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# Factorial

In Mathematics, factorial is an important function, which is used to find how many ways things can be arranged or the ordered set of numbers. The well known interpolating function of the factorial function was discovered by Daniel Bernoulli. The factorial concept is used in many mathematical concepts such as probability, permutations and combinations, sequences and series, etc. In short, a factorial is a function that multiplies a number by every number below it till 1. For example, the factorial of 3 represents the multiplication of numbers 3, 2, 1, i.e. 3! = 3 × 2 × 1 and is equal to 6. In this article, you will learn the mathematical definition of the factorial, its notation, formula, examples and so on in detail.

Also, Check: Factorial Calculator

## What is Factorial?

In Mathematics, factorial is a simple thing. Factorials are just products. An exclamation mark indicates the factorial. Factorial is a multiplication operation of natural numbers with all the natural numbers that are less than it. In this article, let’s discuss the factorial definition, formula and examples.

## Factorial Notation

The multiplication of all positive integers, say “n”, that will be smaller than or equivalent to n is known as the factorial. The factorial of a positive integer is represented by the symbol “n!”.

## Factorial Formula

The formula to find the factorial of a number is

n! = n × (n-1) × (n-2) × (n-3) × ….× 3 × 2 × 1

For an integer n ≥ 1, the factorial representation in terms of pi product notation is:

$$n! = \prod_{i=1}^{n}i$$

From the above formulas, the recurrence relation for the factorial of a number is defined as the product of the factorial number and factorial of that number minus 1. It is given by:

n! = n. (n-1) !

### Factorial of a Number

To find the factorial of any given number, substitute the value for n in the above given formula. The expansion of the formula gives the numbers to be multiplied together to get the factorial of the number.

### Factorial of 10

For example, the factorial of 10 is written as

10! = 10. 9 !

10! = 10 (9 × 8 × 7 × 6 × 5× 4 × 3 × 2 × 1)

10! = 10 (362,880)

10! = 3,628,800

Therefore, the value of 10 factorial is 3,628,800

The factorial operation is encountered in many areas of Mathematics such as algebra, permutation and combination, and mathematical analysis. Its primary use is to count “n” possible distinct objects.

For example,  the number of ways in which 4 persons can be seated in a row can be found using the factorial. That means, the factorial of 4 gives the required number of ways, i.e. 4! = 4 × 3 × 2 × 1 = 24. Hence, 4 persons can be seated in a row in 24 ways.

## Factorials of Numbers 1 to 10 Table

The list of factorial values from 1 to 10 are:

 n Factorial of a Number n! Expansion Value 1 1! 1 1 2 2! 2 × 1 2 3 3! 3 × 2 × 1 6 4 4! 4 × 3 × 2 × 1 24 5 5! 5 × 4 × 3 × 2 × 1 120 6 6! 6 × 5 × 4 × 3 × 2 × 1 720 7 7! 7 × 6 × 5 × 4 × 3 × 2 × 1 5,040 8 8! 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 40,320 9 9! 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 362,880 10 10! 10 × 9 ×8 × 7 × 6 × 5 ×4 × 3 × 2 × 1 3,628,800

### What is Sub factorial of a Number?

A mathematical term “sub-factorial”, defined by the term “!n”, is defined as the number of rearrangements of n objects. It means that the number of permutations of n objects so that no object stands in its original position. The formula to calculate the sub-factorial of a number is given by:

$$!n = n!\sum_{k=0}^{n}\frac{(-1)^{k}}{k!}$$

### Factorial of 5

Finding the factorial of 5 is quite simple and easy. This can be found using formula and expansion of numbers. This is given below with detailed steps.

We know that,

n! = 1 × 2 × 3 …… × n

Factorial of 5 can be calculated as:

5! = 1 × 2 × 3 × 4 × 5

5! = 120

Therefore, the value of factorial of 5 is 120.

## Video Lesson

### Factorial Examples

Example 1:

What is the factorial of 6?

Solution:

We know that the factorial formula is

n! = n × (n – 1) × (n – 2) × (n – 3) × ….× 3 × 2 × 1

So the factorial of 6 is

6! = 6 × (6 -1) × (6 – 2) × (6 – 3) ×  (6 – 4) × 1

6! = 6 × 5 × 4 × 3 × 2 ×1

6! = 720

Therefore, the factorial of 6 is 720.

Example 2:

What is the factorial of 0?

Solution:

The factorial of 0 is 1

i.e., 0 ! = 1

According to the convention of empty product, the result of multiplying no factors is a nullary product. It means that the convention is equal to the multiplicative identity.

### Practice Problems

Practice the problems given below to understand the concept.

1. Evaluate 7! – 5!.
2. What is the value of 12!/(10! 4!)
3. If (1/6!) = (x/8!) – (1/7!), then what is the value of x?
4. Is 4! + 5! = 9!?

Visit BYJU’S – The Learning App for more information on factorial of numbers and explore Maths-related videos to learn with ease.

## Frequently Asked Questions on Factorial

### What is a factorial of 10?

The value of factorial of 10 is 3628800, i.e. 10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 3628800.

### What is the meaning of 5 factorial?

The meaning of 5 factorial is that we need to multiply the numbers from 1 to 5. That means, 5! = 5 × 4 × 3 × 2 × 1 = 120.

### What is the symbol of factorial?

The factorial function is a mathematical formula represented by an exclamation mark “!”. For example, the factorial of 8 can be represented as 8! and it is read as eight factorial.

### What is a factorial of 0?

The value of factorial of 0 is 1, i.e. 0! = 1.

### What is the value of 7!?

The value of 7! is 5040, i.e. 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040.

Test your knowledge on Factorial


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# Controlling the Quantum State with a time varying potential

## Abstract

The problem of controlling the quantum state of a system is investigated using a time varying potential. As a concrete example we study the problem of a particle in a box with a periodically oscillating infinite square-well potential, from which we obtain results that can be applied to systems with periodically oscillating boundary conditions. We derive an analytic expression for the frequencies of resonance between states, and against standard intuition, we show how to use this behavior to control the quantum state of the system at will. In particular, we offer as an example the transition from the ground state to the first excited state of the square well potential. At first sight, it may be counter intuitive that we can control the state of such a quantum Hamiltonian, as the Schrödinger equation conserves the norm of the wave function. In this manuscript, we show how that can be achieved.

## Introduction

Since the development of Quantum Theory, the study of time dependent quantum systems has been of wide interest1,2,3,4, and in particular, the control the quantum phenomena has been an implicit task. As Dong and Petersen wrote, “One of the main goals in quantum control theory is to establish a firm theoretical footing and develop a series of systematic methods for the active manipulation and control of quantum systems”. Such capability could be used to store information in the system, or in an array of them, using many states to store information as the experimental limits allow us5.

With this in mind, in this letter we revisit the problem of a particle in a one-dimensional infinite square-well potential, but with a moving wall which we use to drive the system systematically from one quantum state to another. Furthermore, we will derive analytic expressions that characterize this process. Our scheme may be characterized as a bi-linear quantum control strategy6 that can be used to develop similar strategies in the context of other systems where the dynamics follows a Schrodinger equation; not necessary in the quantum regime, such as quantum optics or in Bose-Einstein condensates; but also in water waves, optical fibers, planar waveguides, among others.

To do so, we start by showing that by oscillating the wall at certain frequencies we can produce an oscillation between two predefined states, in a manner that is similar to the results obtained by Lenz et al.7 in a different system. It is important to mention that Glasser et al.8 also reported a similar behavior for the infinite square-well potential but they didn’t found the rule to do so since they focused their study on low, intermediate, and high frequencies of the wall motion; finding adiabatic, chaotic, and periodic behavior respectively. We will see, as a by-product of our quest, that the classical macroscopic effects of the Fermi acceleration process9 and chaotic dynamics for large amplitude oscillating walls10 could also be associated with the respective dynamics in the quantum domain. Our problem of interest is a quantum variant of the famous Fermi acceleration problem11, but with impenetrable walls instead of inhomogeneous magnetic fields. In this sense, the idea behind this equivalent Fermi acceleration process can be illustrated by making a ping pong ball bounce between a table and a paddle moving towards the table.

The first work about our problem at hand, as far as we can tell, was made by Doescher and Rice12, where they solved the Schrödiner equation for the case of a moving wall at constant magnitude of velocity motion. After that article, the problem has been discussed in several publications using different approaches, such as the adiabatic approximation, sudden expansion or contraction of the wall, rapidly varying square well width, etc.1,2,3,4.

Following our quest, we now demonstrate that is it possible to construct a sort of effective stroboscopic Hamiltonian for this time-dependent periodic problem, as it is done for the classical particle with the oscillating wall9,10. This operator emerges naturally when we study the time evolution operator that can give us the oscillatory response of the system, so that in essence we are constructing a simile of Bloch’s Theorem13, but for the time domain, in the case of oscillating boundary conditions. This stroboscopic Hamiltonian determines the interaction of the accessible states and the long term evolution of the system. It is worth mentioning that recently a work by Di Martino et al.14 suggested that a unitary transformation to the fixed wall problem can be made. Before, Pereshogin and Pronin15 showed that this moving wall problem can be cast to an effective Hamiltonian and a Berry phase. Even earlier, Munier et al.16 constructed a generalized canonical transformation, which suggested why this effective Hamiltonian could be constructed.

## Method

### How to use the wall to connect quantum states of a particle in a square-well potential

To simplify the analysis, we will first study the time evolution of a particle in an infinite square well potential with an oscillating wall moving with a constant magnitude of velocity. Later on, we will show that the results also apply to more general oscillations. For the case of a particle of mass m between two rigid walls at positions x = 0 and x = L 0, we can choose a Hilbert space with the basis {|φ n〉} given by the solution of the time-dependent Schrödinger equation, namely

$$\langle x|{\phi }^{n}(t)\rangle =\sqrt{\frac{2}{{L}_{0}}}{e}^{-i\frac{{n}^{2}{\pi }^{2}\hslash }{2m{L}_{0}^{2}}t}\,\sin \,(\frac{n\pi x}{{L}_{0}}),$$
(1)

where n is the enumeration index of the base vector, and ħ is Planck’s constant. We can follow the time evolution of every admissible state by simply projecting that state into this basis at for example t = 0.

Similarly, Doescher and Rice12 found a solution for the problem where the right wall moves with constant magnitude of velocity v. In that case, we can choose as the complete time-dependent Hilbert space basis the set of eigenstates $$\{|{\varphi }_{v,L(t)}^{n}\}$$ given by

$$\langle x|{\varphi }_{v,L(t)}^{n}(t)\rangle =\sqrt{\frac{2}{L(t)}}{e}^{i\frac{{n}^{2}{\pi }^{2}\hslash }{2mvL(t)}}{e}^{\frac{ivm{x}^{2}}{2\hslash L(t)}}\,\sin \,(\frac{n\pi x}{L(t)}).$$
(2)

As before, we can follow the dynamics of every admissible state by simply projecting that state into this basis at for example t = 0. It becomes important to note that the time evolution of this eigenstate is simply parameterized by the change of L(t) = L 0 + vt, so we can forget the time and think the evolution in terms of the box length. At the same time, we note that for v < 0, the kinetic energy of the particle in a given eigenstate goes as E n  ~ L(t)−2 plus a correction which is function of v, so that the energy increases as the wall shrink the system as L = L 0 − vt, in a similar way a ping pong paddle accelerates the bouncing ball in a Fermi acceleration process.

