is a morphism in V from this product to `C(a, c)`. A morphism in V is defined as the greater-or-equal relation. In other words, the sum of distances from `a` to `b` and from `b` to `c` is greater than or equal to the distance from `a` to `c`. But that’s just the standard triangle inequality. Check!

By re-casting the metric space in terms of an enriched category, we get the triangle inequality and the zero self-distance “for free.”

## Enriched Functors

The definition of a functor involves the mapping of morphisms. In the enriched setting, we don’t have the notion of individual morphisms, so we have to deal with hom-objects in bulk. Hom-objects are objects in a monoidal category V, and we have morphisms between them at our disposal. It therefore makes sense to define enriched functors between categories when they are enriched over the same monoidal category V. We can then use morphisms in V to map the hom-objects between two enriched categories.

An enriched functor `F` between two categories C and D, besides mapping objects to objects, also assigns, to every pair of objects in C, a morphism in V:

`Fa b :: C(a, b) -> D(F a, F b)`

A functor is a structure-preserving mapping. For regular functors it meant preserving composition and identity. In the enriched setting, the preservation of composition means that the following diagram commute:

The preservation of identity is replaced by the preservation of the morphisms in V that “select” the identity:

## Self Enrichment

A closed symmetric monoidal category may be self-enriched by replacing hom-sets with internal homs (see the definition above). To make this work, we have to define the composition law for internal homs. In other words, we have to implement a morphism with the following signature:

`[b, c] ⊗ [a, b] -> [a, c]`

This is not much different from any other programming task, except that, in category theory, we usually use point free implementations. We start by specifying the set whose element it’s supposed to be. In this case, it’s a member of the hom-set:

`V([b, c] ⊗ [a, b], [a, c])`

This hom-set is isomorphic to:

`V(([b, c] ⊗ [a, b]) ⊗ a, c)`

I just used the adjunction that defined the internal hom `[a, c]`. If we can build a morphism in this new set, the adjunction will point us at the morphism in the original set, which we can then use as composition. We construct this morphism by composing several morphisms that are at our disposal. To begin with, we can use the associator `α[b, c] [a, b] a` to reassociate the expression on the left:

`([b, c] ⊗ [a, b]) ⊗ a -> [b, c] ⊗ ([a, b] ⊗ a)`

We can follow it with the co-unit of the adjunction `εa b`:

`[b, c] ⊗ ([a, b] ⊗ a) -> [b, c] ⊗ b`

And use the counit `εb c` again to get to `c`. We have thus constructed a morphism:

`εb c . (id[b, c] ⊗ εa b) . α[b, c] [a, b] a`

that is an element of the hom-set:

`V(([b, c] ⊗ [a, b]) ⊗ a, c)`

The adjunction will give us the composition law we were looking for.

Similarly, the identity:

`ja :: i -> [a, a]`

is a member of the following hom-set:

`V(i, [a, a])`

which is isomorphic, through adjunction, to:

` V(i ⊗ a, a)`

We know that this hom-set contains the left identity `λa`. We can define `ja` as its image under the adjunction.

A practical example of self-enrichment is the category Set that serves as the prototype for types in programming languages. We’ve seen before that it’s a closed monoidal category with respect to cartesian product. In Set, the hom-set between any two sets is itself a set, so it’s an object in Set. We know that it’s isomorphic to the exponential set, so the external and the internal homs are equivalent. Now we also know that, through self-enrichment, we can use the exponential set as the hom-object and express composition in terms of cartesian products of exponential objects.

## Relation to 2-Categories

I talked about 2-categories in the context of Cat, the category of (small) categories. The morphisms between categories are functors, but there is an additional structure: natural transformations between functors. In a 2-category, the objects are often called zero-cells; morphisms, 1-cells; and morphisms between morphisms, 2-cells. In Cat the 0-cells are categories, 1-cells are functors, and 2-cells are natural transformations.

But notice that functors between two categories form a category too; so, in Cat, we really have a hom-category rather than a hom-set. It turns out that, just like Set can be treated as a category enriched over Set, Cat can be treated as a category enriched over Cat. Even more generally, just like every category can be treated as enriched over Set, every 2-category can be considered enriched over Cat.

Next: Topoi.


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# The Legolas testing framework

This page gives a detailed overview on the various Legolas testing suites, how to run them and what they are testing.

As the Legolas codebase becomes larger when new features are implemented, it is important to test everything thoroughly to make sure that new or modified code does not break backwards compatibility or introduces bugs in pieces of code that previously worked.

Legolas has two levels of testing suites: unit tests and regression tests, all of which run on both the master and develop branches for every commit or pull request. The Pylbo framework has its own dedicated testing suite. All of these test suites are decouples from each other and test distinctly different things: the unit tests are low-level and test individual subroutines and functions in the Legolas source code using pFUnit. The regression tests on the other hand are high-level and compare code output from a recent commit to previously known results, while the Pylbo tests check data loading and management, interfacing with Legolas and visualisations.

The entire testing framework is supplemented with code coverage indications.

## Our testing policy

When developing new features or modifying existing code we have a few general guidelines that we follow:

1. All functions/subroutines in the Legolas or Pylbo source codes should be accompanied by one or more unit tests.
2. Adding a new equilibrium means adding at least one regression test.
3. Code coverage should not decrease (although in some cases a minor decrease is unavoidable).

## Style checks

All Python files in the legolas repository (so Pylbo, setup files and test files) adhere to Black formatting standards, which automatically take care of trailing whitespaces, improper indents etc. We also run flake8 as style enforcement.

We explicitly check for both Black and flake8 formatting and style during the automated tests and strictly enforce this. Many IDE’s support Black/flake8 plugins which can automatically format the file on save, so it may be useful to set this up if you are making edits to the source code.

## The Legolas unit tests

### Running the tests

To run the unit tests you will need to have pFUnit installed, which has some prerequisites. Make sure Legolas is compiled, then navigate to the tests folder and then down one level to the unit tests folder. In there, compile the tests and run:

mkdir build
cd build
cmake ..
make
cd ..
./test_legolas

This will execute the entire unit test suite.

### Creating a new unit test

All similar tests are grouped together in dedicated .pf files. Either add a new test to an already existing file, or create a new file. The module mod_suite_utils.f90 has a few convenient routines which can be used to set up or tear down new tests. As a general rule of thumb all arrays should be deallocated and all global variables reset when a test finishes (on either failure or success), to ensure that every test can run independently of the others. If you added a new .pf file you will have to add it to the list in the CMakeLists.txt file in the unit tests folder.

Note that unit tests should test only a single (part of a) subroutine/function and should finish quickly. If you want to test multiple things, just split them in multiple tests as much as possible.

## The Legolas regression tests

The regression tests run the various pre-implemented equilibria and compare the results with previously stored datfiles. Contrary to the Legolas unit tests this suite is Python-based and uses pytest together with the Pylbo-Legolas interface to set up and run everything. These tests mainly compare eigenvalues and eigenfunctions between new and previously stored results using an image-based comparison and the root-mean-squared difference.

### How the regression tests work

During the regression tests we check both the eigenvalue spectrum and (only in some cases) the eigenfunctions.

#### Spectrum tests

These are done for all pre-implemented equilibria, and every spectrum test typically consists of four steps:

1. Generate the datfile for the corresponding equilibrium
2. For both the “new” datfile and the stored datfile:
• for every region in the complex plane generate a spectrum (usually 3-4 images)
• save the images to a temporary folder
3. Do a root-mean-squared (RMS) image comparison to check spectral similarity, fail the test if the RMS is above the tolerance (which is 2 by default).
4. Save the difference between the two images as a new image on failure, otherwise delete the images.

An example of a failing test is shown here, where the background flow parameter for the Suydam cluster modes was slightly changed: The left image is the expected baseline result, the center image the test result and the right image the difference between both. In all these images there is no background alpha present, so all deviations in the RMS is purely due to shifting eigenvalues.

We also test multispectra, which is a composed spectrum for the same equilibrium but run with varying parameters. The testing principle remains the same: once the composed spectrum is created the baseline and test result are again compared on an RMS-based criterion.

#### Eigenfunction tests

These tests use a similar background as the spectrum tests and are also RMS-image-based. We do not check the eigenfunctions for all pre-implemented equilibria, only those that have clear, isolated eigenvalues and associated eigenfunctions that are properly resolved at low resolution. When the test datfile is generated some pre-set eigenvalues are selected and the eigenfunctions are queried. Then, for each eigenvalue we plot the sum of the real and imaginary parts of each eigenfunction as a function of the grid and save the images on a 3x3 grid (since we have 8 panels per image, per eigenvalue). The same principle as above is then applied, where the RMS value of the curves is checked against each other.

### Running the tests

To run the regression tests make sure you have pytest installed, along with Pylbo and all its prerequisites. Then, navigate to the tests folder and move down one level to the regression test directory, where you execute pytest.

cd tests/regression_tests
pytest

This will automatically create a directory tree called test_results inside the regression_tests folder. Every subdirectory corresponds to a single regression test, with in turn multiple subdirectories for generated datfiles, spectrum images and eigenfunction images. The above command will run the entire test suite. If you want to only run a specific test, provide the filename instead:

pytest test_uni_adiabatic.py

You can add as many filenames as you want, separated by a whitespace.

Note: tests that are passing will clean up after themselves by deleting the generated datfile and images. You can ask the test suite to keep the images instead (for visual inspections, for example) by simply adding the --keep-files flag when running the tests.

