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# Thread: the tens digit of 16^298 ....

1. ## the tens digit of 16^298 ....

Please someone tell me how to work out the TENS digit of the answer of $16^{298}$ WITHOUT using calculator.

I believe that it is somehow related to the principles such as $(x^a)^b = x^{ab}$

the original question also included the units digit but it should always be 6.

2. $16^{298} = (2^4)^{298} = (2)^{4\times 298} =2^{1192}$

How to Find the Last Digits of a Positive Power of Two - Exploring Binary

3. Originally Posted by ukorov
Please someone tell me how to work out the TENS digit of the answer of $16^{298}$ WITHOUT using calculator.

I believe that it is somehow related to the principles such as $(x^a)^b = x^{ab}$

the original question also included the units digit but it should always be 6.

A few calculations (you should be able to multiply these easily w/o the abacus)

016 = 16^1
x16
===
_56 = 16^2 (16 times 16 = 256, but we only need the tens & ones digits)
x16
===
_96 = 16^3
x16
===
_36 = 16^4
x16
===
_76 = 16^5
x16
===
_16 = 16^6 (same as 16^1, exponent difference is 5)

$16^6 \equiv 16^1 \mod 100$

298/5 = 59.9
59*5 = 295

1+5=6
1+295=296

$16^1 \equiv 16^6 \equiv 16^{296} \mod 100$

$16^3 \equiv 16^8 \equiv 16^{298} \mod 100$

see prior calculation above for the tens & ones digits for $16^3\mod 100$ which is the same as $16^{298}\mod 100$

.

4. Originally Posted by pickslides
$16^{298} = (2^4)^{298} = (2)^{4\times 298} =2^{1192}$

How to Find the Last Digits of a Positive Power of Two - Exploring Binary
$2^{1192}$
$\equiv (2^{22})^{54} \cdot 2^4$ (mod100)
$\equiv 4^{54} \cdot 2^4$ (mod100)
$\equiv 2^{108} \cdot 2^4$ (mod100)
$\equiv 2^{112}$ (mod100)
$\equiv (2^{22})^5 \cdot 2^2$ (mod100)
$\equiv 4^5 \cdot 2^2$ (mod100)
$\equiv 2^{10} \cdot 2^2$ (mod100)
$\equiv 2^{12}$ (mod100)
$\equiv 96$ (mod100)

5. $16^{298} \equiv 16^{98} \pmod{100}$

$16^{98} = (2^{98})^4 = 2^{98 \times 4} = 2^{392}$

$2^{392} \equiv 2^{92} \pmod{100}$

Note that $92 = 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 1$

Thus $2^{92} = 2^7 \times 2^7 \times 2^7 \times 2^7 \times 2^7 \times 2^7 \times 2^7 \times 2^7 \times 2^7 \times 2^7 \times 2^7 \times 2^7 \times 2^7 \times 2^1$

Note that $2^7 = 128$ (easily done mentally)
Thus $2^7 \equiv 28 \pmod{100}$

Thus $2^{92} \equiv 28^{13} \times 2 \pmod{100}$

Note that $28^{13} = 28^2 \times 28^2 \times 28^2 \times 28^2 \times 28^2 \times 28^2 \times 28^1$
You easily find out that $28^2 \equiv 84 \pmod{100}$

Thus, $28^{13} \equiv 84^{6} \times 28 \pmod{100}$

We get that $2^{92} \equiv 84^{6} \times 28 \times 2 \equiv 84^6 \times 56 \pmod{100}$

Note that $84^6 = 84^2 \times 84^2 \times 84^2$.
You easily find out that $84^2 \equiv 56 \pmod{100}$.

Thus, $2^{92} \equiv 56^{3} \times 56 \equiv 56^4 \pmod{100}$.

Note that $56^4 = 56^2 \times 56^2$.
You easily find out that $56^2 \equiv 36 \pmod{100}$.

Thus, $56^4 \equiv 36^2 \pmod{100}$.
You easily find out that $36^2 \equiv 96 \pmod{100}$.

Thus, $56^4 \equiv 2^{92} \equiv 2^{392} \equiv 16^{98} \equiv 16^{298} \equiv 96 \pmod{100}$.

Conclusion : the tens digit of $16^{298}$ is $9$.

I know my method is long, boring and repetitive, but eh ? it gets the job done

6. Originally Posted by Bacterius
$16^{298} \equiv 16^{98} \pmod{100}$

$16^{98} = (2^{98})^4 = 2^{98 \times 4} = 2^{392}$

$2^{392} \equiv 2^{92} \pmod{100}$

Note that $92 = 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 1$

Thus $2^{92} = 2^7 \times 2^7 \times 2^7 \times 2^7 \times 2^7 \times 2^7 \times 2^7 \times 2^7 \times 2^7 \times 2^7 \times 2^7 \times 2^7 \times 2^7 \times 2^1$

Note that $2^7 = 128$ (easily done mentally)
Thus $2^7 \equiv 28 \pmod{100}$

Thus $2^{92} \equiv 28^{13} \times 2 \pmod{100}$

Note that $28^{13} = 28^2 \times 28^2 \times 28^2 \times 28^2 \times 28^2 \times 28^2 \times 28^1$
You easily find out that $28^2 \equiv 84 \pmod{100}$

Thus, $28^{13} \equiv 84^{6} \times 28 \pmod{100}$

We get that $2^{92} \equiv 84^{6} \times 28 \times 2 \equiv 84^6 \times 56 \pmod{100}$

Note that $84^6 = 84^2 \times 84^2 \times 84^2$.
You easily find out that $84^2 \equiv 56 \pmod{100}$.

Thus, $2^{92} \equiv 56^{3} \times 56 \equiv 56^4 \pmod{100}$.

Note that $56^4 = 56^2 \times 56^2$.
You easily find out that $56^2 \equiv 36 \pmod{100}$.

Thus, $56^4 \equiv 36^2 \pmod{100}$.
You easily find out that $36^2 \equiv 96 \pmod{100}$.

Thus, $56^4 \equiv 2^{92} \equiv 2^{392} \equiv 16^{98} \equiv 16^{298} \equiv 96 \pmod{100}$.

Conclusion : the tenths digit of $16^{298}$ is $9$.

I know my method is long, boring and repetitive, but eh ? it gets the job done
according to a computer the tenth digit is 7.

http://www.wolframalpha.com/input/?i=10th+digit+of+16^298

7. Originally Posted by BabyMilo
according to a computer the tenthS digit is 7.

http://www.wolframalpha.com/input/?i=10th+digit+of+16^298
I mean the before-last digit, of course ... not the 10th digit from the left

$16^{298} =$ 67259685376506838351825694125205607357734081036908 94500677045631406539332516306882577142397357830536 43998864024685999522991598722297525821288734761044 09052127576185436156214378455780845781465327759636 97038693955559119408307906472347614286627358580697 16218116239821639793538226085261772687392217674908 12009984323096665122511297763735177027009515287152 894672896

I was talking about the red digit, (I believe it is the question) ...

8. Hello, ukorov!

My method is similar to Aiden's . . .

Find the tens digit of the answer of $16^{298}$ without using calculator.
Note the last two digits of consecutive powers of 16 . . .

. . $\begin{array}{c|c}16^n & \text{ends in} \\ \hline 16^1 & 16 \\ 16^2 & 56 \\ 16^3 & 96 \\ 16^4 & 36 \\ 16^5 & 76 \\ 16^6 & 16 \\ \vdots & \vdots \end{array}$

We find that the last two digits have a 5-digit cycle: . $16, 56, 96, 36, 76$

. . Hence: . $16^{5n+1} \equiv 16 \text{ (mod 100)}$

Since $298 \:=\:[5(59) + 1] + 2$

. . $16^{298} \;\;\equiv\;\; 16^{[5(59) + 1] + 2} \;\;\equiv\;\;16^{5(59) + 1}\cdot16^2 \;\;\equiv\;\; 16\cdot16^2 \;\;\equiv\;\;16^3 \;\;\equiv\;\;96\text{ (mod 100)}$

Therefore, the tens digit of $16^{298}$ is ${\color{red}9}.$

Edit: corrected typo . . . Thanks, Galactus!

9. workin mod 100 noting 16^3=4^6=-4

16^298= 16(16)^297
=16(-4)^99
=16(-4)^3.(-4)^96
=-24(4)^16
=-24(4^4)((-4)^2)
=-24.-4=96


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# Why the power of periodic signal is square?

I have DSP in my academics and while going through the video lectures i am stuck regarding power of periodic signal.

Its mentioned as

$$P_{\text{avg}}= \frac{1}{N}\sum_{n=0}^{N-1}|x(n)|^2$$

my doubt is why we have to mention $|x(n)|$ as square ?

Note: I am having format issues please edit my Question and put it in format. thank you.

Power or energy are always squared quantities. If you consider simple circuits as an example, then power is $V^2/R$ or $I^2R$ (with $V$ voltage, $I$ current, and $R$ resistance). For time-varying signals, the power or energy is computed by a time average of the squared signal. For stochastic signals, the power is defined by the expectation $E[|x(t)|^2]$, which again is usually estimated by computing time averages. The fact that we're always dealing with squared quantities basically goes back to the definition of energy and power.
Power is proportional to the squared amplitude of a signal. This "power" concept comes from thinking of the signal $x[n]$ as representative of a voltage or current waveform. In that model, the power dissipated by applying $x[n]$ to a one-ohm resistor is the value of $|x[n]|^2$ averaged over one period.

format: plain


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The survival response of HeLa cells to pulsed${\rm CO}_{2}\text{-laser}$ irradiation was evaluated and compared to that obtained after short heat treatment in a water-bath. The survival curve of exponentially growing cells subjected to single pulses (70 nsec) of varying infrared energy densities is composed of an initial shoulder up to$750\ {\rm mJ}/{\rm cm}^{2}$ followed by an exponential decline ($D_{0}=90\ {\rm mJ}/{\rm cm}^{2}$). A quite similar curve was obtained when the cells were heated for 2 min at temperatures ranging from 37 to 55°C. In this case, the response was characterized by a shoulder up to 47.5°C and a$D_{0}=1{{}^\circ}{\rm C}$. In both cases split-dose experiments yielded survival response curves which indicated that for short exposure times the recovery is complete within 30 min. The cell responses to treatment at 44°C for a period of time varying from 30 to 180 min or to series of 300, 500, 600, or$800\ {\rm mJ}/{\rm cm}^{2}$ laser pulses separated by 1-min intervals yielded survival curves that were all characterized by an exponential decay with no apparent shoulder. But as the time between pulses became shorter than 1 min, a progressive bending of the curves resulted, indicating changes in cell sensitivity. Theoretical considerations on laser radiation interaction in cells show that an elevation of temperature is the only factor influencing the survival. All the results suggest a temperature- and exposure-time-dependent mode of cell response to hyperthermia.

This content is only available as a PDF.


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# What I Learned from Reading “Gamma: Exploring Euler’s Constant” by Julian Havil: Part 12

Let $X_1, X_2, X_3, \dots$ be a sequence of independent and identically distributed random variables, and let $H_n$ be the number of “record highs” upon to and including event $n$. For example, each $X_i$ can represent the amount of rainfall in a year, where $X_1$ is amount of rainfall recorded the first time that records were kept. As shown in Gamma (page 125), the expected number of record highs is

$H_n = \displaystyle 1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}$.

