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# Wigner–Eckart theorem

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The Wigner–Eckart theorem is a theorem of representation theory and quantum mechanics. It states that matrix elements of spherical tensor operators in the basis of angular momentum eigenstates can be expressed as the product of two factors, one of which is independent of angular momentum orientation, and the other a Clebsch–Gordan coefficient. The name derives from physicists Eugene Wigner and Carl Eckart, who developed the formalism as a link between the symmetry transformation groups of space (applied to the Schrödinger equations) and the laws of conservation of energy, momentum, and angular momentum.[1]

Mathematically, the Wigner–Eckart theorem is generally stated in the following way. Given a tensor operator ${\displaystyle T^{(k)}}$ and two states of angular momenta ${\displaystyle j}$ and ${\displaystyle j'}$, there exists a constant ${\displaystyle \langle j\|T^{(k)}\|j'\rangle }$ such that for all ${\displaystyle m}$, ${\displaystyle m'}$, and ${\displaystyle q}$, the following equation is satisfied:

${\displaystyle \langle j\,m|T_{q}^{(k)}|j'\,m'\rangle =\langle j'\,m'\,k\,q|j\,m\rangle \langle j\|T^{(k)}\|j'\rangle ,}$

where

• ${\displaystyle T_{q}^{(k)}}$ is the q-th component of the spherical tensor operator ${\displaystyle T^{(k)}}$ of rank k,[2]
• ${\displaystyle |jm\rangle }$ denotes an eigenstate of total angular momentum J2 and its z component Jz,
• ${\displaystyle \langle j'm'kq|jm\rangle }$ is the Clebsch–Gordan coefficient for coupling j with k to get j,
• ${\displaystyle \langle j\|T^{(k)}\|j'\rangle }$ denotes[3] some value that does not depend on m, m, nor q and is referred to as the reduced matrix element.

The Wigner–Eckart theorem states indeed that operating with a spherical tensor operator of rank k on an angular momentum eigenstate is like adding a state with angular momentum k to the state. The matrix element one finds for the spherical tensor operator is proportional to a Clebsch–Gordan coefficient, which arises when considering adding two angular momenta. When stated another way, one can say that the Wigner–Eckart theorem is a theorem that tells how vector operators behave in a subspace. Within a given subspace, a component of a vector operator will behave in a way proportional to the same component of the angular momentum operator. This definition is given in the book Quantum Mechanics by Cohen–Tannoudji, Diu and Laloe.

## Background and overview

### Motivating example: position operator matrix elements for 4d → 2p transition

Let's say we want to calculate transition dipole moments for an electron transition from a 4d to a 2p orbital of a hydrogen atom, i.e. the matrix elements of the form ${\displaystyle \langle 2p,m_{1}|r_{i}|4d,m_{2}\rangle }$, where ri is either the x, y, or z component of the position operator, and m1, m2 are the magnetic quantum numbers that distinguish different orbitals within the 2p or 4d subshell. If we do this directly, it involves calculating 45 different integrals: there are 3 possibilities for m1 (−1, 0, 1), 5 possibilities for m2 (−2, −1, 0, 1, 2), and 3 possibilities for i, so the total is 3 × 5 × 3 = 45.

The Wigner–Eckart theorem allows one to obtain the same information after evaluating just one of those 45 integrals (any of them can be used, as long as it is nonzero). Then the other 44 integrals can be inferred from that first one—without the need to write down any wavefunctions or evaluate any integrals—with the help of Clebsch–Gordan coefficients, which can be easily looked up in a table or computed by hand or computer.

### Qualitative summary of proof

The Wigner–Eckart theorem works because all 45 of these different calculations are related to each other by rotations. If an electron is in one of the 2p orbitals, rotating the system will generally move it into a different 2p orbital (usually it will wind up in a quantum superposition of all three basis states, m = +1, 0, −1). Similarly, if an electron is in one of the 4d orbitals, rotating the system will move it into a different 4d orbital. Finally, an analogous statement is true for the position operator: when the system is rotated, the three different components of the position operator are effectively interchanged or mixed.

If we start by knowing just one of the 45 values (say, we know that ${\displaystyle \langle 2p,m_{1}|r_{i}|4d,m_{2}\rangle =K}$) and then we rotate the system, we can infer that K is also the matrix element between the rotated version of ${\displaystyle \langle 2p,m_{1}|}$, the rotated version of ${\displaystyle r_{i}}$, and the rotated version of ${\displaystyle |4d,m_{2}\rangle }$. This gives an algebraic relation involving K and some or all of the 44 unknown matrix elements. Different rotations of the system lead to different algebraic relations, and it turns out that there is enough information to figure out all of the matrix elements in this way.

(In practice, when working through this math, we usually apply angular momentum operators to the states, rather than rotating the states. But this is fundamentally the same thing, because of the close mathematical relation between rotations and angular momentum operators.)

### In terms of representation theory

To state these observations more precisely and to prove them, it helps to invoke the mathematics of representation theory. For example, the set of all possible 4d orbitals (i.e., the 5 states m = −2, −1, 0, 1, 2 and their quantum superpositions) form a 5-dimensional abstract vector space. Rotating the system transforms these states into each other, so this is an example of a "group representation", in this case, the 5-dimensional irreducible representation ("irrep") of the rotation group SU(2) or SO(3), also called the "spin-2 representation". Similarly, the 2p quantum states form a 3-dimensional irrep (called "spin-1"), and the components of the position operator also form the 3-dimensional "spin-1" irrep.

Now consider the matrix elements ${\displaystyle \langle 2p,m_{1}|r_{i}|4d,m_{2}\rangle }$. It turns out that these are transformed by rotations according to the direct product of those three representations, i.e. the spin-1 representation of the 2p orbitals, the spin-1 representation of the components of r, and the spin-2 representation of the 4d orbitals. This direct product, a 45-dimensional representation of SU(2), is not an irreducible representation, instead it is the direct sum of a spin-4 representation, two spin-3 representations, three spin-2 representations, two spin-1 representations, and a spin-0 (i.e. trivial) representation. The nonzero matrix elements can only come from the spin-0 subspace. The Wigner–Eckart theorem works because the direct product decomposition contains one and only one spin-0 subspace, which implies that all the matrix elements are determined by a single scale factor.

Apart from the overall scale factor, calculating the matrix element ${\displaystyle \langle 2p,m_{1}|r_{i}|4d,m_{2}\rangle }$ is equivalent to calculating the projection of the corresponding abstract vector (in 45-dimensional space) onto the spin-0 subspace. The results of this calculation are the Clebsch–Gordan coefficients. The key qualitative aspect of the Clebsch–Gordan decomposition that makes the argument work is that in the decomposition of the tensor product of two irreducible representations, each irreducible representation occurs only once. This allows Schur's lemma to be used.[4]

## Proof

Starting with the definition of a spherical tensor operator, we have

${\displaystyle [J_{\pm },T_{q}^{(k)}]=\hbar {\sqrt {(k\mp q)(k\pm q+1)}}T_{q\pm 1}^{(k)},}$

which we use to then calculate

{\displaystyle {\begin{aligned}&\langle j\,m|[J_{\pm },T_{q}^{(k)}]|j'\,m'\rangle =\hbar {\sqrt {(k\mp q)(k\pm q+1)}}\,\langle j\,m|T_{q\pm 1}^{(k)}|j'\,m'\rangle .\end{aligned}}}

If we expand the commutator on the LHS by calculating the action of the J± on the bra and ket, then we get

{\displaystyle {\begin{aligned}\langle j\,m|[J_{\pm },T_{q}^{(k)}]|j'\,m'\rangle ={}&\hslash {\sqrt {(j\pm m)(j\mp m+1)}}\,\langle j\,(m\mp 1)|T_{q}^{(k)}|j'\,m'\rangle \\&-\hslash {\sqrt {(j'\mp m')(j'\pm m'+1)}}\,\langle j\,m|T_{q}^{(k)}|j'\,(m'\pm 1)\rangle .\end{aligned}}}

We may combine these two results to get

{\displaystyle {\begin{aligned}{\sqrt {(j\pm m)(j\mp m+1)}}\langle j\,(m\mp 1)|T_{q}^{(k)}|j'\,m'\rangle =&{\sqrt {(j'\mp m')(j'\pm m'+1)}}\,\langle j\,m|T_{q}^{(k)}|j'\,(m'\pm 1)\rangle \\&+{\sqrt {(k\mp q)(k\pm q+1)}}\,\langle j\,m|T_{q\pm 1}^{(k)}|j'\,m'\rangle .\end{aligned}}}

This recursion relation for the matrix elements closely resembles that of the Clebsch–Gordan coefficient. In fact, both are of the form c ab, c xc = 0. We therefore have two sets of linear homogeneous equations:

{\displaystyle {\begin{aligned}\sum _{c}a_{b,c}x_{c}&=0,&\sum _{c}a_{b,c}y_{c}&=0.\end{aligned}}}

one for the Clebsch–Gordan coefficients (xc) and one for the matrix elements (yc). It is not possible to exactly solve for xc. We can only say that the ratios are equal, that is

${\displaystyle {\frac {x_{c}}{x_{d}}}={\frac {y_{c}}{y_{d}}}}$

or that xcyc, where the coefficient of proportionality is independent of the indices. Hence, by comparing recursion relations, we can identify the Clebsch–Gordan coefficient j1 m1 j2 (m2 ± 1)|j m with the matrix element jm′|T(k)q ± 1|j m, then we may write

${\displaystyle \langle j'\,m'|T_{q\pm 1}^{(k)}|j\,m\rangle \propto \langle j\,m\,k\,(q\pm 1)|j'\,m'\rangle .}$

## Alternative conventions

There are different conventions for the reduced matrix elements. One convention, used by Racah[5] and Wigner,[6] includes an additional phase and normalization factor,

${\displaystyle \langle j\,m|T_{q}^{(k)}|j'\,m'\rangle ={\frac {(-1)^{2k}\langle j'\,m'\,k\,q|j\,m\rangle \langle j\|T^{(k)}\|j'\rangle _{\mathrm {R} }}{\sqrt {2j+1}}}=(-1)^{j-m}{\begin{pmatrix}j&k&j'\\-m&q&m'\end{pmatrix}}\langle j\|T^{(k)}\|j'\rangle _{\mathrm {R} }.}$

where the 2 × 3 array denotes the 3-j symbol. (Since in practice k is often integral, the (−1)2 k factor is sometimes omitted in literature.) With this choice of normalization, the reduced matrix element satisfies the relation:

${\displaystyle \langle j\|T^{\dagger (k)}\|j'\rangle _{\mathrm {R} }=(-1)^{k+j'-j}\langle j'\|T^{(k)}\|j\rangle _{\mathrm {R} }^{*},}$

where the Hermitian adjoint is defined with the kq convention. Although this relation is not affected by the presence or absence of the (−1)2 k phase factor in the definition of the reduced matrix element, it is affected by the phase convention for the Hermitian adjoint.