To add a new regression test create a new file in the regression test folder, and make sure to prepend the name with test_ so it can be picked up by pytest. Below is an example for a regression test of the taylor_couette equilibrium, with eigenfunctions. For additional inspiration there are plenty of tests set up in the regression_tests folder.

from .regression import RegressionTest
import numpy as np
import pytest

class TestExample(RegressionTest):
name = "a descriptive name"
filename = "a_descriptive_filename"
equilibrium = "taylor_couette"
geometry = "cylindrical"
x_start = 1
x_end = 2

parameters = {
"k2": 0,
"k3": 1,
"cte_rho0": 1.0,
"alpha": 1.0,
"beta": 2.0,
}
physics_settings = {
"flow": True,
"coaxial": True,
"viscosity": True,
"viscosity_value": 1e-3,
}
eigenfunction_settings = {
"write_eigenfunctions": True,
"write_eigenfunction_subset": True,
"eigenfunction_subset_center": 0.5 - 0.3j,
}
eigenvalues_are_real = False

spectrum_limits = [
{"xlim": (-18, 18), "ylim": (-3, 0.3)},
{"xlim": (-1.5, 1.5), "ylim": (-1.25, 0.2), "RMS_TOLERANCE": 2.9},
]
eigenfunctions = [
{"eigenvalue": 0.87784 - 0.04820j},
{"eigenvalue": 0.49861 - 0.08124j},
]

@pytest.mark.parametrize("limits", spectrum_limits)
def test_spectrum(self, limits, ds_test, ds_base):
super().run_spectrum_test(limits, ds_test, ds_base)

@pytest.mark.parametrize("eigenfunction", eigenfunctions)
def test_eigenfunction(self, eigenfunction, ds_test, ds_base):
super().run_eigenfunction_test(eigenfunction, ds_test, ds_base)

A few things of note:

• Every regression test should subclass RegressionTest, and have their name start with Test such that it can be discovered automatically by pytest.
• The name, filename, parameters, geometry and equilibrium properties are required to be set.
• The properties x_start and x_end are optional, and default to 0 and 1 respectively if not set.
• The parameters, physics_settings and eigenfunction_settings dictionaries follow the exact same conventions as used in parfile generation. An additional property called solver_settings (also a dict) can be used to specify the solver to be used, if not given this defaults to the QR algorithm.
• The eigenvalues_are_real property can be set to True, which will run an additional test to check that the imaginary part of every eigenvalue is numerically zero. If False (default), complex values are allowed.

#### Generating spectrum image tests

As in the above example there is a list spectrum_limits with dictionaries as items. Every dictionary has the following keys:

• "xlim": this is a tuple representing the range in Re$(\omega)$ of the spectrum
• "ylim": this is a tuple representing the range in Im$(\omega)$ of the spectrum
• "RMS_TOLERANCE": [optional], this is a float representing a custom RMS tolerance to be used for this specific test. If not given the default value is used.

You can add as many items to this list as you want. In order to tell the testing suite to run the tests, provide the method test_spectrum as shown above:

@pytest.mark.parametrize("limits", spectrum_limits)
def test_spectrum(self, limits, ds_test, ds_base):
super().run_spectrum_test(limits, ds_test, ds_base)

The method arguments should be the same as shown above, and within the method you call the run_spectrum_test method from the super class. This will automatically setup everything and run the tests. The decorator @pytest.mark.parametrize("limits", spectrum_limits) is important, this tells pytest to parametrise the spectrum_limits list given earlier, such that a test will run for every item in this list.

#### Generating eigenfunction image tests

For eigenfunction tests eigenfunctions should be set up using the eigenfunction_settings property. Then, add the desired eigenvalues to an eigenfunctions list, which is similar to spectrum_limits. Every item is a dictionary with the following keys:

• "eigenvalue": the (approximate) eigenvalue of which to test the eigenfunctions. This does not have to be exact, but should be as close as possible to the desired eigenvalue.
• "RMS_TOLERANCE": [optional], this is a float representing a custom RMS tolerance to be used for this specific test. If not given the default value is used.

Again, you can add as many eigenvalues to this list as you want, and similar as before you should tell pytest to parametrise this list whilst calling the superclass method:

@pytest.mark.parametrize("eigenfunction", eigenfunctions)
def test_eigenfunction(self, eigenfunction, ds_test, ds_base):
super().run_eigenfunction_test(eigenfunction, ds_test, ds_base)

#### Generating the baseline file

Once everything is set up properly and you run your newly created regression test it will probably fail due to the baseline file not being present. Once you are sure everything is correct you can (re)generate it by simply calling pytest with an additional argument:

pytest test_my_example.py --generate

This will automatically create a file BASE_{filename}.dat in the appropriate folder. If this file already exists you will be prompted to either overwrite or keep the existing file.

## The Pylbo unit tests

To run the Pylbo unit tests you also need pytest, along with the timeout plugin which you can install as

pip install pytest-timeout

Then, ro tun the tests navigate to the tests folder, down one level to the Pylbo unit tests and run

pytest -v

format: plain


text:
# Tag Info

### Mathematica is messing around with absolute values and sqrts of complex numbers?

Perhaps this? Eigenvalues[ ComplexExpand[{{0, z}, {Conjugate[z], 0}}, z, TargetFunctions -> {Abs, Arg}]] (* {-Abs[z], Abs[z]} *)
• 213k

### Change style of the above cell

You can do it this way : Cell to be transformed : I want to change this cell (*the cell that I want to change cell-style*) Cell to transform the previous cell : <...
• 16.9k
Accepted

### Change style of the above cell

First we move the selection to the previous cell: SelectionMove[EvaluationCell[], Previous, Cell]; We now read the selected cell, change its style e.g. to "...
• 24.3k
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### How to type special characters used in $\LaTeX$?

Just take a look at the fonts that you have installed, as "Euklid Fraktur" is not available for everyone. ...
• 12.3k

### How to type special characters used in $\LaTeX$?

Not sure if this is an answer or what particular use you have in mind, but a font that resembles such $\LaTeX$ typography closely is the "Euclid Fraktur". Not sure if there are better ...
• 14.9k
Accepted

...
• 7,036
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### How to copy Mathematica code into Microsoft Word?

Use MathML format: Newer version of Office 365 support direct $\LaTeX$ input as described here.
• 14.9k

### Formatting the output of Abs[x] to look like $|x|$

First off, I would strongly recommend you don't redefine Abs and Conjugate, but obviously that's your perogative. Second, it's difficult to understand what you want. You say that you want the output ...
• 3,777
1 vote

### A bug in copy&paste of plain text

As pointed out by @ihojnicki in the comments, this behavior is most likely caused by using an external clipboard history tool. For some reason Mathematica is unable to work with these tools, and you ...
• 62.7k
1 vote

### Text cell numbering

The 'labels' for input and output are generated upon evaluation by the kernel and are not notebook specific. I do not think that this is what OP is after with text cells (I might be wrong). I would ...
• 2,240
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### How to type special characters used in $\LaTeX$?

Probably the closest are \[GothicCapitalR] and \[GothicCapitalI] which can be also typed as EscgoREsc and EscgoIEsc. The ...
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Only top scored, non community-wiki answers of a minimum length are eligible


text:
# HELP! Parallel plate Capacitors

1. Jan 30, 2005

### NotaPhysicsMan

Here's the question.

The drawing (attached) shows an electron entering the lower left side of a parallel plate capacitor and exiting at the upper right side. The inital speed of the electron is 7.00x 10^6m/s. The capacitor is 2.00cm long, and its plates are separated by 0.150cm. Assume that the electric field between the plates is uniform everywhere and find its magnitude.

Here's what I got.

Seems simple enough. I use E=q/(A Eo), where (permittivity of space) Eo=8.85x10^-12C^2/N.

Ok so I plug in the charge of an electron divided by the area of the rectangle and Eo. So 1.60x10^-19/((2.00cm/100 x 0.150cm/100)(8.85x10^-12)). I get something like 6.03x10^-8 N/C.

See, I didn't use the velocity they gave me, that's where I think the problem is, but I don't see where I could use it! Maybe to find acceleration using kinematics, nope not that...

Any ideas?

Thanks

#### Attached Files:

• ###### parplcap.JPG
File size:
5 KB
Views:
155
2. Jan 30, 2005

### Staff: Mentor

This formula allows you to calculate the field between the plates given the charge on the capacitor and the area of the plates. But that won't help you here, since you have neither the charge nor the area! (1) The charge on the plates does not equal the charge of the electron! (2) You used the area of a sideways cross-section, not the area of the capacitor plates.

That's exactly what you want to do: find the acceleration, then the force, then the field.

3. Jan 30, 2005

### NotaPhysicsMan

That's odd, then what are the length's and width of the cross-section used for? In terms of acceleration, I don't have time, or distance, but the initial velocity, so I can't use kinematics.
So wait, let's see. Only the horizontal distance is important. And so the d travelled in that velocity is 2.00cm.

so t= 0.02m/7.00x10^6m/s = 2.857x 10^-9s(wow that's small)

Ok, so I have time, I will plug into x=Vt+1/2at^2.

solve for a= 2(x-vot)/t^2.
a= 2^-6/(2.857x10^-9s)
a=700m/s^2.

OK sounds good so far.

Now I use F=ma to get force.

9.11x10^-31(700m/s^2)=6.377x10^-28 N

and Now I use F=qE, solve for E=F/q.

6.377x10^-28N/(1.60x10^-19)

E=1.02x10^-46 N/C... Huh, this doesn't make sense.

4. Jan 30, 2005

### NotaPhysicsMan

What the, now I get E=F/q and E=3.985x10^-9 N/C...AHHHH

5. Jan 30, 2005

### NotaPhysicsMan

6. Jan 30, 2005

### NotaPhysicsMan

No one eh..

