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# Tag Info

14

Well, it can't (float), since a Black Hole is not a solid object that has any kind of surface. When someone says that a super massive black hole has less density than water, one probably means that since the density goes like $\frac{M}{R^3}$ where M is the mass and R is the typical size of the object, then for a black hole the typical size is the ...

13

A popular assumption about black holes is that their gravity grows beyond any limit so it beats all repulsive forces and the matter collapses into a singularity. [...] Is there any evidence for this assumption? It's not an assumption, it's a calculation plus a theorem, the Penrose singularity theorem. The calculation is the Tolman-Oppenheimer-Volkoff ...

8

Black holes and "anti"-black holes are the same objects. A black hole resulting from the collapse of normal matter, and a black hole resulting from the collapse of antimatter, are indistinguishable. Recall that black holes only have charge, mass, and spin and there is no way to tell that a black hole originally was matter or not (e.g., we can't measure B or ...

7

It is true that, from an outside perspective, nothing can ever pass the event horizon. I will attempt to describe the situation as best I can, to the best of my knowledge. First, let's imagine a classical black hole. By "classical" I mean a black-hole solution to Einstein's equations, which we imagine not to emit Hawking radiation (for now). Such an ...

7

Suppose you have some collection of matter that is so dense it has an event horizon where the escape velocity is greater than the speed of light. The escape velocity is obviously due to the strong gravitational field of the matter inside the event horizon, and equally obviously that matter is also pulled by it's own gravity towards it's centre of mass. Also ...

7

As dmckee says in his comment, the answer is no, a stationary spherical shell isn't possible. This is because not even the interparticle forces in neutronium are strong enough to support it. The problem is that once inside the event horizon there is no way to travel away from the singularity, or even maintain your distance from it, without travelling faster ...

6

Indeed, nothing can get under the horizon. The stuff close to the event horizon does move outwards as the BH radius increases. Even more with any BH deformations such as waves on its surface, the tidal deformations or the change of the rotation speed, all the oblects close enough to the horizon remain "sticked" to it and follow all the changes of the BH ...

6

No, there's no detectable dispersion in gravitational lensing, at least not when the wavelength is much shorter than the curvature radius. The reason is simple to see: one may approximate the light by rays propagating along geodesics. They have to be null geodesics because the photons are massless. And given the location of the source and initial ...

6

Spacetime is made out of events. An event simply means a moment in time plus a point in space. Events can cause other events, e.g., if a spaceship flies from event A to event B, or a radio signal travels from A to B. It's also possible to have events that can't be causally linked, e.g., if B is 10 light-years away from A and 5 years in A's future, then A ...

6

There are actually several types of black-hole horizons with different definitions, each of which are sometimes called the event horizon. But the one that scientists typically mean when they use this phrase is more precisely called the "absolute horizon". Wikipedia's page is accurate, but not very complete. There is also a nice overview given by the ...

6

The final stages of star collapse include various stages, but three common ones to consider are white dwarfs, neutron stars, and black holes. White Dwarfs are formed when gravitational forces of the mass of the remnants of the star cannot overcome the repulsion of the electron degeneracy pressure. So think of gravity competing with the electromagnetic ...

6

I suspect you're not asking the question you're really interested in, because the answer to your question is really boring. If you jump into a black hole you'll see the event horizon retreating before you, and you'll never cross it. The distance you've travelled is an ambiguous quantity since of course in your frame you're stationary and have travelled no ...

5

Whatever happens, one should always remember that for any observer gravity manifests itself only through second order effecs in the distance to the observer. In other words, in the coordinates, comoving with any observer, metric is always flat along the observer's world line and is quadratic in spatial distance to the world line (see comoving Fermi ...

5

We are already living in a nearly empty de Sitter space - the cosmological constant already represents 73% of the energy density in the Universe - and the Universe won't experience any qualitative change in the future: the percentage will just approach 100%. However, once the space may be approximated as an empty de Sitter space, all moments of time are ...

5

The Hubble length $c/H_0$ does not coincide with the radius of the observable universe. Your calculation assumes a Hubble parameter that doesn't change over time. This is not correct: the Hubble parameter $H$ changes over time, and $H_0$ (the Hubble constant) indicates the current value of $H$. To refer to $H_0$ as a 'constant' is a bit of a misnomer, it ...

5

This is one of the common fallacies when it comes to both special and general relativity. A lot of people, when encountering SR for the first time, think that causality can be violated by using a long pole to send messages. Similarly, it is a common thought that one can "dip" a pole into a black hole and then pull it out. After all, solid things are solid, ...

4

This is a case of an unwisely chosen simile taken waaaay too far. This idea, that the entire universe could be inside the event horizon of not a supermassive, but rather a superduperultrahypermegastupendouslymassive black hole, is usually introduced in introductory classes about general relativity. The instructor in this case is trying to make clear that, ...

4

Yes indeed, in the circumstances you describe a horizon does form, and it's called a cosmological event horizon. Googling for this term will lots of articles on the subject, though for once Wikipedia has let me down and does not have a good article on the subject. However each galaxy wouldn't be behind it's own horizon as groups of galaxies tend to be ...

4

An event horizon is the theoretical boundary around a black hole beyond which an observer cannot be in contact (i.e) the region from which light or any other radiation cannot (classically) escape from the inside. In my opinion, it's NO because once an object has reached beyond the event horizon, we don't even know what happens to it inside. It can't even ...

4

You've heard a slightly garbled account of the physics inside a black hole event horizon, but what you've heard is not so far from the truth. The physics of (stationary) black holes is described by the Schwarzschild metric, though unless you're a GR nerd you'll find this a bit opaque. I'll try to describe what is going on in everyday terms, but you need to ...

4

A stationary uncharged black hole is described the the Schwarzschild metric: $$ds^2 = -\left(1-\frac{2GM}{c^2r}\right)dt^2 + \frac{dr^2}{\left(1-\frac{2GM}{c^2r}\right)} + r^2 (d\theta^2 + sin^2\theta d\phi^2)$$ The event horizon is at $r = 2GM/c^2$, where the $dr^2$ term goes to infinity, so it is a surface of constant $r$ i.e. it is indeed a sphere. ...

4

To add to John's answer: black hole with nonzero angular momentum is represented by Kerr metric. It's horizon is a spherical surface, but it also has a special surface: ergosphere that is oblate spheroid touching horizon at two 'poles'. The no-hair theorem of black hole physics precludes them from having more complicated shapes, because such shape would have ...

3

When matter and antimatter meets we see an explosion but no additional mass/energy is created. What happens is that matter is converted to energy while the total mass/energy is constant. So when the "matter" black hole and the "antimatter" black hole merge, the total mass/energy of the resultant black hole will be equal to the sum of the mass/energy of the ...

3

I would like to add a fact that, perhaps, is not controversial. Namely, that all the information about any infalling object will be available for the outside observer at any time. The information cannot get lost under the horizon, otherwise we have the information loss paradox. This means that it is theoretically possible for an outside observer to restore ...

3

There is no uniquely defined time axis is relativity. Locally, observers in different states of motion have time axes that don't coincide. Even if we fix a state of motion for the observer, there is both a forward and a backward time axis. A more meaningful question would be phrased in terms of the past and future light cones: ...

3

It seems to me that this question is quite trivial, so I hope I am not overlooking anything. For a horizon to form, the observer needs to be "as light-like as possible" (otherwise the slope of his world-line will be more than 45 degrees and union of his past light-cones will cover all of universe). And because there is no other way of achieving that than ...

3

Yes, P will observe a thermal radiation that is locally identical to the Hawking radiation and it is called the Unruh radiation. Its temperature is $a/2\pi$ in the $\hbar=c=k_B=1$ units. http://en.wikipedia.org/wiki/Unruh_radiation A few more words about the relationship of Unruh and Hawking radiation. Unruh found his derivation after Hawking found ...

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The BLAS Interface¶

The cvxopt.blas module provides an interface to the double-precision real and complex Basic Linear Algebra Subprograms (BLAS). The names and calling sequences of the Python functions in the interface closely match the corresponding Fortran BLAS routines (described in the references below) and their functionality is exactly the same. Many of the operations performed by the BLAS routines can be implemented in a more straightforward way by using the matrix arithmetic of the section Arithmetic Operations, combined with the slicing and indexing of the section Indexing and Slicing. As an example, C = A * B gives the same result as the BLAS call gemm(A, B, C). The BLAS interface offers two advantages. First, some of the functions it includes are not easily implemented using the basic matrix arithmetic. For example, BLAS includes functions that efficiently exploit symmetry or triangular matrix structure. Second, there is a performance difference that can be significant for large matrices. Although our implementation of the basic matrix arithmetic makes internal calls to BLAS, it also often requires creating temporary matrices to store intermediate results. The BLAS functions on the other hand always operate directly on their matrix arguments and never require any copying to temporary matrices. Thus they can be viewed as generalizations of the in-place matrix addition and scalar multiplication of the section Arithmetic Operations to more complicated operations.

• C. L. Lawson, R. J. Hanson, D. R. Kincaid, F. T. Krogh, Basic Linear Algebra Subprograms for Fortran Use, ACM Transactions on Mathematical Software, 5(3), 309-323, 1975.
• J. J. Dongarra, J. Du Croz, S. Hammarling, R. J. Hanson, An Extended Set of Fortran Basic Linear Algebra Subprograms, ACM Transactions on Mathematical Software, 14(1), 1-17, 1988.
• J. J. Dongarra, J. Du Croz, S. Hammarling, I. Duff, A Set of Level 3 Basic Linear Algebra Subprograms, ACM Transactions on Mathematical Software, 16(1), 1-17, 1990.

Matrix Classes¶

The BLAS exploit several types of matrix structure: symmetric, Hermitian, triangular, and banded. We represent all these matrix classes by dense real or complex matrix objects, with additional arguments that specify the structure.

Vector
A real or complex $$n$$-vector is represented by a matrix of type 'd' or 'z' and length $$n$$, with the entries of the vector stored in column-major order.
General matrix
A general real or complex $$m$$ by $$n$$ matrix is represented by a real or complex matrix of size ($$m$$, $$n$$).
Symmetric matrix

A real or complex symmetric matrix of order $$n$$ is represented by a real or complex matrix of size ($$n$$, $$n$$), and a character argument uplo with two possible values: 'L' and 'U'. If uplo is 'L', the lower triangular part of the symmetric matrix is stored; if uplo is 'U', the upper triangular part is stored. A square matrix X of size ($$n$$, $$n$$) can therefore be used to represent the symmetric matrices

$\begin{split}\left[\begin{array}{ccccc} X[0,0] & X[1,0] & X[2,0] & \cdots & X[n-1,0] \\ X[1,0] & X[1,1] & X[2,1] & \cdots & X[n-1,1] \\ X[2,0] & X[2,1] & X[2,2] & \cdots & X[n-1,2] \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ X[n-1,0] & X[n-1,1] & X[n-1,2] & \cdots & X[n-1,n-1] \end{array}\right] \quad \mbox{(uplo = 'L')},\end{split}$$\begin{split}\left[\begin{array}{ccccc} X[0,0] & X[0,1] & X[0,2] & \cdots & X[0,n-1] \\ X[0,1] & X[1,1] & X[1,2] & \cdots & X[1,n-1] \\ X[0,2] & X[1,2] & X[2,2] & \cdots & X[2,n-1] \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ X[0,n-1] & X[1,n-1] & X[2,n-1] & \cdots & X[n-1,n-1] \end{array}\right] \quad \mbox{(uplo = U')}.\end{split}$
Complex Hermitian matrix

A complex Hermitian matrix of order $$n$$ is represented by a matrix of type 'z' and size ($$n$$, $$n$$), and a character argument uplo with the same meaning as for symmetric matrices. A complex matrix X of size ($$n$$, $$n$$) can represent the Hermitian matrices

$\begin{split}\left[\begin{array}{ccccc} \Re X[0,0] & \bar X[1,0] & \bar X[2,0] & \cdots & \bar X[n-1,0] \\ X[1,0] & \Re X[1,1] & \bar X[2,1] & \cdots & \bar X[n-1,1] \\ X[2,0] & X[2,1] & \Re X[2,2] & \cdots & \bar X[n-1,2] \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ X[n-1,0] & X[n-1,1] & X[n-1,2] & \cdots & \Re X[n-1,n-1] \end{array}\right] \quad \mbox{(uplo = 'L')},\end{split}$$\begin{split}\left[\begin{array}{ccccc} \Re X[0,0] & X[0,1] & X[0,2] & \cdots & X[0,n-1] \\ \bar X[0,1] & \Re X[1,1] & X[1,2] & \cdots & X[1,n-1] \\ \bar X[0,2] & \bar X[1,2] & \Re X[2,2] & \cdots & X[2,n-1] \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \bar X[0,n-1] & \bar X[1,n-1] & \bar X[2,n-1] & \cdots & \Re X[n-1,n-1] \end{array}\right] \quad \mbox{(uplo = 'U')}.\end{split}$
Triangular matrix

