text: Contribute to erwanp/pytexit development by creating an account on GitHub. Convert a Python expression to a LaTeX formula. Installation. It is based on very simple idea. LaTeX using psfrag¶ Note: This section is obsolete. Lets look at an example. Next time when visitor opens that web page in the browser formula is shown. This package includes the following binaries: 1. katex-cli-0.10.2-linux-x64 2. katex-cli-0.10.2-macos-x64 3. katex-cli-0.10.2-win-x64 If you are on a different platform, or want to use a more recent version of katex-cli, you will need to install it via npm. pip install latex-render Download the file for your platform. Plot Histogram for List of Data in Matplotlib, Plot Numpy Linear Fit in Matplotlib Python, Create Reverse Colormap in Python Matplotlib. Input LaTeX, Tex, AMSmath or ASCIIMath notation (Click icon to switch to ASCIIMath mode) to make formula. Have you ever asked yourself, how they write complex maths and physics equations using computer? I am familiar with using LaTeX on online websites (like MathOverflow or Physicsforums). This Inkscape extension searches all text elements in a document and renders them with PdfLatex. vscode: 1.41.1 vscode-python-ext: 2020.1.58038. recurrent. The only issue is the default Jinja block, variable, and comment identification strings can conflict with the LaTeX commands. I've been using sage math a lot and the latex generation is really solid. The addon is called Math Anywhere and it should render every MathML/LaTeX formulas on any page, also outside Medium. There are various config values you can set to influence how the images are built: pngmath_latex. User inserts URL into web page in place where formula should be. My input is rendered using MathJax. It will render the LaTex formulae in the Matplotlib figure.eval(ez_write_tag([[300,250],'delftstack_com-medrectangle-4','ezslot_1',112,'0','0'])); To render a LaTex formula in Matplotlib, we must set 'text.usetex' to True. To install latex_render, you'll just need to run: Once you've installed latex_render, you can run the command-line interface either with the latex_render command. latex to png image converter. Examples: Ubuntu sudo apt-get install texlive-pictures texlive-science texlive-latex â¦ Donate today! We can also render Greek alphabets and many more symbols in Matplotlib using the Tex format. For engineers, data scientists, educators, and students. In this Python tutorial, We'll look at a new Python package handcalcs. Please try enabling it if you encounter problems. PyLaTeX has two quite different usages: generating full pdfs and generating LaTeX snippets. Different libraries are used from Data preprocessing to data visualisation, etc.Through this article, we will discuss a new python library HandCalcs that is used to render different calculation code in Latex that is dependent on the writing of calculations. Status: Enclose LaTeX code in dollar signs $...$ to display math inline. It is on the menu bar: Extensions->Render->LaTeX Formula. Convert Latex equations into beautiful, transparency-correct PNGs. sphinx.ext.pngmath â Render math as PNG images. If 'text.usetex' is set to False, we can use the following script to set 'text.usetex' to True: We also need to have LaTex, dvipng and Ghostscript(Version 9.0 or later) to render the LaTex formulae and add all the installations dependencies to the PATH. However, a LaTeX environment is a very heavy dependence that is unlikely to be installed by the users. There are various configuration values you can set to influence how the images are built: LaTeX. GitHub Gist: instantly share code, notes, and snippets. Print quality: KaTeXâs layout is based on Donald Knuthâs TeX, the gold standard for math typesetting. It only renders greek letters correctly if you type them out though, it won't transform unicode greek letters to the correct latex equivalent. Or enter in display math mode by writing between. Recent versions of Inkscape (since 0.48?) This of course means that the computer where the docs are built must have both programs available. This extension renders math via LaTeX and dvipng into PNG images. The goal of this library is being an easy, but extensible interface between Python and LaTeX. How to generate latex first. This of course means that the computer where the docs are built must have both programs available. Recent versions matplotlib break the psfrag functionality (see for example this discussion. Installation. IPython notebook uses MathJax to render LaTeX inside html/markdown. For example: renders as fâ²(a)=limxâaf(x)âf(a)xâa See the LaTeX WikiBook for more information (especially the section on mathematics). Simply install using pip: pip install pylatex And then install a relevant LaTeX processor and other dependencies. With handcalcs, use the %%render cell magic and your Python calculation is rendered with the symbolic formula, followed by the numeric substitution, and then the result, just as though you had written it by hand. This is the python file test.py: To make use of the inline math feature, simply write your text and if you need to typeset a single math symbol or formula, surround it with dollar signs:Output equation: This formula f(x)=x2 is an example.This formula f(x)=x2 is an example. This menu might be hidden if Inkscape cannot find these helper programs: latexâ¦ Invaluable for creating content for presentations in powerpoint and keynote. Developed and maintained by the Python community, for the Python community. all systems operational. This tutorial explains how we can render the LaTex formulas or equations in Matplotlib. Latex Render. That being said, one can use the usetex feature to render the LaTeX text directly with very good results (if you are careful about choosing fonts). Make sure that you have Python 3.6.1 or above and pip installed. Text handling with matplotlib's LaTeX support is slower than matplotlib's very capable mathtext, but is more flexible, since different LaTeX packages (font packages, math packages, etc.) Will it render my matrices/vectors? L a T e X allows two writing modes for mathematical expressions: the inline mode and the display mode. The Jinja templating engine works well to create LaTeX templates for generating PDF files. HandCalcs is a library in Python to make calculations automatically in Latex, but in a way that imitates how the equation might be formatted when handwritten, write the mathematical formula, supported by numerical substitutions, and then the output. We have seen several python libraries and packages that support various multiple tasks in the field of Data Science. It is well known that if , then . Site map. I have downloaded MacTeX, and all I would like to do is make tiny â¦ Here is an example. The goal of this library is to be an easy, but extensible interface between Python and LaTeX. But it is far from supporting all of the LaTeX constructs and depends on the availability of glyphs in the chosen font. Created: December-29, 2020 . have built-in support for including a LaTeX formula inside a drawing. The Jupyter Notebook uses MathJax to render LaTeX inside HTML / Markdown. The rendered text will appear at the same position as the original text and may contain Latex commands and math formula. render latex formula with matplotlib and pillow. r "\end {document} ") from latex import build_pdf # this builds a pdf-file inside a temporary directory pdf = build_pdf (min_latex) # look at the first few bytes of the header print bytes (pdf)[: 10] Also comes with support for using Jinja2 templates to generate LaTeX files. Renders LaTeX for markdown file, include Github README. The code that was used to render this formula is just $$\frac{n!}{k!(n-k)!} Fast: KaTeX renders its math synchronously and doesnât need to reflow the page. Try this simple one upmath â Latex editor online. Open an example in Overleaf. Modifications of the source text will be considered when the extension is executed again. As you see, the way the equations are displayed depends on the delimiter, in this case  and . A solution is to change the Jinja environment to mimic the LaTeX environment. Enclose LaTeX code in double dollar signs$$ ... $$to display expressions in a centered paragraph. make_env can be used to create an Environment that plays well with LaTex: See python -m readme2tex --help for a list of what you can do in readme2tex. Since HandCalcs indicates the numerical replacement, the equations become much easier to manually check and verify. Here's an inline formula. This extension will always use the locally installed version of KaTeX if it is available, instead of using the implementation bundled with this package. Works as both a cell magic and as a decorator on your functions. QuickLaTeX renders it to the image stored on the server and returns URL of that image. = {n \choose k}$$ Note: you can escape $so that they don't render. Mathematical modes. I suppose since Python doesn't have static typing that is unlikely. During the editing, you just need to put your formulas in appropriate delimiters like dollar symbols (the addon should be turned off) and during the reading, once activated the addon, the equations will be converted. If you're not sure which to choose, learn more about installing packages. can be used. Just put your LaTeX math inside $$. Some features may not work without JavaScript. © 2020 Python Software Foundation User feeds QuickLaTeX.com with the piece of LaTeX code.$$c = \sqrt{a^2 + b^2}$$Or you can display LaTeX / Math output from Python, as seen towards the end of the notebook tour: from IPython.display import display, Math, Latex display(Math(r'F(k) = \int_{-\infty}^{\infty} f(x) e^{2\pi i k} dx')) Self contained: KaTeX has no dependencies and can easily be bundled with your website resources. Once you've installed latex_render, you can run the command-line interface either with the latex_render command We can use any online Latex editor to generate Latex. make a blank ipynb file; paste the following code as markdown into the first cell No JavaScript is required to render the resulting HTML, so it can be used with more limited renderers (which don't suppoâ¦ The results can be striking, especially when you take care to use the same fonts in your figures as in the main document. Matplotlib has a simple pure python LaTeX interpreter that uses system fonts to render the output. How to embed a LaTeX equation inside Inkscape. Just put your LaTeX math inside$ $. Examples: Here's a display level formula. from io import BytesIO import matplotlib.pyplot as plt def renderLatex(formula, fontsize=12, dpi=300, format='svg', file=None): """Renders LaTeX formula into image or prints to file. However, I do not understand the LaTeX environment in its native OS form. For example, the code$\int_a^b f(x) = F(b) - F(a)$renders inline as â«abf(x)dx=F(b)âF(a). Help the Python Software Foundation raise$60,000 USD by December 31st! Make sure that you have Python 3.6.1 or above and pip installed. This tutorial explains how we can render the LaTex formulas or equations in Matplotlib.. Write LaTex Formulae in Matplotlib Python import math import numpy as np import matplotlib.pyplot as plt x=np.linspace(0,2*math.pi,100) y=np.sin(x) plt.plot(x,y) plt.xlabel("x") plt.ylabel(r'$\sin (x)$') plt.title("Plot of sinx") plt.show() Well, itâs all about LaTeX. Copy PIP instructions, View statistics for this project via Libraries.io, or by using our public dataset on Google BigQuery, Renders LaTeX for markdown file, include Github README. All you have to do is include MathJax in your page and change config options if required. The first one is used to write formulas â¦ We can use the following script to check the value of 'text.usetex':eval(ez_write_tag([[728,90],'delftstack_com-medrectangle-3','ezslot_2',113,'0','0'])); You may get True as the output if 'text.usetex' to True for your system. environment. To install latex_render, you'll just need to run: pip install latex_render Usage. This extension renders math via LaTeX and dvipng or dvisvgm into PNG or SVG images. 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# Hermitian symmetric space

In mathematics, a Hermitian symmetric space is a Hermitian manifold which at every point has as an inversion symmetry preserving the Hermitian structure. First studied by Élie Cartan, they form a natural generalization of the notion of Riemannian symmetric space from real manifolds to complex manifolds.