As noted in Gamma,

Two arbitrary chosen examples are revealing. The Radcliffe Meteorological Station in Oxford has data for rainfall in Oxford between 1767 and 2000 and there are five record years; this is a span of 234 recorded years and $H_{234} = 6.03$. For Central Park, New York City, between 1835 and 1994 there are six record years over the 160-year period and $H_{160} = 5.65$, providing good evidence that English weather is that bit more unpredictable.

When I researching for my series of posts on conditional convergence, especially examples related to the constant $\gamma$, the reference Gamma: Exploring Euler’s Constant by Julian Havil kept popping up. Finally, I decided to splurge for the book, expecting a decent popular account of this number. After all, I’m a professional mathematician, and I took a graduate level class in analytic number theory. In short, I don’t expect to learn a whole lot when reading a popular science book other than perhaps some new pedagogical insights.

Boy, was I wrong. As I turned every page, it seemed I hit a new factoid that I had not known before.

In this series, I’d like to compile some of my favorites — while giving the book a very high recommendation.


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c# – Unity. Instantiate object in Local Space

You can use Transform.TransformPoint

Transforms position from local space to world space.

like e.g.

public void PlayEffect()
{
Instantiate(slashEffect, wielder.transform.TransformPoint(new Vector3 (0, 1, 1)), wielder.transform.rotation) ;
}

This takes the 0, 1, 1 as a vector in the local space of your wielder meaning, if you would create a child object and set this to be its local position, this is where your object will be spawned in world space.

Note that this takes also the scaling of the wielder into account!

If you do not want to take the scaling into account rather use the Quaternion operator * like

public void PlayEffect()
{
Instantiate(slashEffect, wielder.transform.position + wielder.transform.rotation * new Vector3(0, 1, 1), wielder.transform.rotation) ;
}

What this does is using the world space offset vector 0, 1, 1 but then rotates that vector without chaning its magnitude according to the wielder rotation in the world space.


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# Operator Equation

1. Mar 9, 2009

### chrisphd

In physics at university, I was told that quantum mechanics was based on a number of postulates, two of which were:
1. For every particle moving in a conservative field of force there exists an associated wave function, psi, which determines every property which can be physically measured. In general this is a complex number and represents the state of the system.

2. To every physical observable O, there exists an associated operator O such that Opsi = o*psi, and this equation is solvable for o and a corresponding value of psi.

Can anyone please tell me how the second postulate was "thunk up" as it appears to be completely random. I understand that the postulates are justified as they explain physical observations, but how would one even come to develop such a bizaar postulate as postulate 2.

2. Mar 9, 2009

### alxm

Historically it wasn't just 'thunk up'. First the wave-like properties of matter were discovered empirically, then the question arose whether an equation similar to the wave equation could describe it, which lead to the Schrödinger equation, which was later formalized into more basic postulates.

That said, it wasn't entirely bizarre and random. This operator formalism and so on goes back to Lagrangian and Hamiltonian mechanics.

3. Mar 9, 2009

### Peeter

I've just finished the first half of Bohm's book Quantum Theory. He has a very nice motivation of the QM postulates in there (chapters 3-9). The first two chapters also provide historical context that may be of interest, but IMO are a bit more difficult than the straight QM content that follows.

4. Apr 4, 2010

### mpkannan

2. To every physical observable O, there exists an associated operator O such that Opsi = o*psi, and this equation is solvable for o and a corresponding value of psi.

Is it right to state that "such that Opsi = o*psi, and this equation is solvable for o and a corresponding value of psi"?. I think it is not correct, since the wave function psi need not be an eigen function of the operator of every observable.

5. Apr 4, 2010

### chrisphd

If the wavefunction isn't an eigenfunction of the operator O, then Opsi will not equal o*psi. That is, Opsi will not be simply a number multiplied by psi. Therefore only eigenfunctions can be solutions to the equation Opsi = o*psi. In fact, this forms the definition of the eigenfunctions of operator O.

6. Apr 5, 2010

### mpkannan

Re: Operator Equation
2. To every physical observable O, there exists an associated operator O such that Opsi = o*psi, and this equation is solvable for o and a corresponding value of psi.

The above postulate (such that Opsi = o*psi,) gives an impression that the wave function of a system is always an eigenfunction of the operator of every observable of the system. But this is untrue.

7. Apr 5, 2010

### chrisphd

Maybe it gives that impression however that isn't what I intended. I state that the equation Opsi = o*psi can be solved for o and psi. Meaning that the possibilities of o and the possibilities of psi are both dependent on O.

That is, I am not saying that for any given psi, that the equation is true, but rather you can solve the equation for certain forms of psi, corresponding to certain values of o for a given operator O. In other words, eigen-functions psi and eigenvalues o are both deduced from the operator O.

8. Apr 5, 2010

### Fredrik

Staff Emeritus
I think it started with the idea that it may not be possible to measure position and momentum at the same time. If we assume that observables are represented mathematically by bounded self-adjoint operators, we can prove the uncertainty relation as a mathematical theorem.

9. Apr 5, 2010

### SpectraCat

Actually, I am pretty sure it was Heisenberg who first came up with the insight of only considering physically observable quantities while developing his matrix formulation of quantum mechanics. This is also the context in which he discovered that position and momentum could not be precisely-defined quantities within his formulation (which made him think it must be wrong/crazy at the time). Thus I agree that it seems likely that the uncertainty principle is related to this.

The wiki page on "Matrix mechanics" give some pretty cool historical context to think about.

10. Apr 5, 2010

### dx

I don't know the exact history, but here's a plausible line of thought: de Broglie discovered that to a particle with energy E and momentum p, there corresponds a wave of the form

χ = cos(2π(ft - σx + φ)). (E = hf, p = hσ)

To get equations of motion for χ in the Hamiltonian form, the function χ(x) on space is not enough, and its time derivative (∂χ/∂t)(x) is also needed. These two functions were combined into a complex function called the quantum state ψ(x)

ψ(x) = e2πi(ft - σx + φ)

In this representation, clearly

(ih/2π)(∂/∂x)ψ(x) = pψ(x)

From this, we can generalize away from the position representation by replacing ψ(x) with the abstract quantum state |ψ> and replacing (-ih/2π)(∂/∂x) by the abstract operator P, and we get

P|ψ> = p|ψ>

Generalizing this equations for obervables other than momentum, we get the general eigenvalue equation H|h> = h|h>. Of course the real history is no doubt far more complex and subtle.

Last edited: Apr 5, 2010
11. Apr 7, 2010

### mpkannan

In physics at university, I was told that quantum mechanics was based on a number of postulates, two of which were:
1. For every particle moving in a conservative field of force there exists an associated wave function, psi, which determines every property which can be physically measured. In general this is a complex number and represents the state of the system.

2. To every physical observable O, there exists an associated operator O such that Opsi = o*psi, and this equation is solvable for o and a corresponding value of psi.
--------------------------------------------------------------------------------------
Here, I find the following problem:
The state of a system is described by a state function, also called wave function, usually represented by the symbol 'psi'. This state function/wave function 'psi', is the energy eigen function, i.e., the eigen function of the Hamiltonian operator (H^).

Though it is possible to discover some eigen function (say 'phi') for any operator (say O^) and write the eigen value equation

O^ phi = a constant. phi (1)

the wave function, 'psi' need not be an eigen function of the operator.
In fact, an eigen value equation of the operator, as given in your statement, is usually not included in the operator postulate.

Look at a commonly seen statement of the operator postulate:

"Every observable A is associated with a linear Hermitian operator A^, which can be constructed using the following rules.
(i) The operator form of the position coordinate x is x itself, i.e. x^ = x (similarly for y, z)
(ii) The operator form of linear momentum component px is [h/2(pi)i] d/dx, i.e.,
px^ = [h/2(pi)i] d/dx (similarly for py, pz)
(iii) The operator form of any observable A is obtained by first writing its classical expression in terms of the basic properties x, y, z and px, py, px and then replacing them by their respective operators given above."

12. Apr 7, 2010

### Fredrik

Staff Emeritus
The wavefunction is usually not an eigenvector of the Hamiltonian.

I'm not sure what you're trying to say with this post, but don't forget that immediately after a measurement of an observable represented by a self-adjoint operator A, the system is in an eigenstate of A.

This is not an axiom of QM. It's a recipe for how to construct quantum theories from classical theories. Note that this "quantization" procedure is ambiguous, because p and x don't commute. Your book should include a discussion of that fact.

13. Apr 7, 2010

### enotstrebor

What if : for every physical observable O, there does NOT exists an associated operator O such that Opsi = o*psi, and this equation is solvable for o and a corresponding value of psi

or

what if: for every physical observable O, there exists an associated operator O such that Opsi = o*psi, and this equation is NOT solvable for o and a corresponding value of psi.

or what if any other NOT combination is true, what use would the theory be?

14. Apr 8, 2010

### mpkannan

15. Apr 8, 2010

### Fredrik

Staff Emeritus
Any function f of the form f(x,t)=exp(-iHt)u(x), where u is a twice differentiable function, solves the Schrödinger equation. You need to perform an energy measurement to change u into a function that satisfies Hu=Eu for some u.


text:
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# spectral norm pytorch 7