Let us consider a wall oscillation from L 0 to L 0 + d and back at a constant magnitude of velocity defining an effective frequency Ω. In that case, we must consider three different basis, {|φ n〉} and $$\{|{\varphi }_{\pm v,L(t)}^{n}\rangle \}$$, where v ± = ±ωd/π are the speeds of the right wall when it moves right and left, respectively. This is a simple example that illustrates a more general result, as we will explain later in this letter. Note that all these basis are orthogonal, consequently, the change of basis is unitary.

With this in mind, we can study the time evolution of every state. Initially, the quantum state of the system should be a linear combination of {|φ n〉}, but when the wall moves right the system will evolve according to the basis $$\{|{\varphi }_{+v,L(t)}^{n}\rangle \}$$, so that we project the state in this basis to give account of that. In a similar way, when the box size becomes L 0 + d the system will evolve according to the basis $$\{|{\varphi }_{-v,L(t)}^{n}\rangle \}$$ until the box size becomes L 0 again, and then we can go back to the basis {|φ n〉}. After these three projections, we can encounter the system in a different state. The dynamics evolves the system in the same way for every oscillation because the eigenstates are parameterized only by L(t).

We can give account of this process by a discrete time evolution operator, that evolves the system over one oscillation, namely, $${\check{S} }|\psi (t=\mathrm{0)}\rangle =|\psi (t=2\pi /\omega )\rangle$$. The matrix elements of this operators can be expressed on the {|ϕ n〉} basis as follows

$$\langle {\varphi }^{n}|{\check{S} }|{\varphi }^{p}\rangle =\sum _{k=1}^{\infty }\,\sum _{l=1}^{\infty }\,\langle {\varphi }^{n}|{\varphi }_{-v,{L}_{0}}^{l}\rangle \langle {\varphi }_{-v,{L}_{0}+d}^{l}|{\varphi }_{+v,{L}_{0}+d}^{k}\rangle \langle {\varphi }_{+v,{L}_{0}}^{k}|{\varphi }^{p}\rangle .$$
(3)

Note that $${\check{S} }$$ is unitary because the product of two or more unitary operators is unitary. Furthermore, this discrete stroboscopic time evolution operator can have its own set of eigenstates and eigenvalues, where the eigenvalues are complex numbers of modulus equal to one, hence for N oscillations may be written as $${\lambda }_{n}^{N}={e}^{i2\pi N{\alpha }_{n}/\hslash \omega }$$. Inspired by the form of the eigenvalues, it may be useful to think of this stroboscopic operator as coming from a Hamiltonian $${\mathop{H}\limits^{\check{}}}_{s}$$, that we call the stroboscopic Hamiltonian, with eigenvalues α n . Note that $${\mathop{H}\limits^{\check{}}}_{s}$$ is an Hermitian operator. It’s worth to mention that this decomposition presented here of the evolution operator is similar to the one presented in the work of Novo et al.17.

Hence, the time evolution can be described by the effective Hamiltonian $${\mathop{H}\limits^{\check{}}}_{s}$$, and it’s expected that $$[{\mathop{H}\limits^{\check{}}}_{s},{\mathop{H}\limits^{\check{}}}_{0}]\ne 0$$, where $${\mathop{H}\limits^{\check{}}}_{0}$$ is the Hamiltonian of the static problem. In that case, we will have oscillations on the value $$\langle {\mathop{H}\limits^{\check{}}}_{0}\rangle$$, as a result of Ehrenfest theorem18, so that we can define Bohr frequencies that determine the long time evolution of the system, and the mixing of the states. Below, we will analyze the conditions that produce $$[{\mathop{H}\limits^{\check{}}}_{s},{\mathop{H}\limits^{\check{}}}_{0}]\ne 0$$. Furthermore, we must recall from classical dynamics that the Hamiltonian is not necessarily equal to the energy so that the eigenvalues of $${\mathop{H}\limits^{\check{}}}_{s}$$ don’t need to correspond to the energy of the state.

It’s important to point out that we can follow the same argument for any periodic oscillation because every oscillation of the wall can be understood as many steps with d → 0.

So far this treatment is exact but difficult to track analytically so that in order to make explicit calculations let us consider a particular case where $$d/{L}_{0}\ll 1$$ and the oscillations are made with the constant magnitude of velocity strategy discussed above. In this case, we can solve (3) and later compute the matrix elements in the basis of the static problem, namely,

$${S}_{k,n}=\langle {\phi }^{k}|{\check{S} }|{\phi }^{n}\rangle ={\delta }_{kn}{e}^{-2i\frac{{k}^{2}\pi }{\bar{\omega }}}+i{C}_{k,n}{e}^{-i\frac{({k}^{2}+{n}^{2})\pi }{\bar{\omega }}}\,(1-{e}^{-i\frac{({k}^{2}-{n}^{2})\pi }{\bar{\omega }}})+{\mathscr{O}}(\bar{d}{}^{2})\,$$
(4)

where $$\bar{\omega }=2m\omega {L}_{0}^{2}/{\pi }^{2}\hslash$$, $$\bar{d}=d/{L}_{0}$$, and

$${C}_{k,n}=\frac{\bar{\omega }\bar{d}}{{\pi }^{2}}\frac{4{(-1)}^{k+n}kn\pi }{{({k}^{2}-{n}^{2})}^{2}}.$$
(5)

It’s worth noting that at least at first order of $$\bar{d}$$, all the phenomenology is given by this two dimensionless parameters, $$\bar{d}$$ and $$\bar{\omega }$$. Another point is that, as we established before, the operator is unitary as can be seen by computing $${{\check{S} }}^{\dagger }{\check{S} }=1$$ to first order.

The probability transference between states as defined by the eigenstates of the evolution operator are slightly different from the static ones. That must happen if the non-diagonal elements are not zero. Indeed, the squared modulus of the non-diagonal elements are proportional to

$${|{S}_{k,n}|}^{2}\propto \frac{1}{2}-\frac{1}{2}\,\cos \,(\frac{({k}^{2}-{n}^{2})\pi }{\bar{\omega }}).$$
(6)

For a given value of $${\bar{\omega }}_{k,n}^{\ell }$$, this quantity has a maximum for a certain value of k and n, so the state k and n becomes connected. Note that this condition is symmetric under change of k and n, as $${\bar{\omega }}_{k,n}^{\ell }$$ can be rewritten as

$${\bar{\omega }}_{k,n}^{\ell }=\frac{{k}^{2}-{n}^{2}}{2\ell -1}\,{\rm{for}}\,{\rm{some}}\,\ell \in {\mathbb{N}},$$
(7)

which is the resonance condition.

It becomes also important to note that the dynamics is determined by the non-diagonal elements, and hence the phenomenology should become noticeable after many oscillations because the non-diagonal elements are proportional to $$\bar{d}$$. Therefore, it makes sense to think in terms of a Stroboscopic Hamiltonian for small $$\bar{d}$$ values. We will analyze below what happens for larger $$\bar{d}$$ values.

Before we end this section of the manuscript, we must point out that our discrete time evolution operator could be obtained via a Floquet approach19, which is completely equivalent to the one used here. In fact, the Floquet exponents λ n are related to the eigenvalues of the stroboscopic Hamiltonian, namely, λ n  =  n /ħ. However, in the case of a wall moving with a constant magnitude velocity, we have an expression for the intermediate states, so that constructing the evolution operator in terms of the inner products of these states is straight forward.

### How to use the connection between states to control the state of the system at will

We now turn to our most fundamental task, that is to drive the system from one quantum state to another, and keep it there. At first sight, it might be quite counter intuitive that it is possible to control the state of the system represented by the stroboscopic Hamiltonian, as the Schrödinger equation conserves the norm of the states. The key for our quest is to understand how the system manages to remember when a probability is increasing and when is decreasing on a resonance process. The information is stored in the phase of the amplitude of every eigenstate, information that vanishes when we take the squared moduli. As the Schrödinger equation doesn’t have inertia, it is the only place where it can be stored. Therefore, if we can control the phase of every state, we can control which probability is increasing and which is not. To do so we can let the system evolve certain amount of time with no oscillation until we have the correct relative phase between the states we want to affect, and then restart the oscillation of the wall to drive the system to any asymptotic state we want to obtain.

## Results

### Connecting states

Let us use our analysis to connect the state |φ 1〉 with the state |φ 2〉. As we determined in Eq. (6) the wider maximum is at $$\bar{\omega }={\bar{\omega }}_{\mathrm{1,2}}^{1}=3$$, so, this frequency should provides the easiest way to tune this resonance. We compute the dynamics applying N times the evolution operator calculated numerically at all orders of $$\bar{d}$$, and then compute the probability of finding the system at the states |φ 1〉 and |φ 2〉, as we observe in Fig. 1, where we use |φ 1〉 at t = 0 as the initial state. In this Figure, we find oscillations between the two states of the static box. Note that the time scale of the phenomena is almost three orders of magnitude slower than the oscillation of the box, in accordance with the size of $$\bar{d}$$.

We can explain the curve if we suppose that only the block of the operator $${\mathop{H}\limits^{\check{}}}_{s}$$ that involves the first and second state is non-diagonal on the basis of eigenstates of $${\mathop{H}\limits^{\check{}}}_{0}$$, namely $$|{\lambda }_{1}={e}^{i{\beta }_{1}}\rangle =\,\cos \,\theta |{\phi }_{1}\rangle +{e}^{i\varphi }\,\sin \,\theta |{\phi }_{2}\rangle$$, $$|{\lambda }_{2}={e}^{i{\beta }_{2}}\rangle =-{e}^{-i\varphi }\,\sin \,\theta \,|{\phi }_{1}\rangle +\,\cos \,\theta \,|{\phi }_{2}\rangle$$, and $$|{\lambda }_{j}={e}^{i{\beta }_{j}}\rangle =|{\phi }_{j}\rangle$$ for j > 2. If we follow the dynamics of the state |φ 1〉 through N oscillations, the probability of finding the system on the eigenstate |φ 1〉 and |φ 2〉 are

$${P}_{1}(N)={\cos }^{4}\,\theta +{\sin }^{4}\,\theta +2\,{\cos }^{2}\,\theta \,{\sin }^{2}\,\theta \,\cos \,[({\beta }_{1}-{\beta }_{2})N]$$
(8)

and P 2(N) = 1 − P 1(N), respectively. Now, if we compute numerically the matrix elements of $${\mathop{H}\limits^{\check{}}}_{s}$$ on the basis of eigenstates of $${\mathop{H}\limits^{\check{}}}_{0}$$, we can solve for θ and make a prediction of the curve, that we see as the dashed lines of Fig. 1. Hence, we find that our reduced model works quite accurately, which suggest that we can split the matrix in blocks for our analysis as long as $$\bar{d}$$ is small, as we did. In the same way, a similar analysis can be made for three or more eigenstates, in the case of larger $$\bar{d}$$ values.