A real or complex triangular matrix of order $$n$$ is represented by a real or complex matrix of size ($$n$$, $$n$$), and two character arguments: an argument uplo with possible values 'L' and 'U' to distinguish between lower and upper triangular matrices, and an argument diag with possible values 'U' and 'N' to distinguish between unit and non-unit triangular matrices. A square matrix X of size ($$n$$, $$n$$) can represent the triangular matrices

$\begin{split}\left[\begin{array}{ccccc} X[0,0] & 0 & 0 & \cdots & 0 \\ X[1,0] & X[1,1] & 0 & \cdots & 0 \\ X[2,0] & X[2,1] & X[2,2] & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ X[n-1,0] & X[n-1,1] & X[n-1,2] & \cdots & X[n-1,n-1] \end{array}\right] \quad \mbox{(uplo = 'L', diag = 'N')},\end{split}$$\begin{split}\left[\begin{array}{ccccc} 1 & 0 & 0 & \cdots & 0 \\ X[1,0] & 1 & 0 & \cdots & 0 \\ X[2,0] & X[2,1] & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ X[n-1,0] & X[n-1,1] & X[n-1,2] & \cdots & 1 \end{array}\right] \quad \mbox{(uplo = 'L', diag = 'U')},\end{split}$$\begin{split}\left[\begin{array}{ccccc} X[0,0] & X[0,1] & X[0,2] & \cdots & X[0,n-1] \\ 0 & X[1,1] & X[1,2] & \cdots & X[1,n-1] \\ 0 & 0 & X[2,2] & \cdots & X[2,n-1] \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & X[n-1,n-1] \end{array}\right] \quad \mbox{(uplo = 'U', diag = 'N')},\end{split}$$\begin{split}\left[\begin{array}{ccccc} 1 & X[0,1] & X[0,2] & \cdots & X[0,n-1] \\ 0 & 1 & X[1,2] & \cdots & X[1,n-1] \\ 0 & 0 & 1 & \cdots & X[2,n-1] \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1 \end{array}\right] \quad \mbox{(uplo = 'U', diag = 'U')}.\end{split}$
General band matrix

A general real or complex $$m$$ by $$n$$ band matrix with $$k_l$$ subdiagonals and $$k_u$$ superdiagonals is represented by a real or complex matrix X of size ($$k_l + k_u + 1$$, $$n$$), and the two integers $$m$$ and $$k_l$$. The diagonals of the band matrix are stored in the rows of X, starting at the top diagonal, and shifted horizontally so that the entries of column $$k$$ of the band matrix are stored in column $$k$$ of X. A matrix X of size ($$k_l + k_u + 1$$, $$n$$) therefore represents the $$m$$ by $$n$$ band matrix

$\begin{split}\left[ \begin{array}{ccccccc} X[k_u,0] & X[k_u-1,1] & X[k_u-2,2] & \cdots & X[0,k_u] & 0 & \cdots \\ X[k_u+1,0] & X[k_u,1] & X[k_u-1,2] & \cdots & X[1,k_u] & X[0,k_u+1] & \cdots \\ X[k_u+2,0] & X[k_u+1,1] & X[k_u,2] & \cdots & X[2,k_u] & X[1,k_u+1] & \cdots \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \ddots \\ X[k_u+k_l,0] & X[k_u+k_l-1,1] & X[k_u+k_l-2,2] & \cdots & & & \\ 0 & X[k_u+k_l,1] & X[k_u+k_l-1,2] & \cdots & & & \\ \vdots & \vdots & \vdots & \ddots & & & \end{array}\right].\end{split}$
Symmetric band matrix

A real or complex symmetric band matrix of order $$n$$ with $$k$$ subdiagonals, is represented by a real or complex matrix X of size ($$k+1$$, $$n$$), and an argument uplo to indicate whether the subdiagonals (uplo is 'L') or superdiagonals (uplo is 'U') are stored. The $$k+1$$ diagonals are stored as rows of X, starting at the top diagonal (i.e., the main diagonal if uplo is 'L', or the $$k$$-th superdiagonal if uplo is 'U') and shifted horizontally so that the entries of the $$k$$-th column of the band matrix are stored in column $$k$$ of X. A matrix X of size ($$k+1$$, $$n$$) can therefore represent the band matrices

$\begin{split}\left[ \begin{array}{ccccccc} X[0,0] & X[1,0] & X[2,0] & \cdots & X[k,0] & 0 & \cdots \\ X[1,0] & X[0,1] & X[1,1] & \cdots & X[k-1,1] & X[k,1] & \cdots \\ X[2,0] & X[1,1] & X[0,2] & \cdots & X[k-2,2] & X[k-1,2] & \cdots \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \ddots \\ X[k,0] & X[k-1,1] & X[k-2,2] & \cdots & & & \\ 0 & X[k,1] & X[k-1,2] & \cdots & & & \\ \vdots & \vdots & \vdots & \ddots & & & \end{array}\right] \quad \mbox{(uplo = 'L')},\end{split}$$\begin{split}\left[ \begin{array}{ccccccc} X[k,0] & X[k-1,1] & X[k-2,2] & \cdots & X[0,k] & 0 & \cdots \\ X[k-1,1] & X[k,1] & X[k-1,2] & \cdots & X[1,k] & X[0,k+1] & \cdots \\ X[k-2,2] & X[k-1,2] & X[k,2] & \cdots & X[2,k] & X[1,k+1] & \cdots \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \ddots \\ X[0,k] & X[1,k] & X[2,k] & \cdots & & & \\ 0 & X[0,k+1] & X[1,k+1] & \cdots & & & \\ \vdots & \vdots & \vdots & \ddots & & & \end{array}\right] \quad \mbox{(uplo='U')}.\end{split}$
Hermitian band matrix

A complex Hermitian band matrix of order $$n$$ with $$k$$ subdiagonals is represented by a complex matrix of size ($$k+1$$, $$n$$) and an argument uplo, with the same meaning as for symmetric band matrices. A matrix X of size ($$k+1$$, $$n$$) can represent the band matrices

$\begin{split}\left[ \begin{array}{ccccccc} \Re X[0,0] & \bar X[1,0] & \bar X[2,0] & \cdots & \bar X[k,0] & 0 & \cdots \\ X[1,0] & \Re X[0,1] & \bar X[1,1] & \cdots & \bar X[k-1,1] & \bar X[k,1] & \cdots \\ X[2,0] & X[1,1] & \Re X[0,2] & \cdots & \bar X[k-2,2] & \bar X[k-1,2] & \cdots \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \ddots \\ X[k,0] & X[k-1,1] & X[k-2,2] & \cdots & & & \\ 0 & X[k,1] & X[k-1,2] & \cdots & & & \\ \vdots & \vdots & \vdots & \ddots & & & \end{array}\right] \quad \mbox{(uplo = 'L')},\end{split}$$\begin{split}\left[ \begin{array}{ccccccc} \Re X[k,0] & X[k-1,1] & X[k-2,2] & \cdots & X[0,k] & 0 & \cdots \\ \bar X[k-1,1] & \Re X[k,1] & X[k-1,2] & \cdots & X[1,k] & X[0,k+1] & \cdots \\ \bar X[k-2,2] & \bar X[k-1,2] & \Re X[k,2] & \cdots & X[2,k] & X[1,k+1] & \cdots \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \ddots \\ \bar X[0,k] & \bar X[1,k] & \bar X[2,k] & \cdots & & & \\ 0 & \bar X[0,k+1] & \bar X[1,k+1] & \cdots & & & \\ \vdots & \vdots & \vdots & \ddots & & & \end{array}\right] \quad \mbox{(uplo='U')}.\end{split}$
Triangular band matrix

A triangular band matrix of order $$n$$ with $$k$$ subdiagonals or superdiagonals is represented by a real complex matrix of size ($$k+1$$, $$n$$) and two character arguments uplo and diag, with similar conventions as for symmetric band matrices. A matrix X of size ($$k+1$$, $$n$$) can represent the band matrices

$\begin{split}\left[ \begin{array}{cccc} X[0,0] & 0 & 0 & \cdots \\ X[1,0] & X[0,1] & 0 & \cdots \\ X[2,0] & X[1,1] & X[0,2] & \cdots \\ \vdots & \vdots & \vdots & \ddots \\ X[k,0] & X[k-1,1] & X[k-2,2] & \cdots \\ 0 & X[k,1] & X[k-1,1] & \cdots \\ \vdots & \vdots & \vdots & \ddots \end{array}\right] \quad \mbox{(uplo = 'L', diag = 'N')},\end{split}$$\begin{split}\left[ \begin{array}{cccc} 1 & 0 & 0 & \cdots \\ X[1,0] & 1 & 0 & \cdots \\ X[2,0] & X[1,1] & 1 & \cdots \\ \vdots & \vdots & \vdots & \ddots \\ X[k,0] & X[k-1,1] & X[k-2,2] & \cdots \\ 0 & X[k,1] & X[k-1,2] & \cdots \\ \vdots & \vdots & \vdots & \ddots \end{array}\right] \quad \mbox{(uplo = 'L', diag = 'U')},\end{split}$$\begin{split}\left[ \begin{array}{ccccccc} X[k,0] & X[k-1,1] & X[k-2,3] & \cdots & X[0,k] & 0 & \cdots\\ 0 & X[k,1] & X[k-1,2] & \cdots & X[1,k] & X[0,k+1] & \cdots \\ 0 & 0 & X[k,2] & \cdots & X[2,k] & X[1,k+1] & \cdots \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \ddots \end{array}\right] \quad \mbox{(uplo = 'U', diag = 'N')},\end{split}$$\begin{split}\left[ \begin{array}{ccccccc} 1 & X[k-1,1] & X[k-2,3] & \cdots & X[0,k] & 0 & \cdots\\ 0 & 1 & X[k-1,2] & \cdots & X[1,k] & X[0,k+1] & \cdots \\ 0 & 0 & 1 & \cdots & X[2,k] & X[1,k+1] & \cdots \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \ddots \end{array}\right] \quad \mbox{(uplo = 'U', diag = 'U')}.\end{split}$

When discussing BLAS functions in the following sections we will omit several less important optional arguments that can be used to select submatrices for in-place operations. The complete specification is documented in the docstrings of the source code, and can be viewed with the pydoc help program.

Level 1 BLAS¶

The level 1 functions implement vector operations.

cvxopt.blas.scal(alpha, x)

Scales a vector by a constant:

$x := \alpha x.$

If x is a real matrix, the scalar argument alpha must be a Python integer or float. If x is complex, alpha can be an integer, float, or complex.

cvxopt.blas.nrm2(x)

Euclidean norm of a vector: returns

$\|x\|_2.$
cvxopt.blas.asum(x)

1-Norm of a vector: returns

$\|x\|_1 \quad \mbox{(x real)}, \qquad \|\Re x\|_1 + \|\Im x\|_1 \quad \mbox{(x complex)}.$
cvxopt.blas.iamax(x)

Returns

$\mathop{\rm argmax}_{k=0,\ldots,n-1} |x_k| \quad \mbox{(x real)}, \qquad \mathop{\rm argmax}_{k=0,\ldots,n-1} |\Re x_k| + |\Im x_k| \quad \mbox{(x complex)}.$

If more than one coefficient achieves the maximum, the index of the first $$k$$ is returned.

cvxopt.blas.swap(x, y)

Interchanges two vectors:

$x \leftrightarrow y.$

x and y are matrices of the same type ('d' or 'z').

cvxopt.blas.copy(x, y)

Copies a vector to another vector:

$y := x.$

x and y are matrices of the same type ('d' or 'z').

cvxopt.blas.axpy(x, y[, alpha = 1.0])

Constant times a vector plus a vector:

$y := \alpha x + y.$

x and y are matrices of the same type ('d' or 'z'). If x is real, the scalar argument alpha must be a Python integer or float. If x is complex, alpha can be an integer, float, or complex.

cvxopt.blas.dot(x, y)

Returns

$x^Hy.$

x and y are matrices of the same type ('d' or 'z').

cvxopt.blas.dotu(x, y)

Returns

$x^Ty.$

x and y are matrices of the same type ('d' or 'z').