7. Jan 30, 2005

### Staff: Mentor

Right!

Redo this calculation: $a = 2 d / t^2$. (d = 0.0015m)

8. Jan 30, 2005

### NotaPhysicsMan

Ok,

a=2(0.0015m)/(2.857x10^-9)^2
a=3.675x10^14

F=ma
=9.11x10^-31kg x 3.675x10^14
=3.345x10^-16 N.

E=F/q
=3.345x10^-16N/1.60x10^-19C
E=2093 N/C or 2.09x10^3 N/C.

NOW that looks like a more sensible answer!

Thanks.


text:
# Tree Version of Hechler Forcing

In their recent (2009) paper Eventually Different Functions and Inaccessible Cardinals, Brendle and Löwe consider a 'tree version' of the Hechler forcing. This forcing $\mathbb{D}$ consists of nonempty trees $T\subseteq\omega^{<\omega}$ with the property that there is a unique stem $s\in T$ so that for every $t\in T$ extending $s$, $t\frown n\in T$ for all but finitely many $n\in\omega$. The forcing is ordered by inclusion. This forcing allows for a very elegant rank analysis, and many properties about Hechler forcing that are proved to hold using rank arguments can also be proved to hold for $\mathbb{D}$ in a conceptually simpler way. I do not know if the two notions of forcing are equivalent, and based on what is said in the paper I suspect neither do the authors. This is not my real question, though I would be happy to see an answer.

My question is simply whether this specific tree forcing has appeared anywhere in the literature previously. No reference is given in the paper but I ask because I do think I remember seeing it somewhere and now I can't seem to find mention of it in likely places.

-


text:
## Random Matrices in Three Short Stories – III

#### This is the third and final part of the series. For those who haven’t already, going through the first two posts, which can be found here and here, will be a good idea.

Three

Why do we want a quantum theory of gravity? We just want it, okay?

Hold on, this isn’t me. This was the American theoretical physicist John Preskill at a conference earlier this year [1]. And though, in almost all probability, he was trying to be funny, this does give an idea about how difficult (and often, frustrating) it is for theoretical physicists to answer why-do-we-want-this or what-use-doing-that questions. (By the way, apart from his seminal contributions to the fields of quantum information and quantum gravity, Preskill is also famous for winning a black hole bet against Stephen Hawking which Hawking conceded by offering him a baseball encyclopedia.)

But something about the notion of universality in random matrix theory before I go on to talk about the theory of quantum gravity, cosmological inflation, black holes, wormholes and time travel. (Okay, I was kidding about the last two – no wormholes and time travel here.)

Large matrices with random entries have some very intricate (and interesting) statistical properties. And the reason they come in so handy in our attempts to answer questions of different sorts is the applicability of the statistical laws of random matrix ensembles to all those systems which have the same symmetries as those of the ensemble.

These statistical laws often involve eigenvalues of the matrices. Eigenvalues are certain quantities associated with matrices. In fact, all square matrices (arrays of numbers with the same number rows and columns) can be characterised by their eigenvalues. And in most cases, computing the eigenvalues is not a very difficult task.

##### For a matrix H, all the possible values of λ satisfying the above equation are its eigenvalues. I is a square matrix the size of H with zeros along the diagonal going from left to right. And det() represents computing the determinant which again is a number associated with a matrix.

For a set of random numbers, it is very natural and useful to talk about the probability distribution – how probable the occurrence of each of the numbers is. Now, if your matrices have random elements, their eigenvalues will be random as well, which makes it useful to talk about the eigenvalue distributions of the ensembles.

These distributions in random matrix theory are universal in the sense that they don’t depend on the underlying structures as long as they have a common overall symmetry. In the context of a physical system, these eigenvalues correspond to the energy levels of the system and, in essence, contain the information about its dynamical properties. Superficially, this is what a lot of random matrix analyses is about. The energy spectra of systems which are difficult to interpret otherwise find meaning in the language of random matrix ensembles.

But understanding the underlying dynamics of many systems is not easy. And as we have come to realise in the case of black holes – it is certainly not.

Black holes are interesting; though calling them interesting might be an understatement. They are formed in a number of different ways all over the universe. Within the framework of the theory of general relativity, one can, in quite a straightforward manner, show that a black hole has a gravitational pull so huge that nothing can escape it, not even light.

But everything is not that straightforward. Though general relativity explains the force of gravitation, we believe our universe to be inherently quantum mechanical, that is, we expect every object in the universe to follow the laws of quantum theory. And taking quantum mechanical laws into consideration, one can show (as Hawking did for the first time in the early 1970s) that black holes radiate stuff [2]. Now, this may seem puzzling; in fact, it is puzzling. But what this radiation also indicates is that black holes are thermal objects – they have thermodynamic properties (say, temperature, for example) in a manner similar to how your daily cup of coffee does.

It is evident that black holes demand a better understanding than the one at present. And this can possibly be achieved by constructing a unified framework incorporating both gravity and quantum mechanical principles. This is somewhere random matrix analysis turns out to be useful – understanding energy spectra of black holes.

Depending on the theoretical framework in which the calculations are being done, black holes can be studied using different models. In principle, the details of the energy spectra can be worked out for each of these models. But as it turns out, not all black hole models are soluble. The trick is to then use random matrix models which can possibly mimic the expected properties.

Black holes have posed some of the most interesting challenges to those working in physics for the last hundred years. They have also motivated a quest for quantum gravity. However, a theory of quantum gravity will possibly also explain many other mysteries. One of them is cosmological inflation. Based on a large amount of astronomical data, it has been conjectured that our universe underwent a phase of extremely rapid expansion for a small fraction of second just after the big bang. This conjecture also explains the origin of the large scale structures in the universe pretty well. However, what is missing is a concrete theoretical underpinning of inflation itself. Among various proposals to explain inflation, there are a few which employ random matrix techniques as well [3]. But again, a complete understanding of inflation, like that of black holes, still belongs to the large set of open problems waiting to be solved!

#### References

[1] John Preskill, Quantum Information and Spacetime (I)https://youtu.be/td1fz5NLjQs, Tutorial at the 20th Annual Conference on Quantum Information Processing, 2017.
[2] Leonard Susskind, Black Holes and the Information Paradox, Scientific American, April 1997.
[3] M.C. David Marsh, Liam McAllister, Enrico Pajer and Timm Wrase, Charting an Inflationary Landscape with Random Matrix Theory, JCAP 11, 2013,  [arXiv:hep-th/1307.3559].


text:
# The sales for Best Buy were approximately $35.9 billion in 2006 and$40.0 billion in 2007. Using...

## Question:

The sales for Best Buy were approximately $35.9 billion in 2006 and$40.0 billion in 2007.

Using only this information, write a linear equation that gives the sales (in billions of dollars) in terms of the year. Then predict the sales for 2010.

## Linear Equation:

We have to predict the sales for 2010. It is an example of linear equation. With the help of the given information we can get two points. Use these points to derived the linear function to get the desired result.

Let the sales represent on the y-axis and year represent on x-axis. Let 2006 represent x=0. The sales for Best Buy were approximately $35.9 billion in 2006 and$40.0 billion in 2007 which gives us two points {eq}\left( {{x_1},{y_1}} \right) = \left( {0,35.9} \right),\left( {{x_2},{y_2}} \right) = \left( {1,40} \right). {/eq}

Now, find the slope of the sale function with the help of the derived points and we have \begin{align*} m &= \frac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\\ &= \frac{{40 - 35.9}}{{1 - 0}}\\ m &= 4.1. \end{align*}

Use point slope equation to get the linear equation of sale so, we have \begin{align*} y - {y_2} &= m\left( {x - {x_2}} \right)\\ y - 40 &= 4.1\left( {x - 1} \right)\\ y &= 4.1x - 4.1 + 40\\ y &= 4.1x + 35.9. \end{align*}

For 2010 plug x=4 into the derived linear equation and we have \begin{align*} y &= 4.1x + 35.9\\ y &= 4.1\left( 4 \right) + 35.9\\ y &= 52.3 \text{ billion}. \end{align*}


text:
## Cauchy Riemann Equations (basic doubt)

Lets say we have a function of a complex variable z , f(z).

I read that for the function to be differentiable at a point z0 , the CR equations are a necessary condition but not a sufficient condition.

Can someone give me an example where the CR equations hold but the function is not differentiable at that point , thus justifying that the CR equations holding true aren't sufficient test.

I am unable to visualise.

 PhysOrg.com science news on PhysOrg.com >> Heat-related deaths in Manhattan projected to rise>> Dire outlook despite global warming 'pause': study>> Sea level influenced tropical climate during the last ice age
 Recognitions: Gold Member Science Advisor Staff Emeritus Let g(x, y)= 1 if xy is not 0, 0 if xy= 0 and let f(z)= g(x,y)(1+ i)= g(x, y)+ ig(x,y) where z= x+ iy. Then $$\frac{g(x,y)}{\partial x}= \frac{\partial g(x,y)}{\partial y}= 0$$ for (x,y)= (0, 0) so the Riemann-Cauchy equations are satisfied there but the function is not even continuous at (0, 0).
 Thanks.Doubt resolved.

 Tags cauchy riemann


text:
# nLab stable homotopy theory

## Theorems

#### Stable Homotopy theory

stable homotopy theory

# Contents

## Idea

One may think of classical homotopy theory as the study of the (∞,1)-category Top of topological spaces, or rather of its homotopy category $\mathrm{Ho}\left(\mathrm{Top}\right)$.