In order to test Eq. (7) we study the average probability of the first 7 states, starting from the state |φ 1〉 for different values of $$\bar{\omega }$$ and $$\bar{d}$$, as we show in Fig. 2a. We observe that the position of all the maxima fit well with our predictions, at least for $$\ell =1,\,2,\,3$$; and $$\bar{d}=0.001$$. It is interesting to note that the maxima move a little for $$\bar{d}=0.01$$, but this can be expected since Eq. (7) is only valid at first order. It is worth noticing that different resonances have different widths but all widths increase with the value of $$\bar{d}$$. Furthermore, when we use bigger values of $$\bar{d}$$, we can observed oscillations (interactions) between three or more states, which may be related to the chaotic behavior that appears in the classical regime10, remembering that chaotic dynamics is expected to appear quite naturally when be observe three wave resonances20. We will analyze this in a future manuscript. Finally, we observe that not all peaks have the same maximum average probability.

We claim that our analysis is valid for different types of oscillations, no just magnitude constantly moving walls. Indeed, we can numerically solve the Schrödinger equation for a particle in a periodically varying box, but with a more continuously varying oscillation. To do so, we use a similar algorithm to the one proposed by Doescher et al.12 Let us introduce the ansatz

$$\psi (x,\,t)=\sum _{n=1}^{\infty }\,{A}_{n}(t)\sqrt{\frac{2}{L(t)}}\,\sin \,(\frac{n\pi x}{L(t)}),$$
(9)

in the time-dependent Schrödinger equation. We find that

$$\frac{{\hslash }^{2}{k}^{2}{\pi }^{2}}{2m{L}^{2}}{A}_{k}=i\hslash \frac{\partial {A}_{k}}{\partial t}-i\hslash \frac{\dot{L}(t)}{2L(t)}{A}_{k}-i\hslash \,\sum _{n=1}^{\infty }\,\frac{2n\dot{L}(t){A}_{n}}{\pi L(t)}\,{\int }_{0}^{\pi }\,\sin \,(kx)\,\cos \,(nx)\,x\,dx,$$
(10)

where L(t) is the varying position of the right wall. If we truncate the sum to m terms, we can solve this numerically. We can construct a figure equivalent to Fig. 2a, but for a harmonic oscillation of the form,

$$L(t)={L}_{0}+d\,[1-\,\cos \,(\omega t)],$$
(11)

as we see in Fig. 2b. In this case, we find the same resonances between states at the same frequencies predicted by our approach with the constantly moving wall. Hence, in this Figure we corroborate that our analysis, as described by Eq. (7) holds for such continuous type of oscillations, at least with one frequency. It’s important to recall that this explains in an analytic manner the results of Glasser et al.8. Oscillations with multiple frequencies will be studied in detail in a future publication.

For large amplitudes, we expect the system to become more classical, as states with larger energy become involved, and eventually reach the chaotic regime of the Fermi-Ulam problem9. This transition is of course not trivial, as the Floquet approach becomes more involved. We plan to study this problem in detail in a future manuscript.

### Controlling the quantum state

In Fig. 3 we use that procedure to obtain the state |φ 2〉, asymptotically, starting from the state |φ 1〉. To do so, we start with no wall oscillation and at the first vertical black line we start oscillating the wall using a constant magnitude of velocity motion with $$\bar{\omega }=2.9998$$, as predicted by our previous analysis. At the blue vertical line the probability of finding the system in the state |φ 1〉 has a minimum. At that time, we stop the wall oscillation and wait, for a time equivalent to 9.94 oscillations of the wall, until we acquire the necessary relative phase. At that point, shown as the vertical red dashed line, we start oscillating the wall again until we get to another minimum, shown as the last vertical black line. Here, we stop the wall oscillation and find the system in a state that has the probability P 2 = 0.99918 of being in the state |φ 2〉. This process can be optimized to increase the probability P 2 if necessary.

## Conclusions and Discussion

In summary, we have studied the phenomenology of a particle in a squared-well potential with an oscillating wall. We found a previously reported oscillation between states for this system and we constructed an analytic description by considering small and constant magnitude of velocity oscillations. Additionally, we found the frequencies that appear in this phenomenon using a simple two level model, at least for $$\bar{d}\ll 1$$. We then go even further and showed that our results are even more general since they apply to more continuous oscillations.

Using our previous results, we developed a method to use this phenomenon to control systematically the quantum state of the system by controlling the motion of the oscillating wall. This procedure may be used to store information in the system, or in an array of them, using many states to store information as the experimental limits allow us.

At first sight, it may be very surprising that this time-dependent problem can be studied in terms of a time-independent problem using the stroboscopic Hamiltonian. Similarly, it is quite counter intuitive that we can control the state of the system using only this Hamiltonian, because an important property of the Schrödinger equation is that the time evolution doesn’t change the norm of the states. But in this case, the eigenstates are a combination of “non-moving wall” states so that an appropriate combination with two Hamiltonian can make this happen.

Finally, we plan to study in a future work the chaotic transition that we expect to happen at bigger values of $$\bar{d}$$, where several frequencies resonate at the same time, and that may be related to the transition to the chaotic dynamics that appears in the classical regime, which has been studied in previous works. Although the general setting of this manuscript is about the fundamentals of quantum mechanics, we can imagine possible applications in quantum optics or Bose-Einstein condensates, where it could be possible to submit these type of systems to a control strategy by varying the boundary conditions; for example, in the later case, by tuning the laser trapping system21. Of course in such systems, we would expect to see the appearance of nonlinear terms, that could also have applicability in systems like water waves, optical fibers, planar waveguides, among others22,23.

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Martino, S. D. et al. A quantum particle in a box with moving walls. J. Phys. A: Math. Theor. 46, 365301 (2013).

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Pereshogin, P. & Pronin, P. Effective hamiltonian and berry phase in a quantum mechanical system with time dependent boundary conditions. Phys. Lett. A 156, 12–16 (1991).

16. 16.

Munier, A., Burgan, J. R., Feix, M. & Fijalkow, E. Schrödinger equation with time-dependent boundary conditions. J. Math. Phys. 22, 1219–1223 (1981).

17. 17.

Novo, L. & Berry, D. Improved hamiltonian simulation via a truncated taylor series and corrections. Quantum Inf. Comput. 17, 623–635 (2017).

18. 18.

Ehrenfest, P. Bemerkungüber die angenäherte gültigkeit der klassischen mechanik innerhalb der quantenmechanik. Zeitschrift für Physik 45, 455–457 (1927).

19. 19.

Asenjo, F., Toledo, B. A., Muñoz, V., Rogan, J. & Valdivia, J. A. Optimal control in a noisy system. Chaos: An Interdiscip. J. Nonlinear Sci. 18, 033106 (2008).

20. 20.

C. Grebogi, E. Ott, and J. A. Yorke. Attractors on an n-torus: Quasiperiodicity versus chaos. Phys. D: Nonlinear Phenom. 15, 354–373 (1985).

21. 21.

Anderson, M. H., Ensher, J. R., Matthews, M. R., Wieman, C. E. & Cornell, E. A. Observation of bose-einstein condensation in a dilute atomic vapor. Sci. 269, 198–201 (1995).

22. 22.

Peregrine, D. H. Water waves, nonlinear schrödinger equations and their solutions. The J. Aust. Math. Soc. Ser. B. Appl. Math. 25, 16–43 (1983).

23. 23.

Kaup, D. J. Perturbation theory for solitons in optical fibers. Phys. Rev. A 42, 5689–5694 (1990).

## Acknowledgements

The authors acknowledge the support by CONICYT-PCHA/Doctorado Nacional/2016-2116103 (S.C.) and the Fondo Nacional de Investigaciones Cientficas y Tecnológicas (FONDECYT, Chile) under grants #1120399, #1150718 (J.A.V.), and CEDENNA through “Financiamiento Basal para Centros Cientficos y Tecnológicos de Excelencia-FB0807” (J.R. and J.A.V.).

## Author information

Authors

### Contributions

S.C. made all the calculations both analytical and numerical. S.C., J.R. and J.A.V. contributed equally to the explication of this physical phenomena. S.C., J.R. and J.A.V. contributed to the writing of the manuscript.

### Corresponding author

Correspondence to Sebastián Carrasco.

## Ethics declarations

### Competing Interests

The authors declare that they have no competing interests.

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

## Rights and permissions

Reprints and Permissions

Carrasco, S., Rogan, J. & Valdivia, J.A. Controlling the Quantum State with a time varying potential. Sci Rep 7, 13217 (2017). https://doi.org/10.1038/s41598-017-13313-3

• Accepted:

• Published:

• ### Photodetachment dynamics of negative ion confined in a time-dependent quantum well

• Tong Shi
• , De-Hua Wang
•  & Xin-Yue Sun

Indian Journal of Physics (2020)

• ### Speeding up maximum population transfer in periodically driven multi-level quantum systems

• Sebastián Carrasco
• , José Rogan
•  & Juan Alejandro Valdivia

Scientific Reports (2019)

• ### Resonant Transitions Due to Changing Boundaries

• Fabio Anzà
• , Antonino Messina
•  & Benedetto Militello

Open Systems & Information Dynamics (2019)

format: plain


text:
# Good variance test when the distributions have different shapes but I really want to test the variance

Consider $$X\sim N(1,1)$$ and $$Y\sim exp(1)$$. The distributions are totally different, but they do have equal variance, so if I were to sample from these and test for unequal variance, I would not want to reject a null hypothesis of variance equality.

I am finding that the usual suspects for variance testing reject too often. In my Monte Carlo simulations, the F-test (var.test in R), Brown-Forsythe (car::leveneTest(…, center="median")), Levene (car::leveneTest(…, center="mean")), Ansari-Bradley (ansari.test), and permutation testing (my own custom function) all rejected considerably more often that my chosen $$\alpha$$. These tests are overpowered for this situation. I suspect (particularly for Ansari-Bradley) that this is due to the tests examining some kind of surrogate for variance (some kind of "scale" parameter) rather than variance itself, and they are finding that $$X$$ and $$Y$$ do differ on that surrogate.

What would be a test that has appropriate power for my $$X$$ and $$Y?$$

• The choice of test ought to depend on the amount of data and on any additional assumptions you can make about the distributions. Could you provide some specifics of your intended application? – whuber Dec 1 '20 at 15:12
• For amount of data, I won’t be working with 3-4 observations. I would expect (usually) at least dozens of not hundreds of points. For intended application, I have bizarre empirical distributions that don’t seem to follow the named distributions. Mostly, I’m content to use something like Ansari-Bradley (a collaborator adores it), but I’d sure like to see something that really examines variance and not the “scale” that Ansari-Bradley examines, since that results in a false rejection of variance equality for $N(1,1)$ and $exp(1)$. – Dave Dec 1 '20 at 15:29
• For comparing variances you typically need much more data than for comparing means. In this light, even a few hundred values is not a large dataset. The A-B test is a rank sum test, which means it may be appropriate for your broad nonparametric assumption: but it also implies it may lack power. The power can be regained to the extent you can narrow your characterization of the distributions your are comparing. I don't follow your distinction between a scale parameter and variance, because the two are interchangeable in any scale family (which this is). – whuber Dec 1 '20 at 15:51
• The normal $X$ and exponential $Y$ are in the same scale family? – Dave Dec 1 '20 at 16:01
• Yes, because you have implicitly formulated this as a nonparametric problem. If that's not what you mean--if instead you literally mean you know you have sampled from this Normal and this Exponential distribution--then there's nothing left to say. In fact, you don't even need a sample to determine whether these distributions have equal variances! – whuber Dec 1 '20 at 16:06

In this case the means of the two distributions are equal. So instead of using a variance test, you can test the hypothesis $$E[X^2]=E[Y^2]$$.
• Could you please explain how Wilcoxon's test handles this? Just square the values and throw them in wilcox.test in R? – Dave Dec 23 '20 at 20:59


text:
# Import and merge PDF files

I have a folder with several pdf no more than 50, I would like to import them to be disordered and that the output was a single pdf (you can imagine it as a proof of form A and we want to produce form B, where the files are the questions)

files = {"file1.pdf", "file2.pdf", "file3.pdf", "file4.pdf", "file5.pdf",
"file6.pdf", "file7.pdf", "file8.pdf", "file9.pdf", "file10.pdf",
"file11.pdf", "file12.pdf", "file13.pdf", "file14.pdf", "file15.pdf",
"file16.pdf", "file17.pdf", "file18.pdf", "file19.pdf", "file20.pdf",
"file21.pdf", "file22.pdf", "file23.pdf", "file24.pdf", "file25.pdf"};

Sort[files]

the previous order, but as disorderly?