Level 2 BLAS¶

The level 2 functions implement matrix-vector products and rank-1 and rank-2 matrix updates. Different types of matrix structure can be exploited using the conventions of the section Matrix Classes.

cvxopt.blas.gemv(A, x, y[, trans = 'N', alpha = 1.0, beta = 0.0])

Matrix-vector product with a general matrix:

$\begin{split}y & := \alpha Ax + \beta y \quad (\mathrm{trans} = \mathrm{'N'}), \\ y & := \alpha A^T x + \beta y \quad (\mathrm{trans} = \mathrm{'T'}), \\ y & := \alpha A^H x + \beta y \quad (\mathrm{trans} = \mathrm{'C'}).\end{split}$

The arguments A, x, and y must have the same type ('d' or 'z'). Complex values of alpha and beta are only allowed if A is complex.

cvxopt.blas.symv(A, x, y[, uplo = 'L', alpha = 1.0, beta = 0.0])

Matrix-vector product with a real symmetric matrix:

$y := \alpha A x + \beta y,$

where $$A$$ is a real symmetric matrix. The arguments A, x, and y must have type 'd', and alpha and beta must be real.

cvxopt.blas.hemv(A, x, y[, uplo = 'L', alpha = 1.0, beta = 0.0])

Matrix-vector product with a real symmetric or complex Hermitian matrix:

$y := \alpha A x + \beta y,$

where $$A$$ is real symmetric or complex Hermitian. The arguments A, x, y must have the same type ('d' or 'z'). Complex values of alpha and beta are only allowed if A is complex.

cvxopt.blas.trmv(A, x[, uplo = 'L', trans = 'N', diag = 'N'])

Matrix-vector product with a triangular matrix:

$\begin{split}x & := Ax \quad (\mathrm{trans} = \mathrm{'N'}), \\ x & := A^T x \quad (\mathrm{trans} = \mathrm{'T'}), \\ x & := A^H x \quad (\mathrm{trans} = \mathrm{'C'}),\end{split}$

where $$A$$ is square and triangular. The arguments A and x must have the same type ('d' or 'z').

cvxopt.blas.trsv(A, x[, uplo = 'L', trans = 'N', diag = 'N'])

Solution of a nonsingular triangular set of linear equations:

$\begin{split}x & := A^{-1}x \quad (\mathrm{trans} = \mathrm{'N'}), \\ x & := A^{-T}x \quad (\mathrm{trans} = \mathrm{'T'}), \\ x & := A^{-H}x \quad (\mathrm{trans} = \mathrm{'C'}),\end{split}$

where $$A$$ is square and triangular with nonzero diagonal elements. The arguments A and x must have the same type ('d' or 'z').

cvxopt.blas.gbmv(A, m, kl, x, y[, trans = 'N', alpha = 1.0, beta = 0.0])

Matrix-vector product with a general band matrix:

$\begin{split}y & := \alpha Ax + \beta y \quad (\mathrm{trans} = \mathrm{'N'}), \\ y & := \alpha A^T x + \beta y \quad (\mathrm{trans} = \mathrm{'T'}), \\ y & := \alpha A^H x + \beta y \quad (\mathrm{trans} = \mathrm{'C'}),\end{split}$

where $$A$$ is a rectangular band matrix with $$m$$ rows and $$k_l$$ subdiagonals. The arguments A, x, y must have the same type ('d' or 'z'). Complex values of alpha and beta are only allowed if A is complex.

cvxopt.blas.sbmv(A, x, y[, uplo = 'L', alpha = 1.0, beta = 0.0])

Matrix-vector product with a real symmetric band matrix:

$y := \alpha Ax + \beta y,$

where $$A$$ is a real symmetric band matrix. The arguments A, x, y must have type 'd', and alpha and beta must be real.

cvxopt.blas.hbmv(A, x, y[, uplo = 'L', alpha = 1.0, beta = 0.0])

Matrix-vector product with a real symmetric or complex Hermitian band matrix:

$y := \alpha Ax + \beta y,$

where $$A$$ is a real symmetric or complex Hermitian band matrix. The arguments A, x, y must have the same type ('d' or 'z'). Complex values of alpha and beta are only allowed if A is complex.

cvxopt.blas.tbmv(A, x[, uplo = 'L', trans = 'N', diag = 'N'])

Matrix-vector product with a triangular band matrix:

$\begin{split}x & := Ax \quad (\mathrm{trans} = \mathrm{'N'}), \\ x & := A^T x \quad (\mathrm{trans} = \mathrm{'T'}), \\ x & := A^H x \quad (\mathrm{trans} = \mathrm{'C'}).\end{split}$

The arguments A and x must have the same type ('d' or 'z').

cvxopt.blas.tbsv(A, x[, uplo = 'L', trans = 'N', diag = 'N'])

Solution of a triangular banded set of linear equations:

$\begin{split}x & := A^{-1}x \quad (\mathrm{trans} = \mathrm{'N'}), \\ x & := A^{-T} x \quad (\mathrm{trans} = \mathrm{'T'}), \\ x & := A^{-H} x \quad (\mathrm{trans} = \mathrm{'T'}),\end{split}$

where $$A$$ is a triangular band matrix of with nonzero diagonal elements. The arguments A and x must have the same type ('d' or 'z').

cvxopt.blas.ger(x, y, A[, alpha = 1.0])

General rank-1 update:

$A := A + \alpha x y^H,$

where $$A$$ is a general matrix. The arguments A, x, and y must have the same type ('d' or 'z'). Complex values of alpha are only allowed if A is complex.

cvxopt.blas.geru(x, y, A[, alpha = 1.0])

General rank-1 update:

$A := A + \alpha x y^T,$

where $$A$$ is a general matrix. The arguments A, x, and y must have the same type ('d' or 'z'). Complex values of alpha are only allowed if A is complex.

cvxopt.blas.syr(x, A[, uplo = 'L', alpha = 1.0])

Symmetric rank-1 update:

$A := A + \alpha xx^T,$

where $$A$$ is a real symmetric matrix. The arguments A and x must have type 'd'. alpha must be a real number.

cvxopt.blas.her(x, A[, uplo = 'L', alpha = 1.0])

Hermitian rank-1 update:

$A := A + \alpha xx^H,$

where $$A$$ is a real symmetric or complex Hermitian matrix. The arguments A and x must have the same type ('d' or 'z'). alpha must be a real number.

cvxopt.blas.syr2(x, y, A[, uplo = 'L', alpha = 1.0])

Symmetric rank-2 update:

$A := A + \alpha (xy^T + yx^T),$

where $$A$$ is a real symmetric matrix. The arguments A, x, and y must have type 'd'. alpha must be real.

cvxopt.blas.her2(x, y, A[, uplo = 'L', alpha = 1.0])

Symmetric rank-2 update:

$A := A + \alpha xy^H + \bar \alpha yx^H,$

where $$A$$ is a a real symmetric or complex Hermitian matrix. The arguments A, x, and y must have the same type ('d' or 'z'). Complex values of alpha are only allowed if A is complex.

As an example, the following code multiplies the tridiagonal matrix

$\begin{split}A = \left[\begin{array}{rrrr} 1 & 6 & 0 & 0 \\ 2 & -4 & 3 & 0 \\ 0 & -3 & -1 & 1 \end{array}\right]\end{split}$

with the vector $$x = (1,-1,2,-2)$$.

>>> from cvxopt import matrix
>>> from cvxopt.blas import gbmv
>>> A = matrix([[0., 1., 2.],  [6., -4., -3.],  [3., -1., 0.],  [1., 0., 0.]])
>>> x = matrix([1., -1., 2., -2.])
>>> y = matrix(0., (3,1))
>>> gbmv(A, 3, 1, x, y)
>>> print(y)
[-5.00e+00]
[ 1.20e+01]
[-1.00e+00]

The following example illustrates the use of tbsv.

>>> from cvxopt import matrix
>>> from cvxopt.blas import tbsv
>>> A = matrix([-6., 5., -1., 2.], (1,4))
>>> x = matrix(1.0, (4,1))
>>> tbsv(A, x)  # x := diag(A)^{-1}*x
>>> print(x)
[-1.67e-01]
[ 2.00e-01]
[-1.00e+00]
[ 5.00e-01]

Level 3 BLAS¶

The level 3 BLAS include functions for matrix-matrix multiplication.

cvxopt.blas.gemm(A, B, C[, transA = 'N', transB = 'N', alpha = 1.0, beta = 0.0])

Matrix-matrix product of two general matrices:

$\newcommand{\op}{\mathop{\mathrm{op}}} C := \alpha \op(A) \op(B) + \beta C$

where

$\begin{split}\newcommand{\op}{\mathop{\mathrm{op}}} \op(A) = \left\{ \begin{array}{ll} A & \mathrm{transA} = \mathrm{'N'} \\ A^T & \mathrm{transA} = \mathrm{'T'} \\ A^H & \mathrm{transA} = \mathrm{'C'} \end{array} \right. \qquad \op(B) = \left\{ \begin{array}{ll} B & \mathrm{transB} = \mathrm{'N'} \\ B^T & \mathrm{transB} = \mathrm{'T'} \\ B^H & \mathrm{transB} = \mathrm{'C'}. \end{array} \right.\end{split}$

The arguments A, B, and C must have the same type ('d' or 'z'). Complex values of alpha and beta are only allowed if A is complex.

cvxopt.blas.symm(A, B, C[, side = 'L', uplo = 'L', alpha =1.0, beta = 0.0])

Product of a real or complex symmetric matrix $$A$$ and a general matrix $$B$$:

$\begin{split}C & := \alpha AB + \beta C \quad (\mathrm{side} = \mathrm{'L'}), \\ C & := \alpha BA + \beta C \quad (\mathrm{side} = \mathrm{'R'}).\end{split}$

The arguments A, B, and C must have the same type ('d' or 'z'). Complex values of alpha and beta are only allowed if A is complex.

cvxopt.blas.hemm(A, B, C[, side = 'L', uplo = 'L', alpha = 1.0, beta = 0.0])

Product of a real symmetric or complex Hermitian matrix $$A$$ and a general matrix $$B$$:

$\begin{split}C & := \alpha AB + \beta C \quad (\mathrm{side} = \mathrm{'L'}), \\ C & := \alpha BA + \beta C \quad (\mathrm{side} = \mathrm{'R'}).\end{split}$

The arguments A, B, and C must have the same type ('d' or 'z'). Complex values of alpha and beta are only allowed if A is complex.

cvxopt.blas.trmm(A, B[, side = 'L', uplo = 'L', transA = 'N', diag = 'N', alpha = 1.0])

Product of a triangular matrix $$A$$ and a general matrix $$B$$:

$\begin{split}\newcommand{\op}{\mathop{\mathrm{op}}} \begin{split} B & := \alpha\op(A)B \quad (\mathrm{side} = \mathrm{'L'}), \\ B & := \alpha B\op(A) \quad (\mathrm{side} = \mathrm{'R'}) \end{split}\end{split}$

where

$\begin{split}\newcommand{\op}{\mathop{\mathrm{op}}} \op(A) = \left\{ \begin{array}{ll} A & \mathrm{transA} = \mathrm{'N'} \\ A^T & \mathrm{transA} = \mathrm{'T'} \\ A^H & \mathrm{transA} = \mathrm{'C'}. \end{array} \right.\end{split}$

The arguments A and B must have the same type ('d' or 'z'). Complex values of alpha are only allowed if A is complex.

cvxopt.blas.trsm(A, B[, side = 'L', uplo = 'L', transA = 'N', diag = 'N', alpha = 1.0])

Solution of a nonsingular triangular system of equations:

$\begin{split}\newcommand{\op}{\mathop{\mathrm{op}}} \begin{split} B & := \alpha \op(A)^{-1}B \quad (\mathrm{side} = \mathrm{'L'}), \\ B & := \alpha B\op(A)^{-1} \quad (\mathrm{side} = \mathrm{'R'}), \end{split}\end{split}$

where

$\begin{split}\newcommand{\op}{\mathop{\mathrm{op}}} \op(A) = \left\{ \begin{array}{ll} A & \mathrm{transA} = \mathrm{'N'} \\ A^T & \mathrm{transA} = \mathrm{'T'} \\ A^H & \mathrm{transA} = \mathrm{'C'}, \end{array} \right.\end{split}$

$$A$$ is triangular and $$B$$ is a general matrix. The arguments A and B must have the same type ('d' or 'z'). Complex values of alpha are only allowed if A is complex.

cvxopt.blas.syrk(A, C[, uplo = 'L', trans = 'N', alpha = 1.0, beta = 0.0])

Rank-$$k$$ update of a real or complex symmetric matrix $$C$$:

$\begin{split}C & := \alpha AA^T + \beta C \quad (\mathrm{trans} = \mathrm{'N'}), \\ C & := \alpha A^TA + \beta C \quad (\mathrm{trans} = \mathrm{'T'}),\end{split}$

where $$A$$ is a general matrix. The arguments A and C must have the same type ('d' or 'z'). Complex values of alpha and beta are only allowed if A is complex.