To every (∞,1)-category is associated its corresponding stable (∞,1)-category of spectrum objects. For Top this is the stable (∞,1)-category of spectra, $\mathrm{Sp}\left(\mathrm{Top}\right)$. Stable homotopy theory is the study of $\mathrm{Sp}\left(\mathrm{Top}\right)$, or rather of its homotopy category, the stable homotopy category $\mathrm{StHo}\left(\mathrm{Top}\right):=\mathrm{Ho}\left(\mathrm{Sp}\left(\mathrm{Top}\right)\right)$.

A tool of major importance in stable homotopy theory and its application to higher algebra is the symmetric monoidal smash product of spectra which allows us to describe A-∞ rings and E-∞ rings as ordinary monoid objects in a model category that presents $\mathrm{Sp}\left(\mathrm{Top}\right)$.

## Variations

When the spaces and spectra in question carry an action of a group $G$ the theory refines to

## References

An excellent general survey is

• A. Elmendorf, I. Kriz, P. May, Modern foundations for stable homotopy theory (pdf)

A survey of formalisms used in stable homotopy theory – tools to present the triangulated homotopy category of a stable (infinity,1)-category – is in

An account in terms of (∞,1)-category theory is in section 7 of

Revised on April 24, 2013 19:39:54 by Urs Schreiber (131.174.42.61)


text:
# Fit_Normal_2P¶

class reliability.Fitters.Fit_Normal_2P(failures=None, right_censored=None, show_probability_plot=True, print_results=True, CI=0.95, percentiles=None, optimizer=None, CI_type='time', method='MLE', force_sigma=None, **kwargs)

Fits a 2-parameter Normal distribution (mu,sigma) to the data provided. Note that it will return a fit that may be partially in the negative domain (x<0). If you need an entirely positive distribution that is similar to Normal then consider using Weibull.

Inputs: failures - an array or list of failure data right_censored - an array or list of right censored data show_probability_plot - True/False. Defaults to True. print_results - True/False. Defaults to True. Prints a dataframe of the point estimate, standard error, Lower CI and Upper CI for each parameter. CI - confidence interval for estimating confidence limits on parameters. Must be between 0 and 1. Default is 0.95 for 95% CI. CI_type - time, reliability, None. Default is time. This is the confidence bounds on time or on reliability. Use None to turn off the confidence intervals. percentiles - percentiles to produce a table of percentiles failed with lower, point, and upper estimates. Default is None which results in no output. True or ‘auto’ will use default array [1, 5, 10,…, 95, 99]. If an array or list is specified then it will be used instead of the default array. method - ‘MLE’ (maximum likelihood estimation), ‘LS’ (least squares estimation), ‘RRX’ (Rank regression on X), ‘RRY’ (Rank regression on Y). LS will perform both RRX and RRY and return the better one. Default is ‘MLE’. optimizer - ‘L-BFGS-B’, ‘TNC’, or ‘powell’. These are all bound constrained methods. If the bounded method fails, nelder-mead will be used. If nelder-mead fails then the initial guess will be returned with a warning. For more information on optimizers see https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.minimize.html#scipy.optimize.minimize force_sigma - Use this to specify the sigma value if you need to force sigma to be a certain value. Used in ALT probability plotting. Optional input. kwargs are accepted for the probability plot (eg. linestyle, label, color)

Outputs: mu - the fitted Normal_2P mu parameter sigma - the fitted Normal_2P sigma parameter loglik - Log Likelihood (as used in Minitab and Reliasoft) loglik2 - LogLikelihood*-2 (as used in JMP Pro) AICc - Akaike Information Criterion BIC - Bayesian Information Criterion AD - the Anderson Darling (corrected) statistic (as reported by Minitab) distribution - a Normal_Distribution object with the parameters of the fitted distribution mu_SE - the standard error (sqrt(variance)) of the parameter sigma_SE - the standard error (sqrt(variance)) of the parameter Cov_mu_sigma - the covariance between the parameters mu_upper - the upper CI estimate of the parameter mu_lower - the lower CI estimate of the parameter sigma_upper - the upper CI estimate of the parameter sigma_lower - the lower CI estimate of the parameter results - a dataframe of the results (point estimate, standard error, Lower CI and Upper CI for each parameter) goodness_of_fit - a dataframe of the goodness of fit values (Log-likelihood, AICc, BIC, AD). percentiles - a dataframe of the percentiles with bounds on time. This is only produced if percentiles is ‘auto’ or a list or array. Since percentiles defaults to None, this output is not normally produced. probability_plot - the axes handle for the probability plot (only returned if show_probability_plot = True)

static LL(params, T_f, T_rc)
static LL_fs(params, T_f, T_rc, force_sigma)
static logR(t, mu, sigma)
static logf(t, mu, sigma)


text:
##### Venue: Department of Mathematics, Lecture Hall 3

In this thesis we study some Questions on vector bundles over hypersurfaces. More precisely, for hypersurfaces of dimension $\\geq 2$, we study the extension problem of Vector bundles. We try to find some conditions under which a vector bundle over an ample divisor of non-singular projective variety, extends as a vector bundle to an open set containing that ample divisor.

Contact: +91 (80) 2293 2711, +91 (80) 2293 2265 ;     E-mail: chair.math[at]iisc[dot]ac[dot]in
Last updated: 06 Mar 2020


text:
# An ideal diatomic gas, with rotation but no oscillation, undergoes an adiabatic compression. Its initial pressure...

###### Question:

An ideal diatomic gas, with rotation but no oscillation, undergoes an adiabatic compression. Its initial pressure and volume are 1.8 atm and 0.40 m3. It's final pressure is 2.7 atm. How much work is done by the gas? NumberTT-2.50 Units the tolerance is +/-2% Open Show Work Click if you would like to Show Work for this question:

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text:
Center for Theoretical Physics of the Universe (CTPU)

# Interpretations of the DAMPE electron data

## by Lei Feng

Wednesday, 3 March 2021 from to (Asia/Seoul)
at zoom meeting
 Description The DArk Matter Particle Explorer (DAMPE), a high energy cosmic ray and γ-ray detector in space, had reported the new measurement of the total electron plus positron flux between 25 GeV and 4.6 TeV. A spectral softening at ∼0.9 TeV and a tentative peak at ∼1.4 TeV have been reported. We discuss the physical implications of the DAMPE data in this lecture. The tentive peak at ∼1.4 TeV implies local sources of electrons/positrons with quasi-monochromatic injection spectrum and caused extensive discussions. The annihilation of DM particles (mχ∼1.5 TeV) into e+e− pairs in a nearby clump or an over-density region could explain the data. Otherwise, cold, ultra-relativistic e+e− winds from pulsars may also give rise to such a structure Material:


text:
# Prove countable set function: natural numbers and pairs of natural numbers

Can someone please explain me the proof, where there is a
1-1 Correspondence between the set of natural numbers and the set of all pairs of natural numbers How can the below data be one-to-one function that is total over the natural numbers and onto the pairs of natural numbers?

The following printed list defines a relation between the natural numbers and the natural number pairs.

0:(0,0)
1:(0,1)
2:(1,0)
3:(0,2)
4:(1,1)
5:(2,0)
6:(0,3)
7:(1,2)
8:(2,1)
9:(3,0)
10:(0,4)
11:(1,3)
12:(2,2)
13:(3,1)
14:(4,0)
15:(0,5)
16:(1,4)
17:(2,3)
18:(3,2)
19:(4,1)
20:(5,0)
21:(0,6)
22:(1,5)
23:(2,4)
24:(3,3)
25:(4,2)
26:(5,1)
27:(6,0)
28:(0,7)
29:(1,6)
30:(2,5)
... and so on ...

• Welcome to Math.SE! Can you explain a little more explicitly why you don't understand that this defines a one-to-one function? – Hrodelbert Oct 2 '15 at 9:18
• Thank you. How two different types (one is a natural number and the other is a pair of natural numbers) are related and how do they have one-to-one function. How are they connected? – user3337714 Oct 2 '15 at 9:20
• math.stackexchange.com/questions/444447/… – Stravog Oct 2 '15 at 9:52
• Can you see the pattern? Can you see that "... and so on ..." continues to assign each number a pair? Can you figure out a way to from a given pair deduce which number it corresponds to? – skyking Oct 2 '15 at 9:52
• See the info about the Cantor pairing function, you may have to switch the coordinates (I didn't check). en.wikipedia.org/wiki/Pairing_function – coffeemath Oct 2 '15 at 9:56

Using $(m,n)\mapsto\dfrac{(m+n)(m+n+1)}2+m$, we can go the other way round by considering $k=T(n)+m$, where $T(n)=\dfrac{n(n+1)}{2}$, $n$ is maximal and $m\ge0$ to give $(m,n-m)$ as the pairing.