Another convention for reduced matrix elements is that of Sakurai's Modern Quantum Mechanics:

${\displaystyle \langle j\,m|T_{q}^{(k)}|j'\,m'\rangle ={\frac {\langle j'\,m'\,k\,q|j\,m\rangle \langle j\|T^{(k)}\|j'\rangle _{\mathrm {S} }}{\sqrt {2j'+1}}}.}$

## Example

Consider the position expectation value n j m|x|n j m. This matrix element is the expectation value of a Cartesian operator in a spherically symmetric hydrogen-atom-eigenstate basis, which is a nontrivial problem. However, the Wigner–Eckart theorem simplifies the problem. (In fact, we could obtain the solution quickly using parity, although a slightly longer route will be taken.)

We know that x is one component of r, which is a vector. Since vectors are rank-1 spherical tensor operators, it follows that x must be some linear combination of a rank-1 spherical tensor T(1)q with q ∈ {−1, 0, 1}. In fact, it can be shown that

${\displaystyle x={\frac {T_{-1}^{(1)}-T_{1}^{(1)}}{\sqrt {2}}},}$

where we define the spherical tensors as[7]

${\displaystyle T_{q}^{(1)}={\sqrt {\frac {4\pi }{3}}}rY_{1}^{q}}$

and Ylm are spherical harmonics, which themselves are also spherical tensors of rank l. Additionally, T(1)0 = z, and

${\displaystyle T_{\pm 1}^{(1)}=\mp {\frac {x\pm iy}{\sqrt {2}}}.}$

Therefore,

{\displaystyle {\begin{aligned}&\langle n\,j\,m|x|n'\,j'\,m'\rangle =\\&=\left\langle n\,j\,m\left|{\frac {T_{-1}^{(1)}-T_{1}^{(1)}}{\sqrt {2}}}\right|n'\,j'\,m'\right\rangle =\\&={\frac {1}{\sqrt {2}}}\langle n\,j\|T^{(1)}\|n'\,j'\rangle \,{\big (}\langle j'\,m'\,1\,(-1)|j\,m\rangle -\langle j'\,m'\,1\,1|j\,m\rangle {\big )}.\end{aligned}}}

The above expression gives us the matrix element for x in the |n j m basis. To find the expectation value, we set n′ = n, j′ = j, and m′ = m. The selection rule for m and m is m ± 1 = m for the T(1)±1 spherical tensors. As we have m′ = m, this makes the Clebsch–Gordan Coefficients zero, leading to the expectation value to be equal to zero.

## References

1. ^ Eckart Biography – The National Academies Press.
2. ^ The parenthesized superscript (k) provides a reminder of its rank. However, unlike q, it need not be an actual index.
3. ^ This is a special notation specific to the Wigner–Eckart theorem.
4. ^ Hall 2015 Appendix C.
5. ^ Racah, G. (1942). "Theory of Complex Spectra II". Physical Review. 62 (9–10): 438–462. Bibcode:1942PhRv...62..438R. doi:10.1103/PhysRev.62.438.
6. ^ Wigner, E. P. (1951). "On the Matrices Which Reduce the Kronecker Products of Representations of S. R. Groups". In Wightman, Arthur S. (ed.). The Collected Works of Eugene Paul Wigner. 3. p. 614. doi:10.1007/978-3-662-02781-3_42.
7. ^ J. J. Sakurai: "Modern quantum mechanics" (Massachusetts, 1994, Addison-Wesley).

### General

• Hall, Brian C. (2015), Lie groups, Lie algebras, and representations: An elementary introduction, Graduate Texts in Mathematics, 222 (2nd ed.), Springer, ISBN 978-3319134666


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# Mathematical Sciences Research Institute

Home » Conformal Field Theory and Supersymmetry

# Workshop

Conformal Field Theory and Supersymmetry April 15, 2002 - April 26, 2002
 Registration Deadline: April 26, 2002 about 12 years ago January 15, 2002 over 12 years ago
Parent Program: Infinite-Dimensional Algebras and Mathematical Physics
Organizers G. Felder, D. Freed, E. Frenkel, V. Kac, T. Miwa, I. Penkov, V. Serganova, I. Singer and G. Zuckerman
Speaker(s)

## Show List of Speakers

Description

Week 1: "Infinite-dimensional Algebras, Conformal Field Theory and Integrable Systems" Some of the most spectacular applications of representation theory of infinite-dimensional algebras are related to recent developments in mathematical physics. Conformal quantum field theory (CFT) in two dimensions and string theory have a multitude of deep connections to the representation theory of infinite algebras. In a sense the two subjects have been joined at the hip since their inception in the late 1960's. The early physics papers on string theory introduced representations of the Virasoro Lie algebra and quantum fields known as vertex operators. By the early 1980's it was realized that the Virasoro algebra is a universal symmetry algebra of a 2D CFT, and that affine Kac-Moody algebras (central extensions of loop algebras) are symmetry algebras of CFTs of a more special type (Wess-Zumino-Witten models). Eventually, Borcherds succeeded in formalizing the general notion of a symmetry algebra of a CFT in the concept of vertex algebra. More recently, Beilinson and Drinfeld have introduced a geometric version of vertex algebras which they call chiral algebras. Chiral algebras give rise to some novel concepts and techniques in algebraic geometry, in particular, in the geometric Langlands correspondence. Physicists have recently begun to study perturbations of conformal field theories in the hopes of using the rich structure of CFT for understanding more general integrable quantum field theories. The study of conformal field theory in dimensions greater than two is also under way, and surprising connections have been found between these theories and string theory (AdS/CFT correspondence). Infinite-dimensional algebras have also been used in the recent years in the theory of exactly solved models. It turned out that quantum deformations of the enveloping algebras of the affine Kac-Moody algebras, the Virasoro algebra, and the W-algebras may be realized as dynamical symmetry algebras of various lattice models, and hence their representation theory has important applications. Infinite-dimensional Lie algebras also play a central role in the study of integrable systems, such as those associated to the KdV equation and other soliton hierarchies. These integrable systems appear in the study of the Gromov-Witten invariants of algebraic manifolds. The Virasoro conjecture, in particular, indicates the relevance of infinite-dimensional algebras in this area. In this workshop we will discuss the areas mentioned above and interactions between them. Week 2: "Supersymmetry in Mathematics and Physics" Supersymmetry is now considered as one of the most fundamental concepts in physics. Lie superalgebras in particular have increasingly been playing central role in recent studies of quantum field theory and string theory. There has also been an important development in mathematics, like structure and representation theory of Lie superalgebras, "supersymmetric" proofs of the index theorems, the theory of superconnections, etc. In this workshop, we hope to discuss the recent advances in "superphysics" and "supermathematics."          Group photo of participants

Funding & Logistics

## Show Funding

To apply for funding, you must register by the funding application deadline displayed above.

Students, recent Ph.D.'s, women, and members of underrepresented minorities are particularly encouraged to apply. Funding awards are typically made 6 weeks before the workshop begins. Requests received after the funding deadline are considered only if additional funds become available.

## Show Lodging

MSRI has preferred rates at the Rose Garden Inn, depending on room availability. Reservations may be made by calling 1-800-992-9005 OR directly on their website. Click on Corporate at the bottom of the screen and when prompted enter code MATH (this code is not case sensitive). By using this code a new calendar will appear and will show the MSRI rate on all room types available.

MSRI has preferred rates at the Hotel Durant. Reservations may be made by calling 1-800-238-7268. When making reservations, guests must request the MSRI preferred rate. If you are making your reservations on line, please go to this link and enter the promo/corporate code MSRI123. Our preferred rate is $129 per night for a Deluxe Queen/King, based on availability. MSRI has preferred rates of$149 - $189 plus tax at the Hotel Shattuck Plaza, depending on room availability. Guests can either call the hotel's main line at 510-845-7300 and ask for the MSRI- Mathematical Science Research Inst. discount; or go to www.hotelshattuckplaza.com and click Book Now. Once on the reservation page, click “Promo/Corporate Code“ and input the code: msri. MSRI has preferred rates of$110 - \$140 at the Berkeley Lab Guest House, depending on room availability. Reservations may be made by calling 510-495-8000 or directly on their website. Select “I am an individual traveler affiliated with MSRI”.