cvxopt.blas.herk(A, C[, uplo = 'L', trans = 'N', alpha = 1.0, beta = 0.0])

Rank-$$k$$ update of a real symmetric or complex Hermitian matrix $$C$$:

$\begin{split}C & := \alpha AA^H + \beta C \quad (\mathrm{trans} = \mathrm{'N'}), \\ C & := \alpha A^HA + \beta C \quad (\mathrm{trans} = \mathrm{'C'}),\end{split}$

where $$A$$ is a general matrix. The arguments A and C must have the same type ('d' or 'z'). alpha and beta must be real.

cvxopt.blas.syr2k(A, B, C[, uplo = 'L', trans = 'N', alpha = 1.0, beta = 0.0])

Rank-$$2k$$ update of a real or complex symmetric matrix $$C$$:

$\begin{split}C & := \alpha (AB^T + BA^T) + \beta C \quad (\mathrm{trans} = \mathrm{'N'}), \\ C & := \alpha (A^TB + B^TA) + \beta C \quad (\mathrm{trans} = \mathrm{'T'}).\end{split}$

$$A$$ and $$B$$ are general real or complex matrices. The arguments A, B, and C must have the same type. Complex values of alpha and beta are only allowed if A is complex.

cvxopt.blas.her2k(A, B, C[, uplo = 'L', trans = 'N', alpha = 1.0, beta = 0.0])

Rank-$$2k$$ update of a real symmetric or complex Hermitian matrix $$C$$:

$\begin{split}C & := \alpha AB^H + \bar \alpha BA^H + \beta C \quad (\mathrm{trans} = \mathrm{'N'}), \\ C & := \alpha A^HB + \bar\alpha B^HA + \beta C \quad (\mathrm{trans} = \mathrm{'C'}),\end{split}$

where $$A$$ and $$B$$ are general matrices. The arguments A, B, and C must have the same type ('d' or 'z'). Complex values of alpha are only allowed if A is complex. beta must be real.

format: plain


text:
# What is the symmetry behind this degeneracy?

I was working on a quantum mechanics problem, involving the perturbation of the 3D cubical potential well:

Suppose we perturb the infinite cubical well

$$$$V(x,y,z)=\begin{cases} 0, \text{ if } 0

by the potential

$$$$V'(x,y,z)=\alpha\delta(x-a/4)\delta(y-a/2)\delta(z-3a/4),\hspace{10pt} \alpha>0.$$$$

I found the matrix representation of the perturbation in the unperturbed basis to be:

$$$$W=\frac{\alpha}{2}\left(\frac{2}{a}\right)^3\begin{pmatrix} 1 & 0 & -1\\ 0 & 0& 0\\ -1 & 0 & 1 \end{pmatrix},$$$$

with eigenvalues

$$$$E_1=2, E_2=E_3=0.$$$$

My question is as follows: Since two of the "good states" (by the way, does anybody have a good explanation on good states, or some resources? Griffiths is horrible at explaining this.) do not get an energy correction, they are still degenerate, and I think that there must be some kind of symmetry which causes the degeneracy. I know that the "good states" for those two energy corrections are

$$$$|E_2\rangle=\frac{1}{\sqrt{2}}(|211\rangle+|112\rangle), \hspace{10pt}|E_3\rangle=|121\rangle,$$$$

where the three numbers refer to $$n_x,n_y,n_z$$. By looking at these states expressed in terms of sines, I see that they are zero at the position where the perturbation is located, thus yielding no energy correction. But this, at least for me, is not the symmetry of the Hamiltonian, or at least I'm afraid of calling this the symmetry.

Can anybody help me figuring this out?

• Didn't you already ask this question today? I can't find it anymore, but I'm positive that I read this same post earlier today. – march Apr 15 at 22:27
• There's clearly a $\mathbb{Z}_2$ symmetry $(x,y,z) \mapsto (a-z,\ldots)$. (Leaving the rest as an exercise.) – Hans Moleman Apr 15 at 22:33
• @HansMoleman Could you explain to me what a $\mathbb{Z}_2$ symmetry is, as my understanding of more abstract quantum mechanics is rather limited? (Note that this is a Griffiths problem, which should say a lot about my progression in this topic) – pjHart1000 Apr 15 at 22:37
• It's an action such that if you do it twice, you're back to where you started. Mathematicians call this group $\mathbb{Z}/2\mathbb{Z}$ but in physics we just write $\mathbb{Z}_2$. – Hans Moleman Apr 15 at 22:42
• Actually there's one more symmetry! So the theory with the $V'$ potential has at least a $V_4$ symmetry group. – Hans Moleman Apr 16 at 1:01


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1. CJM Online first

Folha, Abigail; Penafiel, Carlos
 Weingarten type surfaces in $\mathbb{H}^2\times\mathbb{R}$ and $\mathbb{S}^2\times\mathbb{R}$ In this article, we study complete surfaces $\Sigma$, isometrically immersed in the product space $\mathbb{H}^2\times\mathbb{R}$ or $\mathbb{S}^2\times\mathbb{R}$ having positive extrinsic curvature $K_e$. Let $K_i$ denote the intrinsic curvature of $\Sigma$. Assume that the equation $aK_i+bK_e=c$ holds for some real constants $a\neq0$, $b\gt 0$ and $c$. The main result of this article state that when such a surface is a topological sphere it is rotational. Keywords:Weingarten surface, extrinsic curvature, intrinsic curvature, height estimate, rotational Weingarten surfaceCategories:53C42, 53C30

2. CJM 2016 (vol 68 pp. 1096)

Smith, Benjamin H.
 Singular $G$-Monopoles on $S^1\times \Sigma$ This article provides an account of the functorial correspondence between irreducible singular $G$-monopoles on $S^1\times \Sigma$ and $\vec{t}$-stable meromorphic pairs on $\Sigma$. A theorem of B. Charbonneau and J. Hurtubise is thus generalized here from unitary to arbitrary compact, connected gauge groups. The required distinctions and similarities for unitary versus arbitrary gauge are clearly outlined and many parallels are drawn for easy transition. Once the correspondence theorem is complete, the spectral decomposition is addressed. Keywords:connection, curvature, instanton, monopole, stability, Bogomolny equation, Sasakian geometry, cameral coversCategories:53C07, 14D20

3. CJM Online first

Xiao, Jie; Ye, Deping
 Anisotropic Sobolev Capacity with Fractional Order In this paper, we introduce the anisotropic Sobolev capacity with fractional order and develop some basic properties for this new object. Applications to the theory of anisotropic fractional Sobolev spaces are provided. In particular, we give geometric characterizations for a nonnegative Radon measure $\mu$ that naturally induces an embedding of the anisotropic fractional Sobolev class $\dot{\Lambda}_{\alpha,K}^{1,1}$ into the $\mu$-based-Lebesgue-space $L^{n/\beta}_\mu$ with $0\lt \beta\le n$. Also, we investigate the anisotropic fractional $\alpha$-perimeter. Such a geometric quantity can be used to approximate the anisotropic Sobolev capacity with fractional order. Estimation on the constant in the related Minkowski inequality, which is asymptotically optimal as $\alpha\rightarrow 0^+$, will be provided. Keywords:sharpness, isoperimetric inequality, Minkowski inequality, fractional Sobolev capacity, fractional perimeterCategories:52A38, 53A15, 53A30

4. CJM 2016 (vol 69 pp. 220)

Zheng, Tao
 The Chern-Ricci Flow on Oeljeklaus-Toma Manifolds We study the Chern-Ricci flow, an evolution equation of Hermitian metrics, on a family of Oeljeklaus-Toma (OT-) manifolds which are non-KÃ¤hler compact complex manifolds with negative Kodaira dimension. We prove that, after an initial conformal change, the flow converges, in the Gromov-Hausdorff sense, to a torus with a flat Riemannian metric determined by the OT-manifolds themselves. Keywords:Chern-Ricci flow, Oeljeklaus-Toma manifold, Calabi-type estimate, Gromov-Hausdorff convergenceCategories:53C44, 53C55, 32W20, 32J18, 32M17

5. CJM 2016 (vol 68 pp. 445)

Martins, Luciana de Fátima; Saji, Kentaro
 Geometric Invariants of Cuspidal Edges We give a normal form of the cuspidal edge which uses only diffeomorphisms on the source and isometries on the target. Using this normal form, we study differential geometric invariants of cuspidal edges which determine them up to order three. We also clarify relations between these invariants. Keywords:cuspidal edge, curvature, wave frontsCategories:57R45, 53A05, 53A55

6. CJM 2016 (vol 68 pp. 655)

 Discrete Curvature and Abelian Groups We study a natural discrete Bochner-type inequality on graphs, and explore its merit as a notion of curvature'' in discrete spaces. An appealing feature of this discrete version of the so-called $\Gamma_2$-calculus (of Bakry-Ãmery) seems to be that it is fairly straightforward to compute this notion of curvature parameter for several specific graphs of interest -- particularly, abelian groups, slices of the hypercube, and the symmetric group under various sets of generators. We further develop this notion by deriving Buser-type inequalities (Ã  la Ledoux), relating functional and isoperimetric constants associated with a graph. Our derivations provide a tight bound on the Cheeger constant (i.e., the edge-isoperimetric constant) in terms of the spectral gap, for graphs with nonnegative curvature, particularly, the class of abelian Cayley graphs -- a result of independent interest. Keywords:Ricci curvature, graph theory, abelian groupsCategories:53C21, 57M15

7. CJM 2015 (vol 68 pp. 463)

 The Weak b-principle: Mumford Conjecture In this note we introduce and study a new class of maps called oriented colored broken submersions. This is the simplest class of maps that satisfies a version of the b-principle and in dimension $2$ approximates the class of oriented submersions well in the sense that every oriented colored broken submersion of dimension $2$ to a closed simply connected manifold is bordant to a submersion. We show that the Madsen-Weiss theorem (the standard Mumford Conjecture) fits a general setting of the b-principle. Namely, a version of the b-principle for oriented colored broken submersions together with the Harer stability theorem and Miller-Morita theorem implies the Madsen-Weiss theorem. Keywords:generalized cohomology theories, fold singularities, h-principle, infinite loop spacesCategories:55N20, 53C23

8. CJM 2015 (vol 67 pp. 1091)

Mine, Kotaro; Yamashita, Atsushi
 Metric Compactifications and Coarse Structures Let $\mathbf{TB}$ be the category of totally bounded, locally compact metric spaces with the $C_0$ coarse structures. We show that if $X$ and $Y$ are in $\mathbf{TB}$ then $X$ and $Y$ are coarsely equivalent if and only if their Higson coronas are homeomorphic. In fact, the Higson corona functor gives an equivalence of categories $\mathbf{TB}\to\mathbf{K}$, where $\mathbf{K}$ is the category of compact metrizable spaces. We use this fact to show that the continuously controlled coarse structure on a locally compact space $X$ induced by some metrizable compactification $\tilde{X}$ is determined only by the topology of the remainder $\tilde{X}\setminus X$. Keywords:coarse geometry, Higson corona, continuously controlled coarse structure, uniform continuity, boundary at infinityCategories:18B30, 51F99, 53C23, 54C20

9. CJM 2015 (vol 67 pp. 1109)

Nohara, Yuichi; Ueda, Kazushi
 Goldman Systems and Bending Systems We show that the moduli space of parabolic bundles on the projective line and the polygon space are isomorphic, both as complex manifolds and symplectic manifolds equipped with structures of completely integrable systems, if the stability parameters are small. Keywords:toric degenerationCategories:53D30, 14H60

10. CJM 2015 (vol 67 pp. 1411)

Kawakami, Yu
 Function-theoretic Properties for the Gauss Maps of Various Classes of Surfaces We elucidate the geometric background of function-theoretic properties for the Gauss maps of several classes of immersed surfaces in three-dimensional space forms, for example, minimal surfaces in Euclidean three-space, improper affine spheres in the affine three-space, and constant mean curvature one surfaces and flat surfaces in hyperbolic three-space. To achieve this purpose, we prove an optimal curvature bound for a specified conformal metric on an open Riemann surface and give some applications. We also provide unicity theorems for the Gauss maps of these classes of surfaces. Keywords:Gauss map, minimal surface, constant mean curvature surface, front, ramification, omitted value, the Ahlfors island theorem, unicity theorem.Categories:53C42, 30D35, 30F45, 53A10, 53A15