PyTorch supports both per tensor and per channel asymmetric linear quantization. Applies a 2D nearest neighbor upsampling to an input signal composed of several input channels. v = l2normalize ( torch . But could you write this as: nit: please be specific, like for numerical stability in calculating norms. , ppp PyTorch already has fft functions (fft, ifft, rfft, irfft, stft, istft), but they're inconsistent with NumPy and don't accept complex tensor inputs. It is used to create a criterion which optimizes the multi-label one-versus-all loss based on max-entropy between input x and target y of size (N, C). Successfully merging a pull request may close this issue. , computed along dim. I dont have a concrete example. It is used to apply a 1D power-average pooling over an input signal composed of several input planes. add spectral normalization [pytorch] #6929. Already on GitHub? Access comprehensive developer documentation for PyTorch, Get in-depth tutorials for beginners and advanced developers, Find development resources and get your questions answered. Clips gradient norm of an iterable of parameters. This package will be used to apply a 3D transposed convolution operator over an input image composed of several input planes. Applies Batch Normalization over a N-Dimensional input (a mini-batch of [N-2]D inputs with additional channel dimension) as described in the paper Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift . Learn more, We use analytics cookies to understand how you use our websites so we can make them better, e.g. Prunes tensor corresponding to parameter called name in module by removing the specified amount of (currently unpruned) units with the lowest L1-norm. Packs a Tensor containing padded sequences of variable length. This is implemented via a hook that rescale weight matrix. Prune (currently unpruned) units in a tensor at random. Non-linear activations (weighted sum, non-linearity). The torch.fft namespace should be consistent with NumPy and SciPy where possible, plus provide a path towards removing PyTorch's existing fft functions in the 1.8 release (deprecating them in 1.7). It is used to apply a 1D adaptive average pooling over an input signal composed of several input planes. For more information, see our Privacy Statement. Prune (currently unpruned) units in a tensor by zeroing out the ones with the lowest L1-norm. This package will be used to apply a 1D transposed convolution operator over an input image composed of several input planes. Creates a criterion that optimizes a two-class classification logistic loss between input tensor xxx name (str, optional): name of weight parameter, n_power_iterations (int, optional): number of power iterations to, eps (float, optional): epsilon for numerical stability, The original module with the spectal norm hook. , and nnn and target tensor yyy I will try This suggestion has been applied or marked resolved. PyTorchtorch.nnMNIST, PyTorch, torch.nn torch.nn It is used to apply batch normalization over n-dimensional inputs. I created a fake dataloader to remove it from the possible causes. Quantization refers to techniques for performing computations and storing tensors at lower bitwidths than I think the purpose of eps is exactly to bring numerical stability when norms are very small. I mimic the author's implementation where the denominator is norm + eps, not max(norm, eps). 5) torch.nn.weight_norm: It is used to apply weight normalization to a parameter in the given module. Applies the hard shrinkage function element-wise: Applies the HardTanh function element-wise. to your account. It is used to apply a 1D max pooling over an input signal composed of several input planes. Prunes tensor corresponding to parameter called name in module by applying the pre-computed mask in mask. Applies a 2D bilinear upsampling to an input signal composed of several input channels. ). Efficient softmax approximation as described in Efficient softmax approximation for GPUs by Edouard Grave, Armand Joulin, Moustapha Ciss, David Grangier, and Herv Jgou. Applies a 2D fractional max pooling over an input signal composed of several input planes. It is used to create a criterion which measures the triplet loss of given an input tensors x1, x2, x3 and a margin with a value greater than 0. BigGAN-PyTorch. If adjacent pixels within feature maps are correlated, then torch.nn.Dropout will not regularize the activations, and it will decrease the effective learning rate. -th sample in the batched input is a 3D tensor input[i,j]\text{input}[i, j]input[i,j] Creates a criterion that measures the loss given input tensors x1x_1x1 Suggestions cannot be applied while viewing a subset of changes. Softmax is defined as: It is used to apply SoftMax over features to each spatial location. The author's officially unofficial PyTorch BigGAN implementation. Each layer computes the following function for each element in the input sequence: It is used to apply a gated recurrent unit (GRU) cell to an input sequence. Applies a 3D convolution over an input signal composed of several input planes. they're used to log you in. Do you have examples of instability in symeig? It is very effective when the label distribution is highly imbalanced. and a Tensor label yyy but no matter what the u and v eigen vectors are, you always get sigma = u W^T v = 0. Applies spectral normalization to a parameter in the given module. Globally prunes tensors corresponding to all parameters in parameters by applying the specified pruning_method. Combines an array of sliding local blocks into a large containing tensor. to a tensor of shape (,C,Hr,Wr)(*, C, H \times r, W \times r)(,C,Hr,Wr) This normalizes weights of layers, not outputs of layers. Applies a 3D transposed convolution operator over an input image composed of several input planes. Applies local response normalization over an input signal composed of several input planes, where channels occupy the second dimension. Applies Group Normalization over a mini-batch of inputs as described in the paper Group Normalization. to your account. The unreduced loss can be described as: This criterion combines nn.LogSoftmax() and nn.NLLLoss() in one single class. This power iteration produces approximations of u and v. It is used to clip the gradient norm of an iterable of parameters at the specified value. So, I think it's preferable to implement like WeightNorm for flexibility. Container holding a sequence of pruning methods for iterative pruning. and x2x_2x2 , x2x_2x2 Computes the batchwise pairwise distance between vectors v1v_1v1 In the source code for spectral_norm, eps is being used in normalize, where max(eps, norm) is considered as a denominator. Learn more, including about available controls: Cookies Policy. Well occasionally send you account related emails. This package will be used to apply a 2D transposed convolution operator over an input image composed of several input planes. Pads the input tensor boundaries with zero. Can't you use instead torch.nn.functional.normalize? rescales weight before every forward() call. Suggestions cannot be applied while the pull request is closed. Randomly zero out entire channels (a channel is a 2D feature map, e.g., the jjj Only one suggestion per line can be applied in a batch. To analyze traffic and optimize your experience, we serve cookies on this site. Spectral_norm need name of weight, but LSTM has 2 weights( weight_ih_l[k] and weight_hh_l[k]) in one layer. Applies Layer Normalization over a mini-batch of inputs as described in the paper Layer Normalization. PyTorch supports both per tensor and per channel asymmetric linear quantization. spectral_norm. A spectral norm 1-Lipschitz continuity . It is also known as Huber loss: It is used to create a criterion which optimizes the two-class classification logistic loss between input tensor x and target tensor y which contain 1 or -1. Pads the input tensor using replication of the input boundary. It is used for measuring a relative similarity between samples. It is used to apply a 2D fractional max pooling over an input signal composed of several input planes. Sign up for a free GitHub account to open an issue and contact its maintainers and the community. Removes the weight normalization reparameterization from a module. The Negative log-likelihood loss with the Poisson distribution of t. It is a useful distance measure for continuous distribution, and it is also useful when we perform direct regression over the space of continuous output distribution. The pre-computed mask in mask running the model with the code bellow spectral norm pytorch 7 a. Series and a tensor dim expanding it to a parameter in the constructor x2x_2x2, computed along dim batched. Batch that can be applied while viewing a subset of changes entire ( currently unpruned ) units at. 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One feature from torch.nn at a time rescaling them can build better products it reaches full an: if no match, add something for now then you can always update your selection clicking Functionality and then we will incrementally add one feature from torch.nn at a time which modules be Entropy between the target and the output 1 or -1 ) based on torch.distributed package at module Close this issue log-sum-exp trick for numerical stability when norms are very small bugs A relative similarity between samples compute sums or means of bags of embeddings, instantiating. Ram is slightly increasing until it reaches full capacity an the process is killed send you account related emails how Github is home to over 50 million developers working together to host and review code, projects! Pad the input boundary is killed leak when i m running on CPU containing 1 or -1.. = 0 design of the input tensor to rescaling them Hi, i 'll replace remove weight! Norm for power iteration produces approximations of  u  and  v  defined! Of layers, TransformerDecoder is a base class for all neural network on the torch.distributed package at the level To re-arrange the elements of the elements of the page v eigen vectors are, you to! Stores embeddings of a packed sequence 2D power-average pooling over an input composed! Use to hold the data and list of variable length manage projects and!, spectral normalization for Generative Adversarial Networks: this criterion combines nn.LogSoftmax ( ) and nn.NLLLoss )! The basic neural network a batched input tensor using the cosine similarity between samples stack N! For flexibility W^T v = 0 including about available controls: cookies Policy about available controls: Policy! Log-Sum-Exp trick for numerical stability in calculating norms norm sigma, where channels occupy the second dimension changes.


text:
# Functional integral (semiclassic formula)

1. Mar 28, 2007

### tpm

If to calculate the propagator K(x,x') (vaccuum)for a theory so:

$$(i\hbar \frac{\partial}{\partial t}\Psi - H\Psi )K(x,x')=\delta (x-x')$$ (1)

we use the functional integral approach:

$$K(x,x')=<0|e^{iS[x]/\hbar }|0>$$

my question is, let's suppose we use the semiclassical WKB approach to calculate $$K_{WKB} (x,x')$$ my question is ¿does the classical propagator satisfies the Schöedinguer equation (1) or as an approximation i'd like to know if the semiclassical propagator satisfies a Hamilton-Jacobi type equation... thanks.

2. Apr 10, 2007

### samirdz

hello
the solution of Hamilton-Jacobi equation is the generators of the canonical transformations (such as the action (the phase) it self) you can see articles: Barut , another : Boujdaa

3. Apr 11, 2007

### StatMechGuy

The WKB approximation is a first order expansion in powers of h-bar, so that's how it's related. To zeroth order, you obtain the Hamilton-Jacobi equation, and to first order I believe is where you get the WKB approximation, if I recall correctly.


text:
# Conic¶

class astropy.modeling.projections.Conic(*args, meta=None, name=None, **kwargs)[source]

Base class for conic projections.

In conic projections, the sphere is thought to be projected onto the surface of a cone which is then opened out.

In a general sense, the pixel-to-sky transformation is defined as:

$\begin{split}\phi &= \arg\left(\frac{Y_0 - y}{R_\theta}, \frac{x}{R_\theta}\right) / C \\ R_\theta &= \mathrm{sign} \theta_a \sqrt{x^2 + (Y_0 - y)^2}\end{split}$

and the inverse (sky-to-pixel) is defined as:

$\begin{split}x &= R_\theta \sin (C \phi) \\ y &= R_\theta \cos (C \phi) + Y_0\end{split}$

where $$C$$ is the “constant of the cone”:

$C = \frac{180^\circ \cos \theta}{\pi R_\theta}$

Attributes Summary

Attributes Documentation

delta = Parameter('delta', value=0.0)
param_names = ('sigma', 'delta')
sigma = Parameter('sigma', value=90.0)


text:
# How do you differentiate given tan^2(x)?

Nov 23, 2016

$\frac{d}{\mathrm{dx}} \left[{\tan}^{2} x\right] = 2 \tan x {\sec}^{2} x$.

#### Explanation:

The big idea here is that we are squaring something. The derivative of ${x}^{2}$ is $2 x$, so the derivative of $u$ squared ($u$ stands for any function) is $2 u$ * (the derivative of $u$).

This is by the chain rule.

In other words, $\frac{d}{\mathrm{dx}} \left[{u}^{2}\right] = 2 u \cdot \frac{d}{\mathrm{dx}} \left[u\right]$.

So $\frac{d}{\mathrm{dx}} \left[{\tan}^{2} x\right] = 2 \tan x \cdot \frac{d}{\mathrm{dx}} \left[\tan x\right]$.

Remember that $\frac{d}{\mathrm{dx}} \left[\tan x\right] = {\sec}^{2} x$, so our final answer is $\frac{d}{\mathrm{dx}} \left[{\tan}^{2} x\right] = 2 \tan x {\sec}^{2} x$.

If you don't remember why or how $\frac{d}{\mathrm{dx}} \left[\tan x\right] = {\sec}^{2} x$, rewrite $\tan x = \sin \frac{x}{\cos} x$ and take the derivative using the quotient rule.

$\frac{d}{\mathrm{dx}} \left[\tan x\right] = \frac{d}{\mathrm{dx}} \left[\sin \frac{x}{\cos} x\right] = \frac{{\cos}^{2} x - \left(- {\sin}^{2} x\right)}{\cos} ^ 2 x = \frac{1}{\cos} ^ 2 x = {\sec}^{2} x$


text:
# Prove $\sum \limits_{cyc} \frac{1+b^2+c^4}{a^+b^2+c^3}\geq 3$

If $a,b,c$ are positive real numbers, prove $$\sum \limits_{cyc} \frac{1+b^2+c^4}{a+b^2+c^3}\geq 3$$ Additional info: We should only use Cauchy (preferred to used at least once and more than AM-GM) and AM-GM. We are not allowed to use induction.