Every Hermitian symmetric space is a homogeneous space for its isometry group and has a unique decomposition as a product of irreducible spaces and a Euclidean space. The irreducible spaces arise in pairs as a non-compact space that, as Borel showed, can be embedded as an open subspace of its compact dual space. Harish Chandra showed that each non-compact space can be realized as a bounded symmetric domain in a complex vector space. The simplest case involves the groups SU(2), SU(1,1) and their common complexification SL(2,C). In this case the non-compact space is the unit disk, a homogeneous space for SU(1,1). It is a bounded domain in the complex plane C. The one-point compactification of C, the Riemann sphere, is the dual space, a homogeneous space for SU(2) and SL(2,C).

Irreducible compact Hermitian symmetric spaces are exactly the homogeneous spaces of simple compact Lie groups by maximal closed connected subgroups which contain a maximal torus and have center isomorphic to T. There is a complete classification of irreducible spaces, with four classical series, studied by Cartan, and two exceptional cases; the classification can be deduced from Borel–de Siebenthal theory, which classifies closed connected subgroups containing a maximal torus. Hermitian symmetric spaces appear in the theory of Jordan triple systems, several complex variables, complex geometry, automorphic forms and group representations, in particular permitting the construction of the holomorphic discrete series representations of semisimple Lie groups.[1]

## Hermitian symmetric spaces of compact type

### Definition

Let H be a connected compact semisimple Lie group, σ an automorphism of H of order 2 and Hσ the fixed point subgroup of σ. Let K be a closed subgroup of H lying between Hσ and its identity component. The compact homogeneous space H / K is called a symmetric space of compact type. The Lie algebra $\mathfrak{h}$ admits a decomposition

$\displaystyle{\mathfrak{h}=\mathfrak{k}\oplus\mathfrak{m},}$

where $\mathfrak{k}$, the Lie algebra of K, is the +1 eigenspace of σ and $\mathfrak{m}$ the –1 eigenspace. If $\mathfrak{k}$ contains no simple summand of $\mathfrak{h}$, the pair ($\mathfrak{h}$, σ) is called an orthogonal symmetric Lie algebra of compact type.[2]

Any inner product on $\mathfrak{h}$, invariant under the adjoint representation and σ, induces a Riemannian structure on H / K, with H acting by isometries. A canonical example is given by minus the Killing form. Under such an inner product, $\mathfrak{k}$ and $\mathfrak{m}$ are orthogonal. H / K is then a Riemannian symmetric space of compact type.[3]

The symmetric space H / K is called a Hermitian symmetric space if it has an almost complex structure preserving the Riemannian metric. This is equivalent to the existence of a linear map J with J2 = −I on $\mathfrak{m}$ which preserves the inner product and commutes with the action of K.

### Symmetry and center of isotropy subgroup

If ($\mathfrak{h}$,σ) is Hermitian, K has non-trivial center and the symmetry σ is inner, implemented by an element of the center of K.

In fact J lies in $\mathfrak{k}$ and exp tJ forms a one-parameter group in the center of K. This follows because if A, B, C, D lie in $\mathfrak{m}$, then by the invariance of the inner product on $\mathfrak{h}$[4]

$\displaystyle{([[A,B],C],D)=([A,B],[C,D])=([[C,D],B],A).}$

Replacing A and B by JA and JB, it follows that

$\displaystyle{[JA,JB] = [A,B].}$

Define a linear map δ on $\mathfrak{h}$ by extending J to be 0 on $\mathfrak{k}$. The last relation shows that δ is a derivation of $\mathfrak{h}$. Since $\mathfrak{h}$ is semisimple, δ must be an inner derivation, so that

$\displaystyle{\delta(X)=[T + A,X],}$

with T in $\mathfrak{k}$ and A in $\mathfrak{m}$. Taking X in $\mathfrak{k}$, it follows that A = 0 and T lies in the center of $\mathfrak{k}$ and hence that K is non-semisimple. The symmetry σ is implemented by z = exp πT and the almost complex structure by exp π/2 T.[5]

The innerness of σ implies that K contains a maximal torus of H, so has maximal rank. On the other hand the centralizer of the subgroup generated by the torus S of elements exp tT is connected, since if x is any element in K there is a maximal torus containing x and S, which lies in the centralizer. On the other hand it contains K since S is central in K and is contained in K since z lies in S. So K is the centralizer of S and hence connected. In particular K contains the center of H.[2]

### Irreducible decomposition

The symmetric space or the pair ($\mathfrak{h}$, σ) is said to be irreducible if the adjoint action of $\mathfrak{k}$ (or equivalently the identity component of Hσ or K) is irreducible on $\mathfrak{m}$. This is equivalent to the maximality of $\mathfrak{k}$ as a subalgebra.[6]

In fact there is a one-one correspondence between intermediate subalgebras $\mathfrak{l}$ and K-invariant subspaces $\mathfrak{m}_1$ of $\mathfrak{m}$ given by

$\displaystyle{\mathfrak{l}=\mathfrak{k}\oplus \mathfrak{m}_1,\,\,\,\ \mathfrak{m}_1=\mathfrak{l}\cap \mathfrak{m}.}$

Any orthogonal symmetric algebra ($\mathfrak{g}$, σ) of Hermitian type can be decomposed as an (orthogonal) direct sum of irreducible orthogonal symmetric algebras of Hermitian type.[7]

In fact $\mathfrak{h}$ can be written as a direct sum of simple algebras

$\displaystyle{\mathfrak{h}=\oplus_{i=1}^N \mathfrak{h}_i,}$

each of which is left invariant by the automorphism σ and the complex structure J, since they are both inner. The eigenspace decomposition of $\mathfrak{h}_1$ coincides with its intersections with $\mathfrak{k}$ and $\mathfrak{m}$. So the restriction of σ to $\mathfrak{h}_1$ is irreducible.

This decomposition of the orthogonal symmetric Lie algebra yields a direct product decomposition of the corresponding compact symmetric space H / K when H is simply connected. In this case the fixed point subgroup Hσ is automatically connected. For simply connected H, the symmetric space H / K is the direct product of Hi / Ki with Hi simply connected and simple. In the irreducible case, K is a maximal connected subgroup of H. Since K acts irreducibly on $\mathfrak{m}$ (regarded as a complex space for the complex structure defined by J), the center of K is a one-dimensional torus T, given by the operators exp tT. Since each H is simply connected and K connected, the quotient H/K is simply connected.[8]

### Complex structure

if H / K is irreducible with K non-semisimple, the compact group H must be simple and K of maximal rank. From Borel-de Siebenthal theory, the involution σ is inner and K is the centralizer of its center, which is isomorphic to T. In particular K is connected. It follows that H / K is simply connected and there is a parabolic subgroup P in the complexification G of H such that H / K = G / P. In particular there is a complex structure on H / K and the action of H is holomorphic. Since any Hermitian symmetric space is a product of irreducible spaces, the same is true in general.

At the Lie algebra level, there is a symmetric decomposition

$\mathfrak h = \mathfrak k\oplus\mathfrak m,$

where $(\mathfrak m,J)$ is a real vector space with a complex structure J, whose complex dimension is given in the table. Correspondingly, there is a graded Lie algebra decomposition

$\mathfrak g = \mathfrak{m}_{+}\oplus\mathfrak l\oplus\mathfrak{m}_-$

where $\mathfrak m\otimes\mathbb C= \mathfrak m_{-}\oplus\mathfrak m_{+}$ is the decomposition into +i and −i eigenspaces of J and $\mathfrak l=\mathfrak k\otimes\mathbb C$. The Lie algebra of P is the semidirect product $\mathfrak m^{+}\oplus\mathfrak l$. The complex Lie algebras $\mathfrak{m}_\pm$ are Abelian. Indeed if U and V lie in $\mathfrak{m}_\pm$, [U,V] = J[U,V] = [JU,JV] = [±iUiV] = –[U,V], so the Lie bracket must vanish.

The complex subspaces $\mathfrak{m}_\pm$ of $\mathfrak{m}_{\mathbb C}$ are irreducible for the action of K, since J commutes with K so that each is isomorphic to $\mathfrak{m}$ with complex structure ±J. Equivalently the centre T of K acts on $\mathfrak{m}_+$ by the identity representation and on $\mathfrak{m}_-$ by its conjugate.[9]

The realization of H/K as a generalized flag variety G/P is obtained by taking G as in the table (the complexification of H) and P to be the parabolic subgroup equal to the semidirect product of L, the complexification of K, with the complex Abelian subgroup exp $\mathfrak{m}_+$. (In the language of algebraic groups, L is the Levi factor of P.)

### Classification

Any Hermitian symmetric space of compact type is simply connected and can be written as a direct product of irreducible hermitian symmetric spaces Hi / Ki with Hi simple, Ki connected of maximal rank with center T. The irreducible ones are therefore exactly the non-semisimple cases classified by Borel–de Siebenthal theory.[2]

Accordingly the irreducible compact Hermitian symmetric spaces H/K are classified as follows.

G H K complex dimension rank geometric interpretation
$\mathrm{SL}(p+q,\mathbb{C})$ $\mathrm{SU}(p+q)\,$ $\mathrm{S}(\mathrm{U}(p) \times \mathrm{U}(q))$ pq min(p,q) Grassmannian of complex p-dimensional subspaces of $\mathbb{C}^{p+q}$
$\mathrm{SO}(2n,\mathbb C)$ $\mathrm{SO}(2n)\,$ $\mathrm{U}(n)\,$ $\tfrac 12 n(n-1)$ $[\tfrac 12 n]$ Space of orthogonal complex structures on $\mathbb{R}^{2n}$
$\mathrm{Sp}(2n,\mathbb C)\,$ $\mathrm{Sp}(n)\,$ $\mathrm{U}(n)\,$ $\tfrac 12 n(n+1)$ n Space of complex structures on $\mathbb{H}^n$ compatible with the inner product
$\mathrm{SO}(n+2,\mathbb{C})$ $\mathrm{SO}(n+2)\,$ $\mathrm{SO}(n) \times \mathrm{SO}(2)$ n 2 Grassmannian of oriented real 2-dimensional subspaces of $\mathbb{R}^{n+2}$
$E_6^{\mathbb C}\,$ $E_6\,$ $\mathrm{SO}(10)\times\mathrm{SO}(2)$ 16 2 Complexification $(\mathbb{C}\otimes\mathbb{O})P^2$ of the Cayley projective plane $\mathbb{O}P^2$
$E_7^{\mathbb C}$ $E_7\,$ $E_6\times\mathrm{SO}(2)$ 27 3 Space of symmetric submanifolds of Rosenfeld projective plane $(\mathbb{H}\otimes\mathbb{O})P^2$ which are isomorphic to $(\mathbb{C}\otimes\mathbb{O})P^2$

In terms of the classification of compact Riemannian symmetric spaces, the Hermitian symmetric spaces are the four infinite series AIII, DIII, CI and BDI with p = 2 or q = 2, and two exceptional spaces, namely EIII and EVII.