11. CJM 2014 (vol 67 pp. 107)

Chang, Jui-En; Xiao, Ling
 The Weyl Problem With Nonnegative Gauss Curvature In Hyperbolic Space In this paper, we discuss the isometric embedding problem in hyperbolic space with nonnegative extrinsic curvature. We prove a priori bounds for the trace of the second fundamental form $H$ and extend the result to $n$-dimensions. We also obtain an estimate for the gradient of the smaller principal curvature in 2 dimensions. Categories:53A99, 5J15, 58J05

12. CJM 2013 (vol 66 pp. 1413)

Zhang, Xi; Zhang, Xiangwen
 Generalized KÃ¤hler--Einstein Metrics and Energy Functionals In this paper, we consider a generalized KÃ¤hler-Einstein equation on KÃ¤hler manifold $M$. Using the twisted $\mathcal K$-energy introduced by Song and Tian, we show that the existence of generalized KÃ¤hler-Einstein metrics with semi-positive twisting $(1, 1)$-form $\theta$ is also closely related to the properness of the twisted $\mathcal K$-energy functional. Under the condition that the twisting form $\theta$ is strictly positive at a point or $M$ admits no nontrivial Hamiltonian holomorphic vector field, we prove that the existence of generalized KÃ¤hler-Einstein metric implies a Moser-Trudinger type inequality. Keywords:complex Monge--AmpÃ¨re equation, energy functional, generalized KÃ¤hler--Einstein metric, Moser--Trudinger type inequalityCategories:53C55, 32W20

13. CJM 2013 (vol 66 pp. 783)

Izmestiev, Ivan
 Infinitesimal Rigidity of Convex Polyhedra through the Second Derivative of the Hilbert-Einstein Functional The paper is centered around a new proof of the infinitesimal rigidity of convex polyhedra. The proof is based on studying derivatives of the discrete Hilbert-Einstein functional on the space of "warped polyhedra" with a fixed metric on the boundary. The situation is in a sense dual to using derivatives of the volume in order to prove the Gauss infinitesimal rigidity of convex polyhedra. This latter kind of rigidity is related to the Minkowski theorem on the existence and uniqueness of a polyhedron with prescribed face normals and face areas. In the spherical and in the hyperbolic-de Sitter space, there is a perfect duality between the Hilbert-Einstein functional and the volume, as well as between both kinds of rigidity. We review some of the related work and discuss directions for future research. Keywords:convex polyhedron, rigidity, Hilbert-Einstein functional, Minkowski theoremCategories:52B99, 53C24

14. CJM 2013 (vol 66 pp. 31)

Bailey, Michael
 Symplectic Foliations and Generalized Complex Structures We answer the natural question: when is a transversely holomorphic symplectic foliation induced by a generalized complex structure? The leafwise symplectic form and transverse complex structure determine an obstruction class in a certain cohomology, which vanishes if and only if our question has an affirmative answer. We first study a component of this obstruction, which gives the condition that the leafwise cohomology class of the symplectic form must be transversely pluriharmonic. As a consequence, under certain topological hypotheses, we infer that we actually have a symplectic fibre bundle over a complex base. We then show how to compute the full obstruction via a spectral sequence. We give various concrete necessary and sufficient conditions for the vanishing of the obstruction. Throughout, we give examples to test the sharpness of these conditions, including a symplectic fibre bundle over a complex base which does not come from a generalized complex structure, and a regular generalized complex structure which is very unlike a symplectic fibre bundle, i.e., for which nearby leaves are not symplectomorphic. Keywords:differential geometry, symplectic geometry, mathematical physicsCategory:53D18

15. CJM 2013 (vol 66 pp. 400)

Mendonça, Bruno; Tojeiro, Ruy
 Umbilical Submanifolds of $\mathbb{S}^n\times \mathbb{R}$ We give a complete classification of umbilical submanifolds of arbitrary dimension and codimension of $\mathbb{S}^n\times \mathbb{R}$, extending the classification of umbilical surfaces in $\mathbb{S}^2\times \mathbb{R}$ by Souam and Toubiana as well as the local description of umbilical hypersurfaces in $\mathbb{S}^n\times \mathbb{R}$ by Van der Veken and Vrancken. We prove that, besides small spheres in a slice, up to isometries of the ambient space they come in a two-parameter family of rotational submanifolds whose substantial codimension is either one or two and whose profile is a curve in a totally geodesic $\mathbb{S}^1\times \mathbb{R}$ or $\mathbb{S}^2\times \mathbb{R}$, respectively, the former case arising in a one-parameter family. All of them are diffeomorphic to a sphere, except for a single element that is diffeomorphic to Euclidean space. We obtain explicit parametrizations of all such submanifolds. We also study more general classes of submanifolds of $\mathbb{S}^n\times \mathbb{R}$ and $\mathbb{H}^n\times \mathbb{R}$. In particular, we give a complete description of all submanifolds in those product spaces for which the tangent component of a unit vector field spanning the factor $\mathbb{R}$ is an eigenvector of all shape operators. We show that surfaces with parallel mean curvature vector in $\mathbb{S}^n\times \mathbb{R}$ and $\mathbb{H}^n\times \mathbb{R}$ having this property are rotational surfaces, and use this fact to improve some recent results by Alencar, do Carmo, and Tribuzy. We also obtain a Dajczer-type reduction of codimension theorem for submanifolds of $\mathbb{S}^n\times \mathbb{R}$ and $\mathbb{H}^n\times \mathbb{R}$. Keywords:umbilical submanifolds, product spaces $\mathbb{S}^n\times \mathbb{R}$ and $\mathbb{H}^n\times \mathbb{R}$Categories:53B25, 53C40

16. CJM 2012 (vol 65 pp. 266)

Bérard, Vincent
 Les applications conforme-harmoniques Sur une surface de Riemann, l'Ã©nergie d'une application Ã  valeurs dans une variÃ©tÃ© riemannienne est une fonctionnelle invariante conforme, ses points critiques sont les applications harmoniques. Nous proposons ici un analogue en dimension supÃ©rieure, en construisant une fonctionnelle invariante conforme pour les applications entre deux variÃ©tÃ©s riemanniennes, dont la variÃ©tÃ© de dÃ©part est de dimension $n$ paire. Ses points critiques satisfont une EDP elliptique d'ordre $n$ non-linÃ©aire qui est covariante conforme par rapport Ã  la variÃ©tÃ© de dÃ©part, on les appelle les applications conforme-harmoniques. Dans le cas des fonctions, on retrouve l'opÃ©rateur GJMS, dont le terme principal est une puissance $n/2$ du laplacien. Quand $n$ est impaire, les mÃªmes idÃ©es permettent de montrer que le terme constant dans le dÃ©veloppement asymptotique de l'Ã©nergie d'une application asymptotiquement harmonique sur une variÃ©tÃ© AHE est indÃ©pendant du choix du reprÃ©sentant de l'infini conforme. Categories:53C21, 53C43, 53A30

17. CJM 2012 (vol 65 pp. 1164)

Vitagliano, Luca
 Partial Differential Hamiltonian Systems We define partial differential (PD in the following), i.e., field theoretic analogues of Hamiltonian systems on abstract symplectic manifolds and study their main properties, namely, PD Hamilton equations, PD Noether theorem, PD Poisson bracket, etc.. Unlike in standard multisymplectic approach to Hamiltonian field theory, in our formalism, the geometric structure (kinematics) and the dynamical information on the phase space'' appear as just different components of one single geometric object. Keywords:field theory, fiber bundles, multisymplectic geometry, Hamiltonian systemsCategories:70S05, 70S10, 53C80

18. CJM 2012 (vol 65 pp. 1401)

Zhao, Wei; Shen, Yibing
 A Universal Volume Comparison Theorem for Finsler Manifolds and Related Results In this paper, we establish a universal volume comparison theorem for Finsler manifolds and give the Berger-Kazdan inequality and SantalÃ³'s formula in Finsler geometry. Being based on these, we derive a Berger-Kazdan type comparison theorem and a Croke type isoperimetric inequality for Finsler manifolds. Keywords:Finsler manifold, Berger-Kazdan inequality, Berger-Kazdan comparison theorem, SantalÃ³'s formula, Croke's isoperimetric inequalityCategories:53B40, 53C65, 52A38

19. CJM 2012 (vol 65 pp. 634)

Mezzetti, Emilia; Miró-Roig, Rosa M.; Ottaviani, Giorgio
 Laplace Equations and the Weak Lefschetz Property We prove that $r$ independent homogeneous polynomials of the same degree $d$ become dependent when restricted to any hyperplane if and only if their inverse system parameterizes a variety whose $(d-1)$-osculating spaces have dimension smaller than expected. This gives an equivalence between an algebraic notion (called Weak Lefschetz Property) and a differential geometric notion, concerning varieties which satisfy certain Laplace equations. In the toric case, some relevant examples are classified and as byproduct we provide counterexamples to Ilardi's conjecture. Keywords:osculating space, weak Lefschetz property, Laplace equations, toric threefoldCategories:13E10, 14M25, 14N05, 14N15, 53A20

20. CJM 2012 (vol 65 pp. 655)

Shemyakova, E.
 Proof of the Completeness of Darboux Wronskian Formulae for Order Two Darboux Wronskian formulas allow to construct Darboux transformations, but Laplace transformations, which are Darboux transformations of order one cannot be represented this way. It has been a long standing problem on what are other exceptions. In our previous work we proved that among transformations of total order one there are no other exceptions. Here we prove that for transformations of total order two there are no exceptions at all. We also obtain a simple explicit invariant description of all possible Darboux Transformations of total order two. Keywords:completeness of Darboux Wronskian formulas, completeness of Darboux determinants, Darboux transformations, invariants for solution of PDEsCategories:53Z05, 35Q99

21. CJM 2012 (vol 65 pp. 757)

Delanoë, Philippe; Rouvière, François
 Positively Curved Riemannian Locally Symmetric Spaces are Positively Squared Distance Curved The squared distance curvature is a kind of two-point curvature the sign of which turned out crucial for the smoothness of optimal transportation maps on Riemannian manifolds. Positivity properties of that new curvature have been established recently for all the simply connected compact rank one symmetric spaces, except the Cayley plane. Direct proofs were given for the sphere, an indirect one via the Hopf fibrations) for the complex and quaternionic projective spaces. Here, we present a direct proof of a property implying all the preceding ones, valid on every positively curved Riemannian locally symmetric space. Keywords:symmetric spaces, rank one, positive curvature, almost-positive $c$-curvatureCategories:53C35, 53C21, 53C26, 49N60

22. CJM 2012 (vol 65 pp. 553)

Godinho, Leonor; Sousa-Dias, M. E.
 Addendum and Erratum to "The Fundamental Group of $S^1$-manifolds" This paper provides an addendum and erratum to L. Godinho and M. E. Sousa-Dias, "The Fundamental Group of $S^1$-manifolds". Canad. J. Math. 62(2010), no. 5, 1082--1098. Keywords:symplectic reduction; fundamental groupCategories:53D19, 37J10, 55Q05

23. CJM 2012 (vol 65 pp. 467)

Wilson, Glen; Woodward, Christopher T.
 Quasimap Floer Cohomology for Varying Symplectic Quotients We show that quasimap Floer cohomology for varying symplectic quotients resolves several puzzles regarding displaceability of toric moment fibers. For example, we (i) present a compact Hamiltonian torus action containing an open subset of non-displaceable orbits and a codimension four singular set, partly answering a question of McDuff, and (ii) determine displaceability for most of the moment fibers of a symplectic ellipsoid. Keywords:Floer cohomology, Hamiltonian displaceabilityCategory:53Dxx

24. CJM 2012 (vol 65 pp. 66)

Deng, Shaoqiang; Hu, Zhiguang
 On Flag Curvature of Homogeneous Randers Spaces In this paper we give an explicit formula for the flag curvature of homogeneous Randers spaces of Douglas type and apply this formula to obtain some interesting results. We first deduce an explicit formula for the flag curvature of an arbitrary left invariant Randers metric on a two-step nilpotent Lie group. Then we obtain a classification of negatively curved homogeneous Randers spaces of Douglas type. This results, in particular, in many examples of homogeneous non-Riemannian Finsler spaces with negative flag curvature. Finally, we prove a rigidity result that a homogeneous Randers space of Berwald type whose flag curvature is everywhere nonzero must be Riemannian. Keywords:homogeneous Randers manifolds, flag curvature, Douglas spaces, two-step nilpotent Lie groupsCategories:22E46, 53C30

25. CJM 2011 (vol 64 pp. 778)

Calvaruso, Giovanni; Fino, Anna
 Ricci Solitons and Geometry of Four-dimensional Non-reductive Homogeneous Spaces We study the geometry of non-reductive $4$-dimensional homogeneous spaces. In particular, after describing their Levi-Civita connection and curvature properties, we classify homogeneous Ricci solitons on these spaces, proving the existence of shrinking, expanding and steady examples. For all the non-trivial examples we find, the Ricci operator is diagonalizable. Keywords:non-reductive homogeneous spaces, pseudo-Riemannian metrics, Ricci solitons, Einstein-like metricsCategories:53C21, 53C50, 53C25
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Projects

## World Magnetic Model

The World Magnetic Model (WMM) is a data model, which describes the Earth’s magnetic field.