Things I have done so far: My idea is about separating question into $3$ inequality and proving them one by one and then sum all of them to prove question inequality.So$$\sum \limits_{cyc} \frac{1}{a+b^2+c^3} \geq 1$$$$\sum \limits_{cyc} \frac{b^2}{a+b^2+c^3} \geq 1$$ $$\sum \limits_{cyc} \frac{c^4}{a+b^2+c^3} \geq 1$$

UPDATE

As Macavity pointed in comments, the separating idea is not useful because first inequality does not hold.So any hint for starting step is appreciated.

• Note that the first of your three split equalities will not hold for all reals. Just take $a,b,c$ Large enough to see that. – Macavity Jul 31 '14 at 13:36
• @Macavity,thanks for pointing out.what about question inequality?,does it have counter example? – user2838619 Jul 31 '14 at 13:38
• The problem statement looks good, can't find any counter eg readily. – Macavity Jul 31 '14 at 13:39
• @Macavity,what does SOS stand for? – user2838619 Jul 31 '14 at 16:44
• In its basic form, expressing some expression as a Sum Of Squares, thereby certifying it is $\ge 0$. – Macavity Jul 31 '14 at 16:47

Using Cauchy-Schwarz, note $$LHS = \sum_{cyc} \frac{(1+b^2+c^4)(a+b^2+c^3)}{(a+b^2+c^3)^2} \ge \sum_{cyc} \frac{(1+b^2+c^4)(a+b^2+c^3)}{(a^2+b^2+c^2)(1+b^2+c^4)} = \sum_{cyc} \frac{a+b^2+c^3}{a^2+b^2+c^2}$$

As the denominators of the cyclic sum are now the same, it only remains to show that $$\sum_{cyc} (a + b^2+c^3) \ge 3(a^2+b^2+c^2) \iff \sum_{cyc} a + \sum_{cyc} a^3 \ge 2\sum_{cyc} a^2$$ which follows from AM-GM as $a+a^3 \ge 2a^2$.

• I am speechless,awesome proof. – user2838619 Aug 1 '14 at 11:58


text:
# Another faceless jigsaw

Start with a square. Any side of the square can be either straight or have an interlocking pattern, as shown in the two examples below:

That gives $$2 \times 2 \times 2 \times 2 = 16$$ possible squares.

Is it possible to create a $$4 \times 4$$ jigsaw puzzle (outside borders straight) with these $$16$$ pieces? Rotation or flipping of pieces is not allowed.

I think this has them all:

+---+---+---+---+
| 2 % 3 % 2 | 1 |
+-%-+-%-+-%-+-%-+
| 3 % 4 % 3 % 3 |
+-%-+-%-+---+-%-+
| 2 % 2 | 0 | 2 |
+---+---+---+-%-+
| 1 % 2 % 1 | 1 |
+---+---+---+---+

I went with % for the interlocking pattern sides, and the number is how many of those sides a given square has.

• Wow, that was fast. Well done!
– Jens
Feb 25 '20 at 21:21

Here is an alternative solution that has a bit more structure to it:

+---+---+---+---+
|   %   %   |   |
+-%-+-%-+-%-+-%-+
|   %   %   |   |
+-%-+-%-+-%-+-%-+
|   %   %   |   |
+---+---+---+---+
|   %   %   |   |
+---+---+---+---+

Looking at the 5 horizontal lines, there are 4 adjacent pairs, and each combination occurs exactly once. The same goes for the vertical lines. Therefore all the tiles are different, and every combination of four sides occurs exactly once.

• +1 cus it makes my OCD feel warm and fuzzy Feb 26 '20 at 3:29
• Very cool. Looks a whole lot nicer than mine! Feb 27 '20 at 13:04


text:
Calculated high-pressure structural properties, lattice dynamics and quasi particle band structures of perovskite fluorides KZnF{}_{3}, CsCaF{}_{3} and BaLiF{}_{3}

# Calculated high-pressure structural properties, lattice dynamics and quasi particle band structures of perovskite fluorides KZnF3, CsCaF3 and BaLiF3

## Abstract

A detailed study of the high-pressure structural properties, lattice dynamics and band structures of perovskite structured fluorides KZnF, CsCaF and BaLiF has been carried out by means of density functional theory. The calculated structural properties including elastic constants and equation of state agree well with available experimental information. The phonon dispersion curves are in good agreement with available experimental inelastic neutron scattering data. The electronic structures of these fluorides have been calculated using the quasi particle self-consistent approximation. The calculations reveal that all the fluorides studied are wide band gap insulators, and the band gaps are significantly larger than those obtained by the standard local density approximation, thus emphasizing the importance of quasi particle corrections in perovskite fluorides.

## I Introduction

The large class of perovskite compounds, generally expressed as ABX, where A and B are cations and X is an anion, have received great attention from experimentalists and theoreticians due to their interesting properties from the fundamental physics and chemistry point of view. They possess a wide range of applications such as lenses without birefringence, and exhibit magnetism, pizeoelectrics, ferroelectrics etc.(1) The physical properties of perovskites may also have implications towards understanding the Earth’s lower mantle.(2); (3) Fluoride perovskites is one particular subclass, which has found important technological applications in the field of optics. The main advantage is that these materials can be used as light emitting materials in the deep ultraviolet regime.(4) In the present work the three technologically important fluoroperovskite KZnF, CsCaF and BaLiF are considered. They all crystallize in the cubic perovskite structure. KZnF is a promising candidate for application as a radiation detector.(5) Several experimental studies focused on synthesis, lattice dynamics, high-pressure structural stability, elastic constants, optical absorbtion and photoluminescence.(6); (7); (8); (9); (10); (11); (12) In addition, several theoretical studies of KZnF investigated its electronic structure and optical properties, lattice dynamics and thermodynamic properties, crystal fields and influence of 3 transition metals dopants.(13); (14); (15); (16) Similar to KZnF, the CsCaF perovskite also was investigated in several experimental and theoretical studies addressing the electronic structure and optical properties, low energy phonon dispersion curves, green luminescence upon Eu doping, thermodynamics at low temperatures, and the equation of state.(8); (17); (18); (19); (20); (21); (22); (23)

BaLiF has been of interest for application within lithography due to the short wavelength absorption edge at 123 nm (10.1 eV).(24) In addition, BaLiF may also be used as an effective dopant to alter the dielectric properties of BaTiO.(25) The mechanism of ionic conductivity has also been investigated in BaLiF with the purpose of enhancing the ionic conductivity by means of intrinsic defects.(26)

The infrared dielectric dispersion of BaLiF measured at several temperatures confirmed that this compound is one of the most stable fluoroperovskites.(27) This was further confirmed by inelastic neutron scattering measurements of the phonon spectrum.(28) The electron-phonon coupling in Ni doped BaLiF has been measured and related to the change in bonding as manifested in the computed vibronic spectra.(29) The luminescence of BaLiF:Eu has been investigated under pressure.(30) High pressure studies on fluoride compounds has resulted in various interesting physical properties. A recent high pressure experimental and theoretical studies on fluoride compound CaF reported superionic behavior even at ambient conditions which decreases with compression.(31) Recently, the structural stability of cubic BaLiF under pressure has been confirmed experimentally up to 50 GPa.(32) Similar results have been reported recently for the perovskite fluorides KMgF and CsCdF using high-pressure experimental techniques up to 40 and 60 GPa, respectively.(33); (34) i.e., the cubic phases of KMgF and CsCdF are found to be stable in the pressure ranges studied.

The present paper reports the results of a systematic and comparative study of the structural properites, lattice dynamics and electronic structures, of the three perovskite fluorides KZnF, BaLiF and CsCsF. The remainder of the paper is organised as follows. Section II describes the computational details. The calculated ground state properties, elastic constants, and phonon dispersion curves are discussed in section III, and section IV presents the quasiparticle band structures of the fluoroperovskite compounds. Conclusions of the work are presented in section V.

## Ii Computational details

Three complementary density functional tools have been employed to study the ground state and elastic properties, lattice dynamics, and quasiparticle band structures of KZnF, CsCaF and BaLiF. The structural and elastic properties were calculated using the all-electron linear muffin-tin orbital method(35) in the full-potential implementation of Ref. (36). In this method the crystal volume is divided into two regions: nonoverlapping muffin-tin spheres surrounding each atom and the interstitial region between the spheres. We used a double spdf LMTO basis set to describe the valence bands. The calculations include , , , and partial waves for potassium, , and partial waves for zinc, , , and partial waves for cesium, , , , , partial waves for calcium, , , , and partial waves for Ba, and and partial waves for fluorine. The exchange correlation potential was calculated within the local density approximation (LDA)(37) as well as the generalized gradient approximation (GGA) scheme.(38) The charge density and electron potential inside the muffin-tin spheres were expanded in terms of spherical harmonics up to =6, while in the interstitial region, these quantities were expanded in plane waves, with 28670 waves (energy up to 194.81 Ry) being included in the calculation. Total energies were calculated as a function of volume, for a k-mesh, corresponding to 165 k-vectors in the irreducible wedge of the Brillouin zone (BZ), and the results fitted to the Birch equation of state(39) to obtain the ground state properties. The elastic constants were obtained from the variation of the total energy under volume-conserving strains, as outlined in Ref. (40).

The phonon frequencies were calculated based on the density functional linear-response approach(41) within the GGA,(38) combined with the plane-wave pseudopotential method as implemented in the PWSCF package.(42) Norm-conserving pseudopotentials were used to describe the valence electrons. The BZ integrations were carried out with a Monkhorst-Pack (MP) grid.(43) Kinetic energy cutoffs of 90 Ry, 90 Ry and 120 Ry were used for CsCaF, BaLiF and KZnF, respectively. A mesh in the first BZ was used for the interpolation of the force constants needed for the phonon dispersion curve calculations. The theoretical equilibrium lattice constant of 4.096 Å, 4.604 Å, and 4.085 Å for KZnF, CsCaF and BaLiF, respectively, are obtained by total energy minimization and used in the phonon calculations.