**Edit How do I import all the files from a c: \ directory?

how do I merge all those files into the same file called file and record it in the same directory as c: \?**

• Do you mean randomly sorted? RandomSample@files – Rohit Namjoshi May 26 '19 at 23:55
• @ Rohit Namjoshi , it works very well that order thanks, now as I do to import the files from the ws directory or obtain the output merged into pdf in the same directory with the name say fileB , I imagine something like that but it does not work list = FileNames[FileNameJoin[dir, "filename*"]] – wally May 27 '19 at 0:18
• Not really sure what you are asking. Are you trying to get a list of all files that match some pattern in a specific directory? If so, to get a list of all files with the pdf extension in directory dir: FileNames["*.pdf", dir] – Rohit Namjoshi May 27 '19 at 0:35
• I explain to you with 3 files for example, file1.pdf, file2.pdf, file3.pdf housed in c: \ ws, I need to import them, undo them and merge them into a single file called fileB, and this fileB (file1.pdf + file2.pdf + disordered file3.pdf) and burn it to the same directory c: \ ws – wally May 27 '19 at 1:27
• I do not think you can do this well in M. This question looks duplicate of merge-pdfs-to-one and combine-multiple-pdf-files-into-one-document so it is better just to use an external tool. There are many such tools to choose from. – Nasser May 27 '19 at 2:19


text:
IPython - single line spacing possible?

By default, the IPython shell shows a blank line between inputs:

In [1]: 1 + 1
Out[1]: 2

In [2]: import math

In [3]: math.sqrt(2)
Out[3]: 1.4142135623730951

In [4]:

Is there any way to configure the IPython shell so that it does not produce a blank line between each command history? This would significantly increase the amount of history I would be able to see on my screen.

-
The magic command hist (short for history) prints input history without blank lines. –  Steve Tjoa Feb 28 '10 at 17:54

I'm working on Windows, Alok's excerp didn't differ from the default content of these three lines in my %USERPROFILE%/_ipython/ipythonrc.ini

I rather had to change the line:

separate_in \n

to:

separate_in 0

-
That works. Or nosep 1, which is a shortcut for separate_in 0, separate_out 0 and separate_out2 0 (the last two are 0 by default anyway). –  weronika May 3 '12 at 1:33

You can use the --nosep option. e.g.

C:\Python32\Scripts\ipython3.exe --nosep

-

For Mac users, the profile config to permanently edit this can be found in: /Users/(your username)/.ipython/profile_default/ipython_config.py

For linux users, it's probably a nearly identical path, but with /home/ instead of /Users/. Using '~' in the command line will work in either case, (e.g. ~/.ipython/...).

Un-comment line 164 and remove the newline, so it says:

c.TerminalInteractiveShell.separate_in = ''

While this is an old question, it's still relevant for other users (such as myself) who find it via Google, so I wanted to add this answer for Mac & Linux users.

-


text:
# How to calculate time-segmented volume? [closed]

amibrokers has this calculation for TSV:

_SECTION_BEGIN(" Time Segmented Volume ");

t=(Sum(IIf( C>Ref(C,-1), (V*(C-Ref(C,-1))),IIf(C<Ref(C,-1),(V*(C-Ref(C,-1))),0)),13));
m=MA(t ,7 );
Plot(t, "Time Segmented Volume 13", colorRed);
Plot(m, " Simple Moving Average 7" , colorYellow);

_SECTION_END();

Key: t is calculated TSV; read IIF as IF (happily, IIF has been defenestrated); C is the close on the current bar; (C,-1) is the close on the just-previous bar. @chrisaycock points out below that the usual trading day in the US has 13 half-hour periods, represented perhaps by the constant 13.

Can you comment on the correctness of the calculation? Or suggest a better one?

-

## closed as too localized by SRKX♦, Alexey Kalmykov, Tal FishmanFeb 6 '12 at 21:31

This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

There are 13 half-hour periods in a typical US trading day. –  chrisaycock Jan 22 '12 at 22:01
@ chrisaycock -- Thanks!! <gulp> :-) –  Pete Wilson Jan 23 '12 at 0:31


text:
Inverse matrix notation question

1. Apr 17, 2014

SamanthaYellow

I'm hoping that you can help me settle an argument. For a matrix $\textbf{M}$ with elements $m_{ij}$, is there any sitaution where the notation $(M_{ij})^{-1}$ could be correctly interpreted as a matrix with elements $1/m_{ij}$?

Personally I interpret $(M_{ij})^{-1}$ in the usual sense of an inverse matrix, where it would have the property $\textbf M \textbf M^{-1} = I$, but perhaps there are other interpretations that I don't know about. Thanks!

2. Apr 17, 2014

disregardthat

The only interpretation I can think of is when you define matrix multiplication component-wise. In this case, the invertible matrices would be the ones with non-zero values, and the matrix you describe would be the inverse. Note that the identity matrix would be the one with 1's as values. I don't think this is commonly used.

3. Apr 17, 2014

Fredrik

Staff Emeritus
I'm not a fan of the notation $(M_{ij})^{-1}$, mainly because $M_{ij}$ should refer to the component on row i, column j, not the matrix itself. I'm also not a fan of using a lowercase m for the components, because that prevents us from writing the definition of matrix multiplication in what I consider the obviously best way: $(AB)_{ij}=\sum_k A_{ik}B_{kj}$. I find it very puzzling that some authors go out of their way to avoid this notation, by writing things like "if $C=AB$, then $c_{ij}=\sum_k a_{ik}b_{kj}$".

If M is a diagonal matrix, for example
\begin{pmatrix}2 & 0 & 0\\ 0 & 3 & 0\\ 0 & 0 & 4\end{pmatrix} then its inverse is simply
\begin{pmatrix}\frac 1 2 & 0 & 0\\ 0 & \frac 1 3 & 0\\ 0 & 0 & \frac 1 4\end{pmatrix} But even if M is diagonal, and we use horrible notation, we still don't quite have $(M_{ij})^{-1}=1/m_{ij}$ because the off-diagonal elements of $M^{-1}$ aren't 1/0.

Last edited: Apr 17, 2014
4. Apr 18, 2014

SamanthaYellow

This is helpful. The matrix in question isn't diagonal, and that's a good point about 1/0. Hopefully I can convince this other person to change their notation!


text:
Properties

 Label 8619h Number of curves $1$ Conductor $8619$ CM no Rank $0$

Related objects

Show commands for: SageMath
sage: E = EllipticCurve("h1")

sage: E.isogeny_class()

Elliptic curves in class 8619h

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8619.a1 8619h1 $$[0, -1, 1, -9520, -632838]$$ $$-692224/867$$ $$-119523312433083$$ $$[]$$ $$68016$$ $$1.3946$$ $$\Gamma_0(N)$$-optimal

Rank

sage: E.rank()

The elliptic curve 8619h1 has rank $$0$$.

Complex multiplication

The elliptic curves in class 8619h do not have complex multiplication.

Modular form8619.2.a.h

sage: E.q_eigenform(10)

$$q - 2q^{2} - q^{3} + 2q^{4} - 4q^{5} + 2q^{6} + 3q^{7} + q^{9} + 8q^{10} - 4q^{11} - 2q^{12} - 6q^{14} + 4q^{15} - 4q^{16} + q^{17} - 2q^{18} + 4q^{19} + O(q^{20})$$


text:
Inverse Trigonometric and Hyperbolic Functions in the Natural Units Environment - Maple Programming Help

Home : Support : Online Help : Science and Engineering : Units : Environments : Natural : Units/Natural/arctrig

Inverse Trigonometric and Hyperbolic Functions in the Natural Units Environment

Description

 • In the Natural Units environment, the arguments for the inverse trigonometric or hyperbolic functions must have dimension 1. Otherwise, an error is returned.
 • For other properties, see the global functions in the arctrig help page.

Examples

 > $\mathrm{with}\left(\mathrm{Units}\left[\mathrm{Natural}\right]\right):$
 > $\mathrm{arcsin}\left(0.323\right)$
 ${0.3288976902}$ (1)
 > $\mathrm{arcsin}\left(\frac{1.3m}{\mathrm{km}}\right)$
 ${0.001300000366}$ (2)
 > $\mathrm{arccos}\left(\mathrm{cos}\left(172.\mathrm{deg}+40.\mathrm{arcmin}+10.32\mathrm{arcsec}\right)\right)$
 ${3.013651875}$ (3)


text:
DCP-LETT-4053

# To Asa Gray   20 March [1863]1

## March 20th

My dear Gray.