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Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.

The idea of this post is to create a kind of map of the R ecosystem showing dependencies relationships between packages. This is actually quite simple to quickly generate a graph object like that, since all the data we need can be obtained with one call to the function available.packages(). There are a few additional steps to clean the data and arrange them, but with the following code everyone can generate a graph of dependencies in less than 5s.

First, we load the igraph package and we retrieve the data from our current repository.

library(igraph)
dat <- available.packages()

For each package, we produce a character string with all its dependencies (Imports and Depends fields) separated with commas.

dat.imports <- paste(dat[, "Imports"], dat[, "Depends"], sep = ", ")
dat.imports <- as.list(dat.imports)
dat.imports <- lapply(dat.imports, function(x) gsub("$$([^$$]+)\)", "", x))
dat.imports <- lapply(dat.imports, function(x) gsub("n", "", x))
dat.imports <- lapply(dat.imports, function(x) gsub(" ", "", x))

Next step, we split the strings and we use the stack function to get the complete list of edges of the graph.

dat.imports <- sapply(dat.imports, function(x) strsplit(x, split = ","))
dat.imports <- lapply(dat.imports, function(x) x[!x %in% c("NA", "R")])
names(dat.imports) <- rownames(dat)
dat.imports <- stack(dat.imports)
dat.imports <- as.matrix(dat.imports)
dat.imports <- dat.imports[-which(dat.imports[, 1]==""), ]

Finally we create the graph with the list of edges. Here, I select the largest connected component because there are many isolated vertices which will make the graph harder to represent.

g <- graph.edgelist(dat.imports)
g <- decompose.graph(g)
g <- g[[which(sapply(g, vcount) == max(sapply(g, vcount)))]]

Now that we have the graph we can compute many graph-theory related statistics but here I just want to visualize these data. Plotting a large graph like that with igraph is possible but slow and tedious. I prefer to export the graph and open it in Gephi, a free and open-source software for network visualization.
The figure below was created with Gephi. Nodes are sized according to their number of incoming edges. This is pretty useless and uninformative, but still I like it. This looks like a sky map and it is great to feel that we, users and developers are part of this huge R galaxy.

Click on the picture to see the HD version


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I have forgotten

• https://me.yahoo.com
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# time

Calculates the time fraction from the parameters \b hours, \b minutes and \b seconds. Excel: TIME
Controller: CodeCogs
Contents

C++
Excel

## Time

 doubletime( int Hours int Minutes int Seconds )[inline]
Calculates the fraction of a day represented by the specified time (Hours, Minutes and Seconds), using the formula:

Therefore time=0 is midnight, time=0.5 is midday, and time=1 is the start of the next day.

## Example:

double breakfast =time(6,15,0);    // ~0.2604
double midday    =time(12,0,0);    // 0.5
double lunchtime =time(13,30,0);   // ~0.5625
double bedtime   =time(22,40,0);   // ~0.9444

## References:

### Parameters

 Hours is usually any integer number between 0 (12am) and 23 (11pm), though values outside of this range will esculate into forming a positive (or negative) number of elapsed days. Minutes is usually any integer between 0 and 59, where once again values outside of this range will esculate into forming additional hours. Seconds is usually any integer between 0 and 59, with values outside of this range esculating upto to form whole minutes.

### Authors

Will Bateman (Sep 2004)
##### Source Code

Source code is available when you agree to a GP Licence or buy a Commercial Licence.

Not a member, then Register with CodeCogs. Already a Member, then Login.


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# BER/SER and SNR in STBC and Alamouti scheme

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#### basalamah

##### Newbie level 3
hi,

I am trying to understand STBC and Alamouti scheme.
However, i am very confused about the BER of such systems.

Referring to the performance results in Alamouti and Tarokh's papers, for QPSK (for example), the BER/SER shown is against SNR for 2 bit/sec/hz.

1- what does the "per hz" mean here?
2- why do we use SNR here instead of Eb/No?
3- how can i calculate the BER/SER for a 20Khz bandwidth channel for example?

Thanks

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Almost Sure

15 November 16

Optional Processes

The optional sigma-algebra, ${\mathcal{O}}$, was defined earlier in these notes as the sigma-algebra generated by the adapted and right-continuous processes. Then, a stochastic process is optional if it is ${\mathcal{O}}$-measurable. However, beyond the definition, very little use was made of this concept. While right-continuous adapted processes are optional by construction, and were used throughout the development of stochastic calculus, there was no need to make use of the general definition. On the other hand, optional processes are central to the theory of optional section and projection. So, I will now look at such processes in more detail, starting with the following alternative, but equivalent, ways of defining the optional sigma-algebra. Throughout this post we work with respect to a complete filtered probability space ${(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\in{\mathbb R}_+},{\mathbb P})}$, and all stochastic processes will be assumed to be either real-valued or to take values in the extended reals ${\bar{\mathbb R}={\mathbb R}\cup\{\pm\infty\}}$.

Theorem 1 The following collections of sets and processes each generate the same sigma-algebra on ${{\mathbb R}_+\times\Omega}$.

1. {${[\tau,\infty)}$: ${\tau}$ is a stopping time}.
2. ${Z1_{[\tau,\infty)}}$ as ${\tau}$ ranges over the stopping times and Z over the ${\mathcal{F}_\tau}$-measurable random variables.
4. The right-continuous adapted processes.

The optional-sigma algebra was previously defined to be generated by the right-continuous adapted processes. However, any of the four collections of sets and processes stated in Theorem 1 can equivalently be used, and the definitions given in the literature do vary. So, I will restate the definition making use of this equivalence.

Definition 2 The optional sigma-algebra, ${\mathcal{O}}$, is the sigma-algebra on ${{\mathbb R}_+\times\Omega}$ generated by any of the collections of sets/processes in Theorem 1.

A stochastic process is optional iff it is ${\mathcal{O}}$-measurable.

Before giving the proof of Theorem 1 below, I will first show that certain basic constructions automatically lead to optional processes. We know that optional processes are always progressively measurable. In the previous post, it was shown that the running maximum of a progressive process is itself progressive. In fact, it is optional.

Lemma 3 If X is a progressively measurable process then ${X^*_t\equiv\sup_{s\le t}X_s}$ is optional.

Proof: First, as in the proof of Lemma 5 in the previous post, ${X^*_{t-}}$ is left-continuous and adapted. In particular, it is predictable and hence is optional. As the intervals ${(K,\infty]}$ for real K generate the Borel sigma-algebra on ${\bar{\mathbb R}}$, it is enough to show that ${(X^*)^{-1}((K,\infty])}$ is optional.

$\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle X^*((K,\infty])\equiv\{t\colon X^*_t > K\}= A\cup B,\smallskip\\ &\displaystyle A=\{t\colon X^*_{t-} > K\},\smallskip\\ &\displaystyle B= \{t\colon X^*_t > K \ge X^*_{t-}\}. \end{array}$

As ${X^*_{t-}}$ is optional, we know that ${A\in\mathcal{O}}$. Next, suppose that ${(t,\omega)}$ is in B for some ${t\in{\mathbb R}_+}$ and ${\omega\in\Omega}$. Then,

$\displaystyle X^*_s\le X^*_{t-}\le K$

for ${s < t}$ and

$\displaystyle X^*_{s-}\ge X^*_t > K$

for ${s > t}$. In either case, ${(s,\omega)\not\in B}$. So, we see that there can be at most a single time lying in B for each ${\omega\in\Omega}$ and, hence, B is the graph ${[\tau]}$ of a map ${\tau\colon\Omega\rightarrow\bar{\mathbb R}_+}$. As ${X^*_t}$ and ${X^*_{t-}}$ are progressive, B is also progressive. By Lemma 7 from the previous post, this means that ${\tau}$ is a stopping time. In particular, using the first statement of Theorem 1,

 $\displaystyle [\tau]=[\tau,\infty)\setminus\bigcup_{n=1}^\infty[\tau+1/n,\infty)\in\mathcal{O}.$ (1)

So, ${(X^*)^{-1}((K,\infty])\in\mathcal{O}}$ as required. ⬜

It was also shown that the limit supremum and left/right limit supremums of progressive processes are again progressive. Actually, the stronger property of optionality holds for most of these.

Lemma 4 If X is a progressively measurable process then

$\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle t&\displaystyle\mapsto\limsup_{s\uparrow t}X_s,\smallskip\\ \displaystyle t&\displaystyle\mapsto\limsup_{s\uparrow\uparrow t}X_s \end{array}$

are optional. If, furthermore, the filtration ${\mathcal{F}_\cdot}$ is right-continuous then,

$\displaystyle t\mapsto\limsup_{s\rightarrow t}X_s$

is optional.

Proof: For each positive integer n, choose a sequence of times ${0=t^n_0 < t^n_1 < \cdots}$ increasing to infinity, and such that ${\sup_k(t^n_k-t^n_{k-1})}$ goes to zero as n goes to infinity. For example, ${t^n_k=k/n}$. Define processes ${Y^n}$ by ${Y^n_0=X_0}$ and

$\displaystyle Y^n_t=\sup\left\{X_s\colon s\in[t^n_{k-1},t)\right\}$

for ${t\in(t^n_{k-1},t^n_k]}$. This is left-continuous and, by Lemma 4 of the previous post, is adapted. So, ${Y^n}$ and, hence, the limit

$\displaystyle \limsup_{s\uparrow\uparrow t}X_s=\lim_{n\rightarrow\infty}Y^n_t$

Next, define ${Z^n_0=X_0}$ and

$\displaystyle Z^n_t=\sup\left\{X_s\colon s\in[t^n_{k-1},t]\right\}$

for ${t\in(t^n_{k-1},t^n_k]}$. Applying Lemma 3 over each of the intervals ${(t^n_{k-1},t^n_k]}$ shows that this optional. So, the limit

$\displaystyle \limsup_{s\uparrow t}X_s=\lim_{n\rightarrow\infty}Z^n_t$

is also optional.