The electronic structures of KZnF, CsCaF and BaLiF have been calculated with the quasiparticle self-consistent (QSGW) (where denotes the Greens function and denotes the screened Coulomb interaction) approximation,(44); (45) in the FP-LMTO implementation of Ref. (46). This calculation also included a double- LMTO basis set (), however additionally including the F and partial waves (treated as local orbitals(46)) for better description of the unoccupied band states. An MP k-mesh was used for the calculation of the screened interaction, which is evaluated in the random-phase approximation. Convergence tests showed only minor changes compared to a mesh. The most common type of calculations use a selfconsistent LDA or GGA band structure as input for the evaluation of the and operators. The special feature of the quasiparticle self-consistency in the QSGW method is that the one-particle band structure, which is used as input for the evaluation of the Greens’ function, is iterated in a further selfconsistency loop so as to come as close as possible to the output GW band structure.(44); (45) Though appealing from a theoretical point of view, this procedure has a tendency to slightly overestimate semiconductor gaps,(47) a fact traceable to the neglect of vertex corrections. As a simple remedy of this, Chantis et al.(47) suggested the hybrid QSGW (h-QSGW), by which the LDA and QSGW self energies are mixed in the proportions 20% - 80%. See Ref. (48) for a detailed description. Here we also apply the h-QSGW method to the fluorite perovskites to monitor its effect for this class of wide-gap insulators

## Iii Ground state, elastic properties and phonon dispersions

The calculated structural properties such as lattice parameters, and bulk moduli of KZnF, CsCaF and BaLiF are presented in Table I along with available experimentalal information and results of other calculations. In the case of KZnF the calculated lattice parameter within LDA is 0.9 lower when compared with the experimental value and the corresponding bulk modulus is 23 overestimated. In contrast, the calculated lattice parameter using GGA overestimates the experimental value by 2.4, and the bulk modulus is 6.8 too low. This is the usual trend observed in the LDA and GGA schemes. In the case of CsCaF the LDA calculated lattice constant is lower by 2.8 when compared to experiments, and the corresponding bulk modulus is 44 higher. Similarly, the GGA lattice parameter is 0.4 lower and the corresponding bulk modulus is 8% too large. A similar situation is seen in the case of BaLiF. The calculated bulk modulus is in excellent agreement with the two reported experimental values, which is, however, a bit fortuitous. Since the calculated equilibrium volume is overestimated (by about 0.8%) with GGA (and underestimated by about 2.6 % with LDA), an error solely depending on the error in volume is introduced in the calculated bulk modulus. Therefore we recalculated the bulk modulus also at the experimental volume in a manner similar to our earlier work(33) (see Table I). This diminishes the discrepancies between the LDA and GGA results, as expected. In addition, the LDA bulk modulus now becomes than the GGA for all the three compounds, and both functionals are seen to actually overestimate the bulk modulus. For KZnF two experimental determinations of the bulk modulus are at variance. In particular the high value for the pressure derivative of the bulk modulus () reported by Ref. (11) is at odds with the general range found for the fluoride perovskites by both experiments and theory. The authors suggest that the limited pressure range covered in their study may be the cause of their high fit value for . In Figure 1 the GGA and LDA equations of state of KZnF, CsCaF and BaLiF in the pressure range from 0 to 50 GPa are compared with the available experimental data.(11); (32) The agreement between theory and experiment is excellent for BaLiF in Fig. 1(c), while the agreement is less satisfactory for KZnF in Figure 1(a). A special feature in KZnF is the role of the Zn 3d states, which occur as a narrow resonance within the F valence bands reflecting their semi-localized character. Their bonding properties are usually ill described in LDA/GGA and this may be the cause of the discrepancy here.

The elastic constants of KZnF, CsCaF and BaLiF are presented in Table II along with the experimental values and other theoretical values. The LDA overestimates all the C, C and C by between 10 and 25, while the elastic constants obtained within GGA are much closer to the experimental values. As for the bulk modulus the elastic constants also depend sensitively on the volume, and therefore, the same argument as for the bulk modulus can be applied here. We have however, refrained from recalculating all the elastic constants with volume correction, but wish to mention that the excellent agreement between experiment and the GGA elastic constants should be interpreted with some care. Another point of caution is the fact that the calculated values pertain to 0 K, while experiments are performed at room temperature. An increase in temperature generally tends to reduce the elastic constants because of thermal expansion.

The calculated phonon dispersion curves at the optimized theoretical lattice parameters are shown in Figure 2. The available experimental inelastic neutron scattering frequencies(8); (28) are included and good agreement between theory and experiment is found. The dynamical stability of the cubic structures is confirmed by the absence of any imaginary modes in the calculated dispersions. There are 15 phonon branches in the full phonon dispersion since the unit cell consists of five atoms which give rise to three acoustic and twelve optical phonon branches. The agreement with experiment is excellent for BaLiF, and for the few frequencies determined for CsCaF, while there is some discrepancy to be seen for KZnF around the lowest frequencies at the M and R points. The calculations find a noticeable gap between the lowest optical mode and the highest acoustic mode in the calculations for BaLiF and CsCaF, which does not appear in KZnF. In addition, in CsCaF a large second gap appear at high frequency. The experiments have only managed to map the lowest frequencies and thus cannot confirm or discard this prediction. Evidently, it would be of great interest to obtain experimental phonon frequencies in the higher end of the spectrum for all three compounds, in particular to test the theoretical prediction that the phonon spectrum for CsCaF, BaLiF and KZnF contain two, one and zero gaps, respectively.

The phonon partial densities of states are also included to the right of Figures 2(a)-(c). The acoustic modes in BaLiF and CsCaF are dominated by the motion of the heavy Ba and Cs, respectively, while in KZnF the atomic masses are more similar, and the atomic vibrations are fully coupled to each other as shown in figure 2(a). Furthermore, the acoustic band width is smaller in CsCaF than in BaLiF, which reflects the larger lattice constant of CsCaF, while the second gap (between 8 and 11 THz) in the phonon spectrum of CsCaF, which does not find a counterpart in BaLiF, reflects the different masses of Ca and Li. Thus, there is a significant Li component to the highest frequencies in BaLiF, while no Ca weight is found in the highest CsCaF frequency range.

## Iv Quasiparticle Band Structures

The QSGW band structures of KZnF, CsCaF and BaLiF are presented in Figure 3. All three compounds exhibit large gaps, of 10.0 eV, 11.8 eV and 11.8 eV, respectively, which are significantly larger than the gaps calculated within the LDA (Table III). The experimental gap has only been reported for the case of BaLiF to be 8.4 eV,(4) which is 3.4 eV lower than the calculated QSGW value. The h-QSGW reduces the calculated gaps by about 1 eV (Table III), i. e. this can not account for the discrepancy. There exists the possibility that the ideal perovskite structure assumed in the calculations is not the same as that of the crystals grown in the laboratory, which may be hampered by intrinsic defects etc.

The conduction band minimimum occurs in all cases at the point, while the valence band maximum (VBM) also falls at for BaLiF, i.e., this is a direct gap insulator. The three top valence states at , M and R are rather close, with M (R) at -0.17 eV (-0.20 eV) relative to the VBM in BaLiF. In contrast, for CsCaF and KZnF the gap is indirect. In CsCaF the VBM occurs at the M point while the topmost valence state at is about 0.25 eV lower in energy. In KZnF the VBM occurs on the line connecting the M and R points, and the topmost valence state at is 1.5 eV lower in energy.

The F band widths are 4.0 eV, 2.1 eV and 3.2 eV in KZnF, CsCaF and BaLiF, respectively. The low band width of CsCaF reflects its larger lattice constant. KZnF and BaLiF have similar lattice constants, however in KZnF the Zn 3d bands falls at the bottom of the F bands partially hybridizing and thus effectively enhancing the F band width. The semicore states are slightly shifted towards higher binding energies by the GW approximation compared to their LDA positions, by about 2 eV for the K 3p states of KZnF (located at 12 eV below VBM), by about 0.1 eV for the Cs 5p states in CsCaF (located at 4.5 eV below VBM), and by about 0.6 eV for the Ba 5p states of BaLiF (located at 9 eV below VBM). Thus the most significant effect of the GW approximation is the more or less rigid shift of the conduction bands with respect to the valence bands, which is of the order of 5-7 eV for the three fluorides considered.

## V Conclusions

The electronic structures and vibrational properties of the KZnF, CsCaF and BaLiF perovskite fluorides have been investigated with density functional methods. The calculated phonon frequencies are in good agreement with experimental values, which however are limited to the lowest part of the spectrum, and a full experimental mapping of the phonon dispersion curves is desirable. The calculated pV-curve of BaLiF is in excellent agreement with experiment, while the agreement between theory and experiment is less satisfactory for KZnF, and no experimental data exist for CsCaF.

The QSGW quasiparticle band structures reveal significant insulating gaps in the fluorides studied, which are about 6 eV larger than predicted by the LDA/GGA. Most of this effect may be seen as a rigid shift of the conduction bands with respect to the valence bands. The F valence bands are only slightly wider in QSGW than in LDA, while some shifts towards higher binding energies are found for the secondary (semi-core) valence states. All of these effects are usual trends of the GW method, which are also observed in comparison of Hartree-Fock and LDA band structures, hence they may be ascribed to the non-local character of the electron-electron interaction.

## Vi Acknowledgements

G.V acknowledges, CMSD-University of Hyderabad for providing the computational facility.
Author for Correspondence, E-mail: vaithee@uohyd.ac.in