I have very little to say, but will amuse myself by scribbling a few lines to thank you for information in last note of Feb. 9th 2 & to thank you in my dear little man’s name for two precious stamps.3 He told me with joyful triumph that in American stamps he equalled all other collections put together in the school. He exchanged a duplicate Blood’s stamp for a whole lot of treasures.4 He says there is an envelope of same value as the stamps you put on your letter, which would be of value to him.—

I have one request to make for myself, viz seed of Campanula perfoliata: I have tried in vain at Kew & elsewhere for some.—5

I am very glad you like Bates’ paper;6 I expect his Amazonian Travels will be good.7 If you read Lyell’s book, tell me what you think of it;8 I (& Hooker) have told him that we regret much that he did not speak more boldly out about Species.9 He answers that his belief in change fluctuates.—10 His Book has made me reread your essay; & I admire it as much as ever.11 What a dead hand you are in parrying a lounge & transfixing your adversary! You ask about Sprengels “Dichogamy”:12 he means by this a plant in which each flower first matures & sheds its pollen & then has its stigma mature; & much more rarely matures its stigma first & subsequently its pollen: so that these plants are in function monoœcious. I am sure his observations are to large extent correct, & the case is very common.13 In the Primula-like cases the plants are in function Diœcious.—14

A couple of days ago I had an interesting letter from Dr. Cruger of Bot. Gardens of Trinidad,15 & he tells me odd facts of native species (& only native species) of Cattleya &c which never open their flowers, & yet set seed-capsules. Happy man he has actually seen crowds of Bees flying round Catasetum with the pollinia sticking to their backs! I wrote to him to ask him to observe what insects did in flowers of Melastomaceæ;16 he says not proper season yet, but that on one species a small Bee seemed busy about the horn-like appendages to the anthers. It will be too good luck if my study of the flowers in the green-house has led me to right interpretation of these queer appendages.17 By the way, I have just built a hot-house & got some orchids, & it amuses me much.—18 Some plants of Amsinckia spectabilis, at least the seed was so named (small dark orange flowers, elongated hairy leaves) have just begun to flower, & I find in two plants that stigma stands on exact level with anthers; hence I fear they cannot be dimorphic.—19

Your Mitchellas look healthy: I hope they will not flower very soon;20 for my health (& that of my youngest Boy) has been of late so bad, that we have resolved all to go about middle of April for 6 or 8 weeks to Malvern for Water-Cure for me & change for my Boy.—21 It breaks my heart: I shall never get my present Book on Variation under Domestication finished; yet it interests me much & I am now in middle of long chapter on Inheritance Reversion &c, giving results of my own & other Breeders’ Experiments.—22

Good Night.— | My dear Gray | Yours most truly | Ch. Darwin

Many thanks for Pamphlet Chapters on History of war & newspaper just arrived.23

## Footnotes

1
The year is established by the relationship between this letter and the letter from Asa Gray, 13 April 1863.
2
Gray’s letter has not been found; however, a portion of the letter is quoted in CD’s letter to H. W. Bates, 4 March [1863] (see n. 6, below).
3
CD refers to his twelve-year-old son, Leonard Darwin; at CD’s request, Gray had been sending United States postage stamps for Leonard’s collection since the summer of 1862 (see Correspondence vol. 10).
4
The reference is to a stamp issued by the Philadelphia carriers, D. O. Blood & Co. (see Sutton 1966, p. 41). See also letter to Asa Gray, 2 January [1863].
5
Campanula perfoliata (also called Specularia perfoliata) is native to the eastern United States (Bailey and Bailey 1976). CD had been interested in the plant since learning from Gray that it bore unopened flowers in which self-pollination occurred, a phenomenon later known as cleistogamy (see, for example, Correspondence vol. 9, letter from Asa Gray, 11 October 1861, and Correspondence vol. 10, letters to Asa Gray, 10–20 June [1862] and 26[–7] November [1862], and letters from Asa Gray, 18–19 August 1862 and 29 December 1862).
6
Gray evidently discussed Bates 1861 in a letter to CD written on 9 February 1863 that has not been found; CD quoted Gray’s comments in a letter to Henry Walter Bates of 4 March [1863]. Bates sent a copy of his paper to Gray in January, after CD had persuaded Gray to attempt to have it reviewed in the American Journal of Science and Arts, of which he was one of the contributing editors (see letter to H. W. Bates, 12 January [1863], and letter from H. W. Bates, 17 January [1863]).
7
Bates’s account of his eleven years as a naturalist in the Amazon region of South America (Bates 1863) was published between 1 and 14 April 1863 (Publishers’ Circular 26 (1863): 193).
8
C. Lyell 1863a; see letter from Asa Gray, 20 April 1863.
9
See letters to Charles Lyell, 6 March [1863] and 17 March [1863], and letter from J. D. Hooker, [15 March 1863].
10
See letter from Charles Lyell, 11 March 1863. See also letter from Charles Lyell, 15 March 1863.
11
A. Gray 1861a.
12
See Correspondence vol. 10, letter from Asa Gray, 29 December 1862. The reference is to Sprengel 1793, in which the term ‘dichogamie’ was first used (Baillon et al. 1876–92).
13
CD cited Sprengel 1793 extensively in Orchids, adding (p. 340 n.): I am aware that this author’s curious work … has often been spoken lightly of. No doubt he was an enthusiast, and probably carried some of his ideas to an extreme length. I feel sure, from my own observations, that his work contains a large body of truth. Although little regarded by botanists, the work was notable for its statement of two related doctrines, namely, that flowers were generally adapted to be cross-pollinated, and that floral structures were often adapted for insect visitation (DSB). These doctrines related closely to CD’s own belief that it was a ‘law of nature … that no organic being self-fertilises itself for an eternity of generations; but that a cross with another individual is occasionally … indispensable’ (Origin, p. 97).
14
In his letter to Gray of 26[–7] November [1862] (Correspondence vol. 10), CD objected to Gray’s use of the term ‘Diœcio-dimorphism’ to describe cases of flower dimorphism like those occurring in several species of Primula, on the grounds that he doubted the implied evolutionary transition between such dimorphism and dioeciousness. In his reply of 29 December 1862 (Correspondence vol. 10), Gray drew on CD’s account of the functional similarity of dimorphic and dichogamous plants, suggesting that CD extend the meaning of the word ‘dichogamy’ to include such cases of dimorphism as that in Primula.
15
Letter from Hermann Crüger, 23 February 1863.
16
See letter to Hermann Crüger, 25 January [1863] and n. 2.
17
See letter to Hermann Crüger, 25 January [1863] and n. 6.
18
CD’s new hothouse was completed by 15 February 1863 (see letter to J. D. Hooker, 15 February [1863] and n. 5); Joseph Dalton Hooker had sent CD orchids and other plants for cultivation from the Royal Botanic Gardens, Kew (see letter to J. D. Hooker, [21 February 1863]). See also Appendix VI.
19
Gray had suggested that Amsinckia spectabilis might be dimorphic, a view CD endorsed, having seen dried specimens sent by Gray (see Correspondence vol. 9, letter from Asa Gray, 9 November 1861, and letter to J. D. Hooker, 18 [December 1861]). CD subsequently grew plants from seed in order to experiment on the species (see Correspondence vol. 10, letter to Asa Gray, 15 March [1862], and this volume, letter to Asa Gray, 2 January [1863]). He recorded his observations on these plants, and on one belonging to his neighbour, George Henry Turnbull, in notes dated 23 March and 1 April 1863 (DAR 110: B2), concluding that the plant was not dimorphic, but that the length of the stigma was very variable, and the first-formed flowers tended to have stamens somewhat arrested in development. See also Forms of flowers, pp. 110–11.
20
Gray sent CD live specimens of the dimorphic plant Mitchella repens in December 1862, for use in crossing experiments (see Correspondence vol. 10, letter from Asa Gray, 9 December 1862, and this volume, letter to Asa Gray, 2 January [1863]). CD deferred his experiments until 1864, because the plants did not flower abundantly in 1863 (see letters to Asa Gray, 31 May [1863] and 26 June [1863], and Forms of flowers, p. 125).
21
CD had been suffering ill health since the end of February (see, for example, letter to J. D. Hooker, 5 March [1863]). Horace Darwin had been ill for much of 1862 (see letter from G. V. Reed, 12 January 1863 and n. 2); on 17 March 1863, Emma Darwin recorded in her diary (DAR 242) that he was again unwell. CD refers to James Manby Gully’s hydropathic establishment in Great Malvern, Worcestershire.
22
According to his ‘Journal’ (see Correspondence vol. 11, Appendix II), CD wrote a draft of the section of Variation dealing with inheritance (Variation 2: 1–84) between 23 January and 1 April 1863, noting: ‘took me 6$\frac{1}{2}$ weeks   time lost by illness & London’. Variation, which CD began writing early in 1860, was not published until 1868.
23
The pamphlet has not been identified. Gray periodically sent CD American newspapers, although CD told Hooker that he never read them (see Correspondence vol. 10, letter to J. D. Hooker, 29 [December 1862]).

## Summary

Discusses the meaning of C. K. Sprengel’s term "dichogamy". Dichogamous plants are functionally monoecious; Primula is functionally dioecious.

Reports Hermann Crüger’s observations of Cattleya and of bees pollinating Catasetum. Crüger will observe Melastomataceae.

Has built a hothouse.

Fears Amsinckia cannot be dimorphic.

Ill health slows his work on Variation.

## Letter details

Letter no.
DCP-LETT-4053
From
Darwin, C. R.
To
Gray, Asa
Sent from
Down
Source of text
Gray Herbarium of Harvard University (58)
Physical description
8pp

## Subjects

### Scientific Terms

Darwin Correspondence Project, “Letter no. 4053,” accessed on 25 September 2016, http://www.darwinproject.ac.uk/DCP-LETT-4053

letter


text:
## 5.3.1 Directory tree for transfer functions

The directory tree of a transfer function file is as follows. Note that a name ends with / means a directory.

transferFunction/ --- whatisthis.txt
|- microphone.xml
|- source.xml
|- localization/ --- tf00000.mat
|                 |- tf00001.mat ...
|
|- separation/   --- tf00000.mat
|- tf00001.mat ...

The root directory is transferFunction. whatisthis.txt represents what the file is. In this case, the content of the file is transfer function. microphone.xml represents positions of the microphone array used for the file. source.xml represents positions of transfer function measurement. The directories localization/ and separation contains transfer functions for localization and separation for each position. The file name format is tf%05d.mat, where the format string "with the ID of the position written in source.xml.

Note that localization/ and/or separation/ can be empty. For example, if localization/ is empty and the other is not, this is equivalent to a transfer function file for localization (M2PG). If both directories are not empty, this is equivalent to the combination of the transfer function files for localization and separation.


text:
# A circular body motion around its axis

1. Apr 26, 2009

### Mahmoud Ebied

Hello physicists
I have a circular body goes around its axis very fast.
I am seaching for answer for these questions
1- How can I calculate its momentum or the energy that it has?
2- What is the forces that limit its motion?

thanks a lot

2. Apr 26, 2009

### tiny-tim

Welcome to PF!

Hi Mahmoud! Welcome to PF!

You need to know the moment of inertia (I) of the body.

Then the angular momentum is Iω, and the energy is 1/2 Iω2, where ω is the angular velocity.

And the torque (moment of force) needed to produce an angular acceleration of α is Iα.


text:
# Quippian

In mathematics, a quippian is a degree 5 class 3 contravariant of a plane cubic introduced by Arthur Cayley (1857) and discussed by Igor Dolgachev (2012, p.157). In the same paper Cayley also introduced another similar invariant that he called the pippian, now called the Cayleyan.


text:
# Homework Help: Two masses and two pulleys problem

Tags:
1. Sep 9, 2016

### Mr Davis 97

1. The problem statement, all variables and given/known data
Masses M1 and M2 are connected to a system of strings and pulleys as shown. The strings are massless and inextensible, and the pulleys are massless and frictionless. Find the acceleration of M1.

2. Relevant equations
Newton's 2nd Law of motion

3. The attempt at a solution

So here is my line of reasoning. We have four objects with which we can use Newton's second law to derive relationships which will get us an explicit expression of the acceleration of block 1.

Newton's law for block 1:
$T_1 - m_1 g = m_1 a_1$

Newton's law for block 2:
$T_2 - m_2 g = m_2 a_2$

Newton's law for pulley with block 2:
$T_1 - 2T_2 = 0$
$T_1 = 2T_2$

Next, if we combine the equation for block 1 with the equation for block 2, and if we assume $a_1 = a_2$, then we get,

$\displaystyle a_1 = \frac{g(2m_2 - m_1)}{m_1 - 2m_2}$

However, this is not the correct expression, because we must have that if M1 = M2 then $\displaystyle a_1 = \frac{g}{5}$

Where am I going wrong with my reasoning?

2. Sep 10, 2016

### ehild

a1 is not equal to a2. Think: the moving pulley sinks by dx, how much does m2 move?