Finally, define ${W^n_0=\limsup_{s\downarrow0}X_s}$ and

$\displaystyle W^n_t=\lim_{s\downarrow\downarrow t}\sup\left\{X_u\colon u\in[t^n_{k-1},s)\right\}$

for ${t\in(t^n_{k-1},t^n_k]}$. By Lemma 4 of the previous post, ${W^n_t}$ is ${\mathcal{F}_s}$-measurable for each ${s > t}$. So, ${W^n_t}$ is ${\mathcal{F}_{t+}}$-measurable and, if the filtration is right-continuous, W is adapted. As ${W^n}$ is right-continuous over each of the intervals ${(t^n_{k-1},t^n_k]}$, this implies that it is optional. So,

$\displaystyle \limsup_{s\rightarrow t}X_s=\lim_{n\rightarrow\infty}W^n_t$

is optional. ⬜

Notable by their absence in the statement of Lemma 4 are the right-limits ${\limsup_{s\downarrow t}X_s}$ and ${\limsup_{s\downarrow\downarrow t}X_s}$. In fact, these need not be optional even when the filtration is right-continuous and X is itself optional.

Next, we saw in the previous post that stopping times can be characterised by the progressive measurability of their graphs. Optional measurability can equivalently be used. The term `optional time’ is sometimes used as a synonym for stopping time.

Lemma 5 A map ${\tau\colon\Omega\rightarrow\bar{\mathbb R}_+}$ is a stopping time if and only if its graph ${[\tau]}$ is optional.

Proof: If ${\tau}$ is a stopping time then (1) shows that ${[\tau]\in\mathcal{O}}$. Conversely, if ${[\tau]}$ is optional then, in particular, it is progressive. So, Lemma 7 of the previous post shows that ${\tau}$ is a stopping time. ⬜

Proof of Theorem 1

Let ${\mathcal{O}_k}$, ${k=1,2,3,4}$, be the sigma-algebra generated by each of the respective collections of sets/processes of the Theorem 1. The inclusions ${\mathcal{O}_1\subseteq\mathcal{O}_2\subseteq\mathcal{O}_3\subseteq\mathcal{O}_4}$ are immediate. The first inclusion follows by taking ${Z=1}$ in the second statement. The second inclusion holds because the processes in statement 2 are cadlag and adapted, and the third inclusion holds since cadlag processes are, by definition, right-continuous. It only remains to show that ${\mathcal{O}_4\subseteq\mathcal{O}_1}$. I will break this down into the inclusions ${\mathcal{O}_4\subseteq\mathcal{O}_2}$ and ${\mathcal{O}_2\subseteq\mathcal{O}_1}$.

Proof of ${\mathcal{O}_2\subseteq\mathcal{O}_1}$: We need to show that ${Z1_{[\tau,\infty)}}$ is ${\mathcal{O}_1}$-measurable for stopping times ${\tau}$ and ${\mathcal{F}_\tau}$-measurable Z. By the monotone class theorem, it is enough to consider ${Z=1_A}$ for ${A\in\mathcal{F}_\tau}$. Then,

$\displaystyle Z1_{[\tau,\infty)}=1_{[\tau_A,\infty)}$

where ${\tau_A(\omega)}$ equals ${\tau(\omega)}$ when ${\omega\in A}$ and ${\infty}$ otherwise. As this is a stopping time, ${Z1_{[\tau,\infty)}}$ is ${\mathcal{O}_1}$-measurable. ⬜

To complete the proof of Theorem 1, it only remains to show that right-continuous adapted processes are ${\mathcal{O}_2}$-measurable. Although it is not strictly necessary, I will first prove the result for cadlag adapted processes. There are a few reasons for doing this. Firstly, the proof is more straightforward and, in any case, we are often interested in processes which are known to be cadlag rather than merely right-continuous. Secondly, we have not made use of completeness of the filtration yet, and completeness is not actually required for the equivalence of the first three sigma-algebras of the theorem. It is required, though, for the proof that right-continuous adapted processes are ${\mathcal{O}_2}$-measurable. For this reason, in some situations, it may be more appropriate to use either of the first three definitions of the optional sigma-algebra given by Theorem 1 rather than defining it as generated by the right-continuous adapted processes.

Proof of ${\mathcal{O}_3\subseteq\mathcal{O}_2}$: We show that any cadlag adapted process X is ${\mathcal{O}_2}$-measurable. Fixing ${\epsilon > 0}$, it is enough to construct an ${\mathcal{O}_2}$-measurable process Y such that ${\lvert X-Y\rvert < \epsilon}$. By letting ${\epsilon}$ tend to zero, this will show that X is ${\mathcal{O}_2}$-measurable.

Define an increasing sequence of random times ${\{\tau_n\}}$ by ${\tau_0=0}$ and, for each ${n\ge0}$,

 $\displaystyle \tau_{n+1}=\inf\left\{t\ge\tau_n\colon\lvert X_t-X_{\tau_n}\rvert\ge\epsilon\right\}.$ (2)

Then, define the process Y as,

$\displaystyle Y=\sum_{n=1}^\infty X_{\tau_n}1_{[\tau_n,\tau_{n+1})}.$

By the debut theorem for right-continuous processes, ${\tau_n}$ are stopping times. Then, as ${X_{\tau_n}}$ is ${\mathcal{F}_{\tau_n}}$-measurable, writing

$\displaystyle X_{\tau_n}1_{[\tau_n,\tau_{n+1})}=X_{\tau_n}1_{[\tau_n,\infty)}-X_{\tau_n}1_{[\tau_{n+1},\infty)}$

expresses Y as a sum of processes in the form of the second statement of the theorem. So, Y is ${\mathcal{O}_2}$-measurable.

Next, the times ${\tau_n}$ increase to a limit ${\tau}$ and, by construction, ${\lvert X-Y\rvert < \epsilon}$ on the interval ${[0,\tau)}$. So, the result follows so long as ${\tau = \infty}$. On the event ${\{\tau <\infty\}}$, the condition that X has left limits implies that ${X_{\tau_n}}$ converges to a limit ${X_{\tau-}}$ as n goes to infinity. This contradicts the inequality ${\lvert X_{\tau_{n+1}}-X_{\tau_n}\rvert\ge\epsilon}$ and, hence, ${\tau=\infty}$ as required. ⬜

The proof above only made use of completeness of the filtration at one point, when the debut theorem was invoked to show that ${\tau_n}$ are stopping times. However, this was just for convenience and, if we replace (2) by

$\displaystyle \tau_{n+1}=\inf\left\{t\ge\tau_n\colon\lvert X_t-X_{\tau_n}\rvert\ge\epsilon{\rm\ or\ }\lvert X_{t-}-X_{\tau_n}\rvert\ge\epsilon\right\}$

then it can be seen directly that ${\tau_n \le t}$ if and only if

$\displaystyle \sup_{s\in S}\lvert X_s-X_{\tau_n}\rvert\ge\epsilon,$

for any countable ${S\subset[0,t]}$ containing t. As this is ${\mathcal{F}_t}$-measurable, this alternative definition also gives a sequence of stopping times, and the proof does not require completeness of the filtered probability space.

I now complete the proof of Theorem 1 by showing that right-continuous adapted processes are ${\mathcal{O}_2}$-measurable.

Proof of ${\mathcal{O}_4\subseteq\mathcal{O}_2}$: We show that any right-continuous adapted process X is ${\mathcal{O}_2}$-measurable. As above, it is enough to find an ${\mathcal{O}_2}$-measurable process Y satisfying ${\lvert X-Y\rvert < \epsilon}$, for any given ${\epsilon > 0}$.

Define ${\mathcal{T}}$ to be the set of all stopping times such that there exists an ${\mathcal{O}_2}$-measurable process Y with ${\lvert1_{[0,\tau)}X-Y\rvert < \epsilon}$. For any ${\sigma,\tau\in\mathcal{T}}$, there are ${\mathcal{O}_2}$-measurable Y and Z satisfying

$\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle\lvert 1_{[0,\sigma)}X-Y\rvert < \epsilon,\smallskip\\ &\displaystyle\lvert 1_{[0,\tau)}X-Z\rvert < \epsilon. \end{array}$

Then

$\displaystyle U = 1_{[0,\sigma)}Y+1_{[\sigma\wedge\tau,\tau)}Z$

is ${\mathcal{O}_2}$-measurable and satisfies

$\displaystyle \lvert 1_{[0,\sigma\vee\tau)}X-U\rvert < \epsilon.$

Hence, the maximum ${\sigma\vee\tau}$ is again in ${\mathcal{T}}$. Similarly, for an increasing sequence ${\{\tau_n\}}$ in ${\mathcal{T}}$, there is a sequence ${\{Y^n\}}$ of ${\mathcal{O}_2}$-measurable processes satisfying

$\displaystyle \lvert 1_{[0,\tau_n)}X-Y^n\rvert < \epsilon.$

Then, with ${\tau_0=0}$,

$\displaystyle Y=\sum_{n=1}^\infty1_{[\tau_{n-1},\tau_n)}Y^n$

is an ${\mathcal{O}_2}$-measurable process with

$\displaystyle \lvert1_{[0,\lim_n\tau_n)}X-Y\rvert < \epsilon,$

and the limit of ${\tau_n}$ as n goes to infinity is again in ${\mathcal{T}}$.