text:
--- a
+++ b/thys/SATSolverVerification/document/root.tex
@@ -0,0 +1,82 @@
+\documentclass[11pt,a4paper]{article}
+\usepackage{isabelle,isabellesym}
+
+% isabellesym.sty), use only when needed
+
+%\usepackage{amssymb}
+  %for \<leadsto>, \<box>, \<diamond>, \<sqsupset>, \<mho>, \<Join>,
+  %\<lhd>, \<lesssim>, \<greatersim>, \<lessapprox>, \<greaterapprox>,
+  %\<triangleq>, \<yen>, \<lozenge>
+
+%\usepackage[greek,english]{babel}
+  %option greek for \<euro>
+  %option english (default language) for \<guillemotleft>, \<guillemotright>
+
+%\usepackage[latin1]{inputenc}
+  %for \<onesuperior>, \<onequarter>, \<twosuperior>, \<onehalf>,
+  %\<threesuperior>, \<threequarters>, \<degree>
+
+%\usepackage[only,bigsqcap]{stmaryrd}
+  %for \<Sqinter>
+
+%\usepackage{eufrak}
+  %for \<AA> ... \<ZZ>, \<aa> ... \<zz> (also included in amssymb)
+
+%\usepackage{textcomp}
+  %for \<cent>, \<currency>
+
+% this should be the last package used
+\usepackage{pdfsetup}
+
+% urls in roman style, theory text in math-similar italics
+\urlstyle{rm}
+\isabellestyle{it}
+
+%make a bit more space
+
+\begin{document}
+
+\title{SAT Solver verification}
+\author{By Filip Mari\' c}
+\maketitle
+\abstract{
+This document contains formal correctness proofs of modern SAT solvers.
+Following \cite{KrsticGoel,NieuwenhuisOliverasTinelli}, solvers are described
+using state-transition systems. Several different SAT solver descriptions are given
+and their partial correctness and termination is proved. These include:
+\begin{enumerate}
+\item a solver based on classical DPLL procedure (based on backtrack-search with unit propagation),
+\item a very general solver with backjumping and learning (similar to the description given in
+      \cite{NieuwenhuisOliverasTinelli}), and
+\item a solver with a specific conflict analysis algorithm (similar to the description given
+      in \cite{KrsticGoel}).
+\end{enumerate}
+Within the SAT solver correctness proofs, a large number of lemmas
+about propositional logic and CNF formulae are proved. This theory is
+self-contained and could be used for further exploring of properties of
+CNF based SAT algorithms.
+}
+
+\tableofcontents
+
+% sane default for proof documents
+\parindent 0pt\parskip 0.5ex
+
+% generated text of all theories
+\input{session}
+
+% optional bibliography
+\bibliographystyle{abbrv}
+\bibliography{root}
+
+\end{document}
+
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: t
+%%% End:


text:
How do I know if the linear system has a line of intersection?

I was wondering how can I determine if there is a line of intersection with any matrix?

For example, if I have the following matrix:

$$\left(\begin{array}{rrr|r} 1 & -3 & -2 & -9 \\ 2 & -5 & 1 & 3 \\ -3 & 6 & 2 & 8 \\ \end{array} \right)$$

What does the solution have to look like for me to conclude that there is a line of intersection?

P.S. I know this matrix has a point of intersection but I used this as an example because I didn't know an example for a matrix that had a line of intersection.

-
If there are infinitely many solutions, then you are guaranteed to find a line (more generally subspace) of solutions. An equivalent condition is if the rank of the matrix is not full, and at least 1 solution exists. –  Calvin Lin Jun 11 '13 at 14:03

The existence of a "point of intersection" is the existence of a point $(x, y, z)$ that satisfies the system of equations: a point lying on each line represented by the corresponding system of equations:

\begin{align} x - 3y -2z & = -9 \\ 2x - 5y + z & = 3 \\ -3x + 6y + 2z & = 8 \end{align}

A unique solution exists (a single point of intersection exists) if the augmented matrix does not reduce to a row of all zeros, and no row has all zeros, augmented by a non-zero entry.

If you obtain a row of all zeros, through row reduction, then infinitely many points of intersection occur (two or more lines will be concurrent): the entries in one or more lines will be a scalar multiple of the entries of another. Put differently, there will be a line of intersection.

If the matrix reduces to a row of three zeros, with a non-zero entry in the last column of that row, no solution exists (i.e., no point of intersection exists.)

All these possibilities can be determined by reducing the matrix to row echelon form.

We can reduce your example matrix:

$$\left(\begin{array}{rrr|r} 1 & -3 & -2 & -9 \\ 2 & -5 & 1 & 3 \\ -3 & 6 & 2 & 8 \\ \end{array}\right)$$

$$\left(\begin{array}{rrr|r} 1 & -3 & -2 & -9 \\ 0 & 1 & 5 & 21 \\ 0 & 0 & 11 & 42 \\ \end{array}\right)$$

If reduced further, you'd see that the system represented by the system of equations has a unique point of intersection, hence no common line of intersection.

Note that if the last row were $(0\;\;2\;\;10\;\;42)$, we could "zero it" by taking $-2R_2 + R_3 \rightarrow R_3$, and obtain a row of all zeros, since the third row would be a scalar multiple of the second row. In that event, there would indeed be a line of intersection. If any row is a linear combination of the other rows, we have a linearly dependent system of equations: this shows when a row-reduced matrix has a row of all zeros. And in that case, there is at least a line of intersection.

-
So if the third row of the system is a multiple of the second row or the first row then there is a line of intersection? I'm confused about how it would look in terms of $x$'s and $y$'s and $z$'s. Because for a the final answer for a P.O.I I would just write the solution but what would I write for a L.O.I? –  Jeel Shah Jun 11 '13 at 14:43
Very nice Amy and good morning $\ast$ –  Babak S. Jun 11 '13 at 14:44
@gekkostate What one typically does is set, say, $z = \alpha$, where $\alpha$ is any real number. Then express $x, y$ as values in terms of $z = \alpha$. So say you reduce the matrix to the rows the first two rows in the above row-reduced matrix, but the last row is all zeros. Then the solution is given by $z = \alpha, y = 21 - 5 \alpha, x = -9 + 2\alpha + 3(21 - 5\alpha)$, simplifying x, of course. $\alpha$ is then called a parameter. –  amWhy Jun 11 '13 at 14:56
Some use the parameter $t$ instead of $\alpha$. This post gives an example of such a system: infinite number of solutions. –  amWhy Jun 11 '13 at 15:02

If you are familiar with ranks and Rouché-Capelli (RC) theorem, the following reasoning can be used to answer. The system $A\mathbf x=\mathbf b$, where $A$ is an $n\times m$ matrix, has solutions if and only if $$r(A)=r(A|\mathbf b)=r,$$ and in this case you have "$\infty^{n-r}$" solutions. This really means that your solutions depend on $n-r$ free parameters. So, if the rank $r=n-1$, your solution set depends on $n-(n-1)=1$ free parameter or, in other words, it's monodimensional. This helps the intuition that the solution set is indeed a line in this case. Hence, you'll get a line of solutions when the rank of the complete matrix $r$ is equal to the number of unknowns $n$ less 1. As an example, consider the system \begin{align} x - y & = 2 \\ -2x +2y &=-4 \end{align} The rank $r(A)=r(A|\mathbf b)=1$ and you can indeed write the solutions set as $x=2+t,y=t$, which is equal to the set of points on the line $y=x-2$.

A more interesting example (I like a lot $A$!) is \begin{align} x +y +z& = 6 \\ 4x +5y +6z&=15\\ 7x+8y+10z&=24 \end{align} The reduced complete matrix is $$\left(\begin{array}{rrr|r} 1 & 2 & 3 & 6 \\ 0 & -3 & -6 & -9 \\ 0 & 0 & 0 & 0 \\ \end{array}\right)$$ from which you get the solutions $(x=t,y=3-2t,z=t), \forall t$. The solution set can be described in terms of a unique parameter $t$ and it is a straight line in $\mathbf R^3$.

-


text:
building-blocks edge drape command

Syntax

building-blocks edge drape keyword <novalidate> <range>

Move the points of edges onto an external geometric set. The points to be moved are defined by the <range>. Movement of each point is done along the beam-line defined by keyword until it intersects the external geometric set. The <novalidate> option skips validation of all blocks moved as part of the drape operation.

inwards v

Extend the beam-line from each point to the focus point v.

outwards v

Extend the beam-line from the focus point v to each point.

direction v

Set the beam-line as a constant direction vector v.


text:
## College Algebra 7th Edition

(a) linear, $f(x)=ax+b$ (b) line
Let $f$ be a function with constant rate of change. Then (a) $f$ is a ___linear___ function of the form ___$f(x)=ax+b$___ (b) The graph of $f$ is a ___line___.


text:
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# Of the 700 members of a certain organization

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Of the 700 members of a certain organization [#permalink]  14 Dec 2015, 06:24
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Question Stats:

63% (01:05) correct 36% (01:08) wrong based on 60 sessions
Of the 700 members of a certain organization, 120 are lawyers. Two members of the organization will be selected at random. Which of the following is closest to the probability that neither of the members selected will be a lawyer?
A. 0.5
B. 0.6
C. 0.7
D. 0.8
E. 0.9

Practice Questions
Question: 10
Page: 152
Difficulty: medium
[Reveal] Spoiler: OA

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Re: members of a certain organization, 120 are lawyers [#permalink]  14 Dec 2015, 06:29
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Expert's post
Solution

Here we have an organization of 700 members. Now there are 120 lawyers in the 700 members thus there is 580 not lawyers. Now we have to choose 2 not lawyers.

So the probability becomes $$\frac{580}{700}*\frac{579}{699}$$.

Now we can use the calculator available at the GRE to find the value and it is close to 0.7.

Note that the second term is $$\frac{(580-1)}{(700-1)}$$ since one not lawyer was already selected in the first attempt.
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Re: Of the 700 members of a certain organization [#permalink]  18 Apr 2020, 05:24
Expert's post
sandy wrote:
Of the 700 members of a certain organization, 120 are lawyers. Two members of the organization will be selected at random. Which of the following is closest to the probability that neither of the members selected will be a lawyer?
A. 0.5
B. 0.6
C. 0.7
D. 0.8
E. 0.9

A student asked me to solve this question (and answer a few other questions). So, here it goes...