### Classical examples

The irreducible Hermitian symmetric spaces of compact type are all simply connected. The corresponding symmetry σ of the simply connected simple compact Lie group is inner, given by conjugation by the unique element S in Z(K) / Z(H) of period 2. For the classical groups, as in the table above, these symmetries are as follows:[10]

• AIII: $S=\begin{pmatrix}-\alpha I_p & 0\\ 0 & \alpha I_q\end{pmatrix}$ in S(U(p)×U(q)), where αp+q=(−1)p.
• DIII: S = iI in U(n) ⊂ SO(2n); this choice is equivalent to $J_n=\begin{pmatrix}0 &I_n \\ -I_n & 0\end{pmatrix}$.
• CI: S=iI in U(n) ⊂ Sp(n) = Sp(n,C) ∩ U(2n); this choice is equivalent to Jn.
• BDI: $S=\begin{pmatrix}I_p & 0\\ 0 & -I_2\end{pmatrix}$ in SO(p)×SO(2).

The maximal parabolic subgroup P can be described explicitly in these classical cases. For AIII

$\displaystyle{P(p,q)= \begin{pmatrix}A_{pp} & B_{pq}\\ 0 & D_{qq}\end{pmatrix}}$

in SL(p+q,C). P(p,q) is the stabilizer of a subspace of dimension p in Cp+q.

The other groups arise as fixed points of involutions. Let J be the n × n matrix with 1's on the antidiagonal and 0's elsewhere and set

$\displaystyle{A=\begin{pmatrix} 0 & J\\ -J & 0\end{pmatrix}.}$

Then Sp(n,C) is the fixed point subgroup of the involution θ(g) = A (gt)−1 A−1 of SL(2n,C). SO(n,C) can be realised as the fixed points of ψ(g) = B (gt)−1 B−1 in SL(n,C) where B = J. These involutions leave invariant P(n,n) in the cases DIII and CI and P(p,2) in the case BDI. The corresponding parabolic subgroups P are obtained by taking the fixed points. The compact group H acts transitively on G / P, so that G / P = H / K.

## Hermitian symmetric spaces of noncompact type

### Definition

As with symmetric spaces in general, each compact Hermitian symmetric space H/K has a noncompact dual H*/K obtained by replacing H with the closed real Lie subgroup H* of the complex Lie group G with Lie algebra

$\mathfrak h^* = \mathfrak k \oplus i\mathfrak m\subset\mathfrak g.$

### Borel embedding

Whereas the natural map from H/K to G/P is an isomorphism, the natural map from H*/K to G/P is only an inclusion onto an open subset. This inclusion is called the Borel embedding after Armand Borel. In fact PH = K = PH*. The images of H and H* have the same dimension so are open. Since the image of H is compact, so closed, it follows that H/K = G/P.[11]

### Cartan decomposition

The polar decomposition in the complex linear group G implies the Cartan decomposition H* = K ⋅ exp $i\mathfrak{m}$ in H*.[12]

Moreover given a maximal Abelian subalgebra $\mathfrak{a}$ in t, A = exp $\mathfrak{a}$ is a toral subgroup such that σ(a) = a−1 on A; and any two such $\mathfrak{a}$'s are conjugate by an element of K. A similar statement holds for $\mathfrak{a}^*=i\mathfrak{a}$. Morevoer if A* = exp $\mathfrak{a}^*$, then

$\displaystyle{H^*=KA^*K.}$

These results are special cases of the Cartan decomposition in any Riemannian symmetric space and its dual. The geodesics emanating from the origin in the homogeneous spaces can be identified with one parameter groups with generators in $i\mathfrak{m}$ or $\mathfrak{m}$. Similar results hold for in the compact case: H= K ⋅ exp $i\mathfrak{m}$ and H = KAK.[13]

The properties of the totally geodesic subspace A can be shown directly. A is closed because the closure of A is a toral subgroup satisfying σ(a) = a−1, so its Lie algebra lies in $\mathfrak{m}$ and hence equals $\mathfrak{a}$ by maximality. A can be generated topologically by a single element exp X, so $\mathfrak{a}$ is the centralizer of X in $\mathfrak{m}$. In the K-orbit of any element of $\mathfrak{m}$ there is an element Y such that (X,Ad k Y) is minimized at k = 1. Setting k = exp tT with T in $\mathfrak{k}$, it follows that (X,[T,Y]) = 0 and hence [X,Y] = 0, so that Y must lie in $\mathfrak{a}$. Thus $\mathfrak{m}$ is the union of the conjugates of $\mathfrak{a}$. In particular some conjugate of X lies in any other choice of $\mathfrak{a}$, which centralizes that conjugate; so by maximality the only possibilities are conjugates of $\mathfrak{a}$. [14]

The decompositions

$\displaystyle{H=KAK,\,\,\,H = K\cdot \exp \mathfrak{m}}$

can be proved directly by applying the slice theorem for compact transformation groups to the action of K on H / K.[15] In fact the space H / K can be identified with

$\displaystyle{M=\{ \sigma(g)g^{-1}:g\in H\},}$

a closed submanifold of H, and the Cartan decomposition follows by showing that M is the union of the kAk−1 for k in K. Since this union is the continuous image of K × A, it is compact and connected. So it suffices to show that the union is open in M and for this it is enough to show each a in A has an open neighbourhood in this union. Now by computing derivatives at 0, the union contains an open neighbourhood of 1. If a is central the union is invariant under multiplication by a, so contains an open neighbourhood of a. If a is not central, write a = b2 with b in A. Then τ = Ad b − Ad b−1 is a skew-adjoint operator on $\mathfrak{h}$ anticommuting with σ, which can be regarded as a Z2-grading operator σ on $\mathfrak{h}$. By an Euler–Poincaré characteristic argument it follows that the superdimension of $\mathfrak{h}$ coincides with the superdimension of the kernel of τ. In other words

$\displaystyle{\mathrm{dim} \,\mathfrak{k} - \mathrm{dim} \,\mathfrak{k}_a = \mathrm{dim} \,\mathfrak{m} - \mathrm{dim} \,\mathfrak{m}_a,}$

where $\mathfrak{k}_a$ and $\mathfrak{m}_a$ are the subspaces fixed by Ad a. Let the orthogonal complement of $\mathfrak{k}_a$ in $\mathfrak{k}$ be $\mathfrak{k}_a^\perp$. Computing derivstives it follows that Ad eX (a eY), where X lies in $\mathfrak{k}_a^\perp$ and Y in $\mathfrak{m}_a$, is an open neighbourhood of a in the union. Here the terms a eY lie in the union by the argument for central a: indeed a is in the center of the identity component of the centralizer of a which is invariant under σ and contains A.

The dimension of $\mathfrak{a}$ is called the rank of the Hermitian symmetric space.

### Strongly orthogonal roots

In the case of Hermitian symmetric spaces, Harish-Chandra gave a canonical choice for $\mathfrak{a}$. This choice of $\mathfrak{a}$ is determined by taking a maximal torus T of H in K with Lie algebra $\mathfrak{t}$. Since the symmetry σ is implemented by an element of T lying in the centre of H, the root spaces $\mathfrak{g}_\alpha$ in $\mathfrak{g}$ are left invariant by σ. It acts as the identity on those contained in $\mathfrak{k}_{\mathbb{C}}$ and minus the identity on those in $\mathfrak{m}_{\mathbb{C}}$.

The roots with root spaces in $\mathfrak{k}_{\mathbb{C}}$ are called compact roots and those with root spaces in $\mathfrak{m}_{\mathbb{C}}$ are called noncompact roots. (This terminology originates from the symmetric space of noncompact type.) If H is simple, the generator Z of the centre of K can be used to define a set of positive roots, according to the sign of α(Z). With this choice of roots $\mathfrak{m}_+$ and $\mathfrak{m}_-$ are the direct sum of the root spaces $\mathfrak{g}_\alpha$ over positive and negative noncompact roots α. Root vectors Eα can be chosen so that

$\displaystyle{X_\alpha=E_\alpha + E_{-\alpha}, \,\,\, Y_\alpha=i(E_\alpha - E_{-\alpha})}$

lie in $\mathfrak{h}$. The simple roots α1, ...., αn are the indecomposable positive roots. These can be numbered so that αi vanishes on the center of $\mathfrak{h}$ for i, whereas α1 does not. Thus α1 is the unique noncompact simple root and the other simple roots are compact. Any positive noncompact root then has the form β = α1 + c2 α2 + ⋅⋅⋅ + cn αn with non-negative coefficients ci. These corfficients lead to a lexicographic order on positive roots. The coefficient of α1 is always one because $\mathfrak{m}_-$ is irreducible for K so is spanned by vectors obtained by successively applying the lowering operators E–α for simple compact roots α.