3. Sep 10, 2016

### Mr Davis 97

But how does the second pulley move if it is massless?

4. Sep 10, 2016

### ehild

How does m2 move if the pulley does not move? The other end of the string is fixed to the ground.

5. Sep 10, 2016

### Mr Davis 97

Okay... But my equation $T_1 - 2T_2 = 0$ is correct since the mass of that pulley is zero?

6. Sep 10, 2016

### ehild

Yes.

7. Sep 10, 2016

### Mr Davis 97

Say for example that M1 is super heavy, and goes down. How does the second pulley accelerate upwards if it is massless?

8. Sep 10, 2016

### ehild

Why not? It pulls m2 upward.

9. Sep 10, 2016

### Mr Davis 97

But it's massless, so the net force on it is zero, which means there would be no acceleration it seems

10. Sep 10, 2016

### ehild

A massless object can accelerate without any force. F=ma. What can be a if both F and m are zero?

11. Sep 10, 2016

### Mr Davis 97

The tension on string 1 is twice the tension on string 2, so would it be fair to suppose that the acceleration of mass 2 would be twice that of mass 1?

12. Sep 10, 2016

### ehild

You get the relation between the accelerations from the constraints that the strings do not change their lengths.

13. Sep 10, 2016

### ehild

14. Sep 10, 2016

### Mr Davis 97

I'm just not seeing how to get the relation out of that... I know that I can say that the length of the string = so and so, and then differentiate twice... But I can't figure out the so and so.

15. Sep 10, 2016

### ehild

See the figure. If the pulley sinks by dx, the left piece of string becomes shorter by dx and the right one must become longer by dx. So the mass m2 moves downward by dx with respect to the pulley. But the pulley has moved downward by dx with respect to the ground, so the mass had to move by 2 dx downward with respect to the ground.

#### Attached Files:

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16. Sep 10, 2016

### Mr Davis 97

How would you derive this from the fact that the length of the string is constant? If we say that $l = x_L + x_R + \pi r$, where r is the radius of the pulley, and the two x's are the left and right sections of the string, then if we differentiate twice we just get that $\ddot{x}_L = - \ddot{x}_R$, which doesn't show that the rate at which the mass moves is twice the rate at which the pulley moves.

17. Sep 10, 2016

### ehild

The acceleration is with respect to the ground. You need to differentiate h twice, instead of xr.

18. Sep 10, 2016

### Mr Davis 97

So is it something like $l = h + \pi r + (h - d_g)$ where $d_g$ is the distance of the block from the ground. So if we differentiate twice we get that $\ddot{d_g} = 2\ddot{h}$, which means that the rate at which the block accelerates is twice the rate at which the pulley accelerates...?

19. Sep 10, 2016

### ehild

Yes, and twice the acceleration of the other mass, only opposite direction.

20. Sep 10, 2016

### Mr Davis 97

Do you get this from looking at the constant length of the other string for the other pulley?

21. Sep 10, 2016

### ehild

Yes. Because of the length of the other string is also constant, and the left pulley is fixed, the accelerations of the hanging pulley and mass m1 are opposite and of equal magnitude.

22. Sep 10, 2016

### Mr Davis 97

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Last edited: Sep 10, 2016
23. Sep 10, 2016

### Mr Davis 97

I have one more question about the nature of massless pulleys. If they are massless, wouldn't their acceleration be infinite according to the equation $a = \frac{F}{m}$? How can we justify letting the masses of pulleys and ropes be zero?

24. Sep 10, 2016

### ehild

That is why the net force on a massless pulley has to be zero. Otherwise the acceleration would be infinite. But we know that the pulley has finite acceleration. Therefore the net force is zero.

25. Sep 10, 2016

### Mr Davis 97

But if the pulley has a finite acceleration, how would the net force be zero?

Last edited: Sep 10, 2016


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# Manpages - printf.1p

## PROLOG

This manual page is part of the POSIX Programmer’s Manual. The Linux implementation of this interface may differ (consult the corresponding Linux manual page for details of Linux behavior), or the interface may not be implemented on Linux.

## NAME

printf — write formatted output

## SYNOPSIS

  printf format [argument...]

## DESCRIPTION

The printf utility shall write formatted operands to the standard output. The argument operands shall be formatted under control of the format operand.

None.

## OPERANDS

The following operands shall be supported:

format
A string describing the format to use to write the remaining operands. See the EXTENDED DESCRIPTION section.
argument
The strings to be written to standard output, under the control of format. See the EXTENDED DESCRIPTION section.

Not used.

None.

## ENVIRONMENT VARIABLES

The following environment variables shall affect the execution of printf:

LANG
Provide a default value for the internationalization variables that are unset or null. (See the Base Definitions volume of POSIX.1‐2017, Section 8.2, Internationalization Variables the precedence of internationalization variables used to determine the values of locale categories.)
LC_ALL
If set to a non-empty string value, override the values of all the other internationalization variables.
LC_CTYPE
Determine the locale for the interpretation of sequences of bytes of text data as characters (for example, single-byte as opposed to multi-byte characters in arguments).
LC_MESSAGES

Determine the locale that should be used to affect the format and contents of diagnostic messages written to standard error.
LC_NUMERIC

Determine the locale for numeric formatting. It shall affect the format of numbers written using the e, E, f, g, and G conversion specifier characters (if supported).
NLSPATH
Determine the location of message catalogs for the processing of LC_MESSAGES.

Default.

## STDOUT

See the EXTENDED DESCRIPTION section.

## STDERR

The standard error shall be used only for diagnostic messages.

None.

## EXTENDED DESCRIPTION

The format operand shall be used as the format string described in the Base Definitions volume of POSIX.1‐2017, Chapter 5, File Format Notation with the following exceptions:

1.
A <space> in the format string, in any context other than a flag of a conversion specification, shall be treated as an ordinary character that is copied to the output.
2.
A ’ character in the format string shall be treated as a ’ character, not as a <space>.
3.
In addition to the escape sequences shown in the Base Definitions volume of POSIX.1‐2017, Chapter 5, File Format Notation ( ’\\’, ’\a’, ’\b’, ’\f’, ’\n’, ’\r’, ’\t’, ’\v’), “\ddd”, where ddd is a one, two, or three-digit octal number, shall be written as a byte with the numeric value specified by the octal number.
4.
The implementation shall not precede or follow output from the d or u conversion specifiers with <blank> characters not specified by the format operand.
5.
The implementation shall not precede output from the o conversion specifier with zeros not specified by the format operand.
6.
The a, A, e, E, f, F, g, and G conversion specifiers need not be supported.
7.

An additional conversion specifier character, b, shall be supported as follows. The argument shall be taken to be a string that can contain <backslash>-escape sequences. The following <backslash>-escape sequences shall be supported:

--
The escape sequences listed in the Base Definitions volume of POSIX.1‐2017, Chapter 5, File Format Notation ( ’\\’, ’\a’, ’\b’, ’\f’, ’\n’, ’\r’, ’\t’, ’\v’), which shall be converted to the characters they represent
--
“\0ddd”, where ddd is a zero, one, two, or three-digit octal number that shall be converted to a byte with the numeric value specified by the octal number
--
’\c’, which shall not be written and shall cause printf to ignore any remaining characters in the string operand containing it, any remaining string operands, and any additional characters in the format operand

The interpretation of a <backslash> followed by any other sequence of characters is unspecified.

Bytes from the converted string shall be written until the end of the string or the number of bytes indicated by the precision specification is reached. If the precision is omitted, it shall be taken to be infinite, so all bytes up to the end of the converted string shall be written.

8.
For each conversion specification that consumes an argument, the next argument operand shall be evaluated and converted to the appropriate type for the conversion as specified below.
9.
The format operand shall be reused as often as necessary to satisfy the argument operands. Any extra b, c, or s conversion specifiers shall be evaluated as if a null string argument were supplied; other extra conversion specifications shall be evaluated as if a zero argument were supplied. If the format operand contains no conversion specifications and argument operands are present, the results are unspecified.
(no term)
If a character sequence in the format operand begins with a ’%’ character, but does not form a valid conversion specification, the behavior is unspecified.
(no term)
The argument to the c conversion specifier can be a string containing zero or more bytes. If it contains one or more bytes, the first byte shall be written and any additional bytes shall be ignored. If the argument is an empty string, it is unspecified whether nothing is written or a null byte is written.

The argument operands shall be treated as strings if the corresponding conversion specifier is b, c, or s, and shall be evaluated as if by the /strtod/() function if the corresponding conversion specifier is a, A, e, E, f, F, g, or G. Otherwise, they shall be evaluated as unsuffixed C integer constants, as described by the ISO C standard, with the following extensions:

*
A leading <plus-sign> or <hyphen-minus> shall be allowed.
*
If the leading character is a single-quote or double-quote, the value shall be the numeric value in the underlying codeset of the character following the single-quote or double-quote.
*
Suffixed integer constants may be allowed.

If an argument operand cannot be completely converted into an internal value appropriate to the corresponding conversion specification, a diagnostic message shall be written to standard error and the utility shall not exit with a zero exit status, but shall continue processing any remaining operands and shall write the value accumulated at the time the error was detected to standard output.

It shall not be considered an error if an argument operand is not completely used for a b, c, or s conversion.

## EXIT STATUS

The following exit values shall be returned:

0
Successful completion.
>0
An error occurred.

## CONSEQUENCES OF ERRORS

Default.

The following sections are informative.

## APPLICATION USAGE

The floating-point formatting conversion specifications of /printf/() are not required because all arithmetic in the shell is integer arithmetic. The awk utility performs floating-point calculations and provides its own printf function. The bc utility can perform arbitrary-precision floating-point arithmetic, but does not provide extensive formatting capabilities. (This printf utility cannot really be used to format bc output; it does not support arbitrary precision.) Implementations are encouraged to support the floating-point conversions as an extension.

Note that this printf utility, like the /printf/() function defined in the System Interfaces volume of POSIX.1‐2017 on which it is based, makes no special provision for dealing with multi-byte characters when using the %c conversion specification or when a precision is specified in a %b or %s conversion specification. Applications should be extremely cautious using either of these features when there are multi-byte characters in the character set.

No provision is made in this volume of POSIX.1‐2017 which allows field widths and precisions to be specified as ’* since the ’* can be replaced directly in the format operand using shell variable substitution. Implementations can also provide this feature as an extension if they so choose.

Hexadecimal character constants as defined in the ISO C standard are not recognized in the format operand because there is no consistent way to detect the end of the constant. Octal character constants are limited to, at most, three octal digits, but hexadecimal character constants are only terminated by a non-hex-digit character. In the ISO C standard, the “##” concatenation operator can be used to terminate a constant and follow it with a hexadecimal character to be written. In the shell, concatenation occurs before the printf utility has a chance to parse the end of the hexadecimal constant.

The %b conversion specification is not part of the ISO C standard; it has been added here as a portable way to process <backslash>-escapes expanded in string operands as provided by the echo utility. See also the APPLICATION USAGE section of /echo / for ways to use printf as a replacement for all of the traditional versions of the echo utility.

If an argument cannot be parsed correctly for the corresponding conversion specification, the printf utility is required to report an error. Thus, overflow and extraneous characters at the end of an argument being used for a numeric conversion shall be reported as errors.