SOLUTION #1: Probability rules

Question: Are we selecting people with replacement or without replacement?
We're told that two members are selected, so we can assume that the two members are different.
In other words, we're selecting people without replacement.
ASIDE: If this were my question, I would have written "two DIFFERENT members will be selected..." to be free from ambiguity. That said, this is an official question, so this is what we have to go with.

Of the 700 members of a certain organization, 120 are lawyers.
So we have 120 lawyers
And we have 580 non-lawyers

Two members of the organization will be selected at random. Which of the following is closest to the probability that neither of the members selected will be a lawyer?
P(neither selection is a lawyer) = P(1st selection is non-lawyer AND 2nd selection is non-lawyer)
= P(1st selection is non-lawyer) x P(2nd selection is non-lawyer)
= 580/700 x 579/699
≈ 0.69

ASIDE: I should also mention that, if we were to treat this question as a replacement question, we'd still get the same answer, since the value of 579/699 is very close to the value of 580/700

Cheers,
Brent
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Re: Of the 700 members of a certain organization [#permalink]  18 Apr 2020, 05:32
Expert's post
sandy wrote:
Of the 700 members of a certain organization, 120 are lawyers. Two members of the organization will be selected at random. Which of the following is closest to the probability that neither of the members selected will be a lawyer?
A. 0.5
B. 0.6
C. 0.7
D. 0.8
E. 0.9

SOLUTION #2: Counting methods

Of the 700 members of a certain organization, 120 are lawyers.
So we have 120 lawyers
And we have 580 non-lawyers

P(neither selection is a lawyer) = (number of ways to select 2 non-lawyers)/(TOTAL number of ways to select two members)

number of ways to select 2 non-lawyers
There are 580 non-lawyers, and we want to select 2 of them.
Since the order in which we select the 2 non-lawyers does not matter, we can use combinations.
580C2 = (580)(579)/2! = (290)(579)

TOTAL number of ways to select two members
There are 700 members in total, and we want to select 2 of them
Since the order in which we select the 2 members does not matter, we can use combinations.
700C2 = (700)(699)/2! =(350)(699)

we get; P(neither selection is a lawyer) = (290)(579)/(350)(699)
≈ 0.69

Cheers,
Brent
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Re: Of the 700 members of a certain organization [#permalink]  27 Apr 2020, 14:16
Can we do this by using this formula?
(1- favorable outcome/total outcome)
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Re: Of the 700 members of a certain organization [#permalink]  27 Apr 2020, 14:24
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Expert's post
Farina wrote:
Can we do this by using this formula?
(1- favorable outcome/total outcome)

You COULD use the complement formula, however this will require a lot more work.

We want P(neither selected person as a lawyer)

The complement of neither selected person as a lawyer is:
1st selection is a lawyer and 2nd selection is not a lawyer.
OR
1st selection is not a lawyer and 2nd selection is not a lawyer.
OR
1st selection is a lawyer and 2nd selection is a lawyer.

So, to apply the complement formula, we must calculate all three probabilities.
_________________

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Re: Of the 700 members of a certain organization   [#permalink] 27 Apr 2020, 14:24
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text:
Tall fescue commonly has been described as poor wildlife habitat because of its dense sod-forming structure and short growth compared with native grasses. The presence of the fungal endophyte in a large proportion of tall fescue stands also can negatively impact wildlife health and productivity (Coley et al., 1995). Alkaloids produced by the endophyte may reduce the prevalence of insect foods necessary for brood chicks (Muegge et al., 1991). Although all of these factors may be true for some avian and mammalian species, there are species that may benefit from tall fescue habitat. Wildlife primarily use grasslands for food and cover. Both mammalian and avian species make use of grasslands for these purposes.

 <--Previous Next-->


text:
# Node Wallet

## Initialize wallet#

Restart the node and go to http://127.0.0.1:9053/panel to access the panel. Then set the API key secret from the previous step. Note that you need to set the secret and not the hash from the config file. In our example, this is the string hello.

Click on Initialize wallet. After the pop-up opens, there are two ways to proceed depending on your scenario.

1. If this is the first time you are running the node then you need to initialize it with a new mnemonic sentence.
2. If you had created a wallet earlier and would like to obtain the same address (possibly because there are funds stored in it), then you have to restore the wallet using the mnemonic sentence you had saved earlier.

### Initialize wallet from scratch#

In the pop-up that opens, you must enter a wallet password. The mnemonic password is optional. After you click send, the wallet will return a mnemonic sentence as shown below.

You must copy this sentence and save it in a safe place. This sentence will be needed to restore the wallet on a different computer.

### Restore wallet from earlier#

Copy the mnemonic sentence from earlier paste it into the "Mnemonic" field in the Restore-wallet form. Enter a secure wallet password. Leave the Mnemonic password empty (it is only for advanced users). Refer to the figure below.

After the wallet has been successfully restored from the mnemonic sentence, you will see a confirmation as shown in the figure below.

This is a test to ensure you have set up the node properly. It will return the current addresses in the wallet. In the panel at http://127.0.0.1:9053/panel click on the Wallet tab on the left and then on Get all wallet addresses to view the addresses currently maintained by the wallet. It should return at least one address if the node is set correctly.


text:
• java.lang.Object
• All Implemented Interfaces:
GeometricLinkage, Linkage

@Reference(authors="J. H. Ward Jr.",title="Hierarchical grouping to optimize an objective function",booktitle="Journal of the American statistical association 58.301",url="https://doi.org/10.1080/01621459.1963.10500845",bibkey="doi:10.1080/01621459.1963.10500845") @Reference(authors="D. Wishart",title="256. Note: An Algorithm for Hierarchical Classifications",booktitle="Biometrics 25(1)",url="https://doi.org/10.2307/2528688",bibkey="doi:10.2307/2528688")
@Alias({"ward","MISSQ"})
@Priority(101)
extends java.lang.Object
implements GeometricLinkage
Ward's method clustering method.

This criterion minimizes the increase of squared errors, and should be used with squared Euclidean distance. Usually, ELKI will try to automatically square distances when you combine this with Euclidean distance. For performance reasons, the direct use of squared distances is preferable!

The distance of two clusters in this method is: $d_{\text{Ward}}(A,B):=\text{SSE}(A\cup B)-\text{SSE}(A)-\text{SSE}(B)$ where the sum of squared errors is defined as: $\text{SSE}(X):=\sum\nolimits_{x\in X} (x-\mu_X)^2 \qquad \text{with } \mu_X=\tfrac{1}{|X|}\sum\nolimits_{x\in X} X$ This objective can be rewritten to $d_{\text{Ward}}(A,B):=\tfrac{|A|\cdot|B|}{|A|+|B|} ||\mu_A-\mu_B||^2 = \tfrac{1}{1/|A|+1/|B|} ||\mu_A-\mu_B||^2$

For Lance-Williams, we can then obtain the following recursive definition: $d_{\text{Ward}}(A\cup B,C)=\tfrac{|A|+|C|}{|A|+|B|+|C|} d(A,C) + \tfrac{|B|+|C|}{|A|+|B|+|C|} d(B,C) - \tfrac{|C|}{|A|+|B|+|C|} d(A,B)$

These transformations rely on properties of the L2-norm, so they cannot be used with arbitrary metrics, unless they are equivalent to the L2-norm in some transformed space.

Because the resulting distances are squared, when used with a non-squared distance, ELKI implementations will apply the square root before returning the final result. This is statistically somewhat questionable, but usually yields more interpretable distances that — roughly — correspond to the increase in standard deviation. With ELKI, you can get both behavior: Either choose squared Euclidean distance, or regular Euclidean distance.

This method is also referred to as "minimize increase of sum of squares" (MISSQ) by Podani.

Reference:

J. H. Ward Jr.
Hierarchical grouping to optimize an objective function
Journal of the American statistical association 58.301

The formulation using Lance-Williams equations is due to:

D. Wishart
256. Note: An Algorithm for Hierarchical Classifications
Biometrics 25(1)

Since:
0.6.0
Author:
Erich Schubert
• ### Nested Class Summary

Nested Classes
Modifier and Type Class Description
static class  WardLinkage.Par
Class parameterizer.
• ### Field Summary

Fields
Modifier and Type Field Description
static WardLinkage STATIC
Static instance of class.
• ### Constructor Summary

Constructors
Constructor Description
WardLinkage()
Deprecated.
use the static instance STATIC instead.
• ### Method Summary

All Methods
Modifier and Type Method Description
double combine(int sizex, double dx, int sizey, double dy, int sizej, double dxy)
Compute combined linkage for two clusters.
double distance(double[] x, int sizex, double[] y, int sizey)
Distance of two aggregated clusters.
double initial(double d, boolean issquare)
Initialization of the distance matrix.
double[] merge(double[] x, int sizex, double[] y, int sizey)
Merge the aggregated vectors.
double restore(double d, boolean issquare)
Restore a distance to the original scale.
• ### Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
• ### Field Detail

• #### STATIC

public static final WardLinkage STATIC
Static instance of class.
• ### Constructor Detail

@Deprecated
public WardLinkage()
Deprecated.
use the static instance STATIC instead.
Constructor.
• ### Method Detail