Two roots α and β are said to be strongly orthogonal if ±α ±β are not roots or zero, written α ≐ β. The highest positive root ψ1 is noncompact. Take ψ2 to be the highest noncompact positive root strongly orthogonal to ψ1 (for the lexicographic order). Then continue in this way taking ψ3i + 1 to be the highest noncompact positive root strongly orthogonal to ψ1, ..., ψi until the process terminates. The corresponding vectors

$\displaystyle{X_i= E_{\psi_i} + E_{-\psi_i}}$

lie in $\mathfrak{m}$ and commute by strong orthogonality. Their span $\mathfrak{a}$ is Harish-Chandra's canonical maximal Abelian subalgebra.[16] (As Sugiura later showed, having fixed T, the set of strongly orthogonal roots is uniquely determined up to applying an element in the Weyl group of K.[17])

Maximality can be checked by showing that if

$\displaystyle{[\sum c_\alpha E_\alpha + \overline{c_\alpha}E_{-\alpha}, E_{\psi_i} + E_{-\psi_i}]=0}$

for all i, then cα = 0 for all positive noncompact roots α different from the ψj's. This follows by showing inductively that if cα ≠ 0, then α is strongly orthogonal to ψ1, ψ2, ... a contradiction. Indeed the above relation shows ψi + α cannot be a root; and that if ψi – α is a root, then it would necessarily have the form β – ψi. If ψi – α were negative, then α would be a higher positive root than ψi, strongly orthogonal to the ψj with j < i, which is not possible; similarly if β – ψi were positive.

### Polysphere and polydisk theorem

Harish-Chandra's canonical choice of $\mathfrak{a}$ leads to a polydisk and polysphere theorem in H*/K and H/K. This result reduces the geometry to products of the prototypic example involving SL(2,C), SU(1,1) and SU(2), namely the unit disk inside the Riemann sphere.

In the case of H = SU(2) the symmetry σ is given by conjugation by the diagonal matrix with entries ±i so that

$\displaystyle{\sigma\begin{pmatrix} \alpha & \beta\\ -\overline{\beta} & \overline{\alpha}\end{pmatrix} = \begin{pmatrix} \alpha & -\beta\\ \overline{\beta} & \overline{\alpha}\end{pmatrix}}$

The fixed point subgroup is the maximal torus T, the diagonal matrices with entries e ±it. SU(2) acts on the Riemann sphere C ∪ ≈ transitively by Möbius transformations and T is the stabilizer of 0. SL(2,C), the complexification of SU(2), also acts by Möbius transformations and the stabiliser of 0 is the subgroup B of lower triangular matrices. The noncompact subgroup SU(1,1) acts with precisely three orbits: the open unit disk |z| < 1; the unit circle z = 1; and its exterior |z| > 1. Thus

$\displaystyle{\mathrm{SU}(1,1)/\mathbf{T} = \{z: |z|<1\} \,\,\, \subset \,\,\, B_+/\mathbf{T}_{\mathbb{C}} = \mathbb{C}\,\,\, \subset \,\,\,\mathrm{SL}(2,\mathbb{C})/B = \mathbb{C}\cup\{\infty\},}$

where B+ and TC denote the subgroups of upper triangular and diagonal matrices in SL(2,C). The middle term is the orbit of 0 under the upper unitriangular matrices

$\displaystyle{\begin{pmatrix} 1 & z\\ 0 & 1\end{pmatrix} =\exp \begin{pmatrix} 0 & z\\ 0 & 0\end{pmatrix}.}$

Now for each root ψi there is a homomorphism of πi of SU(2) into H which is compatible with the symmetries. It extends uniquely to a homomorphism of SL(2,C) into G. The images of the Lie algebras for different ψi's since they are strongly orthogonal. Thus there is a homomorphism π of the direct product SU(2)r into H compatible with the symmetries. It extends to a homomorphism of SL(2,C)r into G. The kernel of π is contained in the center (±1)r of SU(2)r which is fixed pointwise by the symmetry. So the image of the center under π lies in K. Thus there is an embedding of the polysphere (SU(2)/T)r into H / K = G / P and the polysphere contains the polydisk (SU(1,1)/T)r. The polysphere and polydisk are the direct product of r copies of the Riemann sphere and the unit disk. By the Cartan decompositions in SU(2) and SU(1,1), the polysphere is the orbit of TrA in H / K and the polydisk is the orbit of TrA*, where Tr = π(Tr) ⊆ K. On the other hand H = KAK and H* = K A* K.

Hence every element in the compact Hermitian symmetric space H / K is in the K-orbit of a point in the polysphere; and every element in the image under the Borel embedding of the noncompact Hermitian symmetric space H* / K is in the K-orbit of a point in the polydisk.[18]

### Harish-Chandra embedding

H* / K, the Hermitian symmetric space of noncompact type, lies in the image of $\exp \mathfrak m_+$, a dense open subset of H / K biholomorphic to $\mathfrak m_+$. The corresponding domain in $\mathfrak m_+$ is bounded. This is the Harish-Chandra embedding named after Harish-Chandra.

In fact Harish-Chandra showed the following properties of the space $\mathbf{X}=\exp (\mathfrak{m}_+)\cdot K_{\mathbb{C}} \cdot \exp(\mathfrak{m}_-)=\exp (\mathfrak{m}_+)\cdot P$:

1. As a space, X is the direct product of the three factors.
2. X is open in G.
3. X is dense in G.
4. X contains H*.
5. The closure of H* / K in X / P = $\exp \mathfrak{m}_+$ is compact.

In fact $M_\pm=\exp \mathfrak{m}_\pm$ are complex Abelian groups normalised by KC. Moreover $[\mathfrak{m}_+,\mathfrak{m}_-] \subset \mathfrak{k}_{\mathfrak{C}}$ since $[\mathfrak{m},\mathfrak{m}] \subset \mathfrak{k}$.

This implies PM+ = {1}. For if x = eX with X in $\mathfrak{m}_+$ lies in P, it must normalize M- and hence $\mathfrak{m}_-$. But if Y lies in $\mathfrak{m}_-$, then

$\displaystyle{Y=\mathrm{Ad}(X)\cdot Y= Y + [X,Y] + {1\over 2} [X,[X,Y]]\in \mathfrak{m}_+ \oplus \mathfrak{k}_{\mathbb{C}} \oplus \mathfrak{m}_-,}$

so that X commutes with $\mathfrak{m}_-$. But if X commutes with every noncompact root space, it must be 0, so x = 1. It follows that the multiplication map μ on M+ × P is injective so (1) follows. Similarly the derivative of μ at (x,p) is

$\displaystyle{\mu^\prime(X,Y)=\mathrm{Ad}(p^{-1})X + Y =\mathrm{Ad}(p^{-1})(X\oplus\mathrm{Ad}(p)Y),}$

which is injective, so (2) follows. For the special case H = SU(2), H* = SU(1,1) and G = SL(2,C) the remaining assertions are consequences of the identification with the Riemann sphere, C and unit disk. They can be applied to the groups defined for each root ψi. By the polysphere and polydisk theorem H*/K, X/P and H/K are the union of the K-translates of the polydisk, Cr and the polysphere. So H* lies in X, the closure of H*/K is compact in X/P, which is in turn dense in H/K.

Note that (2) and (3) are also consequences of the fact that the image of X in G/P is that of the big cell B+B in the Gauss decomposition of G.[19]

Using results on the restricted root system of the symmetric spaces H/K and H*/K, Hermann showed that the image of H*/K in $\mathfrak{m}_+$ is a generalized unit disk. In fact it is the convex set of X for which the operator norm of ad Im X is less than one.[20]

### Bounded symmetric domains

A bounded domain Ω in a complex vector space is said to be a bounded symmetric domain if for every x in Ω, there is an involutive biholomorphism σx of Ω for which x is an isolated fixed point. The Harish-Chandra embedding exhibits every Hermitian symmetric space of noncompact type H* / K as a bounded symmetric domain. The biholomorphism group of H* / K is equal to its isometry group H*.

Conversely every bounded symmetric domain arises in this way. Indeed given a bounded symmetric domain Ω, the Bergman kernel defines a metric on Ω, the Bergman metric, for which every biholomorphism is an isometry. This realizes Ω as a Hermitian symmetric space of noncompact type.[21]

### Classification

The irreducible bounded symmetric domains are called Cartan domains and are classified as follows.

Type complex dimension geometric interpretation
Ipq pq Complex p × q matrices with operator norm less than 1
IIn (n > 4) n(n − 1)/2 Complex antisymmetric n × n matrices with operator norm less than 1
IIIn (n > 1) n(n + 1)/2 Complex symmetric n × n matrices with operator norm less than 1
IVn n Lie-sphere:$|z|^2< {1\over 2} (1 + (z\cdot z)^2) < 1$
V 16 2 × 2 matrices over the Cayley algebra with operator norm less than 1
VI 27 3 × 3 Hermitian matrices over the Cayley algebra with operator norm less than 1

### Classical domains

In the classical cases (I–IV), the noncompact group can be realized by 2 × 2 block matrices[22]

$\displaystyle{g=\begin{pmatrix} A & B \\ C & D\end{pmatrix}}$

acting by generalized Möbius transformations

$\displaystyle{g(Z)=(AZ+B)(CZ+D)^{-1}.}$

The polydisk theorem takes the following concrete form in the classical cases:[23]

• Type Ipq (pq): for every p × q matrix M there are unitary matrices such that UMV is diagonal. In fact this follows from the polar decomposition for p × p matrices.
• Type IIIn: for every complex symmetric n × n matrix M there is a unitary matrix U such that UMUt is diagonal. This is proved by a classical argument of Siegel. Take V unitary so that V*M*MV is diagonal. Then VtMV is symmetric and its real and imaginary parts commute. Since they are real symmetric matrices they can be simultaneously diagonalized by a real orthogonal matrix W. So UMUt is diagonal if U = WVt.
• Type IIn: for every complex skew symmetric n × n matrix M there is a unitary matrix such that UMUt is made up of diagonal blocks $\begin{pmatrix} 0 & a\\ -a & 0\end{pmatrix}$ and one zero if n is odd. As in Siegel's artgument, this can be reduced to case where the real and imaginary parts of M commute. Any real skew-symmetric matrix can be reduced to the given canonical form by an orthogonal matrix and this can be done simultaneously for commuting matrices.
• Type IVn: by a transformation in SO(n) × SO(2) any vector can be transformed so that all but the first two coordinates are non-zero.