The magnetic field lines originate at the magnetic north pole, which is located somewhere north of Canada. So the magnetic north pole is not identical to the geographic north pole. For this reason a compass does not exactly point to the north pole. For example the deviation of the field lines in central Europe is about 2 degrees east.

The WMM describes the above deviation of the magnetic field lines with respect to the geographic north pole. To make use of the WMM we first download the C source code of the WMM Geomagnetism Library from here (we download the file WMM2015_Linux.tar.gz):

  https://www.ngdc.noaa.gov/geomag/WMM/soft.shtml

After unpacking the tar ball, we compile the “grid” software in the src folder on the Unix command line like so:

gcc -c wmm_grid.c
gcc -c GeomagnetismLibrary.c
gcc wmm_grid.o GeomagnetismLibrary.o -o grid

The grid executable needs to reside in the same directory as the WMM.COF file. Here is an example run of the grid utility:

 Welcome to the World Magnetic Model (WMM) C-Program
of the US National Geophysical Data Center
--- Grid Calculation Program ----
--- Model Release Date: 15 Dec 2014 ---
--- Software Release Date: 08 Dec 2014 ---
This program may be used to generate a grid/volume of magnetic field values
over latitude, longitude, altitude and time axes. To skip an axis,
keep the start and end values the same and enter zero for the step size.

Enter grid parameters

Please Enter Minimum Latitude (in decimal degrees):
-90
Please Enter Maximum Latitude (in decimal degrees):
90
Please Enter Minimum Longitude (in decimal degrees):
-180
Please Enter Maximum Longitude (in decimal degrees):
180
Please Enter Step Size (in decimal degrees):
5
Select height (default : above MSL)
1. Above Mean Sea Level
2. Above WGS-84 Ellipsoid
1
Please Enter Minimum Height above MSL (in km):
0
Please Enter Maximum Height above MSL (in km):
0
Please Enter height step size (in km):
0

Please Enter the decimal year starting time:
2015.5
Please Enter the decimal year ending time:
2015.5
Please Enter the time step size:
0
Enter a geomagnetic element to print. Your options are:
1. Declination	9.   Ddot
2. Inclination	10. Idot
3. F		11. Fdot
4. H		12. Hdot
5. X		13. Xdot
6. Y		14. Ydot
7. Z		15. Zdot
8. GV		16. GVdot
1
Select output :
1. Print to a file
2. Print to Screen
2
-90.00 -180.00   0.0000 2015.50     149.88    0.40
-90.00 -175.00   0.0000 2015.50     144.88    0.40
-90.00 -170.00   0.0000 2015.50     139.88    0.40
...snip...
90.00 170.00   0.0000 2015.50     164.63    2.85
90.00 175.00   0.0000 2015.50     169.63    2.85
90.00 180.00   0.0000 2015.50     174.63    2.85

Press any key to exit...

According to the above output, the declination of the magnetic field at the equator (with latitude zero) varies from −20 to +10 degrees.

Now we textually format the above declination data into a matrix and put it as an array into a C++ module just like we did with the geoid data:

static const int size_x=73;
static const int size_y=37;

static const float data[size_y][size_x] = {
{149.88,144.88,139.88,...
...,164.63,169.63,174.63}
};

And add a linear interpolation function to the above data:

#include <math.h>
#include <limits> // for nan

#ifndef NAN
#   define NAN (std::numeric_limits<double>::quiet_NaN())
#endif

// check whether of not a floating point value is not a number
inline int isNAN(const double v) {return(v!=v);}

// bi-linearily sample data matrix at normalized position (x,y)
inline float interpolate(float x, float y,
const float *data, unsigned int width, unsigned int height)
{
unsigned int i,j;

float val1,val2,val3,val4;
const float *floatptr;

if (x>=0.0f && x<=1.0f && y>=0.0f && y<=1.0f)
{
x*=width-1; y*=height-1;
i=(unsigned int)(x+0.5f); j=(unsigned int)(y+0.5f);
x-=i; y-=j;

if (i==width-1 && width>1) {i=width-2; x=1.0f;}
if (j==height-1 && height>1) {j=height-2; y=1.0f;}

floatptr=&data[i+j*width];

val1=floatptr[0];
if (isNAN(val1)) return(NAN);

if (width<2)
return(val1);
else
{
val2=floatptr[1];
if (isNAN(val2)) return(NAN);

if (height<2)
return((1.0f-x)*val1+x*val2);
else
{
val3=floatptr[width];
val4=floatptr[width+1];
if (isNAN(val3) || isNAN(val4)) return(NAN);

return((1.0f-y)*((1.0f-x)*val1+x*val2)+
y*((1.0f-x)*val3+x*val4));
}
}
}

return(NAN);
}

static const int size_x=73;
static const int size_y=37;

static const double LLWGS84_swx=-180;
static const double LLWGS84_swy=-90;
static const double LLWGS84_nwx=-180;
static const double LLWGS84_nwy=90;
static const double LLWGS84_nex=180;
static const double LLWGS84_ney=90;
static const double LLWGS84_sex=180;
static const double LLWGS84_sey=-90;

static const float data[size_y][size_x] = {
{149.88,144.88,139.88,...
...,164.63,169.63,174.63}
};

// sample the magnetic declication at a geographic position
float sample_mag(double latitude, double longitude)
{
// calculate normalized coordinates from geographic position
double nx = (longitude-LLWGS84_swx)/(LLWGS84_sex-LLWGS84_swx);
double ny = (latitude-LLWGS84_swy)/(LLWGS84_nwy-LLWGS84_swy);

nx = nx - floor(nx);

// clamp out-of-range latitudes
if (ny<0.0) ny = 0.0;
else if (ny>1.0) ny = 1.0;

// interpolate magnetic declication
return(interpolate(nx, ny, (const float *)data, size_x, size_y));
}

Full source code is here:

Here is a usage example:

#include <mini/wmm.h>

// example of calculating the wmm declination at Lat/Lon 49/11
double lat = 49.0, lon = 11.0;
double decl = sample_mag(lat, lon);

Note: The above example uses a subsampled coarse grid with 5 degrees spacing to estimate the magnetic declination. The error of the subsampled grid is somewhere in the order of ±1 degree. This is sufficient if you want to correct a magnetic compass of a smartphone, for example, which has a much higher systemic error than the approximation error of the grid. For use cases, where you need a smaller approximation error, the same principle as outlined above applies, but we just use a denser spacing of the samples.


text:
# Total number of possible JPEG's

• June 28th 2011, 03:15 PM
johnghampton
Total number of possible JPEG's
Hi, I'm trying to compute the total number of possible JPEG's that could be created.
JPEGs can be anywhere from 1x1 to 65535x65535 pixels and each pixel has 8bit color x three color channels. This means there is 256x256x256=16777216 possibilities for each pixel.
I'm an amateur at mathematics, but so far what I have come up with is:

(256^3)^(sum_(x=0)^65535(x+sum_(y=0)^65535y))

I figured I had to sum the maximum width of the jpeg (65535) added to the summation of the maximum height (also 65535) to get the total number of possible sizes. Then I would have to use that total (the total amount of possible pixel spaces) as an exponent on the number of possible colors right?

Any help would be greatly appreciated (in solving the equation, verifying it, or correcting it)!
Thanks!

-John G. Hampton
• June 29th 2011, 03:30 AM
SpringFan25
Re: Total number of possible JPEG's
its probable easiest to build up to this. Suppose i fix the dimensions at x and y. Lets find the number of $x \times y$ JPEGs that can be made.

There are xy pixels, and each one can take 256^3 values. So the number is $\left(256^3 \right)^{xy} = 256^{3xy}$

Step 2
Now lets vary one of the dimensions (x), while pretending the other(y) is fixed at some value. We can just add up our formula for all the possible values of x

Combinations if height is fixed at y : $\sum_{x=1}^{65535} 256^{xy}$

Step 3
Now lets vary the other dimension. We can just add up the previous formula for all the possible values of y.

$\sum_{y=1}^{65535} \left( \sum_{x=1}^{65535} 256^{3xy} \right)$
• June 29th 2011, 07:24 AM
chojin996
Re: Total number of possible JPEG's
Those calculations are right for bitmap/raw pictures. But then JPEG uses the DCT to achieve lossy compression, so all possible quantization matrix values combinations affecting the DCT should be taken into account too, shouldn't it? Effectively counting each smallest possible compression factor variation affecting high and low frequencies of the image as the generator of a unique variation of source image to be counted as if it was a new image. Or not?
• June 29th 2011, 09:40 AM
johnghampton
Re: Total number of possible JPEG's
Thanks for the replies!
You bring up a good point Chojin. You're right, this is just a calculation for a bitmap of each possible jpeg image, but I think I can get away with this because this calculation is to figure out how many variations on the JPEG could be printed (assuming uniformity in the printing processes). It should cover all possible artifacting from various levels of compression—at least visually.

Now if only I could solve it! It's far too large for wolfram-alpha which is my usual calculator.


text:
On a weekly forest brigade operates 12 students. After plant one hundred spruces get x CZK, anfter one hundred pines y CZK. How many CZK got one student for one day if planted spruces 25000 and 30000 pines week?

Result

a = Kc (Correct answer is: 2.976x+3.571y)

### Step-by-step explanation:

$a=\left(25000\mathrm{/}100\cdot x+30000\mathrm{/}100\cdot y\right)\mathrm{/}\left(7\cdot 12\right)=2.976x+3.571y$

Did you find an error or inaccuracy? Feel free to write us. Thank you!

Tips to related online calculators
Check out our ratio calculator.

## Related math problems and questions:

• Twelve
Twelve students work on a week forestry brigade. One hundred spruces will receive x CZK, one hundred pine y CZK. How many receive each one students did in one day if they planted a total of 25,000 spruces per week and 30,000 pine trees? Express by express
• Hectares of forest
12 workers plant 24 ha of forest in 6 days. In how many days will 15 people and 12 people plant the same area?
• An average a day
Forest workers planted 96 trees on Monday, 120 trees on Tuesday, and 61 trees on Wednesdays. How many trees did she plant in the last day of Thursday, if she planted an average of 105 trees a day?
• Three friends
Three friends divided the profit by 104,650 CZK, so that for every 4 CZK, which got the first friend equals 5 crowns for second and for every 9 CZK, which got the second equals 16 CZK for third. Question: Who got the most and how much.
• Pines and spruces
They planted 350 spruces and 180 pines in the forest nursery. 46 trees withered. How many trees are ok?
• The gardener
The gardener bought trees for 960 CZK. If every tree were cheaper by 12 CZK, he would have gotten four more trees for the same money. How many trees did he buy?
• Wood harvesting
The harvested wood is bundled from the forest to a sawmill. The driver makes the journey four times a day, and the work takes him 8 days. How many times would he have to go daily to get the job done 2 days earlier?
• Orange colour
In order for the painter to get orange, he has to mix yellow and red in a ratio of 3: 4. How many liters of orange did he mix if he used 8 liters of red?
• Snack
CZK 450 was paid for a snack for 30 pupils. How many CZK will we pay for the same snack for 28 pupils?
• Division
Three siblings Helena, Oliver and George split the bag with candies on merit in the ratio 6:1:4. How many candies should each get if in bag were 88?
• Sides ratio
Calculate the circumference of a triangle with area 84 cm2 if a:b:c = 10:17:21
• Acres
Two hours 24 students harvested chestnuts on 36 acres. How many square meters harvested one student per hour?
• Trees avg
The worker planted 96 trees on Monday, 120 on Tuesday, and 61 trees on Wednesday. How many trees did he plant on Thursday if he averaged 105 trees per day?
• Apples
School kitchen bought 36kg apples for 12 kc/kg. How many kilograms of apples 1/4 cheaper can they buy for the same money?
• Bathroom
How many CZK do we pay for lining the perimeter walls of the bathroom with rectangular shape with dimensions 3.5 m and 4 m, high 1.5 m if 1 square m tile cost 300 CZK?
• A cook
A cook wants to make enough cookies for 159 students. The cookie recipe makes 1 1/4 dozen cookies. A dozen equals 12. How many batches does the cook need to make if each student gets one cookie?
• Birthday
Mother bought 21 desserts on the occasion of Mirka's birthday one tips was 9 CZK and the kremlin cost 12 CZK. For all desserts, she paid 213 CZK. How many kremlins and how many tips mums did buy?