This all shows that the supremum of any countable subset of ${\mathcal{T}}$ is in ${\mathcal{T}}$ and, in particular, its essential supremum, say ${\tau^*}$, is again in ${\mathcal{T}}$. That is, there is an ${\mathcal{O}_2}$-measurable process Y with

 $\displaystyle \lvert1_{[0,\tau^*)}X-Y\rvert < \epsilon.$ (3)

Now, defining the stopping time

$\displaystyle \sigma = \inf\left\{t\ge\tau^*\colon\lvert X_t-X_{\tau^*}\rvert\ge\epsilon\right\}$

which, by right-continuity, is strictly greater than ${\tau^*}$ whenever ${\tau^*}$ is finite. Then

$\displaystyle Z=1_{[0,\tau^*)}Y+X_{\tau^*}1_{[\tau^*,\sigma)}$

is ${\mathcal{O}_2}$-measurable and satisfies

$\displaystyle \lvert 1_{[0,\sigma)}X-Z\rvert < \epsilon.$

This shows that ${\sigma\ge\tau^*}$ is in ${\mathcal{T}}$. By the definition of the essential supremum, ${\sigma=\tau^*}$ almost surely and, hence, ${\tau^*}$ is almost surely infinite. So, (3) shows that X is ${\mathcal{O}_2}$-measurable up to evanescence and, by completeness of the filtration, is then ${\mathcal{O}_2}$-measurable. ⬜

Processes With Right Limits

Occasionally, we may want to consider processes which have right limits but are not necessarily right-continuous everywhere. For example, when the underlying filtered probability space is not right-continuous, then not all martingales will have a cadlag version. However, we were able to show that there is always a version with left and right limits, and which has right limits everywhere outside of a fixed countable set of times. Although they are not cadlag, such processes are still optional. I start by proving the following result relating optional processes with respect to the right-continuous filtration ${\mathcal{F}_+}$ to optional processes under the original filtration.

Lemma 6 Let X be an optional process with respect to the right-continuous filtration ${\{\mathcal{F}_{t+}\}_{t\in{\mathbb R}_+}}$. Then, there exists a countable set ${S\subseteq{\mathbb R}_+}$ such that

 $\displaystyle 1_{S^c}X$ (4)

is optional.

Proof: Let us first consider processes of the form ${X=1_{[\tau,\infty)}}$ for an ${\mathcal{F}_+}$-stopping time ${\tau}$. Set,

$\displaystyle S=\left\{t\in{\mathbb R}_+\colon{\mathbb P}(\tau=t) > 0\right\}.$

As ${{\mathbb P}(\tau=t)}$ is strictly positive on S and sums to a finite value, S is countable. We can also show that (4) is optional. As ${Y\equiv1_{(\tau,\infty)}}$ is left-continuous and adapted, it is optional. Also, the random time

$\displaystyle \sigma=\begin{cases} \tau,&\tau\not\in S,\\ \infty,&\tau\in S \end{cases}$

is an ${\mathcal{F}_+}$-stopping time satisfying ${{\mathbb P}(\sigma=t)=0}$ for all times t. This means that ${\{\sigma\le t\}}$ is equal to ${\{\sigma < t\}\in\mathcal{F}_t}$ up to a zero probability set and, by completeness of the filtered probability space, is a stopping time. Hence,

$\displaystyle 1_{S^c}X=(1_{S^c}Y)\vee(1_{S^c}1_{[\sigma,\infty)})$

is optional.

Let ${\mathcal{S}}$ denote the set of all ${\mathcal{F}_+}$-optional processes for which there exists a countable ${S\subseteq{\mathbb R}_+}$ making (4) optional. From what we have just shown, ${1_{[\tau,\infty)}\in\mathcal{S}}$ for all ${\mathcal{F}_+}$-stopping times ${\tau}$. It is also clear that ${\mathcal{S}}$ is closed under taking finite linear combinations, and under taking limits of sequences. Consider a sequence ${X^n\in\mathcal{S}}$, so that there are countable sets ${S_n}$ for which ${1_{S_n^c}X^n}$ are optional. Taking ${S=\bigcup_nS_n}$ then ${1_{S^c}Y}$ is optional for Y a finite linear combination or limit of the sequence ${X^n}$. So, by the functional monotone class theorem, ${\mathcal{S}}$ includes all ${\mathcal{F}_+}$-optional processes. ⬜

Finally, I show that the existence of right limits together with right-continuity outside of a countable set of times is sufficient to conclude that a process is optional. In particular, this includes the modifications of martingales which we previously constructed without requiring right-continuity of the filtration.

Lemma 7 Let X be an adapted process with right-limits everywhere, and which is right-continuous everywhere outside of a countable set ${S\subset{\mathbb R}_+}$. Then, X is optional.

Proof: The process ${X_{t+}=\lim_{s\downarrow\downarrow t}X_s}$ is right-continuous and is adapted with respect to the right-continuous filtration ${\mathcal{F}_{+}}$. Hence, it is ${\mathcal{F}_{+}}$-optional and, by Lemma 6, there exists a countable ${A\subset{\mathbb R}_+}$ such that ${1_{A^c}X_{+}}$ is optional. By enlarging A, if necessary, to include the set of times at which X is not right-continuous, we see that

$\displaystyle 1_{A^c}X=1_{A^c}X_+$

is optional. Therefore, as ${X_t}$ is ${\mathcal{F}_t}$-measurable for each ${t\in A}$,

$\displaystyle X = 1_{A^c}X+\sum_{t\in A}1_{[t]}X_t$

is optional. ⬜


text:
# ParticleTracker

class plasmapy.simulation.particletracker.ParticleTracker(plasma, particle_type='p', n_particles=1, scaling=1, dt=<Quantity inf s>, nt=inf, integrator='explicit_boris')

Bases: object

Object representing a species of particles: ions, electrons, or simply a group of particles with a particular initial velocity distribution.

Parameters
x

Current position. Shape (n, 3).

Type

astropy.units.Quantity

v

Current velocity. Shape (n, 3).

Type

astropy.units.Quantity

position_history

History of position. Shape (nt, n, 3).

Type

astropy.units.Quantity

velocity_history

History of velocity. Shape (nt, n, 3).

Type

astropy.units.Quantity

q

Charge of particle.

Type

astropy.units.Quantity

m

Mass of particle.

Type

astropy.units.Quantity

eff_q

Total charge of macroparticle.

Type

astropy.units.Quantity

eff_m

Total mass of macroparticle.

Type

astropy.units.Quantity

Examples

Attributes Summary

 integrators kinetic_energy_history Calculate the kinetic energy history for each particle.

Methods Summary

 boris_push([init]) Implement the Boris algorithm for moving particles and updating their velocities. plot_time_trajectories([plot]) Draw position history versus time. Draw trajectory history. Run a simulation instance. Test conservation of kinetic energy.

Attributes Documentation

integrators = {'explicit_boris': <function boris_push>}
kinetic_energy_history

Calculate the kinetic energy history for each particle.

Returns

Array of kinetic energies, shape (nt, n).

Return type

Quantity

Methods Documentation

boris_push(init=False)

Implement the Boris algorithm for moving particles and updating their velocities.

Parameters

init (bool, optional) – If True, does not change the particle positions and sets dt to -dt/2.

Notes

The Boris algorithm is the standard energy conserving algorithm for particle movement in plasma physics. See 1 for more details.

Conceptually, the algorithm has three phases:

1. Add half the impulse from electric field.

2. Rotate the particle velocity about the direction of the magnetic field.

3. Add the second half of the impulse from the electric field.

This ends up causing the magnetic field action to be properly “centered” in time, and the algorithm conserves energy.

References

1

C. K. Birdsall, A. B. Langdon, “Plasma Physics via Computer Simulation”, 2004, p. 58-63

plot_time_trajectories(plot='xyz')

Draw position history versus time.

Parameters

plot (str, optional) – Enable plotting of position component x, y, z for each of these letters included in plot.

plot_trajectories()

Draw trajectory history.

run()

Run a simulation instance.

test_kinetic_energy()

Test conservation of kinetic energy.


text:
# Math Help - Rate of a decreasing spherical balloon

1. ## Rate of a decreasing spherical balloon

I know:
• Formula for a sphere's volume: $V = \frac{4}{3}*\pi*r^3$
• Diameter of balloon is 20cm = radius of balloon is 10cm.
• $\frac{4}{3}*\pi*10^3 = 1333.3*\pi$
• Difference of 5cm in the radius is: $\frac{4}{3}*pi*5^3 = 166.67*\pi$
• 1333.3 - 166.67 = 1166.63
• This is not a choice, so I am obviously wrong.
• Why I think I am wrong: $1166.63*\pi cm^3$ is not in the form of a rate. It needs to be in the form of $\frac{x cm^3}{y seconds}$.
• I can't think of how to do this.
• Lastly, my practice test states that this question will see if I understand the historical development and application of calculus. I clearly did not use Calculus, and quite honestly, I'm don't know how.