## EXAMPLES

To alert the user and then print and read a series of prompts:

printf "\aPlease fill in the following: \nName: "
printf "Phone number: "

To read out a list of right and wrong answers from a file, calculate the percentage correctly, and print them out. The numbers are right-justified and separated by a single <tab>. The percentage is written to one decimal place of accuracy:

while read right wrong ; do
percent=$(echo "scale=1;($right*100)/($right+$wrong)" | bc)
printf "%2d right\t%2d wrong\t(%s%%)\n" \
$right$wrong \$percent
done < database_file

The command:

printf "%5d%4d\n" 1 21 321 4321 54321

produces:

1  21
3214321
54321   0

Note that the format operand is used three times to print all of the given strings and that a ’0’ was supplied by printf to satisfy the last %4d conversion specification.

The printf utility is required to notify the user when conversion errors are detected while producing numeric output; thus, the following results would be expected on an implementation with 32-bit twos-complement integers when %d is specified as the format operand:

TABLE

The diagnostic message format is not specified, but these examples convey the type of information that should be reported. Note that the value shown on standard output is what would be expected as the return value from the /strtol/() function as defined in the System Interfaces volume of POSIX.1‐2017. A similar correspondence exists between %u and /strtoul/() and %e, %f, and %g (if the implementation supports floating-point conversions) and /strtod/().

In a locale using the ISO/IEC 646: 1991 standard as the underlying codeset, the command:

printf "%d\n" 3 +3 -3 \'3 \"+3 "'-3"

produces:

1. Numeric value of constant 3
2. Numeric value of constant 3
3. Numeric value of constant -3
4. Numeric value of the character ’3’ in the ISO/IEC 646: 1991 standard codeset
5. Numeric value of the character ’+’ in the ISO/IEC 646: 1991 standard codeset
6. Numeric value of the character ’-’ in the ISO/IEC 646: 1991 standard codeset

Note that in a locale with multi-byte characters, the value of a character is intended to be the value of the equivalent of the wchar_t representation of the character as described in the System Interfaces volume of POSIX.1‐2017.

## RATIONALE

The printf utility was added to provide functionality that has historically been provided by echo. However, due to irreconcilable differences in the various versions of echo extant, the version has few special features, leaving those to this new printf utility, which is based on one in the Ninth Edition system.

The EXTENDED DESCRIPTION section almost exactly matches the /printf/() function in the ISO C standard, although it is described in terms of the file format notation in the Base Definitions volume of POSIX.1‐2017, Chapter 5, File Format Notation.

Earlier versions of this standard specified that arguments for all conversions other than b, c, and s were evaluated in the same way (as C constants, but with stated exceptions). For implementations supporting the floating-point conversions it was not clear whether integer conversions need only accept integer constants and floating-point conversions need only accept floating-point constants, or whether both types of conversions should accept both types of constants. Also by not distinguishing between them, the requirement relating to a leading single-quote or double-quote applied to floating-point conversions even though this provided no useful functionality to applications that was not already available through the integer conversions. The current standard clarifies the situation by specifying that the arguments for floating-point conversions are evaluated as if by /strtod/(), and the arguments for integer conversions are evaluated as C integer constants, with the special treatment of leading single-quote and double-quote applying only to integer conversions.

## FUTURE DIRECTIONS

None.

/awk, //bc, //echo /

The Base Definitions volume of POSIX.1‐2017, Chapter 5, File Format Notation, Chapter 8, Environment Variables

The System Interfaces volume of POSIX.1‐2017, /fprintf ( )/, /strtod ( )/


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# Section of Hirzebruch Surface Semi Ample

We consider the Hirzebruch surface $$S = \mathbb{P}(\mathcal{E})$$ with locally free sheaf $$\mathcal{E} = \mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(2)$$. By definition of projectivation every section $$i:C \subset S$$ coresponds bijectively to a surjective bundle morphism $$\mathcal{E} \to \mathcal{L}$$ by taking into mind that $$\mathbb{P}(\mathcal{E})$$ represents a functor. Under this condition we have for this section $$C= \mathbb{P}(\mathcal{L})$$.

Futhermore it is known that $$C^2 = deg(\mathcal{L}) - deg(K_{\mathcal{L}})$$ where $$K_{\mathcal{L}} = Ker(\mathcal{E} \to \mathcal{L})$$.

Now back to our situation:

The considerations above provides us two obviously sections:

$$C:= \mathbb{P}(\mathcal{O}_{\mathbb{P}^1}(2))$$ coming from the sequence

$$0 \to \mathcal{O}_{\mathbb{P}^1} \to \mathcal{E} \to \mathcal{O}_{\mathbb{P}^1}(2) \to 0$$

and $$D:= \mathbb{P}(\mathcal{O}_{\mathbb{P}^1})$$ coming from the sequence

$$0 \to \mathcal{O}_{\mathbb{P}^1}(2) \to \mathcal{E} \to \mathcal{O}_{\mathbb{P}^1} \to 0$$

One can show that $$(C \cdot D)=0$$ and $$C^2 = 2$$.

My question is why and how to show that $$C$$ is a semi ample divisor?

Two remarks:

1. by semi ample divisor I mean that there exist a semi ample invertible sheaf $$\mathcal{L}$$ (so for exery $$x \in X$$ there exist $$n \in \mathbb{N}$$ and a global section $$s \in \Gamma(X, \mathcal{L}^{\otimes n}$$) with property $$s(x) \neq 0$$)

2. Here: Divisor with positive Selfintersection Number Semi Ample I asked about a more general case: If we consider a general surface with a nef divisor $$N$$ with $$N^2 >0$$ then - by considering @Asal Beag Dubh's answer - $$D$$ is in general not semi ample.

So my question here is if concretely in the setting above with a Hirzebruch surface the section $$C$$ is semi ample? If yes, why?

• Are you sure you mean $\mathcal{O}_{\mathbb{P}^2}$ etc. and not $\mathcal{O}_{\mathbb{P}^1}$? In the previous case you get a 3-fold and not a surface. – Mohan Oct 4 '18 at 12:05
• @Mohan: Yes, you are right. Thanks. – KarlPeter Oct 4 '18 at 12:18

Yes, in this more specific case $$C$$ is semi-ample, in fact globally generated. To show it:
1. Note that it suffices to show that $$O_S(C)$$ has no base points along $$C$$.
2. Use the twisted ideal sheaf seqeuence \begin{align*} 0 \rightarrow O_S \rightarrow O_S(C) \rightarrow O_C(C) \rightarrow 0 \end{align*} to show that every global section of $$O_C(C)$$ comes from a global section of $$O_S(C)$$.
3. Note that $$C$$ is isomorphic to $$\mathbf P^1$$, and hence $$O_C(C)$$ is just $$O(2)$$, which is clearly globally generated.


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16 views
How long does it take a packet of length 1,000 bytes to propagate over a link of distance 2,500 km, propagation speed 2.5 · 108 m/s, and transmission rate 2 Mbps? More generally, how long does it take a packet of length L to propagate over a link of distance d, propagation speed s, and transmission rate R bps? Does this delay depend on packet length? Does this delay depend on the transmission rate?
| 16 views

$t_{\text{prop}} = \frac{d}{s} = \frac{2500\times {10}^3}{(2.5\times {10}^8)} = 10.00 \text{msec}$
$\text{NO}$
$\text{NO}$


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Category Archives: ring-theory

## Abelian categories: examples and nonexamples

I've been talking a little about abelian categories these days. That's because I've been going over Weibel's An Introduction to Homological Algebra. It's a book I read before, and I still feel pretty confident about the material. This time, though, I think I'm going to explore a few different paths that I haven't really given […]

## Working with group rings in Sage

Let $\Z[\Z/n]$ denote the integral group ring of the cyclic group $\Z/n$. How would you create $\Z[\Z/n]$ in Sage so that you could easily multiply elements? First, if you've already assigned a group to the variable 'A', then

will give you the corresponding group ring and store it in the variable 'R'. The first […]

## Strong Nilpotence and the Jacobson Radical

In the previous post we saw the following definition for a ring $R$: An element $r\in R$ is called strongly nilpotent if every sequence $r = r_0,r_1,r_2,\dots$ such that $r_{n+1}\in r_nRr_n$ is eventually zero. Why introduce this notion? Well, did you know that every finite integral domain is a field? If $R$ is an integral […]

## Nilpotent and Strongly Nilpotent

Let $R$ be an associative ring. An element $r\in R$ is called nilpotent if $r^n = 0$ for some $n$. There is a stronger notion: an element $r\in R$ is called strongly nilpotent if every sequence $r = r_0,r_1,r_2,\dots$ such that $r_{n+1}\in r_nRr_n$ is eventually zero. How are these two related? It is always the […]


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# Rspamd 1.1 has been released

The next major release of rspamd: 1.1.0 is now released. In this version, I did another bunch of architectural rework. This time, I’ve refactored fuzzy storage, regular expressions processing and statistics primarily.

A number of features that I was asked constantly about have been added to rspamd, including such features as:

• Autolearning for BAYES: statistics can learn on good or bad messages automatically
• Redis backend for statistics to enable distributed and fast redis storage for rspamd cluster
• Scalable fuzzy storage: it is now possible to scale hashes storage across multiple processes to process thousands of requests per second

There is also major performance improvement: hyperscan engine support for optimizing regular expressiosn execution.

With this version, I have added a lot of documentation, including tutorials and improved quick start guide.

Here is the full changelog for this version available on github: https://github.com/vstakhov/rspamd/blob/1.1.0/ChangeLog

The new version is almost 100% backward compatible with 1.0 branch but please check the migration document if you are using per-user statistics and rspamd-1.0.

Here are some graph of rspamd performance on scanning:

So you can see that rspamd can scan as much as 200 messages per second consuming less that 50% cpu of a typical scanner machine (Xeon E5405 single CPU).

Rmilter is also upgraded to the version 1.7.0 that brings full IPv6 support, redis cache support and major cleanup of unused and broken stuff. Rmilter changelog is available here: https://github.com/vstakhov/rmilter/blob/1.7.0/ChangeLog

# Rspamd 1.0.11 has been released

The next bugfixes only release 1.0.11 of rspamd is out. This release contains the following changes:

• Fix spf redirects
• Fix domains when parsing mx/ptr/a records in includes/redirects
• Fix unfolded base64 encoding
• Fix GError use-after-free
• Do not rewrite the original url when using redirector
• Fix parsing of fragment in urls
• Fix processing of HTML tags
• Improve empty image rule
• Avoid long double type
• Fix tokens weights in OSB algorithm
• Improve debugging for bayes

This version has backward compatibility with 1.0.0 preserved.

The branch 1.0 is now considered as stable and all development has now been moved to the master branch which is going to be the next 1.1 major release.

# Rspamd 1.0.10 has been released

The next bugfixes only release 1.0.10 of rspamd is out. This release contains the following changes:

• Fix settings application (#416)
• Fix another issue with fixed strings
• Fix hash function invocation
• Use the proper string for make_dns_request in lua_http
• Fix scan time output
• Update webui:
• fix labels for greylisting
• fix dimension of scan time

This version has backward compatibility with 1.0.0 preserved.