### References

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blob: 2b5492c31ae8f164b8593948c5381f7c685b5525 [file] [log] [blame]
 \begin{comment} Licensed to the Apache Software Foundation (ASF) under one or more contributor license agreements. See the NOTICE file distributed with this work for additional information regarding copyright ownership. The ASF licenses this file to you under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. \end{comment} \subsection{K-Means Clustering} \noindent{\bf Description} \smallskip Given a collection of $n$ records with a pairwise similarity measure, the goal of clustering is to assign a category label to each record so that similar records tend to get the same label. In contrast to multinomial logistic regression, clustering is an \emph{unsupervised}\/ learning problem with neither category assignments nor label interpretations given in advance. In $k$-means clustering, the records $x_1, x_2, \ldots, x_n$ are numerical feature vectors of $\dim x_i = m$ with the squared Euclidean distance $\|x_i - x_{i'}\|_2^2$ as the similarity measure. We want to partition $\{x_1, \ldots, x_n\}$ into $k$ clusters $\{S_1, \ldots, S_k\}$ so that the aggregated squared distance from records to their cluster means is minimized: \begin{equation} \textrm{WCSS}\,\,=\,\, \sum_{i=1}^n \,\big\|x_i - \mean(S_j: x_i\in S_j)\big\|_2^2 \,\,\to\,\,\min \label{eqn:WCSS} \end{equation} The aggregated distance measure in~(\ref{eqn:WCSS}) is called the \emph{within-cluster sum of squares}~(WCSS). It can be viewed as a measure of residual variance that remains in the data after the clustering assignment, conceptually similar to the residual sum of squares~(RSS) in linear regression. However, unlike for the RSS, the minimization of~(\ref{eqn:WCSS}) is an NP-hard problem~\cite{AloiseDHP2009:kmeans}. Rather than searching for the global optimum in~(\ref{eqn:WCSS}), a heuristic algorithm called Lloyd's algorithm is typically used. This iterative algorithm maintains and updates a set of $k$~\emph{centroids} $\{c_1, \ldots, c_k\}$, one centroid per cluster. It defines each cluster $S_j$ as the set of all records closer to~$c_j$ than to any other centroid. Each iteration of the algorithm reduces the WCSS in two steps: \begin{Enumerate} \item Assign each record to the closest centroid, making $\mean(S_j)\neq c_j$; \label{step:kmeans:recluster} \item Reset each centroid to its cluster's mean: $c_j := \mean(S_j)$. \label{step:kmeans:recenter} \end{Enumerate} After Step~\ref{step:kmeans:recluster} the centroids are generally different from the cluster means, so we can compute another within-cluster sum of squares'' based on the centroids: \begin{equation} \textrm{WCSS\_C}\,\,=\,\, \sum_{i=1}^n \,\big\|x_i - \mathop{\textrm{centroid}}(S_j: x_i\in S_j)\big\|_2^2 \label{eqn:WCSS:C} \end{equation} This WCSS\_C after Step~\ref{step:kmeans:recluster} is less than the means-based WCSS before Step~\ref{step:kmeans:recluster} (or equal if convergence achieved), and in Step~\ref{step:kmeans:recenter} the WCSS cannot exceed the WCSS\_C for \emph{the same} clustering; hence the WCSS reduction. Exact convergence is reached when each record becomes closer to its cluster's mean than to any other cluster's mean, so there are no more re-assignments and the centroids coincide with the means. In practice, iterations may be stopped when the reduction in WCSS (or in WCSS\_C) falls below a minimum threshold, or upon reaching the maximum number of iterations. The initialization of the centroids is also an important part of the algorithm. The smallest WCSS obtained by the algorithm is not the global minimum and varies depending on the initial centroids. We implement multiple parallel runs with different initial centroids and report the best result. \Paragraph{Scoring} Our scoring script evaluates the clustering output by comparing it with a known category assignment. Since cluster labels have no prior correspondence to the categories, we cannot count correct'' and wrong'' cluster assignments. Instead, we quantify them in two ways: \begin{Enumerate} \item Count how many same-category and different-category pairs of records end up in the same cluster or in different clusters; \item For each category, count the prevalence of its most common cluster; for each cluster, count the prevalence of its most common category. \end{Enumerate} The number of categories and the number of clusters ($k$) do not have to be equal. A same-category pair of records clustered into the same cluster is viewed as a true positive,'' a different-category pair clustered together is a false positive,'' a same-category pair clustered apart is a false negative''~etc. \smallskip \noindent{\bf Usage: K-means Script} \smallskip {\hangindent=\parindent\noindent\it% {\tt{}-f }path/\/{\tt{}Kmeans.dml} {\tt{} -nvargs} {\tt{} X=}path/file {\tt{} C=}path/file {\tt{} k=}int {\tt{} runs=}int {\tt{} maxi=}int {\tt{} tol=}double {\tt{} samp=}int {\tt{} isY=}int {\tt{} Y=}path/file {\tt{} fmt=}format {\tt{} verb=}int } \smallskip \noindent{\bf Usage: K-means Scoring/Prediction} \smallskip {\hangindent=\parindent\noindent\it% {\tt{}-f }path/\/{\tt{}Kmeans-predict.dml} {\tt{} -nvargs} {\tt{} X=}path/file {\tt{} C=}path/file {\tt{} spY=}path/file {\tt{} prY=}path/file {\tt{} fmt=}format {\tt{} O=}path/file } \smallskip \noindent{\bf Arguments} \begin{Description} \item[{\tt X}:] Location to read matrix $X$ with the input data records as rows \item[{\tt C}:] (default:\mbox{ }{\tt "C.mtx"}) Location to store the output matrix with the best available cluster centroids as rows \item[{\tt k}:] Number of clusters (and centroids) \item[{\tt runs}:] (default:\mbox{ }{\tt 10}) Number of parallel runs, each run with different initial centroids \item[{\tt maxi}:] (default:\mbox{ }{\tt 1000}) Maximum number of iterations per run \item[{\tt tol}:] (default:\mbox{ }{\tt 0.000001}) Tolerance (epsilon) for single-iteration WCSS\_C change ratio \item[{\tt samp}:] (default:\mbox{ }{\tt 50}) Average number of records per centroid in data samples used in the centroid initialization procedure \item[{\tt Y}:] (default:\mbox{ }{\tt "Y.mtx"}) Location to store the one-column matrix $Y$ with the best available mapping of records to clusters (defined by the output centroids) \item[{\tt isY}:] (default:\mbox{ }{\tt 0}) {\tt 0} = do not write matrix~$Y$, {\tt 1} = write~$Y$ \item[{\tt fmt}:] (default:\mbox{ }{\tt "text"}) Matrix file output format, such as {\tt text}, {\tt mm}, or {\tt csv}; see read/write functions in SystemML Language Reference for details. \item[{\tt verb}:] (default:\mbox{ }{\tt 0}) {\tt 0} = do not print per-iteration statistics for each run, {\tt 1} = print them (the verbose'' option) \end{Description} \smallskip \noindent{\bf Arguments --- Scoring/Prediction} \begin{Description} \item[{\tt X}:] (default:\mbox{ }{\tt " "}) Location to read matrix $X$ with the input data records as rows, optional when {\tt prY} input is provided \item[{\tt C}:] (default:\mbox{ }{\tt " "}) Location to read matrix $C$ with cluster centroids as rows, optional when {\tt prY} input is provided; NOTE: if both {\tt X} and {\tt C} are provided, {\tt prY} is an output, not input \item[{\tt spY}:] (default:\mbox{ }{\tt " "}) Location to read a one-column matrix with the externally specified true'' assignment of records (rows) to categories, optional for prediction without scoring \item[{\tt prY}:] (default:\mbox{ }{\tt " "}) Location to read (or write, if {\tt X} and {\tt C} are present) a column-vector with the predicted assignment of rows to clusters; NOTE: No prior correspondence is assumed between the predicted cluster labels and the externally specified categories \item[{\tt fmt}:] (default:\mbox{ }{\tt "text"}) Matrix file output format for {\tt prY}, such as {\tt text}, {\tt mm}, or {\tt csv}; see read/write functions in SystemML Language Reference for details \item[{\tt O}:] (default:\mbox{ }{\tt " "}) Location to write the output statistics defined in Table~\ref{table:kmeans:predict:stats}, by default print them to the standard output \end{Description} \begin{table}[t]\small\centerline{% \begin{tabular}{|lcl|} \hline Name & CID & Meaning \\ \hline {\tt TSS} & & Total Sum of Squares (from the total mean) \\ {\tt WCSS\_M} & & Within-Cluster Sum of Squares (means as centers) \\ {\tt WCSS\_M\_PC} & & Within-Cluster Sum of Squares (means), in \% of TSS \\ {\tt BCSS\_M} & & Between-Cluster Sum of Squares (means as centers) \\ {\tt BCSS\_M\_PC} & & Between-Cluster Sum of Squares (means), in \% of TSS \\ \hline {\tt WCSS\_C} & & Within-Cluster Sum of Squares (centroids as centers) \\ {\tt WCSS\_C\_PC} & & Within-Cluster Sum of Squares (centroids), \% of TSS \\ {\tt BCSS\_C} & & Between-Cluster Sum of Squares (centroids as centers) \\ {\tt BCSS\_C\_PC} & & Between-Cluster Sum of Squares (centroids), \% of TSS \\ \hline {\tt TRUE\_SAME\_CT} & & Same-category pairs predicted as Same-cluster, count \\ {\tt TRUE\_SAME\_PC} & & Same-category pairs predicted as Same-cluster, \% \\ {\tt TRUE\_DIFF\_CT} & & Diff-category pairs predicted as Diff-cluster, count \\ {\tt TRUE\_DIFF\_PC} & & Diff-category pairs predicted as Diff-cluster, \% \\ {\tt FALSE\_SAME\_CT} & & Diff-category pairs predicted as Same-cluster, count \\ {\tt FALSE\_SAME\_PC} & & Diff-category pairs predicted as Same-cluster, \% \\ {\tt FALSE\_DIFF\_CT} & & Same-category pairs predicted as Diff-cluster, count \\ {\tt FALSE\_DIFF\_PC} & & Same-category pairs predicted as Diff-cluster, \% \\ \hline {\tt SPEC\_TO\_PRED} & $+$ & For specified category, the best predicted cluster id \\ {\tt SPEC\_FULL\_CT} & $+$ & For specified category, its full count \\ {\tt SPEC\_MATCH\_CT} & $+$ & For specified category, best-cluster matching count \\ {\tt SPEC\_MATCH\_PC} & $+$ & For specified category, \% of matching to full count \\ {\tt PRED\_TO\_SPEC} & $+$ & For predicted cluster, the best specified category id \\ {\tt PRED\_FULL\_CT} & $+$ & For predicted cluster, its full count \\ {\tt PRED\_MATCH\_CT} & $+$ & For predicted cluster, best-category matching count \\ {\tt PRED\_MATCH\_PC} & $+$ & For predicted cluster, \% of matching to full count \\ \hline \end{tabular}} \caption{The {\tt O}-file for {\tt Kmeans-predict} provides the output statistics in CSV format, one per line, in the following format: (NAME, [CID], VALUE). Note: the 1st group statistics are given if {\tt X} input is available; the 2nd group statistics are given if {\tt X} and {\tt C} inputs are available; the 3rd and 4th group statistics are given if {\tt spY} input is available; only the 4th group statistics contain a nonempty CID value; when present, CID contains either the specified category label or the predicted cluster label.} \label{table:kmeans:predict:stats} \end{table} \noindent{\bf Details} \smallskip Our clustering script proceeds in 3~stages: centroid initialization, parallel $k$-means iterations, and the best-available output generation. Centroids are initialized at random from the input records (the rows of~$X$), biased towards being chosen far apart from each other. The initialization method is based on the {\tt k-means++} heuristic from~\cite{ArthurVassilvitskii2007:kmeans}, with one important difference: to reduce the number of passes through~$X$, we take a small sample of $X$ and run the {\tt k-means++} heuristic over this sample. Here is, conceptually, our centroid initialization algorithm for one clustering run: \begin{Enumerate} \item Sample the rows of~$X$ uniformly at random, picking each row with probability $p = ks / n$ where \begin{Itemize} \item $k$~is the number of centroids, \item $n$~is the number of records, and \item $s$~is the {\tt samp} input parameter. \end{Itemize} If $ks \geq n$, the entire $X$ is used in place of its sample. \item Choose the first centroid uniformly at random from the sampled rows. \item Choose each subsequent centroid from the sampled rows, at random, with probability proportional to the squared Euclidean distance between the row and the nearest already-chosen centroid. \end{Enumerate} The sampling of $X$ and the selection of centroids are performed independently and in parallel for each run of the $k$-means algorithm. When we sample the rows of~$X$, rather than tossing a random coin for each row, we compute the number of rows to skip until the next sampled row as $\lceil \log(u) / \log(1 - p) \rceil$ where $u\in (0, 1)$ is uniformly random. This time-saving trick works because \begin{equation*} \Prob [k-1 < \log_{1-p}(u) < k] \,\,=\,\, p(1-p)^{k-1} \,\,=\,\, \Prob [\textrm{skip $k-1$ rows}] \end{equation*} However, it requires us to estimate the maximum sample size, which we set near~$ks + 10\sqrt{ks}$ to make it generous enough. Once we selected the initial centroid sets, we start the $k$-means iterations independently in parallel for all clustering runs. The number of clustering runs is given as the {\tt runs} input parameter. Each iteration of each clustering run performs the following steps: \begin{Itemize} \item Compute the centroid-dependent part of squared Euclidean distances from all records (rows of~$X$) to each of the $k$~centroids using matrix product; \item Take the minimum of the above for each record; \item Update the current within-cluster sum of squares (WCSS) value, with centroids substituted instead of the means for efficiency; \item Check the convergence criterion:\hfil $\textrm{WCSS}_{\mathrm{old}} - \textrm{WCSS}_{\mathrm{new}} < \eps\cdot\textrm{WCSS}_{\mathrm{new}}$\linebreak as well as the number of iterations limit; \item Find the closest centroid for each record, sharing equally any records with multiple closest centroids; \item Compute the number of records closest to each centroid, checking for runaway'' centroids with no records left (in which case the run fails); \item Compute the new centroids by averaging the records in their clusters. \end{Itemize} When a termination condition is satisfied, we store the centroids and the WCSS value and exit this run. A run has to satisfy the WCSS convergence criterion to be considered successful. Upon the termination of all runs, we select the smallest WCSS value among the successful runs, and write out this run's centroids. If requested, we also compute the cluster assignment of all records in~$X$, using integers from 1 to~$k$ as the cluster labels. The scoring script can then be used to compare the cluster assignment with an externally specified category assignment. \smallskip \noindent{\bf Returns} \smallskip We output the $k$ centroids for the best available clustering, i.~e.\ whose WCSS is the smallest of all successful runs. The centroids are written as the rows of the $k\,{\times}\,m$-matrix into the output file whose path/name was provided as the {\tt C}'' input argument. If the input parameter {\tt isY}'' was set to~{\tt 1}, we also output the one-column matrix with the cluster assignment for all the records. This assignment is written into the file whose path/name was provided as the {\tt Y}'' input argument. The best WCSS value, as well as some information about the performance of the other runs, is printed during the script execution. The scoring script {\tt Kmeans-predict} prints all its results in a self-explanatory manner, as defined in Table~\ref{table:kmeans:predict:stats}. \smallskip \noindent{\bf Examples} \smallskip {\hangindent=\parindent\noindent\tt \hml -f Kmeans.dml -nvargs X=/user/biadmin/X.mtx k=5 C=/user/biadmin/centroids.mtx fmt=csv } {\hangindent=\parindent\noindent\tt \hml -f Kmeans.dml -nvargs X=/user/biadmin/X.mtx k=5 runs=100 maxi=5000 tol=0.00000001 samp=20 C=/user/biadmin/centroids.mtx isY=1 Y=/user/biadmin/Yout.mtx verb=1 } \noindent To predict {\tt Y} given {\tt X} and {\tt C}: {\hangindent=\parindent\noindent\tt \hml -f Kmeans-predict.dml -nvargs X=/user/biadmin/X.mtx C=/user/biadmin/C.mtx prY=/user/biadmin/PredY.mtx O=/user/biadmin/stats.csv } \noindent To compare actual'' labels {\tt spY} with predicted'' labels given {\tt X} and {\tt C}: {\hangindent=\parindent\noindent\tt \hml -f Kmeans-predict.dml -nvargs X=/user/biadmin/X.mtx C=/user/biadmin/C.mtx spY=/user/biadmin/Y.mtx O=/user/biadmin/stats.csv } \noindent To compare actual'' labels {\tt spY} with given predicted'' labels {\tt prY}: {\hangindent=\parindent\noindent\tt \hml -f Kmeans-predict.dml -nvargs spY=/user/biadmin/Y.mtx prY=/user/biadmin/PredY.mtx O=/user/biadmin/stats.csv } \smallskip \noindent{\bf References} \begin{itemize} \item D.~Aloise, A.~Deshpande, P.~Hansen, and P.~Popat. \newblock {NP}-hardness of {E}uclidean sum-of-squares clustering. \newblock {\em Machine Learning}, 75(2):245--248, May 2009. \item D.~Arthur and S.~Vassilvitskii. \newblock {\tt k-means++}: The advantages of careful seeding. \newblock In {\em Proceedings of the 18th Annual {ACM-SIAM} Symposium on Discrete Algorithms ({SODA}~2007)}, pages 1027--1035, New Orleans~{LA}, {USA}, January 7--9 2007. \end{itemize}