2. Originally Posted by EMyk01

I know:
• Formula for a sphere's volume: $V = \frac{4}{3}*\pi*r^3$
• Diameter of balloon is 20cm = radius of balloon is 10cm.
• $\frac{4}{3}*\pi*10^3 = 1333.3*\pi$
• Difference of 5cm in the radius is: $\frac{4}{3}*pi*5^3 = 166.67*\pi$
• 1333.3 - 166.67 = 1166.63
• This is not a choice, so I am obviously wrong.
• Why I think I am wrong: $1166.63*\pi cm^3$ is not in the form of a rate. It needs to be in the form of $\frac{x cm^3}{y seconds}$.
• I can't think of how to do this.
• Lastly, my practice test states that this question will see if I understand the historical development and application of calculus. I clearly did not use Calculus, and quite honestly, I'm don't know how.
1. The rate of change of the volume wrt the time t is:

$\dfrac{dV}{dt}=\dfrac{d\left(\frac43 \pi r^3\right)}{dt}$

2. The rate of change of the radius is $\dfrac{dr}{dt} = 5\ \frac{cm^3}{s}$

3. Calculate dV using the equation from 1. and the value of 2.


text:
zbMATH — the first resource for mathematics

Hu, Yongquan

Compute Distance To:
 Author ID: hu.yongquan Published as: Hu, Y.; Hu, Y. Q.; Hu, Yong Quan; Hu, Yongquan External Links: MGP · Wikidata
 Documents Indexed: 20 Publications since 2002 Reviewing Activity: 3 Reviews

Co-Authors

 6 single-authored 1 Morra, Stefano 1 Paskunas, Vytautas 1 Schraen, Benjamin 1 Tan, FuCheng 1 Wang, Haoran
all top 5

Serials

 1 Annales Scientifiques de l’École Normale Supérieure. Quatrième Série 1 Bulletin de la Société Mathématique de France 1 Journal of Algebra 1 Journal of the London Mathematical Society. Second Series 1 Journal für die Reine und Angewandte Mathematik 1 Mathematische Annalen 1 Rendiconti del Seminario Matematico della Università di Padova 1 Mathematical Research Letters 1 Acta Mathematica Sinica. English Series 1 Journal of the Institute of Mathematics of Jussieu

Fields

 8 Topological groups, Lie groups (22-XX) 5 Number theory (11-XX) 1 Commutative algebra (13-XX)

Citations contained in zbMATH

10 Publications have been cited 32 times in 25 Documents Cited by Year
On some supersingular representations of $$\mathrm{GL}_2 (\mathbb Q_{p^f})$$. Zbl 1206.22010
Hu, Yongquan
2010
Canonical diagrams and representations modulo $$p$$ of $$\mathrm{GL}_{2}(F)$$. Zbl 1232.22012
Hu, Yongquan
2012
Invariant norms and the existence of admissible filtrations. Zbl 1214.11132
Hu, Yongquan
2009
The Breuil-Mézard conjecture for non-scalar split residual representations. Zbl 1334.11041
Hu, Yongquan; Tan, Fucheng
2015
An application of a theorem of Emerton to mod $$p$$ representations of $$\mathrm{GL}_2$$. Zbl 1408.22018
Hu, Yongquan
2017
On crystabelline deformation rings of $$\mathrm{Gal}(\overline{\mathbb {Q}}_p/\mathbb {Q}_p)$$ (with an appendix by Jack Shotton). Zbl 07051748
Hu, Yongquan; Paškūnas, Vytautas
2019
Multiplicity one for the $$\mathrm {mod} p$$ cohomology of Shimura curves: the tame case. Zbl 06965424
Hu, Yongquan; Wang, Haoran
2018
On the faithfulness of certain representations of $$\mathrm{GL}_2(\mathbb F)$$ under an Iwasawa algebra. Zbl 1311.22024
Hu, Yongquan; Morra, Stefano; Schraen, Benjamin
2014
Entanglement and decoherence of coupled superconductor qubits in contact with a common environment. Zbl 1245.81010
Ji, Y. H.; Hu, Y.; Yu, Y. X.
2011
Controlling on entangled decoherence by the interaction model between qubit and environment. Zbl 1225.81013
Ji, Y. H.; Lai, H. F.; Hu, Y.
2011
On crystabelline deformation rings of $$\mathrm{Gal}(\overline{\mathbb {Q}}_p/\mathbb {Q}_p)$$ (with an appendix by Jack Shotton). Zbl 07051748
Hu, Yongquan; Paškūnas, Vytautas
2019
Multiplicity one for the $$\mathrm {mod} p$$ cohomology of Shimura curves: the tame case. Zbl 06965424
Hu, Yongquan; Wang, Haoran
2018
An application of a theorem of Emerton to mod $$p$$ representations of $$\mathrm{GL}_2$$. Zbl 1408.22018
Hu, Yongquan
2017
The Breuil-Mézard conjecture for non-scalar split residual representations. Zbl 1334.11041
Hu, Yongquan; Tan, Fucheng
2015
On the faithfulness of certain representations of $$\mathrm{GL}_2(\mathbb F)$$ under an Iwasawa algebra. Zbl 1311.22024
Hu, Yongquan; Morra, Stefano; Schraen, Benjamin
2014
Canonical diagrams and representations modulo $$p$$ of $$\mathrm{GL}_{2}(F)$$. Zbl 1232.22012
Hu, Yongquan
2012
Entanglement and decoherence of coupled superconductor qubits in contact with a common environment. Zbl 1245.81010
Ji, Y. H.; Hu, Y.; Yu, Y. X.
2011
Controlling on entangled decoherence by the interaction model between qubit and environment. Zbl 1225.81013
Ji, Y. H.; Lai, H. F.; Hu, Y.
2011
On some supersingular representations of $$\mathrm{GL}_2 (\mathbb Q_{p^f})$$. Zbl 1206.22010
Hu, Yongquan
2010
Invariant norms and the existence of admissible filtrations. Zbl 1214.11132
Hu, Yongquan
2009
all top 5

Cited by 21 Authors

 5 Morra, Stefano 3 Hu, Yongquan 3 Paskunas, Vytautas 2 Breuil, Christophe 2 Hauseux, Julien 2 Herzig, Florian 2 Sorensen, Claus Mazanti 1 Abdellatif, Ramla 1 Ban, Masashi 1 David, Agnès 1 De Ieso, Marco 1 Dieulefait, Luis Victor 1 Gee, Toby 1 Hu, Jiajun 1 Hu, Yongchang 1 Ji, Youhu 1 Kozioł, Karol 1 Le, Daniel 1 Savitt, David 1 Shotton, Jack 1 Yu, Yinxian
all top 5

Cited in 20 Serials

 2 Journal of Number Theory 2 Transactions of the American Mathematical Society 2 Algebra & Number Theory 1 International Journal of Theoretical Physics 1 Israel Journal of Mathematics 1 Annales de l’Institut Fourier 1 Publications Mathématiques 1 Inventiones Mathematicae 1 Journal of Algebra 1 Manuscripta Mathematica 1 Mathematische Annalen 1 Memoirs of the American Mathematical Society 1 Pacific Journal of Mathematics 1 Journal de Mathématiques Pures et Appliquées. Neuvième Série 1 Documenta Mathematica 1 Annals of Mathematics. Second Series 1 Journal of the European Mathematical Society (JEMS) 1 Acta Mathematica Sinica. English Series 1 Journal of the Institute of Mathematics of Jussieu 1 International Journal of Quantum Information

Cited in 5 Fields

 19 Topological groups, Lie groups (22-XX) 16 Number theory (11-XX) 3 Group theory and generalizations (20-XX) 2 Quantum theory (81-XX) 1 Algebraic geometry (14-XX)

Wikidata Timeline

The data are displayed as stored in Wikidata under a Creative Commons CC0 License. Updates and corrections should be made in Wikidata.


text:
# predict

Classify observations using naive Bayes classifier

## Syntax

``label = predict(Mdl,X)``
``````[label,Posterior,Cost] = predict(Mdl,X)``````

## Description

example

````label = predict(Mdl,X)` returns a vector of predicted class labels for the predictor data in the table or matrix `X`, based on the trained naive Bayes classification model `Mdl`. The trained naive Bayes model can either be full or compact.```

example

``````[label,Posterior,Cost] = predict(Mdl,X)``` also returns the Posterior Probability (`Posterior`) and predicted (expected) Misclassification Cost (`Cost`) corresponding to the observations (rows) in `Mdl.X`. For each observation in `X`, the predicted class label corresponds to the minimum expected classification cost among all classes.```

## Examples

collapse all

Load the `fisheriris` data set. Create `X` as a numeric matrix that contains four petal measurements for 150 irises. Create `Y` as a cell array of character vectors that contains the corresponding iris species.

```load fisheriris X = meas; Y = species; rng('default') % for reproducibility```

Randomly partition observations into a training set and a test set with stratification, using the class information in `Y`. Specify a 30% holdout sample for testing.

`cv = cvpartition(Y,'HoldOut',0.30);`

Extract the training and test indices.

```trainInds = training(cv); testInds = test(cv);```

Specify the training and test data sets.

```XTrain = X(trainInds,:); YTrain = Y(trainInds); XTest = X(testInds,:); YTest = Y(testInds);```

Train a naive Bayes classifier using the predictors `XTrain` and class labels `YTrain`. A recommended practice is to specify the class names. `fitcnb` assumes that each predictor is conditionally and normally distributed.

`Mdl = fitcnb(XTrain,YTrain,'ClassNames',{'setosa','versicolor','virginica'})`
```Mdl = ClassificationNaiveBayes ResponseName: 'Y' CategoricalPredictors: [] ClassNames: {'setosa' 'versicolor' 'virginica'} ScoreTransform: 'none' NumObservations: 105 DistributionNames: {'normal' 'normal' 'normal' 'normal'} DistributionParameters: {3x4 cell} Properties, Methods ```

`Mdl` is a trained `ClassificationNaiveBayes` classifier.

Predict the test sample labels.