### Boundary components

The noncompact group H* acts on the complex Hermitian symmetric space H/K = G/P with only finitely many orbits. The orbit structure is described in detail in Wolf (1972). In particular the closure of the bounded domain H*/K has a unique closed orbit, which is the Shilov boundary of the domain. In general the orbits are unions of Hermitian symmetric spaces of lower dimension. The complex function theory of the domains, in particular the analogue of the Cauchy integral formulas, are described for the Cartan domains in Hua (1979). The closure of the bounded domain is the Baily–Borel compactification of H*/K.[24]

The boundary structure can be described using Cayley transforms. For each copy of SU(2) defined by one of the noncompact roots ψi, there is a Cayley transform ci which as a Möbius transformation maps the unit disk onto the upper half plane. Given a subset I of indices of the strongly orthogonal family ψ1, ..., ψr, the partial Cayley transform cI is defined as the product of the ci's with i in I in the product of the groups πi. Let G(I) be the centralizer of this product in G and H*(I) = H* ∩ G(I). Since σ leaves H*(I) invariant, there is a corresponding Hermitian symmetric space MI H*(I)/H*(I)∩KH*/K = M . The boundary component for the subset I is the union of the K-translates of cI MI. When I is the set of all indices, MI is a single point and the boundary component is the Shilov boundary. Moreover MI is in the closure of MJ if and only if IJ.[25]

## Geometric properties

Every Hermitian symmetric space is a Kähler manifold. They can be defined equivalently as Riemannian symmetric spaces with a parallel complex structure with respect to which the Riemannian metric is Hermitian. The complex structure is automatically preserved by the isometry group H of the metric, and so any Hermitian symmetric space M is a homogeneous complex manifold. Some examples are complex vector spaces and complex projective spaces, with their usual Hermitian metrics and Fubini–Study metrics, and the complex unit balls with suitable metrics so that they become complete and Riemannian symmetric. The compact Hermitian symmetric spaces are projective varieties, and admit a strictly larger Lie group G of biholomorphisms with respect to which they are homogeneous: in fact, they are generalized flag manifolds, i.e., G is semisimple and the stabilizer of a point is a parabolic subgroup P of G. Among (complex) generalized flag manifolds G/P, they are characterized as those for which the nilradical of the Lie algebra of P is abelian. The non-compact Hermitian symmetric spaces can be realized as bounded domains in complex vector spaces.

## Jordan algebras

Although the classical Hermitian symmetric spaces can be constructed by ad hoc methods, Jordan triple systems, or equivalently Jordan pairs, provide a uniform algebraic means of describing all the basic properties connected with a Hermitian symmetric space of compact type and its non-compact dual. This theory is described in detail in Koecher (1969) and Loos (1977) and summarized in Satake (1981). The development is in the reverse order from that using the structure theory of compact Lie groups. It starting point is the Hermitian symmetric space of noncompact type realized as a bounded symmetric domain. It can be described in terms of a Jordan pair or hermitian Jordan triple system. This Jordan algebra structure can be used to reconstruct the dual Hermitian symmetric space of compact type, including in particular all the associated Lie algebras and Lie groups.

The theory is easiest to describe when the irreducible compact Hermitian symmetric space is of tube type. In that case the space is determined by a simple real Lie algebra $\mathfrak{g}$ with negative definite Killing form. It must admit an action of SU(2) which only acts via the trivial and adjoint representation, both types occurring. Since $\mathfrak{g}$ is simple, this action is inner, so implemented by an inclusion of the Lie algebra of SU(2) in $\mathfrak{g}$. The complexification of $\mathfrak{g}$ decomposes as a direct sum of three eigenspaces for the diagonal matrices in SU(2). It is a three-graded complex Lie algebra, with the Weyl group element of SU(2) providing the involution. Each of the ±1 eigenspaces has the structure of a unital complex Jordan algebra explicitly arising as the complexification of a Euclidean Jordan algebra. It can be identified with the multiplicity space of the adjoint representation of SU(2) in $\mathfrak{g}$.

The description of irreducible Hermitian symmetric spaces of tube type starts from a simple Euclidean Jordan algebra E. It admits Jordan frames, i.e. sets of orthogonal minimal idempotents e1, ..., em. Any two are related by an automorphism of E, so that the integer m is an invariant called the rank of E. Moreover if A is the complexification of E, it has a unitary structure group. It is a subgroup of GL(A) preserving the natural complex inner product on A. Any element a in A has a polar decomposition a = u ∑ αi ai with αi ≥ 0. The spectral norm is defined by ||a|| = sup αi. The associated bounded symmetric domain is just the open unit ball D in A. There is a biholomorphism between D and the tube domain T = E + iC where C is the open self-dual convex cone of elements in E of the form a = u ∑ αi ai with u an automorphism of E and αi > 0. This gives two descriptions of the Hermitian symmetric space of noncompact type. There is a natural way of using mutations of the Jordan algebra A to compactify the space A. The compactification X is a complex manifold and the finite-dimensional Lie algebra $\mathfrak{g}$ of holomorphic vector fields on X can be determined explicitly. One parameter groups of biholomorphisms can be defined such that the corresponding holomorphic vector fields span $\mathfrak{g}$. This includes the group of all complex Moebius transformations corresponding to matrices in SL(2,C). The subgroup SU(1,1) leaves invariant the unit ball and its closure. The subgroup SL(2,R) leaves invariant the tube domain and its closure. The usual Cayley transform and its inverse, mapping the unit disk in C to the upper half plane, establishes analogous maps between D and T. The polydisk corresponds to the real and complex Jordan subalgebras generated by a fixed Jordan frame. It admits a transitive action of SU(2)m and this action extends to X. The group G generated by the one-parameter groups of biholomorphisms acts faithfully on $\mathfrak{g}$. The subgroup generated by the identity component K of the unitary structure group and the operators in SU(2)m. It defines a compact Lie group H which acts transitively on X. Thus H / K is the corresponding Hermitian symmetric space of compact type. The group G can be identified with the complexification of H. The subgroup H* leaving D invariant is a noncompact real form of G. It acts transitively on D so that H* / K is the dual Hermitian symmetric space of noncompact type. The inclusions DAX reproduce the Borel and Harish-Chandra embeddings. The classification of Hermitian symmetric spaces of tube type reduces to that of simple Euclidean Jordan algebras. These were classified by Jordan, von Neumann & Wigner (1934) in terms of Euclidean Hurwitz algebras, a special type of composition algebra.

In general a Hermitian symmetric space gives rise to a 3-graded Lie algebra with a period 2 conjugate linear automorphism switching the parts of degree ±1 and preserving the degree 0 part. This gives rise to the structure of a Jordan pair or hermitian Jordan triple system, to which Loos (1977) extended the theory of Jordan algebras. All irreducible Hermitian symmetric spaces can be constructed uniformly within this framework. Koecher (1969) constructed the irreducible Hermitian symmetric space of non-tube type from a simple Euclidean Jordan algebra together with a period 2 automorphism. The −1 eigenspace of the automorphism has the structure of a Jordan pair, which can be deduced from that of the larger Jordan algebra. In the non-tube type case corresponding to a Siegel domain of type II, there is no distinguished subgroup of real or complex Moebius transformations. For irreducible Hermitian symmetric spaces, tube type is characterized by the real dimension of the Shilov boundary S being equal to the complex dimension of D.

## Notes

1. ^ Knapp 1972
2. ^ a b c Wolf 2010
3. ^ See:
4. ^ Kobayashi & Nomizu 1996, pp. 149–150
5. ^ Kobayashi & Nomizu 1996, pp. 261–262
6. ^ See:
7. ^ See:
8. ^ Helgason 1978
9. ^ Mok 1989
10. ^ Helgason 1978, pp. 444–447,451–455
11. ^ See:
12. ^ Dieudonné 1977
13. ^ & Helgason 1978
14. ^ & Helgason 1978, p. 248
15. ^ See:
16. ^ See:
17. ^ Agaoka & Kaneda 2002
18. ^ See: &Mok 1989, pp. 88–94
19. ^ See:
20. ^ See:
21. ^ See:
22. ^ See:
23. ^ See:
24. ^ Borel & Ji 2006, pp. 77–91
25. ^ Wolf 1972, pp. 286–293

## References

• Agaoka, Yoshio; Kaneda, Eiji (2002), "Strongly orthogonal subsets in root systems", Hokkaido Math. J. 31: 107–136, doi:10.14492/hokmj/1350911773
• Arazy, Jonathan (1995), "A survey of invariant Hilbert spaces of analytic functions on bounded symmetric domains", Contemporary Mathematics 185: 7–65, doi:10.1090/conm/185/02147
• Borel, Armand (1952), Les espaces hermitiens symétriques, Exposé No. 62, Séminaire Bourbaki 2
• Borel, Armand; Ji, Lizhen (2006), Compactifications of Symmetric and Locally Symmetric Spaces, Springer, ISBN 0817632476
• Bourbaki, N. (1981), Groupes et Algèbres de Lie (Chapitres 7-8), Éléments de Mathématique, Masson, ISBN 3540339396
• Bourbaki, N. (1982), Groupes et Algèbres de Lie (Chapitre 9), Éléments de Mathématique, Masson, ISBN 354034392X
• Cartan, Élie (1935), "Sur les domaines bornés homogènes de l’espace des variables complexes", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 11: 116–162, doi:10.1007/bf02940719
• Dieudonné, J. (1977), Compact Lie groups and semisimple Lie groups, Chapter XXI, Treatise on analysis 5, Academic Press, ISBN 012215505X
• Duistermaat, J.J.; Kolk, A. (2000), Lie groups, Universitext, Springer, ISBN 3540152938
• Gilmore, Robert (1994), Lie groups, Lie algebras, and some of their applications, Krieger, ISBN 0-89464-759-8
• Helgason, Sigurdur (1978), Differential geometry, Lie groups, and symmetric spaces, Academic Press, ISBN 0-8218-2848-7 The standard book on Riemannian symmetric spaces.
• Helgason, Sigurdur (1994), Geometric Analysis on Symmetric Spaces, American Mathematical Society, ISBN 0-8218-1538-5
• Hua, L. K. (1979), Harmonic analysis of functions of several complex variables in the classical domains, Translations of Mathematical Monographs 6, American Mathematical Society, Providence, ISBN 0-8218-1556-3
• Jordan, P.; von Neumann, J.; Wigner, E. (1934), "On an algebraic generalization of the quantum mechanical formalism", Ann. of Math. 35: 29–64, doi:10.2307/1968117
• Knapp, Anthony W. (1972), "Bounded symmetric domains and holomorphic discrete series", in Boothby, William; Weiss, Guido, Symmetric spaces (Short Courses, Washington University), Pure and Applied Mathematics 8, Dekker, pp. 211–246
• Kobayashi, Shoshichi; Nomizu, Katsumi (1996), Foundations of differential geometry 2, Wiley-Interscience, ISBN 0-471-15732-5
• Koecher, Max (1969), An elementary approach to bounded symmetric domains, Lecture notes in mathematics, Rice University
• Loos, Ottmar (1977), Bounded symmetric domains and Jordan pairs, Mathematical lectures, University of California, Irvine
• Mok, Ngaiming (1989), Metric Rigidity Theorems on Hermitian Locally Symmetric Manifolds, World Scientific, ISBN 9971-5-0802-8
• Satake, Ichiro (1981), Algebraic Structures of Symmetric Domains, Princeton University Press, ISBN 9780691082714
• Siegel, Carl Ludwig (1943), "Symplectic Geometry", American Journal of Mathematics 65: 1–86, doi:10.2307/2371774
• Wolf, Joseph A. (1964), "On the Classification of Hermitian Symmetric Spaces", Indiana Univ. Math. J. 13: 489–495
• Wolf, Joseph A. (2010), Spaces of constant curvature, AMS Chelsea Publishing (6th ed.), American Mathematical Society, ISBN 0821852825. Chapter 8 contains a self-contained account of Hermitian symmetric spaces of compact type.
• Wolf, Joseph A. (1972), "Fine structure of Hermitian symmetric spaces", in Boothby, William; Weiss, Guido, Symmetric spaces (Short Courses, Washington University), Pure and Applied Mathematics 8, Dekker, pp. 271–357. This contains a detailed account of Hermitian symmetric spaces of noncompact type.