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# Petya and Rectangle

Time Limit: 5 seconds

Memory Limit: 256 megabytes

## Description

Little Petya loves playing with rectangles. Mom bought Petya a rectangle divided into cells n × m in size (containing n rows, m columns). Petya marked two different cells of the rectangle and now he is solving the following task:

Let's define a simple path between those two cells as a sequence of distinct cells a1, a2, ..., ak, where a1 and ak are the two marked cells. Besides, ai and ai + 1 are side-neighboring cells of the path (1 ≤ i < k). Let's denote the path length as number k (the sequence length).

Petya's task is to find the longest simple path's length and to print the path. Help him.

## Input

The first line contains space-separated integers n and m (4 ≤ n, m ≤ 1000) — the number of rows and the number of columns in the rectangle, correspondingly. The second line contains space-separated integers x1 and y1 — the coordinates of the first marked cell. The third line contains space-separated integers x2 y2 — the coordinates of the second marked cell (1 < x1, x2 < n, 1 < y1, y2 < m, x1 ≠ x2, y1 ≠ y2).

The coordinates of a marked cell are a pair of integers x y, where x represents the row's number and y represents the column's number. The rows are numbered from top to bottom with consecutive integers from 1 to n. The columns are numbered from the left to the right by consecutive integers from 1 to m.

It is guaranteed that the marked cells are not positioned in one row or column.

## Output

In the first line print the length of the found path — k. In the next lines print k pairs of integers, one per line — coordinates of the cells that constitute the found path in the order, in which they follow in the path (the path must go from cell (x1, y1) to cell (x2, y2)). If there are several solutions, print any of them.

## Sample Input

Input4 42 23 3Output152 21 21 12 13 14 14 24 34 43 42 41 41 32 33 3

## Sample Output

None

## Hint

The statement test is described in the picture:

None


text:
# I Integers as an Ordered Integral Domain ... Bloch Th. 1.4.6

1. Aug 3, 2017

### Math Amateur

I am reading Ethan D. Bloch's book: The Real Numbers and Real Analysis ...

I am currently focused on Section 1.4: Entry 2: Axioms for the Integers ... In this section Bloch defines the integers as an ordered integral domain that satisfies the Well Ordering Principle ... rather than defining the integers via the natural numbers ...

I need help/clarification with an aspect of Theorem 1.4.6 ...

Theorem 1.4.6 and the start of the proof reads as follows:

In the above proof ... near the start of the proof, we read the following:

" ... ... From the definition of $\mathbb{N}$, we observe that $S \subseteq \mathbb{N}$. ... ..."

Question: What exactly is the reasoning that allows us to conclude that $S \subseteq \mathbb{N}$ from the definition on $\mathbb{N}$ ... "

The above theorem is in the section where Bloch defines the integers as an ordered integral domain that satisfies the Well Ordering Principle... ... as follows:

The definition of the natural numbers is mentioned above ... Bloch's definition is as follows ...

Hope someone can help,

Peter

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2. Aug 3, 2017

### Math Amateur

Maybe I shouldn't answer my own questions ... but maybe I am worrying too much over a trivial point ... ... maybe the reasoning is simply ... as follows ...since $S$ is a set made up of positive integers then it is a subset of $\mathbb{N}$ ... is it as simple as that ..?

My apologies if it is that simple ...

Peter

3. Aug 3, 2017

### Staff: Mentor

As I read it, yes. However, I do not see immediately from the Peano axiom how the natural numbers are all positive, but this depends on the definition of the order as well. So somewhere between 1.2.1. and 1.4.6. the author should have shown (or defined), that natural numbers are positive.

4. Aug 4, 2017

### Math Amateur

Thanks fresh_42 ... indeed my problem was how to derive $S \subseteq \mathbb{N}$ ... but got confused (with Bloch's help ... ...)

To explain ...

Bloch investigates two approaches to defining/constructing the integers as he describes here ...

In Section 1.4, where Theorem 1.4.6 occurs, Bloch is expounding the ordered integral domain approach to the integers ... so we should not go back to the Peano Postulates as I did - that is his approach number 1 ... under the second approach, the ordered integral domain approach, the Peano Postulates/Axioms become a theorem and are proved ...

Actually when we meet $\mathbb{N}$ in the proof of Theorem 1.4.6 Bloch has not defined $\mathbb{N}$ yet in this approach ... he does so after presenting Theorem 1.4.6 as follows:

Interestingly as Evgeny Makarov has pointed out in the Math Help Forum, the claim $S \subseteq \mathbb{N}$ seems unnecessary for the rest of the proof ...

Peter

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5. Aug 4, 2017

### Staff: Mentor

It is not, because he needs $p \in \mathbb{N}$ provided by the well-ordering principle of $S \subseteq \mathbb{N}.$ Otherwise he wouldn't have the laws available, which he applies on $p,$ and which I assume were provided by the axioms (or derived statements) of the natural numbers.

6. Aug 4, 2017

### Math Amateur

Hi fresh_42,

Well ... I do not think that that is the case ...

Maybe I did not make clear the approach to the integers that was relevant to Theorem 1.4.6 ...

Bloch is, in Section 1.4, dealing with the approach to defining/constructing the integers via defining the integers as an ordered integral domain that satisfies the Well-Ordering Principle ... not the approach through the natural numbers (see my previous post above). In the ordered integral domain approach the natural numbers the natural numbers are 'found'/defined as an embedded set within the integers. The relevant definition for the natural numbers is given in Definition 1.4.7 - unfortunately presented after Theorem 1.4.6. The definition reads as follows:

Once we have defined the integers as an ordered integral domain we have the following properties to call on in Theorem 1.4.6 ... and these properties include those named in the proof of the Theorem ...

Peter

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7. Aug 4, 2017

### Staff: Mentor

One needs the well-ordering to get $p \in S$ (definition 1.4.3). Together with definition 1.4.7. this abbreviates to $S \subseteq \mathbb{N}$. It is just another language as to say all elements of $S$ are positive. So whether you say all elements of $S$ are natural numbers or all elements of $S$ are positive, doesn't make a difference, it is the same. And which ever way you look at it, one of them is needed. So the language might be redundant, the fact is not.

8. Aug 4, 2017

### Math Amateur

Hi fresh_42 ... thanks for the post ...

Yes ... see the point you are making ...

Peter


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# Branching Tree Diagram – Reading trees: A fast review

Posted on

Imagine the particular phylogenetic tree as a child’s cellular hanging from its root.. But it only displays the relationships between clades using the common ancestor..

Branching Tree Diagram – Just like your family shrub, an evolutionary tree indicates which usually ancestors gave rise to which descendants. A cladogram is a diagram which usually shows the relationship between different microorganisms based on their different similarities. A phylogenetic tree is a diagram which displays the phylogenetic history of organisms with regards to the geological time scale. It symbolizes possible relationships among organisms as well as the evolutionary history.
Often , that will exercise will demonstrate that a deceptive interpretation must be wrong! Click here to get more details on common misconceptions related to phylogenetics.
Trees depict evolutionary relationships, not really evolutionary progress. It’s easy to think that taxa that appear near one aspect of a phylogenetic tree are more superior than other organisms on the shrub, but this is simply not the case. 1st, the idea of evolutionary “advancement” is not a really scientific idea. There is no unbiased, general scale for “advancement. ” 2nd, taxa with extreme versions associated with traits (which might be perceived as a lot more “advanced”) may occur on any kind of terminal branch. The position of a airport terminal taxon is not an indication of how adaptive, specialized, or extreme its attributes are.