text:
Accueil du site > Activités scientifiques et techniques > Physique et matière hadronique > ALICE > Publications > Publications 2012

## Publications 2012

PROCEEDINGS AND confErences 2012 :

• J/psi measurements in ultra-peripheral Pb-Pb collisions with ALICE - Diffraction 2012
Daniel Tapia Takaki, for the ALICE collaboration.
• Charmonia production in ALICE- Hard Probes 2012
Christophe Suire, for the ALICE Collaboration. Proceedings arXiv:1208.5601
• Quarkonia and heavy flavour production measurement in Pb-Pb collisions with the ALICE experiment at the LHC - Quark and Nuclear Physics 2012
Lizardo Valencia for the ALICE Collaboration. Proceeding
• A tentative design for AFTER - SPRING 2012 AFTER meeting, Grenoble (France), juin 2012 - C. Hadjidakis
• Photoproduction of J/psi in ultra-peripheral Pb+Pb collisions at ALICE - 6th International Conference on Quarks and Nuclear Physics : 40 years of development of Quantum Chromodynamics, Palaiseau, France. 16-20 April 2012, D. Tapia-Takaki
• Upgrade of the muon tracking electronics - B. Espagnon - International Muon Workshop 2012
• Physics opportunities of a fixed target experiment (AFTER) using the LHC beams, Seminar SphN Saclay (France), juillet 2012- C. Hadjidakis

Posters

• Quarkonium measurements in pPb collisions at the LHC with the ALICE experiment - Hard Probes, Cagliari (Italie), mai 2012 - C. Hadjidakis

• The Geant4 Virtual Monte Carlo - CHEP 2012 Ivana Hrinacova

• LVDS tester : a systematic test of cable signal transmission at the ALICE experiment - Topical workshop on electronics for Particle Physics (TWEPP). Aachen, Germany. 20-24 September 2010, D. Tapia-Takaki

• J/psi suppression in Pb-Pb collisions at sqrt(s)=2.76 TeV at forward rapidity in ALICE - LHCC student poster session (CERN) - 03/21/2012 - I. Lakomov

• J/Psi analysis in proton+proton at 7 TeV in the ALICE muon spectrometer using a 2D correction method - Quark Matter 2011, Bruno Boyer

Publications (collaboration ALICE) 2012 (26 PAPERS) :

Measurement of electrons from semileptonic heavy-flavour hadron decays in pp collisions at sqrts = 7 TeV
Measurement of the Cross Section for Electromagnetic Dissociation with Neutron Emission in Pb-Pb Collisions at √sNN = 2.76 TeV
Pion, Kaon, and Proton Production in Central Pb-Pb Collisions at sqrt(sNN) = 2.76 TeV
Inclusive J/psi production in pp collisions at sqrt(s)=2.76 TeV
Ds meson production at central rapidity in proton—proton collisions at sqrt(s) = 7 TeV
Coherent J/Psi photoproduction in ultra-peripheral Pb-Pb collisions at
• Article reference : Physics Letters B Volume 718, Issues 4–5, 29 January 2013, Pages 1273–1283 Current publication link
• System : Pb-Pb Energy : 2.76 TeV
Measurement of prompt J/psi and beauty hadron production cross sections at mid-rapidity in pp collisions at √s=7 TeV
Rapidity and transverse momentum dependence of inclusive J/psi production in pp collisions at sqrt(s)=7 TeV
Production of K*(892)0 and phi(1020) in pp collisions at sqrt(s)=7TeV
K0s-K0s correlations in 7 TeV pp collisions from the LHC ALICE experiment
Neutral pion and η meson production in proton-proton collisions at sqrt(s) = 0.9 TeV and 7 TeV
Suppression of high transverse momentum prompt D mesons in central Pb—Pb collisions at sqrt(s_NN)=2.76 TeV
Production of muons from heavy flavour decays at forward rapidity in pp and Pb—Pb collisions at
Transverse sphericity of primary charged particles in minimum bias proton-proton collisions at sqrts=0.9, 2.76 and 7 TeV
J/psi suppression at forward rapidity in Pb-Pb collisions at √sNN = 2.76 TeV
Measurement of charm production at central rapidity in proton proton collisions at sqrts=2.76 TeV
Underlying Event measurements in pp collisions at sqrt(s) = 0.9 and 7 TeV with the ALICE experiment at the LHC
Multi-strange baryon production in pp collisions at √s = 7 TeV with ALICE
J/psi Production as a Function of Charged Particle Multiplicity in pp Collisions at √s = 7 TeV
Light vector meson production in pp collisions at sqrt(s) = 7 TeV


text:
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# Experimental extraction of the quantum effective action for a non-equilibrium many-body system

### Subjects

An Author Correction to this article was published on 12 August 2020

## Abstract

On the fundamental level, quantum fluctuations or entanglement lead to complex dynamical behaviour in many-body systems1 for which a description as emergent phenomena can be found within the framework of quantum field theory. A central quantity in these efforts, containing all information about the measurable physical properties, is the quantum effective action2. Though non-equilibrium quantum dynamics can be exactly formulated in terms of the quantum effective action, finding solutions is in general beyond the capabilities of classical computers3. Here, we present a strategy to determine the non-equilibrium quantum effective action4 using analogue quantum simulators, and demonstrate our method experimentally with a quasi-one-dimensional spinor Bose gas out of equilibrium5,6. Spatially resolved snapshots of the complex-valued transversal spin field7 allow us to infer the quantum effective action up to fourth order in an expansion in one-particle irreducible correlation functions at equal times. We uncover a strong suppression of the irreducible four vertex emerging at low momenta in the highly occupied regime far from equilibrium where perturbative descriptions fail8. Our work constitutes a new realm of large-scale analogue quantum computing9, where highly controlled synthetic quantum systems10 provide the means for solving theoretical problems in high-energy and condensed-matter physics with an experimental approach11,12,13,14.

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## Data availability

Source data and all other data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

## References

1. Polkovnikov, A., Sengupta, K., Silva, A. & Vengalattore, M. Colloquium: nonequilibrium dynamics of closed interacting quantum systems. Rev. Mod. Phys. 83, 863–883 (2011).

2. Weinberg, S.The Quantum Theory of Fields (Cambridge Univ. Press, 1995).

3. Jordan, S. P., Lee, K. S. & Preskill, J. Quantum algorithms for quantum field theories. Science 336, 1130–1133 (2012).

4. Wetterich, C. Nonequilibrium time evolution in quantum field theory. Phys. Rev. E 56, 2687-2690 (1997).

5. Sadler, L., Higbie, J., Leslie, S., Vengalattore, M. & Stamper-Kurn, D. Spontaneous symmetry breaking in a quenched ferromagnetic spinor Bose–Einstein condensate. Nature 443, 312–315 (2006).

6. Stamper-Kurn, D. M. & Ueda, M. Spinor Bose gases: symmetries, magnetism, and quantum dynamics. Rev. Mod. Phys. 85, 1191–1244 (2013).

7. Kunkel, P. et al. Simultaneous readout of noncommuting collective spin observables beyond the standard quantum limit. Phys. Rev. Lett. 123, 063603 (2019).

8. Pitaevskii, L. & Lifshitz, E. Physical Kinetics Vol. 10 (Butterworth-Heinemann, 2012).

9. Horsman, C., Stepney, S., Wagner, R. C. & Kendon, V. When does a physical system compute? Proc. R. Soc. A 470, 20140182 (2014).

10. Bloch, I., Dalibard, J. & Nascimbène, S. Quantum simulations with ultracold quantum gases. Nat. Phys. 8, 267–276 (2012).

11. Zohar, E., Cirac, J. I. & Reznik, B. Quantum simulations of lattice gauge theories using ultracold atoms in optical lattices. Rep. Prog. Phys. 79, 014401 (2015).

12. Lewenstein, M. et al. Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond. Adv. Phys. 56, 243–379 (2007).

13. Loh, E. Jr et al. Sign problem in the numerical simulation of many-electron systems. Phys. Rev. B 41, 9301 (1990).

14. Troyer, M. & Wiese, U.-J. Computational complexity and fundamental limitations to fermionic quantum Monte Carlo simulations. Phys. Rev. Lett. 94, 170201 (2005).

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## Acknowledgements

This Letter comprises parts of the doctoral thesis work of M.P.30 and T.V.Z., submitted to Heidelberg University, Germany. We thank S. Erne, T. Gasenzer, P. Hauke, C.-M. Schmied, J. Schmiedmayer, T. Schweigler and M. Tarpin for discussions and R. Rosa-Medina for experimental assistance. This work was supported by the DFG Collaborative Research Center SFB1225 (ISOQUANT) and the ERC Advanced Grant Horizon 2020 EntangleGen (project ID 694561), by Deutsche Forschungsgemeinschaft (DFG) under Germany’s Excellence Strategy EXC-2181/1 - 390900948 (the Heidelberg STRUCTURES Excellence Cluster) and the Heidelberg Center for Quantum Dynamics. P.K. acknowledges support from the Studienstiftung des deutschen Volkes.

## Author information

Authors

### Contributions

The experimental and theoretical concept was developed in discussion among all authors. M.P., P.K., S.L. and A.B. controlled the experimental apparatus. M.P. and T.V.Z. analysed the data. M.P., T.V.Z., P.K., H.S., J.B. and M.K.O. discussed the measurement results. T.V.Z. and J.B. elaborated the equal-time formalism. All authors contributed to the discussion of the results and the writing of the manuscript.

### Corresponding author

Correspondence to Maximilian Prüfer.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended data

### Extended Data Fig. 1 Observation of the complex valued field for all times.

We show the distribution of F (as in Fig. 1c; for the relative frequency see colorbar) for all evolution times shown in Fig. 4. The blue circle indicates F = 0.7.

### Extended Data Fig. 2 1PI four vertex.

$${\varGamma }_{t}^{(4)}({p}_{1},{p}_{2},{p}_{3},{p}_{4})$$ inferred from the data for the same momenta as shown in Fig. 2c. For comparison we show the results for the Gaussian model.

### Extended Data Fig. 3 Momentum conserving diagonals.

Momentum conserving diagonals of the four-point 1PI correlators shown in Fig. 3 for all evolution times between 9 s and 18 s. Grey shaded area is the finite statistical bias from a Gaussian model. All error bars shown are 1 s.d. calculated from bootstrap resampling.

### Extended Data Fig. 4 Universal dynamics of two-point correlator.

a, Time evolution of two-point correlator for longest times accessible in the experiment. b, A rescaling analysis yields α = β = 0.45 ± 0.05. After rescaling the data collapses on a universal curve. The found dynamics underlines the universal features found in these and previous experiments.

### Extended Data Fig. 5 Rescaling of different momentum conserving diagonals.

Momentum conserving diagonals of the 1PI four-point correlator shown in Fig. 3 for all evolution times between 9 s and 18 s rescaled as in Fig. 4b with γ = 0 and β4 = 1/2. Given the finite statistics, we attribute the quality of the scaling collapse shown in Fig. 4b (compared to the shifted momentum diagonals shown here) to a larger signal-to-noise ratio, which is also evidenced by smaller statistical errors. All error bars shown are 1 s.d. calculated from bootstrap resampling.

## Rights and permissions

Reprints and Permissions

Prüfer, M., Zache, T.V., Kunkel, P. et al. Experimental extraction of the quantum effective action for a non-equilibrium many-body system. Nat. Phys. 16, 1012–1016 (2020). https://doi.org/10.1038/s41567-020-0933-6

• Accepted:

• Published:

• Issue Date:

• DOI: https://doi.org/10.1038/s41567-020-0933-6

• ### Condensation and thermalization of an easy-plane ferromagnet in a spinor Bose gas

• Maximilian Prüfer
• Daniel Spitz
• Markus K. Oberthaler

Nature Physics (2022)

• ### Oscillating scalar dissipating in a medium

• Wen-Yuan Ai
• Marco Drewes
• Jan Hajer

Journal of High Energy Physics (2021)


text:
# RL Environment

Likewise the environment at timestep $$t$$ :

• recieves action $$a_t$$
• emits observation $$o_t$$
• emits reward $$r_t$$

## 1.Environment State

The environment state $$S^{e}_{t}$$ is the environments private representation. This could be all the internal variables inside the Atari game that agent cannot see. The RL algorithms can not depend on these numbers as it does not have access to this information!

This state however is what is used to generate the observation $$o_t$$ and reward $$r_t$$ however.