```idx = randsample(sum(testInds),10); label = predict(Mdl,XTest);```

Display the results for a random set of 10 observations in the test sample.

```table(YTest(idx),label(idx),'VariableNames',... {'TrueLabel','PredictedLabel'})```
```ans=10×2 table TrueLabel PredictedLabel ______________ ______________ {'virginica' } {'virginica' } {'versicolor'} {'versicolor'} {'versicolor'} {'versicolor'} {'virginica' } {'virginica' } {'setosa' } {'setosa' } {'virginica' } {'virginica' } {'setosa' } {'setosa' } {'versicolor'} {'versicolor'} {'versicolor'} {'virginica' } {'versicolor'} {'versicolor'} ```

Create a confusion chart from the true labels `YTest` and the predicted labels `label`.

`cm = confusionchart(YTest,label);`

Estimate posterior probabilities and misclassification costs for new observations using a naive Bayes classifier. Classify new observations using a memory-efficient pretrained classifier.

Load the `fisheriris` data set. Create `X` as a numeric matrix that contains four petal measurements for 150 irises. Create `Y` as a cell array of character vectors that contains the corresponding iris species.

```load fisheriris X = meas; Y = species; rng('default') % for reproducibility```

Partition the data set into two sets: one contains the training set, and the other contains new, unobserved data. Reserve 10 observations for the new data set.

```n = size(X,1); newInds = randsample(n,10); inds = ~ismember(1:n,newInds); XNew = X(newInds,:); YNew = Y(newInds);```

Train a naive Bayes classifier using the predictors `X` and class labels `Y`. A recommended practice is to specify the class names. `fitcnb` assumes that each predictor is conditionally and normally distributed.

```Mdl = fitcnb(X(inds,:),Y(inds),... 'ClassNames',{'setosa','versicolor','virginica'});```

`Mdl` is a trained `ClassificationNaiveBayes` classifier.

Conserve memory by reducing the size of the trained naive Bayes classifier.

```CMdl = compact(Mdl); whos('Mdl','CMdl')```
``` Name Size Bytes Class Attributes CMdl 1x1 5406 classreg.learning.classif.CompactClassificationNaiveBayes Mdl 1x1 12731 ClassificationNaiveBayes ```

`CMdl` is a `CompactClassificationNaiveBayes` classifier. It uses less memory than `Mdl` because `Mdl` stores the data.

Display the class names of `CMdl` using dot notation.

`CMdl.ClassNames`
```ans = 3x1 cell {'setosa' } {'versicolor'} {'virginica' } ```

Predict the labels. Estimate the posterior probabilities and expected class misclassification costs.

`[labels,PostProbs,MisClassCost] = predict(CMdl,XNew);`

Compare the true labels with the predicted labels.

```table(YNew,labels,PostProbs,MisClassCost,'VariableNames',... {'TrueLabels','PredictedLabels',... 'PosteriorProbabilities','MisclassificationCosts'})```
```ans=10×4 table TrueLabels PredictedLabels PosteriorProbabilities MisclassificationCosts ______________ _______________ _________________________________________ ______________________________________ {'virginica' } {'virginica' } 4.0832e-268 4.6422e-09 1 1 1 4.6422e-09 {'setosa' } {'setosa' } 1 3.0706e-18 4.6719e-25 3.0706e-18 1 1 {'virginica' } {'virginica' } 1.0007e-246 5.8758e-10 1 1 1 5.8758e-10 {'versicolor'} {'versicolor'} 1.2022e-61 0.99995 4.9859e-05 1 4.9859e-05 0.99995 {'virginica' } {'virginica' } 2.687e-226 1.7905e-08 1 1 1 1.7905e-08 {'versicolor'} {'versicolor'} 3.3431e-76 0.99971 0.00028983 1 0.00028983 0.99971 {'virginica' } {'virginica' } 4.05e-166 0.0028527 0.99715 1 0.99715 0.0028527 {'setosa' } {'setosa' } 1 1.1272e-14 2.0308e-23 1.1272e-14 1 1 {'virginica' } {'virginica' } 1.3292e-228 8.3604e-10 1 1 1 8.3604e-10 {'setosa' } {'setosa' } 1 4.5023e-17 2.1724e-24 4.5023e-17 1 1 ```

`PostProbs` and `MisClassCost` are `10`-by-`3` numeric matrices, where each row corresponds to a new observation and each column corresponds to a class. The order of the columns corresponds to the order of `CMdl.ClassNames`.

Load the `fisheriris` data set. Create `X` as a numeric matrix that contains petal length and width measurements for 150 irises. Create `Y` as a cell array of character vectors that contains the corresponding iris species.

```load fisheriris X = meas(:,3:4); Y = species;```

Train a naive Bayes classifier using the predictors `X` and class labels `Y`. A recommended practice is to specify the class names. `fitcnb` assumes that each predictor is conditionally and normally distributed.

`Mdl = fitcnb(X,Y,'ClassNames',{'setosa','versicolor','virginica'});`

`Mdl` is a trained `ClassificationNaiveBayes` classifier.

Define a grid of values in the observed predictor space.

```xMax = max(X); xMin = min(X); h = 0.01; [x1Grid,x2Grid] = meshgrid(xMin(1):h:xMax(1),xMin(2):h:xMax(2));```

Predict the posterior probabilities for each instance in the grid.

`[~,PosteriorRegion] = predict(Mdl,[x1Grid(:),x2Grid(:)]);`

Plot the posterior probability regions and the training data.

```h = scatter(x1Grid(:),x2Grid(:),1,PosteriorRegion); h.MarkerEdgeAlpha = 0.3;```

Plot the data.

```hold on gh = gscatter(X(:,1),X(:,2),Y,'k','dx*'); title 'Iris Petal Measurements and Posterior Probabilities'; xlabel 'Petal length (cm)'; ylabel 'Petal width (cm)'; axis tight legend(gh,'Location','Best') hold off```

## Input Arguments

collapse all

Naive Bayes classification model, specified as a `ClassificationNaiveBayes` model object or `CompactClassificationNaiveBayes` model object returned by `fitcnb` or `compact`, respectively.

Predictor data to be classified, specified as a numeric matrix or table.

Each row of `X` corresponds to one observation, and each column corresponds to one variable.

• For a numeric matrix:

• The variables that make up the columns of `X` must have the same order as the predictor variables that trained `Mdl`.

• If you train `Mdl` using a table (for example, `Tbl`), then `X` can be a numeric matrix if `Tbl` contains only numeric predictor variables. To treat numeric predictors in `Tbl` as categorical during training, identify categorical predictors using the `'CategoricalPredictors'` name-value pair argument of `fitcnb`. If `Tbl` contains heterogeneous predictor variables (for example, numeric and categorical data types) and `X` is a numeric matrix, then `predict` throws an error.

• For a table:

• `predict` does not support multicolumn variables or cell arrays other than cell arrays of character vectors.

• If you train `Mdl` using a table (for example, `Tbl`), then all predictor variables in `X` must have the same variable names and data types as the variables that trained `Mdl` (stored in `Mdl.PredictorNames`). However, the column order of `X` does not need to correspond to the column order of `Tbl`. `Tbl` and `X` can contain additional variables (response variables, observation weights, and so on), but `predict` ignores them.

• If you train `Mdl` using a numeric matrix, then the predictor names in `Mdl.PredictorNames` must be the same as the corresponding predictor variable names in `X`. To specify predictor names during training, use the '`PredictorNames`' name-value pair argument of `fitcnb`. All predictor variables in `X` must be numeric vectors. `X` can contain additional variables (response variables, observation weights, and so on), but `predict` ignores them.

Data Types: `table` | `double` | `single`

Notes:

• If `Mdl.DistributionNames` is `'mn'`, then the software returns `NaN`s corresponding to rows of `X` that contain at least one `NaN`.

• If `Mdl.DistributionNames` is not `'mn'`, then the software ignores `NaN` values when estimating misclassification costs and posterior probabilities. Specifically, the software computes the conditional density of the predictors given the class by leaving out the factors corresponding to missing predictor values.

• For predictor distribution specified as `'mvmn'`, if `X` contains levels that are not represented in the training data (that is, not in `Mdl.CategoricalLevels` for that predictor), then the conditional density of the predictors given the class is 0. For those observations, the software returns the corresponding value of `Posterior` as a `NaN`. The software determines the class label for such observations using the class prior probability stored in `Mdl.Prior`.

## Output Arguments

collapse all

Predicted class labels, returned as a categorical vector, character array, logical or numeric vector, or cell array of character vectors.

The predicted class labels have the following:

• Same data type as the observed class labels (`Mdl.Y`). (The software treats string arrays as cell arrays of character vectors.)

• Length equal to the number of rows of `Mdl.X`.

• Class yielding the lowest expected misclassification cost (`Cost`).

Class Posterior Probability, returned as a numeric matrix. `Posterior` has rows equal to the number of rows of `Mdl.X` and columns equal to the number of distinct classes in the training data (`size(Mdl.ClassNames,1)`).

`Posterior(j,k)` is the predicted posterior probability of class `k` (in class `Mdl.ClassNames(k)`) given the observation in row `j` of `Mdl.X`.

Expected Misclassification Cost, returned as a numeric matrix. `Cost` has rows equal to the number of rows of `Mdl.X` and columns equal to the number of distinct classes in the training data (`size(Mdl.ClassNames,1)`).

`Cost(j,k)` is the expected misclassification cost of the observation in row `j` of `Mdl.X` predicted into class `k` (in class `Mdl.ClassNames(k)`).

collapse all

### Misclassification Cost

A misclassification cost is the relative severity of a classifier labeling an observation into the wrong class.