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path: root/src/3rdparty/freetype/src/autofit/afhints.h
blob: 92101de7ca9616b1cae954b686a05d02d8107467 (plain)
 ```1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 ``` ``````/***************************************************************************/ /* */ /* afhints.h */ /* */ /* Auto-fitter hinting routines (specification). */ /* */ /* Copyright 2003-2008, 2010-2012, 2014 by */ /* David Turner, Robert Wilhelm, and Werner Lemberg. */ /* */ /* This file is part of the FreeType project, and may only be used, */ /* modified, and distributed under the terms of the FreeType project */ /* license, LICENSE.TXT. By continuing to use, modify, or distribute */ /* this file you indicate that you have read the license and */ /* understand and accept it fully. */ /* */ /***************************************************************************/ #ifndef __AFHINTS_H__ #define __AFHINTS_H__ #include "aftypes.h" #define xxAF_SORT_SEGMENTS FT_BEGIN_HEADER /* * The definition of outline glyph hints. These are shared by all * writing system analysis routines (until now). */ typedef enum AF_Dimension_ { AF_DIMENSION_HORZ = 0, /* x coordinates, */ /* i.e., vertical segments & edges */ AF_DIMENSION_VERT = 1, /* y coordinates, */ /* i.e., horizontal segments & edges */ AF_DIMENSION_MAX /* do not remove */ } AF_Dimension; /* hint directions -- the values are computed so that two vectors are */ /* in opposite directions iff `dir1 + dir2 == 0' */ typedef enum AF_Direction_ { AF_DIR_NONE = 4, AF_DIR_RIGHT = 1, AF_DIR_LEFT = -1, AF_DIR_UP = 2, AF_DIR_DOWN = -2 } AF_Direction; /* * The following explanations are mostly taken from the article * * Real-Time Grid Fitting of Typographic Outlines * * by David Turner and Werner Lemberg * * http://www.tug.org/TUGboat/Articles/tb24-3/lemberg.pdf * * with appropriate updates. * * * Segments * * `af_{cjk,latin,...}_hints_compute_segments' are the functions to * find segments in an outline. * * A segment is a series of at least two consecutive points that are * approximately aligned along a coordinate axis. The analysis to do * so is specific to a writing system. * * * Edges * * `af_{cjk,latin,...}_hints_compute_edges' are the functions to find * edges. * * As soon as segments are defined, the auto-hinter groups them into * edges. An edge corresponds to a single position on the main * dimension that collects one or more segments (allowing for a small * threshold). * * As an example, the `latin' writing system first tries to grid-fit * edges, then to align segments on the edges unless it detects that * they form a serif. * * * A H * | | * | | * | | * | | * C | | F * +------<-----+ +-----<------+ * | B G | * | | * | | * +--------------->------------------+ * D E * * * Stems * * Stems are detected by `af_{cjk,latin,...}_hint_edges'. * * Segments need to be `linked' to other ones in order to detect stems. * A stem is made of two segments that face each other in opposite * directions and that are sufficiently close to each other. Using * vocabulary from the TrueType specification, stem segments form a * `black distance'. * * In the above ASCII drawing, the horizontal segments are BC, DE, and * FG; the vertical segments are AB, CD, EF, and GH. * * Each segment has at most one `best' candidate to form a black * distance, or no candidate at all. Notice that two distinct segments * can have the same candidate, which frequently means a serif. * * A stem is recognized by the following condition: * * best segment_1 = segment_2 && best segment_2 = segment_1 * * The best candidate is stored in field `link' in structure * `AF_Segment'. * * In the above ASCII drawing, the best candidate for both AB and CD is * GH, while the best candidate for GH is AB. Similarly, the best * candidate for EF and GH is AB, while the best candidate for AB is * GH. * * The detection and handling of stems is dependent on the writing * system. * * * Serifs * * Serifs are detected by `af_{cjk,latin,...}_hint_edges'. * * In comparison to a stem, a serif (as handled by the auto-hinter * module that takes care of the `latin' writing system) has * * best segment_1 = segment_2 && best segment_2 != segment_1 * * where segment_1 corresponds to the serif segment (CD and EF in the * above ASCII drawing). * * The best candidate is stored in field `serif' in structure * `AF_Segment' (and `link' is set to NULL). * * * Touched points * * A point is called `touched' if it has been processed somehow by the * auto-hinter. It basically means that it shouldn't be moved again * (or moved only under certain constraints to preserve the already * applied processing). * * * Flat and round segments * * Segments are `round' or `flat', depending on the series of points * that define them. A segment is round if the next and previous point * of an extremum (which can be either a single point or sequence of * points) are both conic or cubic control points. Otherwise, a * segment with an extremum is flat. * * * Strong Points * * Experience has shown that points not part of an edge need to be * interpolated linearly between their two closest edges, even if these * are not part of the contour of those particular points. Typical * candidates for this are * * - angle points (i.e., points where the `in' and `out' direction * differ greatly) * * - inflection points (i.e., where the `in' and `out' angles are the * same, but the curvature changes sign) [currently, such points * aren't handled specially in the auto-hinter] * * `af_glyph_hints_align_strong_points' is the function that takes * care of such situations; it is equivalent to the TrueType `IP' * hinting instruction. * * * Weak Points * * Other points in the outline must be interpolated using the * coordinates of their previous and next unfitted contour neighbours. * These are called `weak points' and are touched by the function * `af_glyph_hints_align_weak_points', equivalent to the TrueType `IUP' * hinting instruction. Typical candidates are control points and * points on the contour without a major direction. * * The major effect is to reduce possible distortion caused by * alignment of edges and strong points, thus weak points are processed * after strong points. */ /* point hint flags */ typedef enum AF_Flags_ { AF_FLAG_NONE = 0, /* point type flags */ AF_FLAG_CONIC = 1 << 0, AF_FLAG_CUBIC = 1 << 1, AF_FLAG_CONTROL = AF_FLAG_CONIC | AF_FLAG_CUBIC, /* point touch flags */ AF_FLAG_TOUCH_X = 1 << 2, AF_FLAG_TOUCH_Y = 1 << 3, /* candidates for weak interpolation have this flag set */ AF_FLAG_WEAK_INTERPOLATION = 1 << 4 } AF_Flags; /* edge hint flags */ typedef enum AF_Edge_Flags_ { AF_EDGE_NORMAL = 0, AF_EDGE_ROUND = 1 << 0, AF_EDGE_SERIF = 1 << 1, AF_EDGE_DONE = 1 << 2, AF_EDGE_NEUTRAL = 1 << 3 /* set if edge aligns to a neutral blue zone */ } AF_Edge_Flags; typedef struct AF_PointRec_* AF_Point; typedef struct AF_SegmentRec_* AF_Segment; typedef struct AF_EdgeRec_* AF_Edge; typedef struct AF_PointRec_ { FT_UShort flags; /* point flags used by hinter */ FT_Char in_dir; /* direction of inwards vector */ FT_Char out_dir; /* direction of outwards vector */ FT_Pos ox, oy; /* original, scaled position */ FT_Short fx, fy; /* original, unscaled position (in font units) */ FT_Pos x, y; /* current position */ FT_Pos u, v; /* current (x,y) or (y,x) depending on context */ AF_Point next; /* next point in contour */ AF_Point prev; /* previous point in contour */ } AF_PointRec; typedef struct AF_SegmentRec_ { FT_Byte flags; /* edge/segment flags for this segment */ FT_Char dir; /* segment direction */ FT_Short pos; /* position of segment */ FT_Short min_coord; /* minimum coordinate of segment */ FT_Short max_coord; /* maximum coordinate of segment */ FT_Short height; /* the hinted segment height */ AF_Edge edge; /* the segment's parent edge */ AF_Segment edge_next; /* link to next segment in parent edge */ AF_Segment link; /* (stem) link segment */ AF_Segment serif; /* primary segment for serifs */ FT_Pos num_linked; /* number of linked segments */ FT_Pos score; /* used during stem matching */ FT_Pos len; /* used during stem matching */ AF_Point first; /* first point in edge segment */ AF_Point last; /* last point in edge segment */ } AF_SegmentRec; typedef struct AF_EdgeRec_ { FT_Short fpos; /* original, unscaled position (in font units) */ FT_Pos opos; /* original, scaled position */ FT_Pos pos; /* current position */ FT_Byte flags; /* edge flags */ FT_Char dir; /* edge direction */ FT_Fixed scale; /* used to speed up interpolation between edges */ AF_Width blue_edge; /* non-NULL if this is a blue edge */ AF_Edge link; /* link edge */ AF_Edge serif; /* primary edge for serifs */ FT_Short num_linked; /* number of linked edges */ FT_Int score; /* used during stem matching */ AF_Segment first; /* first segment in edge */ AF_Segment last; /* last segment in edge */ } AF_EdgeRec; typedef struct AF_AxisHintsRec_ { FT_Int num_segments; /* number of used segments */ FT_Int max_segments; /* number of allocated segments */ AF_Segment segments; /* segments array */ #ifdef AF_SORT_SEGMENTS FT_Int mid_segments; #endif FT_Int num_edges; /* number of used edges */ FT_Int max_edges; /* number of allocated edges */ AF_Edge edges; /* edges array */ AF_Direction major_dir; /* either vertical or horizontal */ } AF_AxisHintsRec, *AF_AxisHints; typedef struct AF_GlyphHintsRec_ { FT_Memory memory; FT_Fixed x_scale; FT_Pos x_delta; FT_Fixed y_scale; FT_Pos y_delta; FT_Int max_points; /* number of allocated points */ FT_Int num_points; /* number of used points */ AF_Point points; /* points array */ FT_Int max_contours; /* number of allocated contours */ FT_Int num_contours; /* number of used contours */ AF_Point* contours; /* contours array */ AF_AxisHintsRec axis[AF_DIMENSION_MAX]; FT_UInt32 scaler_flags; /* copy of scaler flags */ FT_UInt32 other_flags; /* free for style-specific */ /* implementations */ AF_StyleMetrics metrics; FT_Pos xmin_delta; /* used for warping */ FT_Pos xmax_delta; } AF_GlyphHintsRec; #define AF_HINTS_TEST_SCALER( h, f ) ( (h)->scaler_flags & (f) ) #define AF_HINTS_TEST_OTHER( h, f ) ( (h)->other_flags & (f) ) #ifdef FT_DEBUG_AUTOFIT #define AF_HINTS_DO_HORIZONTAL( h ) \ ( !