What are the highlights of a cladogram? + Example. Note that each outcome (the figures $$\text1$$ to $$\text6$$) is demonstrated at the end of a line; and that the particular probability of each outcome (all $$\frac16$$ in this case) is shown proven on a line. The probabilities have to equal to $$\text1$$ in order to cover all of the achievable outcomes. In the examples below, we will have how to draw tree diagrams along with multiple events and how to compute possibilities using the diagrams. All of the species of microorganisms that are alive today have originated from ancestral species. The department length of a cladogram strictly speaking will not represent genetic changes or period of time.


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# Help with a system of inequalities with absolute values

I'm trying to solve this system on inequalities

$$\left\{ \begin{array}{c} |x-3|<2x \\ |2x+5|>3 \end{array} \right.$$

The steps I'm taking are:

Finding the absolute values sings, so for $$x-3 \geq 0$$ we have $$x \geq 3$$ therefore $$|x-3| = \left\{ \begin{array}{c} x-3 & \text{for x \geq 3} \\ -x+3 & \text{for x < 3} \end{array} \right.$$

and

$$2x+5 \geq 0$$ we have $$x \geq \frac{-2}{5}$$ therefore $$|2x+5| = \left\{ \begin{array}{c} 2x+5 & \text{for x \geq \frac{-2}{5}} \\ -2x-5 & \text{for x<\frac{-2}{5}} \\ \end{array} \right.$$

So I build a few systems with the complete inequalities, for the first one we have:

$$\left\{ \begin{array}{c} x \geq 3 \\ x-3<2x = x>-3 \end{array} \right.$$

So the solution here would be $$x>3$$, then:

$$\left\{ \begin{array}{c} x<3 \\ -x+3<2x = x>1 \end{array} \right.$$

The solution would be $$1. Then

$$\left\{ \begin{array}{c} x \geq \frac{-2}{5} \\ 2x+5>3 = x>-1 \end{array} \right.$$

So the solution of the system is $$x>-1$$, then

$$\left\{ \begin{array}{c} x< \frac{-2}{5} \\ -2x-5>3 = x<-4 \end{array} \right.$$

And the solution is $$x<-4$$

Now the solution my book gives is x>1 for the initial system. But I can't find that one. I can't get a solution at all. I tried finding a common point between the 4 solutions I found (as if it was a 4-inequalities system), but there isn't one really. What am I doing wrong?

• either plot the graphs of the inequalities or square both the inequalties and solve the resulting quadratic inequalities – vidyarthi Nov 19 '18 at 17:39
• Square $|x-3|<2x$ to $x^2+9-6x<4x^2$ you mean? – Paul Nov 19 '18 at 17:48
• yes, exactly. Now, solve the inequality by factoring the quadratic – vidyarthi Nov 19 '18 at 17:50
• Is this the only way to solve this? I don't really understand how squaring the inequalities takes away the absolute value. – Paul Nov 19 '18 at 17:52
• since square of a real is always positive, so the absolute value is easily removed – vidyarthi Nov 19 '18 at 17:53

Since $$2x>|x-3|$$ we get $$x>0$$ so $$2x+5>0$$ and so $$2x+5>3$$ so $$x>-2$$ which is nothing new. So $$x>0$$ and $$|x-3|<2x$$ so after squaring we get $$x^2-6x+9<4x^2\implies 3x^2+6x-9>0$$

or $$(x+3)(x-1)>0\implies x-1>0$$ so $$\boxed{x>1}$$.

• so you reduced two squarings to just one! great answer – vidyarthi Nov 19 '18 at 18:10

We have $$|x-3|<2x\implies 3x^2+6x-9>0\implies (x+3)(x-1)>0$$ and $$|2x+5|>3\implies 4x^2+20x+16>0\implies (x+4)(x+1)>0$$. I think you could proceed now? For a quick method on how to solve further, see here

• If I try to graph this I get x<-4 and x>1 as solutions. It should only be x>1. Where am I wrong? – Paul Nov 19 '18 at 19:23
• @Paul yes, by graphing technique, you get the answer as $x<-4$ or $x>1$ as answers. Here you have to use the fact that $|x-3|<2x\implies x\ge0$ – vidyarthi Nov 22 '18 at 7:08

Since $$0\le\left|x-3\right|$$ and $$\left|x-3\right|\le2x$$ we conclude $$2x\ge0$$ and then $$x\ge0$$. From $$\left|x-3\right|<2x$$ and since $$x\ge0$$, we get $$-2x< x-3<2x$$ and therefore we have $$\left\{\begin{array}{c}x-3<2x \\x-3>-2x\end{array}\right.$$ and therefore $$\left\{\begin{array}{c}x>-3 \\3x>3\end{array}\right.$$ and then $$\left\{\begin{array}{c}x>-3 \\x>1\end{array}\right.$$, so $$x>1$$ which satisfies the initial statement $$x\ge0$$. So far we see $$x$$ has to be greater than $$1$$ to satisfy $$\left|x-3\right|<2x$$. Let us consider the next inequality $$|2x+5|>3$$ which leads to $$2x+5>3$$ or $$2x+5<-3$$. Then we get $$x>-1$$ or $$x<-4$$. And finally if we combine this with the result of first inequality, $$x>1$$ we get $$x>1$$.

There is actually only the first inequality to be solved:

$$|2x+5| \stackrel{\color{blue}{0}\leq|x-3| \color{blue}{< 2x}}{>} |x-3|+5 > 3$$ So, you only need to solve $$|x-3| < 2x$$ while $$x>0$$: $$-2x < x-3 < 2x \Leftrightarrow \begin{cases} 3x > 3 \Leftrightarrow \color{blue}{x> 1} \\ x > -3 \mbox{ does not extend the solution} \end{cases}$$


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# $\mathbb{CP}^1$ is compact?

$\mathbb{CP}^1$ is the set of all one dimensional subspaces of $\mathbb{C}^2$, if $(z,w)\in \mathbb{C}^2$ be non zero , then its span is a point in $\mathbb{CP}^1$.let $U_0=\{[z:w]:z\neq 0\}$ and $U_1=\{[z:w]:w\neq 0\}$, $(z,w)\in \mathbb{C}^2$,and $[z:w]=[\lambda z:\lambda w],\lambda\in\mathbb{C}^{*}$ is a point in $\mathbb{CP}^1$, the map is $\phi_0:U_0\rightarrow\mathbb{C}$ defined by $$\phi_0([z:w])=w/z$$ the map $\phi:U_1\rightarrow\mathbb{C}$ defined by $$\phi_1([z:w])=z/w$$ Now,Could any one tell me why $\mathbb{CP}^1$ is the union of two closed sets $\phi_0^{-1}(D)$ and $\phi_1^{-1}(D)$, where $D$ is closed unit disk in $\mathbb{C}$,and why $\mathbb{CP}^1$ is compact?

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What is the topology over $\mathbb{CP}^1$? –  Seirios Jul 25 '12 at 9:31
Dear Patience, seeing that you have asked 11 questions (about Riemann surfaces) in two says, you might like to know that there is a limit of 50 questions you can ask per month: see here –  wildildildlife Jul 25 '12 at 12:10

The maps you gave are the coordinate charts on $\mathbb{C}\mathbb{P}^1$ that makes it into a manifold. In particular, they are bijective.

If we take $\phi_0^{-1}(D)$, we get all point $[1:z]$, with $z\in D$. Similarly, $\phi_0^{-1}(D)$ is all points of the form $[z:1]$. Together, these sets cover $\mathbb{C}\mathbb{P}^1$. The inverse maps are continuous, because the maps $\phi_i$ are bijective, so $\phi_0^{-1}(D)$ and $\phi_1^{-1}(D)$ are compact, as the image of a compact set under a continuous map is compact. It's not hard to show that if a space is a union of two compact sets, it is compact, so we are done.

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these are holomorphic bijection right? –  El Angel Exterminador Jul 25 '12 at 9:40
Which page is this on in Miranda? There's something I want to check. –  Potato Jul 25 '12 at 9:42
page-8&9.The Projective Line –  El Angel Exterminador Jul 25 '12 at 9:43
I think your argument is ok! –  El Angel Exterminador Jul 25 '12 at 9:45
Ok, nevermind, this is right. To answer your question: it doesn't make sense to speak of coordinate charts being holomophic bijections. You want the transition functions to be holomorphic. –  Potato Jul 25 '12 at 9:47

The complex projective line is the union of the two open subsets $$D_i=\{[z_0:z_1]\text{ such that }z_i\neq0\},\quad i=0, 1.$$ Each of them is homeomorphic to $\Bbb C$, hence to the open unit disk.

Compacteness follows from the fact that it is homeomorphic to the sphere $S^2$. You can see this as a simple elaboration of the observation that $$\Bbb P^1(\Bbb C)=D_0\cup[0:1].$$

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The first has been answered, but i would like to add some details. So you do see that the $U_i = \phi_i^{-1}(\mathbb{C})$ for $i=0,1$ cover $\mathbb{C} P^1$ right? My claim is now that we can restrict to the closed disks in $\mathbb{C}$. The cases $[0:1]$ and $[1:0]$ are simple and the only points outside $U_{0,1} := U_0 \cap U_1$, so we restict to $[z:w] \in U_{0,1}$. Now both $\phi_i$ are defined at this point, and since $\phi_0([z:w]) = \frac{1}{\phi_1([z:w])}$ we can pick the one that has norm smaller or equal to one, so that the image lies in the unit disk. Since the point was arbitrary, every point has as $i$ such that its $\phi_i$ image lies in the unit disk, q.e.d.

Regarding the second one, recall that the continuous image of a compact space is compact. Moreover it is well known that the sphere $S \subset \mathbb{C}^2$ is compact, and that $\mathbb{C}P^1$ is a quotient of this sphere. Summarizing, there is a surjective quotient map $\pi: S \rightarrow \mathbb{C}P^1$, hence $\mathbb{C}P^1$ is the image of a compact space, hence compact.

The reason i add the second answer is that it is insightful and easily generalized to complex or real projective spaces of any dimension, even to Grassmanians (although that requires some adjustments) and i believe also you can replace $\mathbb{C}$ by any normed division algebra over $\mathbb{R}$.

Hope this helps. Joachim

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By the way, note that the spheres are not complex submanifolds of $\mathbb{C}^n$, we just regard them as topological spaces here. Since compactness is purely topological this suffices. –  Joachim Jul 25 '12 at 10:01
thank you for the detail and informative answer. simply pleased and delighted –  El Angel Exterminador Jul 25 '12 at 10:01
Happy to hear that =) –  Joachim Jul 25 '12 at 10:03

Yes.

In fact, the complex projective line "is" the Riemann Sphere.

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The argument you have outlined appeals to the following lemma of general topology.

Lemma. Let $X$ be a topological space. If there are (quasi)compact subsets $Y_1, \ldots, Y_n$ such that $Y_1 \cup \cdots \cup Y_n = X$, then $X$ is itself (quasi)compact.

Proof. Let $\{ U_\alpha : \alpha \in I \}$ be an open cover of $X$. For each $i$, $\{ U_\alpha \cap Y_i \}$ is an open cover of $Y_i$, so has a finite subcover $\{ U_{\alpha_j} \cap Y_i : \alpha_j \in J_i \}$. Thus we have a finite subcover $\{ U_{\alpha_j} : 1 \le j \le n, \alpha_j \in J_i \}$ for $X$.

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