Created: 2022-02-03

Emacs 26.1 (Org mode 9.5)


text:
# Markdown pictures are displayed side by side and the annotation is aligned

Keywords: html5 html css

# preface

According to Markdown, pictures can be placed on the same line, but the format is difficult to control, because the two pictures will be closely connected, even if spaces are added in the middle

The effect we want to achieve is similar to the following figure

Two schemes close to this display are provided below

# Scheme 1: use the
 tag of Html

Scheme I
\qquad Use Html's table
 tag to control the display of pictures side by side

The example md code is as follows:

001.jpg

002.jpg

The example effect is as follows:

 001.jpg 002.jpg

Note:
\qquad ① The CSDN markdown editor will not take effect on some style attribute settings of
 tags, such as invalid border=0 and invalid frame attribute settings
\qquad ② The CSDN markdown editor has the same effect on the rules attribute of the
 tag. Setting any legal value will hide the outer border, but the inner border cannot be hidden
\qquad ③ It is better to set the picture to the same size, otherwise the cell size will be inconsistent (the original size is required to be consistent, and the allocation of the same cell size set through the width attribute is still inconsistent)

\qquad Easy to create annotations and align with pictures

\qquad The table border cannot be completely hidden at this time

# Scheme 2: use  tag of Html

Scheme II
\qquad Use the  tag of Html to control the display of pictures side by side

The example md code is as follows:

001.jpg

002.jpg

The example effect is as follows:

001.jpg                            002.jpg

Note:
\qquad Do not set the align attribute on the  tag. After setting the align attribute on all pictures, the entire  tag height will collapse, resulting in the dislocation of the following contents and poor display effect

\qquad There is no frame like a table, and the effect looks ok

\qquad ① The picture size is too large to be displayed side by side due to automatic line wrapping
\qquad ② The annotation is not easy to align (the display effect of browsers on different clients is different, especially after the browser is resized)
\qquad ③ A lot of html space characters are used in the middle, which will be selected

# Failed idea: use LaTeX to insert pictures and control alignment

As we all know, LaTeX formula can control alignment by using & in \ begin{aligned}         \ end{aligned}

Therefore, in theory, as long as the LaTeX module supported by Markdown can insert pictures, good picture side-by-side display and annotation alignment can be achieved

In fact, referring to the method of inserting pictures in LaTeX on the Internet, it is found that some packages need to be imported, which is not supported by LaTeX in Markdown, so it cannot be implemented

# Attachment: single picture annotation alignment

programme
\qquad Use the  tag of Html to control the alignment of pictures and annotations

The example md code is as follows:

001.jpg

The example effect is as follows:

001.jpg

Posted by Aretai on Sun, 26 Sep 2021 18:13:18 -0700


text:
# Prove $f$ is riemann integrable if $f$ is continuous on closed interval $[a,b]$. (Explanation)

I was given the following proof of a theorem, quick note about notation, the fact that $$f$$ is riemann integrable on $$[a,b]$$ we denote by $$f\in\mathcal{R}([a,b])$$

Theorem: Let $$a,b\in\mathbb{R},a, let $$f$$ be continuous on closed interval $$[a,b]$$, then $$f\in\mathcal{R}([a,b])$$

We use the following lemma:

Lemma: Let $$a,b\in\mathbb{R},a and $$f$$ bounded on $$[a,b]$$. Then the following is equivallent: $$f\in\mathcal{R}([a,b])\tag{1}$$ $$\forall \epsilon >0 \exists \sigma_{[a,b]}:\overline{S}(f,\sigma)-\underline{S}(f,\sigma)<\epsilon$$

where $$\sigma$$ is a partition of $$[a,b]$$. Now, i understand the proof of the lemma and I'm not giving it here.

Proof. (of the theorem) $$f$$ is continuous on $$[a,b]$$, so $$f$$ is bounded on $$[a,b]$$, also because $$f$$ is bounded on bounded interval, f is uniformly continuous. Let $$\epsilon > 0$$ be given, by assumption, we find $$\delta>0$$ s.t. for any pair $$x,y\in[a,b]:|x-y|<\delta \Rightarrow |f(x)-f(y)|<\epsilon$$. Choose partitioning $$\sigma_{[a,b]}$$ s.t. $$\nu(\sigma)<\delta$$. Then by uniform continuity for each $$i\in\{1\ldots,n\}$$ we have that $$\sup_{[x_{i-1},x_i]}f\leq \inf_{[x_{i-1},x_i]}f+\epsilon$$ thus $$\overline{S}(f,\sigma)-\underline{S}(f,\sigma)=\sum_{i=1}^n \bigg(\sup_{[x_{i-1},x_i]}f-\inf_{[x_{i-1},x_i]}f\bigg)(x_i-x_{i-1})\leq \sum_{i=1}^n \epsilon(x_i-x_{i-1})=\epsilon(b-a)$$ which by lemma shows that $$f\in\mathcal{R}([a,b])$$.

where $$\nu(\sigma)$$ is the norm of the partition, that is $$\nu(\sigma)=\max\{x_{i}-x_{i-1}\mid i\in\{1,\ldots,n\}\}$$

Now, the part I don't understand is why uniform continuity implies something like $$\sup_{[x_{i-1},x_i]}f\leq \inf_{[x_{i-1},x_i]}f+\epsilon$$ (bolded part). Would someone explain more thoroughly, what is going on? Do they mean that by continuity $$f$$ attains extreme values (suprema, and infima) on each of $$\mathcal{I}:=[x_{i-1},x_i]$$ and by $$|f(x)-f(y)|<\epsilon$$, this results in $$\sup_\mathcal{I} f - \inf_\mathcal{I} f \leq \epsilon$$ (now, why is there a $$\leq$$ instead of $$<$$ ?).

Suppose that $$|s-r| <\delta$$, and $$|f(x)-f(y)| < \epsilon$$ whenever $$|x-y| <\delta$$. Then, on the interval $$[r,s]$$ (in the domain of $$f$$), $$\epsilon \ge \sup_{x,y\in [r,s]} (f(x)-f(y)) =\sup_{x\in [r,s]}f(x) - \inf_{x\in [r,s]}f(y)$$ That equality is always true, even without continuity; the supremum minus the infimum is an upper bound for the variation, and we can approach arbitrarily close to each of them. What (uniform) continuity gets us is the inequality out front relating to $$\epsilon$$, as we can choose the mesh fine enough that the variation in any one subinterval is small.

• Oh, I see now, also, we can interchange, by symmetry of $x,y$ on $[a,b]$ right? So maybe this could also be $\sup_{x,y\in I}(f(x)-f(y))=\sup_{x,y\in I}(f(y)-f(x))=\sup_{x\in I} f(x) - \inf_{y\in I}f(y)=$... my argument would go as follows: by boundeness suprema $\sup f =: f(M_i)$ and infima $\inf f =: f(m_i)$exist on each of $[x_{i-1},x_i]$ and by the continuity $|f(M_i)-f(m_i)| = f(M_i)-f(m_i) < \epsilon$, would this also be a good reasoning? – Michal Dvořák Dec 24 '18 at 2:02
• I just changed the variable names - my interval is not the same as your $[a,b]$, and now it uses different letters. This argument is all running on one of the small intervals belonging to the partition. Now, your argument, calling the supremum $f(M_i)$? That invokes continuity; you're saying that the function has a maximum, not just a supremum. Be careful with that sort of thing. – jmerry Dec 24 '18 at 2:11
• Well, then we use the continuity again to show that infima are in fact minima and suprema are maxima on the interval, or? – Michal Dvořák Dec 24 '18 at 2:15
• yes, you can, but its not really relevant to the argument. Saying $\sup f = f(M_i)$ is technically correct by continuity on the closed interval, but you may not always have $M_i$ which allows for $f(M_i) = \sup_{x\in I}f(x)$ – rubikscube09 Dec 24 '18 at 3:20


text:
## anonymous one year ago HELP PLEASE:) FAN&MEDAL&TESTIMONIAL:):) A committee of 5 people is to be chosen from a group of 8 women and 10 men. The committee must feature 3 women and two men? The committee must have more women than men?

1. anonymous

Without restrictions all the possibilities are accounted for by 18P5

2. anonymous

but with the above restrictions.....:(

3. ganeshie8

Wrong, does the order in which you pick people matter here ?

4. anonymous

I don't think so.

5. ganeshie8

The order in which you select the people doesn't matter here because, end of the day, all you bother about is "who" are in the committee. Not in which order you picked them.

6. ganeshie8

So this better be a combination problem, yes ?

7. anonymous

Yes.

8. ganeshie8

Without restrictions all the possibilities are accounted for by 18$$\color{red}{C}$$5

9. anonymous

What's the difference between P and C? I am sorry I am not very advanced

10. ganeshie8

Maybe consider a quick example : Suppose you went to juice shop and ask for a salad made up of "bananas, apple, grapes"

11. ganeshie8

does the order in which you ask matter here ?

12. anonymous

No. You can go about picking any

13. anonymous

But what difference does it make if permutation changed to combination?

14. anonymous

Lack of order?

15. ganeshie8

give me a salad made up of "bananas, apple, grapes" is no different from give me a salad made up of "apple, bananas, grapes" order doesn't matter here, so this is a combination problem.

16. anonymous

ok I get it now.

17. ganeshie8

consider another quick example : you want to choose four digits to set a password to your laptop hard disk : "8 7 3 9" does the order matter here ?

18. ganeshie8

sorry to persist, but i think this is really important..

19. anonymous

No

20. ganeshie8

Are you saying "8 7 3 9" is no different from "7 3 8 9" ?

21. ganeshie8

22. anonymous

No

23. anonymous

The computer only accepts a given password you made at the boot up

24. anonymous

So the order would have to remain the same.

25. ganeshie8

Exactly, since the "order" in which you enter the digits matter here, this is clearly a permutation problem.

26. ganeshie8

That is how you differentiate between a permutation and combination problem

27. anonymous

Oh ok. I get it now.

28. anonymous

Ordinal or nominal

29. ganeshie8

 A committee of 5 people is to be chosen from a group of 8 women and 10 men. The committee must feature 3 women and two men?  How many ways can you choose 3 women from 8 women ?

30. anonymous

24

31. ganeshie8

good guess, but no.

32. anonymous

Either this is too hard or I am too stupid:(

33. anonymous

2.7 something

34. anonymous

8/3=2.66

35. ganeshie8

Neither, you just don't happen to have enough practice. You can think of this exactly same as the permutation problem. 3 women can be chosen from 8 women in 8C3 ways

36. ganeshie8

where $$nCr$$ is defined as $nCr = \dfrac{n!}{r!*(n-r)!}$

37. anonymous

8C3=number of possibilities where 3 women are picked from 8 women.

38. ganeshie8

$8C3 = \dfrac{8!}{3!*(8-3)!} = \dfrac{8!}{3!*5!}=\dfrac{8\times 7\times 6\times 5\times 4\times 3\times 2\times 1}{3\times 2\times 1*5\times 4\times 3\times 2\times 1}$

39. ganeshie8

Yes, see if you can simplify above

40. anonymous

(8*7*6*5)/3*2*1=336 1680/6=

41. anonymous

Because most numbers cancel each other out.

42. anonymous

8*7*6/3*2*1

43. ganeshie8

Yes simplifies further

44. anonymous

330/6=55

45. ganeshie8

8*7*6/3*2*1 8*7 56 right ?

46. anonymous

yes.

47. anonymous

so 56 ways in which 3 women can be picked from 8 women.

48. ganeshie8

Yes next find how many ways 2 men can be picked from the available 10 men

49. anonymous

2C10 so

50. ganeshie8

10C2 ok

51. ganeshie8

work it out

52. anonymous

10!/2!(10-2)!

53. anonymous

10!/2!*8!

54. anonymous

10*9*8*7*6*5*4*3*2*1/2*1*8*7*6*5*4*3*2*1*

55. anonymous

I simplify that

56. ganeshie8

yes do it

57. anonymous

90/2=45

58. anonymous

45 ways in which 2 men can be picked from 10 men

59. anonymous

56*45 is my guess.

60. ganeshie8

Excellent! so a committee with 3 women and two men can be formed in : $8C3*10C2 = 56*45~\text{ways}$

61. anonymous

I got it:)

62. anonymous

Do I use the same for the last one?

63. anonymous

where there are more men than women.

64. anonymous

there are 3 cases where men can be more than women 3 men 2 women 4 men 1 woman 5 men 0 women

65. anonymous

Sorry reverse of that

66. anonymous

more women than men. So do I multiply our previous answer by 3?

67. ganeshie8

sry was away.. still here @Robert136

68. ganeshie8

 there are 3 cases where men can be more than women 3 women 2 men 4 women 1 man 5 women 0 man  Yes find the number of ways for each case, then add up.

69. ganeshie8

you should get $8C3*10C2 + 8C4*10C1+5C1*10C0$

70. ganeshie8