_af_debug_disable_horz_hints && \ !AF_HINTS_TEST_SCALER( h, AF_SCALER_FLAG_NO_HORIZONTAL ) ) #define AF_HINTS_DO_VERTICAL( h ) \ ( !_af_debug_disable_vert_hints && \ !AF_HINTS_TEST_SCALER( h, AF_SCALER_FLAG_NO_VERTICAL ) ) #define AF_HINTS_DO_ADVANCE( h ) \ !AF_HINTS_TEST_SCALER( h, AF_SCALER_FLAG_NO_ADVANCE ) #define AF_HINTS_DO_BLUES( h ) ( !_af_debug_disable_blue_hints ) #else /* !FT_DEBUG_AUTOFIT */ #define AF_HINTS_DO_HORIZONTAL( h ) \ !AF_HINTS_TEST_SCALER( h, AF_SCALER_FLAG_NO_HORIZONTAL ) #define AF_HINTS_DO_VERTICAL( h ) \ !AF_HINTS_TEST_SCALER( h, AF_SCALER_FLAG_NO_VERTICAL ) #define AF_HINTS_DO_ADVANCE( h ) \ !AF_HINTS_TEST_SCALER( h, AF_SCALER_FLAG_NO_ADVANCE ) #define AF_HINTS_DO_BLUES( h ) 1 #endif /* !FT_DEBUG_AUTOFIT */ FT_LOCAL( AF_Direction ) af_direction_compute( FT_Pos dx, FT_Pos dy ); FT_LOCAL( FT_Error ) af_axis_hints_new_segment( AF_AxisHints axis, FT_Memory memory, AF_Segment *asegment ); FT_LOCAL( FT_Error) af_axis_hints_new_edge( AF_AxisHints axis, FT_Int fpos, AF_Direction dir, FT_Memory memory, AF_Edge *edge ); FT_LOCAL( void ) af_glyph_hints_init( AF_GlyphHints hints, FT_Memory memory ); FT_LOCAL( void ) af_glyph_hints_rescale( AF_GlyphHints hints, AF_StyleMetrics metrics ); FT_LOCAL( FT_Error ) af_glyph_hints_reload( AF_GlyphHints hints, FT_Outline* outline ); FT_LOCAL( void ) af_glyph_hints_save( AF_GlyphHints hints, FT_Outline* outline ); FT_LOCAL( void ) af_glyph_hints_align_edge_points( AF_GlyphHints hints, AF_Dimension dim ); FT_LOCAL( void ) af_glyph_hints_align_strong_points( AF_GlyphHints hints, AF_Dimension dim ); FT_LOCAL( void ) af_glyph_hints_align_weak_points( AF_GlyphHints hints, AF_Dimension dim ); #ifdef AF_CONFIG_OPTION_USE_WARPER FT_LOCAL( void ) af_glyph_hints_scale_dim( AF_GlyphHints hints, AF_Dimension dim, FT_Fixed scale, FT_Pos delta ); #endif FT_LOCAL( void ) af_glyph_hints_done( AF_GlyphHints hints ); /* */ #define AF_SEGMENT_LEN( seg ) ( (seg)->max_coord - (seg)->min_coord ) #define AF_SEGMENT_DIST( seg1, seg2 ) ( ( (seg1)->pos > (seg2)->pos ) \ ? (seg1)->pos - (seg2)->pos \ : (seg2)->pos - (seg1)->pos ) FT_END_HEADER #endif /* __AFHINTS_H__ */ /* END */ ``````


text:
# chickadee » srfi-127 » lseq=?

##### Identifier search
lseq=? elt=? lseq1 lseq2procedure

Determines lseq equality, given an element-equality procedure. Two lseqs are equal if they are of the same length, and their corresponding elements are equal, as determined by elt=?. When elt=? is called, its first argument is always from lseq1 and its second argument is from lseq2.

The dynamic order in which the elt=? procedure is applied to pairs of elements is not specified.

The elt=? procedure must be consistent with eq?. This implies that two lseqs which are eq? are always lseq=?, as well; implementations may exploit this fact to "short-cut" the element-by-element equality tests.


text:
# How to write this: “------------------\r\n“

When I try to write the above expression I get an error stating: "Undefined control sequence".

The same when I try to write this: ‘\r\n’.

Any ideas?

• You should add more proper data to what you have done so far.. Jun 2 at 7:59
• Welcome to TeX.SX! You can have a look at our starter guide to familiarize yourself further with our format. For future questions, it might help to explain a bit what you're doing and why. For example, I agree with Fran that it looks like you're trying to typeset code of some sort, in which case there are a few package that make that go much easier. Jun 2 at 12:48

Any character sequence (word or single character) preceded by a backslash (\<whatever>) is a command (=control sequence) in LaTeX, but whereas \r is defined (to type characters as \r{a}), \n has not a predefined function (is "undefined"), so your alternatives to remove the error are type it verbatim (\verb|\n|) or as plain text (\textbackslash n), or
defining it to make something, i.e., make a LaTeX macro:

\newcommand\n{\textbackslash n}
\n

If what you show is part of a code listing, then you should consider use a verbatim environment:

\begin{verbatim}
----------\n\t
\end{verbatim}

Or use some package for powered verbatim typesetting or syntax highlight.

You may use \textbackslash{} to printout the \ character without a problem.

\texttt{--------------\textbackslash{r}\textbackslash{n}}


text:
Home > Taylor Series > Taylor Series Error Estimate

# Taylor Series Error Estimate

## Contents

Click on this to open the Tools menu. Furthermore, then the partial derivatives of f exist at a and the differential of f at a is given by d f ( a ) ( v ) = ∂ f Khan Academy 54,407 views 9:18 Loading more suggestions... Taylor's theorem also generalizes to multivariate and vector valued functions f : R n → R m {\displaystyle f\colon \mathbb − 1 ^ − 0\rightarrow \mathbb − 9 ^ − 8} have a peek at this web-site

more hot questions question feed about us tour help blog chat data legal privacy policy work here advertising info mobile contact us feedback Technology Life / Arts Culture / Recreation Science Close the Menu The equations overlap the text! Next, the remainder is defined to be, So, the remainder is really just the error between the function  and the nth degree Taylor polynomial for a given n. For example, if G(t) is continuous on the closed interval and differentiable with a non-vanishing derivative on the open interval between a and x, then R k ( x ) = https://www.khanacademy.org/math/calculus-home/series-calc/taylor-series-calc/v/error-or-remainder-of-a-taylor-polynomial-approximation

## Taylor Series Error Estimation Calculator

Then Cauchy's integral formula with a positive parametrization γ(t)=z + reit of the circle S(z, r) with t ∈ [0, 2π] gives f ( z ) = 1 2 π i Khan Academy 241,634 views 11:27 Lagrange Error Bound - Duration: 4:56. Then there exists a function hk: R → R such that f ( x ) = f ( a ) + f ′ ( a ) ( x − a ) For any k∈N and r>0 there exists Mk,r>0 such that the remainder term for the k-th order Taylor polynomial of f satisfies(*).

This function was plotted above to illustrate the fact that some elementary functions cannot be approximated by Taylor polynomials in neighborhoods of the center of expansion which are too large. Show Answer There are a variety of ways to download pdf versions of the material on the site. The exact content of "Taylor's theorem" is not universally agreed upon. Lagrange Error Bound Calculator Please try the request again.

Hence each of the first k−1 derivatives of the numerator in h k ( x ) {\displaystyle h_{k}(x)} vanishes at x = a {\displaystyle x=a} , and the same is true Taylor Polynomial Approximation Calculator Solution Here are the derivatives for this problem.                                 This Taylor series will terminate after .  This will always happen when we are finding the Taylor Series of a The zero function is analytic and every coefficient in its Taylor series is zero. To find out, use the remainder term: cos 1 = T6(x) + R6(x) Adding the associated remainder term changes this approximation into an equation.

Naturally, in the case of analytic functions one can estimate the remainder term Rk(x) by the tail of the sequence of the derivatives f′(a) at the center of the expansion, but Remainder Estimation Theorem There was a network issue here that caused the site to only be accessible from on campus. Once you have made a selection from this second menu up to four links (depending on whether or not practice and assignment problems are available for that page) will show up Note that these are identical to those in the "Site Help" menu.

## Taylor Polynomial Approximation Calculator

Links to the download page can be found in the Download Menu, the Misc Links Menu and at the bottom of each page. http://tutorial.math.lamar.edu/Classes/CalcII/TaylorSeries.aspx Krista King 59,295 views 8:23 Alternating series error estimation - Duration: 9:18. Taylor Series Error Estimation Calculator With this definition note that we can then write the function as, We now have the following Theorem. Lagrange Error Formula For analytic functions the Taylor polynomials at a given point are finite order truncations of its Taylor series, which completely determines the function in some neighborhood of the point.

Algebra [Notes] [Practice Problems] [Assignment Problems] Calculus I [Notes] [Practice Problems] [Assignment Problems] Calculus II [Notes] [Practice Problems] [Assignment Problems] Calculus III [Notes] [Practice Problems] [Assignment Problems] Differential Equations [Notes] Extras Check This Out Because it was (apparently) working when I left campus yesterday I didn't realize the site was not accessible from off campus until late last night when it was too late to It is especially true for some exponents and occasionally a "double prime" 2nd derivative notation will look like a "single prime". Professor Leonard 99,296 views 3:01:45 Finding a Maclaurin Series Expansion - Another Example 1 - Duration: 4:50. Taylor Series Remainder Calculator