Two types of misclassification cost exist: true and expected. Let K be the number of classes.

• True misclassification cost — A K-by-K matrix, where element (i,j) indicates the cost of classifying an observation into class j if its true class is i. The software stores the misclassification cost in the property `Mdl.Cost`, and uses it in computations. By default, `Mdl.Cost(i,j)` = 1 if `i``j`, and `Mdl.Cost(i,j)` = 0 if `i` = `j`. In other words, the cost is `0` for correct classification and `1` for any incorrect classification.

• Expected misclassification cost — A K-dimensional vector, where element k is the weighted average cost of classifying an observation into class k, weighted by the class posterior probabilities.

`${c}_{k}=\sum _{j=1}^{K}\stackrel{^}{P}\left(Y=j|{x}_{1},...,{x}_{P}\right)Cos{t}_{jk}.$`

In other words, the software classifies observations into the class with the lowest expected misclassification cost.

### Posterior Probability

The posterior probability is the probability that an observation belongs in a particular class, given the data.

For naive Bayes, the posterior probability that a classification is k for a given observation (x1,...,xP) is

`$\stackrel{^}{P}\left(Y=k|{x}_{1},..,{x}_{P}\right)=\frac{P\left({X}_{1},...,{X}_{P}|y=k\right)\pi \left(Y=k\right)}{P\left({X}_{1},...,{X}_{P}\right)},$`

where:

• $P\left({X}_{1},...,{X}_{P}|y=k\right)$ is the conditional joint density of the predictors given they are in class k. `Mdl.DistributionNames` stores the distribution names of the predictors.

• π(Y = k) is the class prior probability distribution. `Mdl.Prior` stores the prior distribution.

• $P\left({X}_{1},..,{X}_{P}\right)$ is the joint density of the predictors. The classes are discrete, so $P\left({X}_{1},...,{X}_{P}\right)=\sum _{k=1}^{K}P\left({X}_{1},...,{X}_{P}|y=k\right)\pi \left(Y=k\right).$

### Prior Probability

The prior probability of a class is the assumed relative frequency with which observations from that class occur in a population.

## Version History

Introduced in R2014b


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Home > Cannot Determine > Cannot Determine Size Of Graphic In No Boundingbox Pdflatex

# Cannot Determine Size Of Graphic In No Boundingbox Pdflatex

How did early mathematicians make it without Set theory? This is my pillow If I receive written permission to use content from a paper without citing, is it plagiarism? share|improve this answer answered Jan 22 '13 at 15:38 Stephan Lukasczyk 1,70521127 add a comment| Not the answer you're looking for? l.10 ...graphics[width=140mm]{xfig/ch1/doubsq.pdf} With \usepackage{graphics} I get the error: Runaway argument? More about the author

No need to spend time to manually set the bounding box with the time-consuming trial-and-errors method (what I did until this day). –user41288 Nov 19 '13 at 15:17 I Does the "bat wing" aircraft paint design have a proper name? Does The Amazing Lightspeed Horse work, RAW? Why is this C++ code faster than my hand-written assembly for testing the Collatz conjecture? http://tex.stackexchange.com/questions/17734/cannot-determine-size-of-graphic

Browse other questions tagged pdf latex eps or ask your own question. I changed one method signature and broke 25,000 other classes. The difference between "an old,old vine" and "an old vine" Are “Referendum” and “Plebiscite” the same in the meaning, or different in the meaning and nuance? I converted the first image to PDF and the compilation happened perfectly.

Please check that there is no inclusion of epsfig, it is deprecated. I am using the following syntax for including the images : \begin{figure}[!h] \begin{center} \scalebox{0.45}{ \includegraphics{tree.jpg}} \end{center} \caption{\small A sample} \label{tree} \end{figure} What should I do ? Solutions? Great help.

jpg, as explained here. more stack exchange communities company blog Stack Exchange Inbox Reputation and Badges sign up log in tour help Tour Start here for a quick overview of the site Help Center Detailed Another question: how can I produce a .pdf file from a .tex file? I changed one method signature and broke 25,000 other classes.

How is it getting the bounding box values ? –euphoria83 Apr 8 '09 at 21:38 MacTex seems to be more robust than other Tex distributions that I've seen. The produced DVI file will only link to the PDF and the DVI-to-PDF converter will need to include it in the final PDF. Is there any known limit for how many dice RPG players are comfortable adding up? How to justify Einstein notation manipulations without explicitly writing sums?

2. Is this different than pdflatex? –zkurtz Jul 17 '13 at 13:11 @dustin, no, [demo] is not there ... –zkurtz Jul 17 '13 at 13:12 1 Can you post
3. Working with DVI is often faster and compatible with more packages.
4. C'est la première fois que j’utilise pstricks et j'ai un problème quand j'essaye d'inclure une image png (évidemment j'utilise le package graphicx) puisque j'obtiens l'erreur "cannot determine size of graphic in...
5. brexis 22 juin 2013 à 15:21:49 Salut.
6. Simply paste the $$\mathrm{\LaTeX}$$ style files (and the two .png files) into the same directory as your Beamer .tex file and add \usetheme{UofA} (or \usetheme{TTU}) before your \begin{document}.
7. share|improve this answer edited Apr 9 '12 at 16:08 answered Apr 25 '10 at 0:09 balaur 9614 add a comment| up vote 4 down vote I use MacTex, and my editor
8. Et j'utilisais aussi l'option width=\textwidth Anonyme 28 juillet 2011 à 21:07:50 Tu ne pourras pas utiliser d'image aux formats jpeg, png et Cie avec un fichier dvi et ps.

Merci. http://stackoverflow.com/questions/731933/error-including-image-in-latex Tried to do pdflatex on my file (under linux) With \usepackage{graphicx} I get the error: ! If those answers do not fully address your question, please ask a new question. 1 Welcome to TeX.sx! –user11232 Jan 22 '13 at 15:06 Your picture may be I have also tried using the option \usepackage[pdftex]{graphicx} but this doesn't help.

share|improve this answer answered Jan 31 at 1:08 IsaacS 21316 add a comment| protected by Community♦ Dec 2 '14 at 13:59 Thank you for your interest in this question. http://scenelink.org/cannot-determine/cannot-determine-size-of-graphic-in-no-boundingbox-latex.php Are “Referendum” and “Plebiscite” the same in the meaning, or different in the meaning and nuance? natwidth=... (not width= as that tries to scale to that size but still needs the natural size. Were the Smurfs the first to smurf their smurfs?

l.67 \includegraphics{mol.png} Some months ago I used the same procedure and it worked. In Latex EPS is best, PDF is a close second. –David Z Apr 8 '09 at 21:46 1 @David: EPS and PDF are essentially equivalent when it comes to image Draw some mountain peaks Are the LMDB files cross-platfrom compatible? http://scenelink.org/cannot-determine/cannot-determine-size-of-graphic-in-no-boundingbox-png.php more stack exchange communities company blog Stack Exchange Inbox Reputation and Badges sign up log in tour help Tour Start here for a quick overview of the site Help Center Detailed

share|improve this answer answered Jul 1 '11 at 11:21 kuszi 1,5681926 1 Compiling into pdf works, thanks. –drozzy Feb 3 '12 at 13:40 1 N.B.: But it's not always neokiller007 28 juillet 2011 à 23:51:30 Citation : pepere250 Soit tu utilises un programme pour réenrengistrer l'image en EPS Tu me recommandes quoi ? This is usually no problem for me.

## C'est exactement comme ça que je procédais sauf que je la mettais dans un environnement figure pour la faire flotter.

You are however able to state the natural size of the images using natwidth and natheight which will make latex compile without error. Otherwise you need to convert your image to EPS using convert and run latex on the TeX file. –Jagath Jul 17 '13 at 3:49 @JagathAR, as I mentioned, I S'inscrire Se connecter Forums des Zéros Une question ? Inscrivez-vous gratuitement en 30s.

Actually latex is pdflatex in DVI mode in modern distributions, but it can read the sizes only in PDF mode for some reason. When I use pdftex, the command: \includegraphics[height=60mm, width=100mm]{number2.png} works fine, but when I use "Tex and Ghostscript", I get the same error as you, about not being able to get the Are the LMDB files cross-platfrom compatible? http://scenelink.org/cannot-determine/cannot-determine-size-of-graphic-no-boundingbox-latex.php Connecting sino japanese verbs Actual meaning of 'After all' Why is using let inside a for loop so slow on Chrome?

But when I submit it online to a scientific journal for publication, it compiles it on the server and sends me the resultant PDF which has all these errors that prevent I've tried several solutions. Existence proof of Lorentz transformation from lightlike to lightlike vectors Does The Amazing Lightspeed Horse work, RAW? How much information can you obtain from a pulsar-black hole system?

The error message is typical when including bitmap graphics, e.g. By changing its extension from .JPG to .jpg (yes, just capital to lower) the error went away. \$ pdflatex --verion pdflatex: unrecognized option '--verion' This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live Are the LMDB files cross-platfrom compatible? so i made an assumption ... –barbara beeton Sep 17 '13 at 21:14 | show 2 more comments 1 Answer 1 active oldest votes up vote 8 down vote Although the

This one and converting the pdf to .eps or .ps all make the error go away, but then there is just a big blank space where the image is supposed to Meaning of double bracket "[[foo()]] type name;" syntax in c++? Your cache administrator is webmaster. E.g., do state how the OP might use the information you've provided to help solve his/her issues. –Mico Oct 14 '14 at 19:45 Welcome to TeX.SX!