• #### initial

public double initial(double d,
boolean issquare)
Description copied from interface: Linkage
Initialization of the distance matrix.
Specified by:
initial in interface Linkage
Parameters:
d - Distance
issquare - Flag to indicate the input values are already squared
Returns:
Initial value
• #### restore

public double restore(double d,
boolean issquare)
Description copied from interface: Linkage
Restore a distance to the original scale.
Specified by:
restore in interface Linkage
Parameters:
d - Distance
issquare - Flag to indicate the input values were already squared
Returns:
Initial value
• #### combine

public double combine(int sizex,
double dx,
int sizey,
double dy,
int sizej,
double dxy)
Description copied from interface: Linkage
Compute combined linkage for two clusters.
Specified by:
combine in interface Linkage
Parameters:
sizex - Size of first cluster x before merging
dx - Distance of cluster x to j before merging
sizey - Size of second cluster y before merging
dy - Distance of cluster y to j before merging
sizej - Size of candidate cluster j
dxy - Distance between clusters x and y before merging
Returns:
Combined distance
• #### merge

public double[] merge(double[] x,
int sizex,
double[] y,
int sizey)
Description copied from interface: GeometricLinkage
Merge the aggregated vectors.
Specified by:
merge in interface GeometricLinkage
Parameters:
x - Center of the first cluster
sizex - Weight of the first cluster
y - Center of the second cluster
sizey - Weight of the second cluster
Returns:
Combined vector
• #### distance

public double distance(double[] x,
int sizex,
double[] y,
int sizey)
Description copied from interface: GeometricLinkage
Distance of two aggregated clusters.
Specified by:
distance in interface GeometricLinkage
Parameters:
x - Center of the first cluster
sizex - Weight of the first cluster
y - Center of the second cluster
sizey - Weight of the second cluster
Returns:
Distance


text:
lilypond-devel
[Top][All Lists]

## Re: shorthand for autoBeam control

 From: Paul Scott Subject: Re: shorthand for autoBeam control Date: Sun, 07 Sep 2008 02:36:31 -0700 User-agent: Mozilla-Thunderbird 2.0.0.16 (X11/20080724)

Werner LEMBERG wrote:
>>> My special example is this where such a shorthand would be quite
>>> convenient:
>>>
>>>   c4 c c \times 2/3 { r8 c16 } c8
>>>
>> This may have nothing to do with your proposal/question but as a
>>
>>     c4 c c \times 2/3 { r8[ c16] } c8
>> or
>>     c4 c c \times 2/3 { r8[ c16 } c8]
>>
>
> Yes.  What I really would like to write is
>
>   c4 c c \times 2/3 { r8 c16[] } c8
>
> and I just demonstrated a case where my proposed notation would be
>
My point is that is it not helpful in this case because it produces a
notation which is IMO harder to read than the two variations that I
gave.  Maybe you can give an example where \noBeam makes something


text:
Love itself became the object of her love.count sadnessesmore quotes

# making poetry out of spam is fun

PNAS Cover: Earth BioGenome Project

# daily quotation server archives

In the late 90’s I started (a good decade for starts) a daily quotation server project at www.quoteserver.ca. The domain is now defunct—some pages are partially viewable at the Way Back Machine.

Below is the list of quotes I had collected by the end of the life of the project. Most are about love—duh—and a few are jolly jests from funny trenches. You know, that place where mustard gas makes your eyes water.

The quotes weren’t scraped from quote archives—each is meaningful and hand-picked.

## the quote archive

And now for full list of 1,700+ other things worth reading. Such as everything Dorothy Parker has written and ... yes, even the Pinky and Brain quotes, which are a special kind of special.

Quote collections about love, heart, desire, life, death, god, mind, science.

Feeling lucky? Read 10 random quotes. Well, will you, punk?

## Quotes about mind

21
Shall any gazer see with mortal eyes,
Or any seeker know by mortal mind?
Veil after veil will lift — but there must be
Veil upon veil behind.
The Light of Asia, VIII
36
The mind has a thousand eyes,
And the heart but one;
Yet the light of a whole life dies,
When love is done.
Light
166
I don’t mind sleeping on an empty stomach provided it isn’t my own.
286
All life is a struggle in the dark ... This dread and
darkness of the mind cannot be dispelled by the
sunbeams, the shining shafts of day, but only by
an understanding of the outward form and inner
workings of nature. And now to business,
I will explain ...
On the Nature of the Universe
345
Serious error.
All shortcuts have disappeared.
Screen. Mind. Both are blank.
390
Friendship is almost always the union of a part of one mind with a part of another; people are friends in spots.
414
And, while with silent, lifting mind I’ve trod
The high untrespassed sanctity of space,
Put out my hand, and touched the face of God.
437
A mind is like a parachute; it only works when it is open.
444
Tact is the ability to tell a man he has an open mind when he has a hole in his head.
450
Get your mind out of the gutter—it’s blocking my view.
488
In the beginner’s mind there are many possibilities, but in the expert’s mind there are a few.
612
There is a lady sweet and kind,
Was never a face so pleased my mind;
I did but see her passing by,
And yet I love her till I die.
617
With women the heart argues, not the mind.
Merope, 1.341
674
In every cry of every Man,
In every Infants cry of fear,
In every voice: in every ban,
The mind-forg’d manacles I hear.
London
696
Measure your mind’s height by the shade it casts.
Paracelsus, pt. iii, i.363
708
I do not mind lying but I hate inaccuracy.
Note Books
743
That out of sight is out of mind
Is true of most we leave behind.
Songs in Absence, That Out of Sight
866
Beauty in things exists in the mind which contemplates them.
Of Tragedy
871
Some experience of popular lecturing had convinced me that the necessity of making things plain to uninstructed people was one of the very best means of clearing up the obscure conrners in one’s own mind.
Man’s Place in Nature
965
A fanatic is one who can’t change his mind and won’t change the subject.
983
Dans les champs de l’observation le hasard ne favorise que les esprits prepares.
[Where observation is concerned, chance favours only the prepared mind.]
1137
Years steal
Fire from the mind as vigor from the limb,
And life’s enchanted cup but sparkles near the brim.
Childe Harold’s Pilgrimage
1182
This have I known always: Love is no more
Than the wide blossom which the wind assails,
Than the great tide that treads the shifting shore,
Strewing fresh wreckage gathered in the gales:
Pity me that the heart is slow to learn
What the swift mind beholds at every turn.
Sonnets, xxix
1195
I know a man that’s a braver man
And twenty men as kind,
And what are you, that you should be
The one man in my mind?
The Philosopher
1317
Her mind lives tidily, apart
From cold and noise and pain,
And bolts the door against her heart,
Out wailing in the rain.
Interior
1508
Nothing contributes so much to tranquilizing the mind as a steady purpose—a point on which the soul may fix its intellectual eye.
1547
So the old tunes float in my mind,
And go from me leaving no trace behind,
Like fragrance borne on the hush of the wind.
but in the instant the airs remain
I know the laughter and the pain
Of times that will not come again.
Old Tunes
1584
Remember me and beare in mind
A truthful friend is hard to find
The path of sorrow and that alone
Leads to a place where sorrow is unknown.
Autograph Albums and Bible of Ella Beaver Calhoun
1634
Something made of nothing, tasting very sweet,
A most delicious compound, with ingredients complete;
But if, as on occasion, the heart and mind are sour,
It has no great significance, and loses half its power.
The Kiss

# Cancer Cell cover

Sat 23-04-2022

My cover design on the 11 April 2022 Cancer Cell issue depicts depicts cellular heterogeneity as a kaleidoscope generated from immunofluorescence staining of the glial and neuronal markers MBP and NeuN (respectively) in a GBM patient-derived explant.

LeBlanc VG et al. Single-cell landscapes of primary glioblastomas and matched explants and cell lines show variable retention of inter- and intratumor heterogeneity (2022) Cancer Cell 40:379–392.E9.

My Cancer Cell kaleidoscope cover (volume 40, issue 4, 11 April 2022). (more)

Browse my gallery of cover designs.

A catalogue of my journal and magazine cover designs. (more)

# Nature Biotechnology cover

Sat 23-04-2022

My cover design on the 4 April 2022 Nature Biotechnology issue is an impression of a phylogenetic tree of over 200 million sequences.

Konno N et al. Deep distributed computing to reconstruct extremely large lineage trees (2022) Nature Biotechnology 40:566–575.

My Nature Biotechnology phylogenetic tree cover (volume 40, issue 4, 4 April 2022). (more)

Browse my gallery of cover designs.

A catalogue of my journal and magazine cover designs. (more)

# Nature cover — Gene Genie

Sat 23-04-2022

My cover design on the 17 March 2022 Nature issue depicts the evolutionary properties of sequences at the extremes of the evolvability spectrum.

Vaishnav ED et al. The evolution, evolvability and engineering of gene regulatory DNA (2022) Nature 603:455–463.

My Nature squiggles cover (volume 603, issue 7901, 17 March 2022). (more)

Browse my gallery of cover designs.

A catalogue of my journal and magazine cover designs. (more)

# Happy 2022 $\pi$ Day—three one four: a number of notes

Mon 14-03-2022

Celebrate $\pi$ Day (March 14th) and finally hear what you've been missing.

âthree one four: a number of notesâ is a musical exploration of how we think about mathematics and how we feel about mathematics. It tells stories from the very beginning (314â¦) to the very (known) end of Ï (...264) as well as math (Wallis Product) and math jokes (Feynman Point), repetition (nn) and zeroes (null).

Listen to $\pi$ in the style of 20th century classical music. (details)

The album is scored for solo piano in the style of 20th century classical music â each piece has a distinct personality, drawn from styles of Boulez, Feldman, Glass, Ligeti, Monk, and Satie.

Each piece is accompanied by a piku (or Ïku), a poem whose syllable count is determined by a specific sequence of digits from Ï.