The branch 1.0 is now considered as stable and all development has now been moved to the master branch which is going to be the next 1.1 major release.


text:
## Braid Word

Any -braid is expressed as a braid word, e.g., is a braid word for the Braid Group . By Alexander's Theorem, any Link is representable by a closed braid, but there is no general procedure for reducing a braid word to its simplest form. However, Markov's Theorem gives a procedure for identifying different braid words which represent the same Link.

Let be the sum of Positive exponents, and the sum of Negative exponents in the Braid Group . If

then the closed braid is not Amphichiral (Jones 1985).

Jones, V. F. R. A Polynomial Invariant for Knots via von Neumann Algebras.'' Bull. Amer. Math. Soc. 12, 103-111, 1985.
Jones, V. F. R. Hecke Algebra Representations of Braid Groups and Link Polynomials.'' Ann. Math. 126, 335-388, 1987.


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# Visualizing & Crossmatching your Light Curve¶

We have introduced two new classes into the eleanor package! This first is eleanor.Visualize(), which provides diagnostic tools for you to use to understand your light curves. The second is eleanor.Crossmatch(), which checks if your target was observed at 2-minute cadence, if the light curve is available from the TASOC pipeline, and if the light curve is available from Oelkers & Stassun (difference imaging).

We’ll run through an example for both of these classes.

## Visualization Tutorial¶

We’ve created an eleanor.Visualization() class to help you understand what is going on with hyour light curves. If you have any additional tools you think would be useful in this class, please submit a pull request!

### 1.1 Setting up the class¶

Initializing the eleanor.visualization() class is very simple! Once you have the light curve you are interested in, created with the eleanor.TargetData() class, you pass this into the visualization class! We’ll run through an example with WASP-100.

[1]:

import eleanor
import numpy as np
import matplotlib.pyplot as plt
plt.rcParams['font.size'] = 15

WARNING: leap-second auto-update failed due to the following exception: RuntimeError('attempted to use clear_download_cache on the path //anaconda3/lib/python3.7/site-packages/astropy/utils/iers/data/Leap_Second.dat outside the data cache directory /Users/arcticfox/.astropy/cache/download/py3') [astropy.time.core]

[2]:

star = eleanor.Source(tic=38846515, sector=1, tc=True)
data = eleanor.TargetData(star)

Downloading URL https://mast.stsci.edu/tesscut/api/v0.1/astrocut?ra=68.95970833333332&dec=-64.02703611111112&y=31&x=31&units=px&sector=1 to ./tesscut_20200114091740.zip ... [Done]
Inflating...
This is the first light curve you have made for this sector. Getting eleanor metadata products for Sector  1...
This will only take a minute, and only needs to be done once. Any other light curves you make in this sector will be faster.
Target Acquired
Quality Flags Assured
Success! Sector  1 now available.

WARNING: The "exact" method is not yet implemented for rectangular apertures -- using "subpixel" method with "subpixels=32" [photutils.aperture.core]

[3]:

vis = eleanor.Visualize(data)

### 1.2 TESS the Movie¶

A great tool often used to understand what’s going on in your light curve is the FFI movies created by Ethan Kruse, hosted on YouTube. We make it very easy to view these videos. If you are calling the class from a Jupyter Notebook, the video will be embedded in the notebook. If you are calling the class from any other editor, the video will open up in a browser tab.

The sector movie for the target in the eleanor.TargetData() object will play by default.

[4]:

vis.tess_the_movie()

[4]:

We provide a very simple function that overplots the aperture you use on the target pixel file. You can also pass in your own aperture and change the color/linewidth of the plotted contour.

[5]:

vis.aperture_contour()

[5]:

Here’s an example of changing the default aperture contour color and linewidth and also passing in a new aperture to plot:

[11]:

ap = np.zeros(data.tpf[0].shape)
ap[5:8,6:9] = 1
fig = vis.aperture_contour(aperture=ap, ap_color='r', ap_linewidth=2)

### 1.4 Pixel-By-Pixel Light Curves¶

Another useful tool is what we call a pixel-by-pixel light curve. It’s exactly what it sounds like: we plot the light curve for a single pixel across the sector. The default light curve that is plotted is the “corrected flux” for each pixel. You can simply call:

[7]:

fig = vis.pixel_by_pixel()

//anaconda3/lib/python3.7/site-packages/numpy/core/fromnumeric.py:734: UserWarning: Warning: 'partition' will ignore the 'mask' of the MaskedArray.
a.partition(kth, axis=axis, kind=kind, order=order)

Or you can specify which rows/columns you want to focus in on within the target pixel file:

[8]:

fig = vis.pixel_by_pixel(colrange=[4,10], rowrange=[4,10])

Or you can plot the raw flux light curve per each pixel (yay Earthshine!):

[9]:

fig = vis.pixel_by_pixel(colrange=[4,10], rowrange=[4,10], data_type="raw")

Or you can plot a Lomb Scargle periodogram for each pixel AND you can color each light curve by the color of the associated pixel in the TPF plotted to the left:

[14]:

fig = vis.pixel_by_pixel(colrange=[4,10], rowrange=[4,10], data_type="periodogram",
color_by_pixel=True)

---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-14-7061decec7c4> in <module>
1 fig = vis.pixel_by_pixel(colrange=[4,10], rowrange=[4,10], data_type="periodogram",
----> 2                         color_by_pixel=True)

TypeError: pixel_by_pixel() got an unexpected keyword argument 'color_by_pixel'

OR OR! Oliver Hall made an awesome pull request that allows you to plot the periodogram per each pixel for all of your asteroseismic needs:

[15]:

vis.pixel_by_pixel(colrange=[4,10], rowrange=[4,10], data_type="amplitude")

[15]:

## Crossmatching¶

On to new things! It’s important to check all of the light curves that are available for your target to make sure your astrophysical signal is indeed real. We’ve made this really easy for you to do by introducting a new eleanor.Crossmatch() class. Within this class, you can check if your target has been observed at either 2-minute cadence or is created by the TASOC or Oelkers & Stassun (2019) pipelines.

The eleanor.Crossmatch() class is initialized using an eleanor.TargetData() object.

[ ]:

cross = eleanor.Crossmatch(data)

The 2-minute function returns a list of lightkurve.search.SearchResult objects if your target has been observed in this mode. Users also have the option to set download=True, which then returns a list of lightkurve.lightcurve.TessLightCurve objects. You also have the option to specify which sectors you want to search/download!

[ ]:

cross.two_minute()

[ ]:

lcs = cross.two_minute(download=True)

### TASOC pipeline¶

The TESS Asteroseismic Consortium (TASC) is providing light curves for the community to use that are tailored towards finding asteroseismic signals. Callingn eleanor.Crossmatch.tasoc_lc() searches MAST to find if these light curves are available. For more information on the TASOC light curves, see https://archive.stsci.edu/hlsp/tasoc.

[ ]:

cross.tasoc_lc()

The TASOC light curves have their own quality flags called eleanor.Crossmatch.tasoc_pixel_quality. We can apply these quality flags to the light curve and see that the WASP-100b transits are recovered really nicely!

[ ]:

plt.figure(figsize=(14,8))

q = cross.tasoc_pixel_quality == 0
plt.plot(cross.tasoc_time[q], cross.tasoc_flux_raw[q], 'k')
plt.xlabel('Time [BJD-2457000]')
plt.ylabel('Flux');

### Oelkers & Stassun pipeline¶

Additionally, the Vanderbilt team is providing light curves for stars in the FFIs via difference imaging.

[ ]:

cross.oelkers_lc()

[ ]:

plt.figure(figsize=(14,8))
plt.plot(cross.os_time, cross.os_mag, 'k')
plt.xlabel('Time [BJD-2457000]')
plt.ylabel('Magnitude');

We can now plot all light curves on the same plot and compare. All of the transits pop out really nicely! This (and the fact that WASP-100 has already been confirmed) is very reassuring that our transits are indeed real.

[ ]:

plt.figure(figsize=(14,8))

e = data.quality == 0
plt.plot(data.time[e], data.corr_flux[e]/np.nanmedian(data.corr_flux[e]),
'k', label='eleanor')

plt.plot(cross.os_time,
10**(-0.4*cross.os_mag)/np.nanmedian(10**(-0.4*cross.os_mag))-0.05,
'r', label='OS19')

plt.plot(cross.tasoc_time[q],
cross.tasoc_flux_raw[q]/np.nanmedian(cross.tasoc_time[q])-8.9,
'b', label='TASOC')

plt.legend(ncol=3)

plt.xlabel('Time [BJD-2457000]')
plt.ylabel('Flux');


text:
# Given that the sum of two numbers is 20 and their difference is 6, find the two numbers.

## Question:

Given that the sum of two numbers is 20 and their difference is 6, find the two numbers.

## Elimination Method:

(i) The elimination method is used to solve a system of two equations in two variables.

(ii) In this method, we add or subtract the given equations and eliminate one variable.

(iii) We solve the resultant equation in one variable using the algebraic operations.

## Answer and Explanation:

Let us assume the two numbers to be {eq}x {/eq} and {eq}y {/eq}.

The sum of the two numbers is 20. So we get: {eq}x+y=20 \,\,\,\rightarrow (1) {/eq}.

The difference between the two numbers is 6. So we get: {eq}x-y=6\,\,\,\rightarrow (2) {/eq}

Adding (1) and (2):

$$x+y+x-y =20 +6 \\ 2x = 26\\ \text{Dividing both sides by 2}, \\ \boxed{\mathbf{x=13}}$$

Substitute this in (1):

$$13+y=20\\ \text{Subtracting 13 from both sides},\\ \boxed{\mathbf{y=7}}$$

Therefore, the numbers are 13 and 7.


text:
# Writing a limit so that the subscript goes directly underneath

When I write $\lim_{x \rightarrow p} f(x) = q$, the subscript x \rightarrow p appears to the right of the limit, instead of directly underneath it.

When I try $\displaystyle \lim_{x \rightarrow p} f(x) = q$, the subscript of the limit takes up extra space, making the line taller than usual and creating an awkward extra space underneath everything else...

Is there a way to write limits in-line with the subscript underneath, and everything squished to the normal height of the line its in?

-

One way would be to use \scalebox from the graphicx package to rescale the version obtained by \displaystyle. Here is a comparison of the usual inline version of \lim and the scaled version:

\documentclass{article}
\usepackage{xcolor}
\usepackage{graphicx}

\newcommand{\Text}{\noindent Lorem ipsum dolor sit amet.
Lorem ipsum dolor sit amet.
Lorem ipsum dolor sit amet }

\newcommand{\Lim}[1]{\raisebox{0.5ex}{\scalebox{0.8}{$\displaystyle \lim_{#1}\;$}}}

\usepackage{graphicx}
\begin{document}
\Text
$\textcolor{blue}{\lim_{x \rightarrow p} f(x) = q}$
\Text
$\textcolor{red}{\Lim{x \rightarrow p} f(x) = q}$
\Text
\end{document}
-
Would it be easy to turn this into a macro which I can use every time I write a limit? –  jamaicanworm Dec 27 '11 at 2:10
@jamaicanworm: Updated solution to use macro version. –  Peter Grill Dec 27 '11 at 2:14
One might ask which one is worse: enlarging the interline spacing or printing an ugly symbol. I'll keep with the standard TeX way of typesetting the limit. :) –  egreg Dec 27 '11 at 11:21
$\underset{x \to \infty}{\lim}$