Therefore, Taylor series of f centered at 0 converges on B(0, 1) and it does not converge for any z ∈ C with |z|>1 due to the poles at i and Do you see a pattern? So, it looks like, Using the third derivative gives, Using the fourth derivative gives, Hopefully by this time you’ve seen the pattern here.  http://accessdtv.com/taylor-series/taylor-and-maclaurin-series-error.html Example 7  Find the Taylor Series for  about .

Example 3  Find the Taylor Series for  about . Taylor's Inequality What can I do to fix this? If we were to write out the sum without the summation notation this would clearly be an nth degree polynomial.  We’ll see a nice application of Taylor polynomials in the next

## Here is the Taylor Series for this function.                                                     Now, let’s work one of the easier examples in this section.  The problem for most students is that it may

Sometimes these constants can be chosen in such way that Mk,r → 0 when k → ∞ and r stays fixed. Graph of f(x)=ex (blue) with its quadratic approximation P2(x) = 1 + x + x2/2 (red) at a=0. Sign in to make your opinion count. Lagrange Error Bound Problems These often do not suffer from the same problems.

Ideally, the remainder term gives you the precise difference between the value of a function and the approximation Tn(x). Remark. You should see an icon that looks like a piece of paper torn in half. have a peek here The Taylor Series and Other Mathematical Concepts - Duration: 1:13:39.

So, we’ve seen quite a few examples of Taylor Series to this point and in all of them we were able to find general formulas for the series.  This won’t always The links for the page you are on will be highlighted so you can easily find them. Player claims their wizard character knows everything (from books). The system returned: (22) Invalid argument The remote host or network may be down.

Krista King 14,459 views 12:03 Taylor's Theorem with Remainder - Duration: 9:00. It may well be that an infinitely many times differentiable function f has a Taylor series at a which converges on some open neighborhood of a, but the limit function Tf Since ex is increasing by (*), we can simply use ex≤1 for x∈[−1,0] to estimate the remainder on the subinterval [−1,0]. This same proof applies for the Riemann integral assuming that f(k) is continuous on the closed interval and differentiable on the open interval between a and x, and this leads to

The strategy of the proof is to apply the one-variable case of Taylor's theorem to the restriction of f to the line segment adjoining x and a.[13] Parametrize the line segment Your cache administrator is webmaster. The same is true if all the (k−1)-th order partial derivatives of f exist in some neighborhood of a and are differentiable at a.[10] Then we say that f is k Put Internet Explorer 10 in Compatibility Mode Look to the right side of the address bar at the top of the Internet Explorer window.

The graph of y = P1(x) is the tangent line to the graph of f at x = a. I am hoping they update the program in the future to address this. Professor Leonard 42,589 views 1:34:10 Using Taylor's Inequality to get an error bound on 3rd degree Taylor Polynomail Ch8R 6 - Duration: 7:23. Solution This is actually one of the easier Taylor Series that we’ll be asked to compute.  To find the Taylor Series for a function we will need to determine a general

The fundamental theorem of calculus states that f ( x ) = f ( a ) + ∫ a x f ′ ( t ) d t . {\displaystyle f(x)=f(a)+\int _{a}^{x}\,f'(t)\,dt.} Show Answer Answer/solutions to the assignment problems do not exist. Unfortunately there were a small number of those as well that were VERY demanding of my time and generally did not understand that I was not going to be available 24 Example 8  Find the Taylor Series for  about .

It's kind of hard to find the potential typo if all you write is "The 2 in problem 1 should be a 3" (and yes I've gotten handful of typo reports If a real-valued function f is differentiable at the point a then it has a linear approximation at the point a. One also obtains the Cauchy's estimates[9] | f ( k ) ( z ) | ⩽ k ! 2 π ∫ γ M r | w − z | k +


text:
# Past the Peak? Excess Deaths in England and Wales

Since my previous post, the Office for National Statistics has posted updated data on weekly deaths, and we’ve updated rcovidUK accordingly. Here’s where things stand:

With the caveat that the date at which a death is reported need not agree with the date at which it actually occurred, excess deaths appear to have peaked and begun to decline in each of the ten regions. (Recently the ONS has started posting data on occurrences rather than reports, but these are unavailable for the historical comparison we’re making here.) As in my earlier post, the red curve is an equally-weighted average of reported weekly deaths in a given region for the past five years, up to and including 2019. Weeks are defined by the ONS to end on Fridays. This means that a given week does not necessarily correspond to the same days of the year across years, but always exactly one of each day of the week: Monday, Tuesday, etc. This eliminates seasonality from day-of-the-week effects, but creates the possibility of neglected seasonality from holidays that move across weeks during different years, e.g. Easter. For each point on the red curve, we have added simple error bars: $$\pm 2 \times \text{SE}$$ where SE is the usual standard error for a mean, ignoring any possible dependence between deaths in the same week across different years. The blue curve depicts weekly deaths in 2020.

We had hoped to include data from Scotland in this update, but it turns out that the NRS, Scotland’s equivalent of the ONS uses a different definition of weeks of the year, so it wouldn’t be easy to line up the respective series for purposes of comparison. We may revisit this later on.

If you’d like to replicate our plots, the R code is as follows:

library(tidyverse)
library(rcovidUK)

#Set the most recent week of data for 2020
week_2020 <- ONSweekly %>%
filter(year == 2020, reg_nm == "North East", !is.na(deaths)) %>%
pull(week) %>%
max

df_2020 <- ONSweekly %>%
filter(year == 2020)

df_prev5 <- ONSweekly %>%
filter(year < 2020 & year >= 2015) # Use last 5 years for average

df_prev5 %>%
group_by(reg_nm, week) %>%
summarise(deaths_mean = mean(deaths),
deaths_sd = sd(deaths),
n = n(),
se = deaths_sd/sqrt(n)) %>%
mutate(year = "2015-2019") %>%
rename(deaths = deaths_mean) %>%
bind_rows(df_2020) %>%
filter(week<=week_2020) %>%
ggplot(aes(x=week, y = deaths, col = year, group = reg_id)) +
geom_errorbar(aes(ymin=deaths - 2 * se, ymax=deaths + 2 * se), width=.2, colour="black")+
geom_line(size = 1) +
facet_wrap(~reg_nm, ncol = 2) +
scale_color_brewer(palette = "Set1",
labels = c("Av. Past 5 Years", "2020")) +
labs(x= "Week from Start of Year",
y = "Deaths per Week") +
theme_bw()+
theme(legend.position = 'top',
legend.title = element_blank(),
legend.key.size = unit(1, "cm"))


text:
1. Other
2. Other
3. if is a linear transformation show that there exists a...

# Question: if is a linear transformation show that there exists a...

###### Question details

If is a linear transformation, show that there exists a number such that   for all .


text:
Driving home from school one day, you spot a ball rolling out into the street (see the figure ). You brake for 1.30 s, slowing your 980-{\rm kg} car from 16.0 \rm m/s to 9.50 \rm m/s.
What was the magnitude of the average force exerted on your car during braking?
F =

\rm N

Part B
What was the direction of the average force exerted on your car during braking?

To the direction of motion.
Opposite to the direction of motion.


text:
## electron density function

https://doi.org/10.1351/goldbook.ET07024
The electron @[email protected] distribution function, $$\rho$$, defined as $\rho (\mathbf{r}) = n\ \int \Psi ^{\text{*}}\left[\mathbf{r}(1),\mathbf{r}(2)\,\text{...}\,\mathbf{r}(n)\right]\ \Psi \left[\mathbf{r}(1),\mathbf{r}(2)\,\text{...}\,\mathbf{r}(n)\right]\text{d}\mathbf{r}(2)\,\text{...}\,\text{d}\mathbf{r}(n)$ where $$\Psi$$ is an electronic wave-function and integration is made over the coordinates of all but the first electron of $$n$$. The physical interpretation of the @[email protected] function is that $$\rho \ \mathrm{d}\mathbf{\mathbf{r}}$$ gives the @[email protected] of finding an electron in a volume element $$\mathrm{d}\mathbf{\mathbf{r}}$$, i.e., @[email protected] in this volume.
Source:
PAC, 1999, 71, 1919. (Glossary of terms used in theoretical organic chemistry) on page 1937 [Terms] [Paper]


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# Using properties of sets, show that (i) A ∪ ( A ∩ B ) = A

Q9. Using properties of sets, show that

(i) A $\cup$ ( A $\cap$ B ) = A

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(i) A $\cup$ ( A $\cap$ B ) = A

We know that  A $\subset$ A

and       A $\cap$ B $\subset$ A

A $\cup$ ( A $\cap$ B )  $\subset$  A

and also , A $\subset$  A $\cup$ ( A $\cap$ B )

Hence,  A $\cup$ ( A $\cap$ B ) = A

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Contents

Idea

For $A$ a commutative ring and $p$ a prime number, then there is the topological completion at $(p)$, the p-completion $A_p^\wedge \coloneqq \underset{\leftarrow}{\lim}_n A/(p^n A)$. The analytic completion is instead the quotient

$A[ [ x ] ]/(x-p)A[ [ x ] ] \,.$

For $A= \mathbb{Z}$ the integers both constructions agree, up to isomorphism, and yield the p-adic integers. In general though they are different.

References

Last revised on July 21, 2014 at 06:22:23. See the history of this page for a list of all contributions to it.