Check out art from previous years: 2013 $\pi$ Day and 2014 $\pi$ Day, 2015 $\pi$ Day, 2016 $\pi$ Day, 2017 $\pi$ Day, 2018 $\pi$ Day, 2019 $\pi$ Day, 2020 $\pi$ Day and 2021 $\pi$ Day.

# PNAS Cover — Earth BioGenome Project

Fri 28-01-2022

My design appears on the 25 January 2022 PNAS issue.

My PNAS cover design captures the vision of the Earth BioGenome Project — to sequence everything. (more)

The cover shows a view of Earth that captures the vision of the Earth BioGenome Project — understanding and conserving genetic diversity on a global scale. Continents from the Authagraph projection, which preserves areas and shapes, are represented as a double helix of 32,111 bases. Short sequences of 806 unique species, sequenced as part of EBP-affiliated projects, are mapped onto the double helix of the continent (or ocean) where the species is commonly found. The length of the sequence is the same for each species on a continent (or ocean) and the sequences are separated by short gaps. Individual bases of the sequence are colored by dots. Species appear along the path in alphabetical order (by Latin name) and the first base of the first species is identified by a small black triangle.

Lewin HA et al. The Earth BioGenome Project 2020: Starting the clock. (2022) PNAS 119(4) e2115635118.


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# Math Help - Question about area under a curve using midpoint rule

1. ## Question about area under a curve using midpoint rule

okay, the question says to use the midpoint rule with n=4 to approximate the area of the region bounded by the graph of the function and the x axis over the indicated interval

function is f(x)= x^2 + 4x interval is [0,4]

midpoint rule

sigma from i=1 to n of f((xi-xi-1)/2) times delta x

thanks very much

2. Originally Posted by doctorgk
okay, the question says to use the midpoint rule with n=4 to approximate the area of the region bounded by the graph of the function and the x axis over the indicated interval

function is f(x)= x^2 + 4x interval is [0,4]

midpoint rule

sigma from i=1 to n of f((xi-xi-1)/2) times delta x

thanks very much
why is this in the pre-calc section?

divide [0,4] into 4 equal sub-intervals, we get the intervals: $[0,1],~[1,2],~[2,3],~[3,4]$

obviously (i hope) we have $\Delta x = 1$ here. if you're not sure, remember $\Delta x = \frac {b - a}{n}$ where [a,b] is the interval we are considering, and n, of course, is the number of sub-intervals

the midpoints of the subintervals respectively are: $x_1 = \frac 12,~x_2 = \frac 32,~ x_3 = \frac 52,~x_4 = \frac 72$

so, by the midpoint rule, our estimate for the integral is given by:

$\int_0^4f(x)~dx = \sum_{i = 1}^{4} f(x_i) \Delta x = \Delta x \left( f \left( \frac 12 \right) + f \left( \frac 32 \right) + f \left( \frac 52 \right) + f \left( \frac 72 \right) \right)$

$= f \left( \frac 12 \right) + f \left( \frac 32 \right) + f \left( \frac 52 \right) + f \left( \frac 72 \right)$

i leave the rest to you


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# The Idele Group

The invertible elements of any commutative topological ring  are a group under multiplication. In general is not a topological group if it is endowed with the subset topology because inversion need not be continuous (only multiplication and addition on are required to be continuous). It is usual therefore to give the following topology. There is an injection

 (19.1)

of into the topological product . We give the corresponding subset topology. Then with this topology is a topological group and the inclusion map is continous. To see continuity of inclusion, note that this topology is finer (has at least as many open sets) than the subset topology induced by , since the projection maps are continuous.

Example 19.1.1   This is a non-example''. The inverse map on is continuous with respect to the -adic topology. If , then , so if , then

Definition 19.1.2 (Idele Group)   The idele group of is the group of invertible elements of the adele ring .

We shall usually speak of as a subset of , and will have to distinguish between the and -topologies.

Example 19.1.3   For a rational prime , let be the adele whose th component is and whose th component, for , is . Then as in , for the following reason. We must show that if is a basic open set that contains the adele , the for all sufficiently large are contained in . Since contains and is a basic open set, it is of the form

where if a finite set, and the , for , are arbitrary open subsets of that contain . If is a prime larger than any prime in , then for , is in . This proves convergence. If the inverse map were continuous on , then the sequence of would converge to . However, if is an open set as above about , then for sufficiently large , none of the adeles are contained in .

Lemma 19.1.4   The group of ideles is the restricted topological project of the with respect to the units , with the restricted product topology.

We omit the proof of Lemma 19.1.4, which is a matter of thinking carefully about the definitions. The main point is that inversion is continuous on for each . (See Example 19.1.1.)

We have seen that is naturally embedded in , so is naturally embedded in  .

Definition 19.1.5 (Principal Ideles)   We call , considered as a subgroup of , the principal ideles.

Lemma 19.1.6   The principal ideles are discrete as a subgroup of .

Proof. For is discrete in , so is embedded in by (19.1.1) as a discrete subset. (Alternatively, the subgroup topology on is finer than the topology coming from being a subset of , and is already discrete in .)

Definition 19.1.7 (Content of an Idele)   The content of is

Lemma 19.1.8   The map is a continuous homomorphism of the topological group into , where we view as a topological group under multiplication. If  is a number field, then is surjective.

Proof. That the content map  satisfies the axioms of a homomorphisms follows from the multiplicative nature of the defining formula for . For continuity, suppose is an open interval in . Suppose is such that . By considering small intervals about each non-unit component of , we find an open neighborhood of such that . It follows the is open.

For surjectivity, use that each archimedean valuation is surjective, and choose an idele that is  at all but one archimedean valuation.

Remark 19.1.9   Note also that the -topology is that appropriate to a group of operators on : a basis of open sets is the , where are, respectively, -compact and -open, and consists of the such that and .

Definition 19.1.10 (-Ideles)   The subgroup of -ideles is the subgroup of ideles such that . Thus is the kernel of , so we have an exact sequence

where the surjectivity on the right is only if is a number field.

Lemma 19.1.11   The subset of is closed as a subset, and the -subset topology on coincides with the -subset topology on .

Proof. Let with . To prove that is closed in , we find an -neighborhood of that does not meet .

1st Case. Suppose that (possibly ). Then there is a finite set  of  such that

1. contains all the with , and
2. .
Then the set can be defined by

for sufficiently small .

2nd Case. Suppose that . Then there is a finite set of such that

1. contains all the with , and
2. if an inequality implies (This is because for a non-archimedean valuation, the largest absolute value less than is , where is the residue characteristic. Also, the upper bound in Cassels's article is instead of , but I think he got it wrong.)
We can choose so small that (for ) implies Then may be defined by

This works because if , then either for all , in which case , so , or for some , in which case

so again .

We next show that the - and -topologies on are the same. If , we must show that every -neighborhood of contains an -neighborhood and vice-versa.

Let be an -neighborhood of . Then it contains an -neighborhood of the type

 (19.2) (19.3)

where is a finite set of valuations . This contains the -neighborhood in which in (19.1.2) is replaced by .

Next let be an -neighborhood. Then it contains an -neighborhood of the form

 (19.4) (19.5)

where the finite set  contains at least all archimedean valuations  and all valuations  with . Since , we may also suppose that is so small that (19.1.4) implies

Then the intersection of (19.1.4) with is the same as that of (19.1.2) with , i.e., (19.1.4) defines an -neighborhood.

By the product formula we have that . The following result is of vital importance in class field theory.

Theorem 19.1.12   The quotient with the quotient topology is compact.

Proof. After the preceeding lemma, it is enough to find an -compact set such that the map

is surjective. We take for the set of with

where is any idele of content greater than the of Lemma 18.4.1.

Let . Then the content of equals the content of , so by Lemma 18.4.1 there is an such that

all $v$

Then , as required.

Remark 19.1.13   The quotient is totally disconnected in the function field case. For the structure of its connected component in the number field case, see papers of Artin and Weil in the Proceedings of the Tokyo Symposium on Algebraic Number Theory, 1955'' (Science Council of Japan) or [AT90]. The determination of the character group of is global class field theory.

William Stein 2012-09-24


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# naginterfaces.library.lapackeig.dsptrd¶

naginterfaces.library.lapackeig.dsptrd(uplo, n, ap)[source]

dsptrd reduces a real symmetric matrix to tridiagonal form, using packed storage.

For full information please refer to the NAG Library document for f08ge

https://www.nag.com/numeric/nl/nagdoc_29/flhtml/f08/f08gef.html

Parameters
uplostr, length 1

Indicates whether the upper or lower triangular part of is stored.

The upper triangular part of is stored.

The lower triangular part of is stored.

nint

, the order of the matrix .

apfloat, array-like, shape

The upper or lower triangle of the symmetric matrix , packed by columns.

Returns
apfloat, ndarray, shape

is overwritten by the tridiagonal matrix and details of the orthogonal matrix .

dfloat, ndarray, shape

The diagonal elements of the tridiagonal matrix .

efloat, ndarray, shape

The off-diagonal elements of the tridiagonal matrix .

taufloat, ndarray, shape

Further details of the orthogonal matrix .

Raises
NagValueError
(errno )

On entry, error in parameter .

Constraint: or .

(errno )

On entry, error in parameter .

Constraint: .

Notes

dsptrd reduces a real symmetric matrix , held in packed storage, to symmetric tridiagonal form by an orthogonal similarity transformation: .

The matrix is not formed explicitly but is represented as a product of elementary reflectors (see the F08 Introduction for details). Functions are provided to work with in this representation (see Further Comments).

References

Golub, G H and Van Loan, C F, 1996, Matrix Computations, (3rd Edition), Johns Hopkins University Press, Baltimore