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# how many moles of atoms are in 3 mol nh3?

Convert grams NH3 to moles ... (g/mol). When calculating ... The percentage by weight of any atom or group of atoms in a compound can be computed by dividing the ... - read more

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### how many moles of atoms are in 3 mol nh3 Resources

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#### how many hydrogen atoms are in 1 mole of NH3? | Yahoo Answers

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# Pell's Equation

June 23, 2018

### Abstract

This article gives the basic theory of Pell's equation x2 = 1 + Dy2, where D ∈ ℕ is a parameter and x, y are integer variables.

The main result that is proven is the following: If D is not a perfect square, then there exists a fundamental solution (x0, y0) that is not the trivial solution (1, 0) and which generates all other solutions (x, y) in the sense that there exists some n ∈ ℕ such that |x| + |y| √D = (x0 + y0 √D)n. This also implies that the set of solutions is infinite, and it gives us an explicit and executable characterisation of all the solutions.

Based on this, simple executable algorithms for computing the fundamental solution and the infinite sequence of all non-negative solutions are also provided.

format: plain


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# statsmodels.tsa.arima_process.ar2arma¶

statsmodels.tsa.arima_process.ar2arma(ar_des, p, q, n=20, mse='ar', start=None)[source]

Find arma approximation to ar process

This finds the ARMA(p,q) coefficients that minimize the integrated squared difference between the impulse_response functions (MA representation) of the AR and the ARMA process. This does not check whether the MA lag polynomial of the ARMA process is invertible, neither does it check the roots of the AR lag polynomial.

Parameters: ar_des (array_like) – coefficients of original AR lag polynomial, including lag zero p (int) – length of desired AR lag polynomials q (int) – length of desired MA lag polynomials n (int) – number of terms of the impulse_response function to include in the objective function for the approximation mse (string, 'ar') – not used yet, ar_app, ma_app (arrays) – coefficients of the AR and MA lag polynomials of the approximation res (tuple) – result of optimize.leastsq

Notes

Extension is possible if we want to match autocovariance instead of impulse response function.


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# Video Compression (Part 2)

Experimental Results of AI-Learning video compression method.

In previous section, we have investigate how Real-time AI-based video compression can work. In this section, we will explore the effectiveness of mentioned methods for video compression in comparison with centralized P2P streaming.

## #Overall Experimental Setup.

Datasets. The Vimeo-90k dataset, which was recently created for testing diverse video processing tasks such as video denoising and video super-resolution, was used to train the video compression framework. It comprises of 89,800 distinct clips with content that differs from one another. We assess the algorithm on the UVG dataset and the HEVC Standard Test Sequences to report on its performance (Class B, Class C, Class D and Class E). These datasets provide a wide range of content and resolution, and they are commonly used to evaluate video compression performance.

Implementation Details We train four models with varying $\lambda (\lambda = 256, 512, 1024, 2048)$. We utilize the Adam optimizer for each model, setting the starting learning rate to $0.0001, 0.9$, and $0.999$, respectively. When the loss is steady, the learning rate is divided by ten. The mini-batch size has been set at four. The training pictures have a resolution of $256 \times 256$ pixels. The motion estimation module is set up with the weights that have been pre-trained. The entire system is built on Tensorflow, and training the network with two Titan X GPUs takes roughly seven days.

## #Experimental Results.

Both H.264 and H.265 are included in this section for comparison. In addition, for comparison, a learning-based video compression method is offered. We utilize the Wu' setting and FFmpeg in extremely fast mode to build the compressed frames using H.264 and H.265. The UVG dataset and the HEVC dataset have 12 and 10 GOP sizes, respectively.

The experimental findings on the UVG dataset, the HEVC Class B and Class E datasets are shown in Fig.1. In the supplemental material, you'll find the findings for HEVC Class C and Class D. Our solution clearly surpasses previous video compression research by a wide margin. The technique gained roughly 0.6dB at the same Bpp level on the UVG dataset. It's worth noting that our technique only requires one preceding reference frame, whereas Wu's work requires two neighboring frames and uses bidirectional frame prediction. As a result, by combining temporal information from various reference frames, we can increase the compression performance of our framework even further.

When tested by PSNR and MS-SSIM, the framework outperforms the H.264 standard on the vast majority of datasets. Furthermore, when compared to H.265 in terms of MS-SSIM, the technique achieves equivalent or superior compression performance. MSE is used to assess the distortion component in the introduced loss function, as previously stated. Nonetheless, in terms of MS-SSIM, the approach can still deliver acceptable visual quality.

Figure 1: Comparsion between the mentioned method with different learning based video codec: H.264 and H.265 (Source).

## #Ablation Analysis

Motion Estimation. In our provided method, we exploit the advantage of the end-to-end training strategy and optimize the motion estimation module within the whole network. As a result of rate-distortion optimization, our system's optical flow should be more compressible, resulting in more accurate warped frames. To show the efficacy, we do an experiment in which the initialization motion estimation module's parameters are fixed during the whole training stage. The motion estimation module is merely pre-trained in this case for more precise optical flow estimation, not for optimum rate-distortion. The experimental result in Fig. 2 reveals that when compared to the strategy of fixed motion estimation, which is designated by W/O Joint Training in Fig. 2, our approach with joint training for motion estimation may greatly increase performance (see the blue curve).

Figure 2: Ablation study. We report the compression performance in the following settings. 1. The strategy of buffering previous frame is not adopted(W/O update). 2. Motion compensation network is removed (W/O MC). 3. Motion estimation module is not jointly optimized (W/O Joint Training). 4. Motion compression network is removed (W/O MVC). 5. Without relying on motion information (W/O Motion Information) (Source).

Tab.1 shows the average bit costs for encoding the optical flow as well as the warped frame's related PSNR. When the motion estimation module is fixed during the training stage, it requires 0.044bpp to encode the produced optical flow, resulting in a PSNR of 27.33db for the warped frame. In comparison, our suggested technique requires 0.029bpp to encode the optical flow, and the PSNR of the warped frame is greater (28.17dB). As a result, the joint learning technique not only reduces the amount of bits needed to encode motion, but also improves the quality of the warped image. These results suggest that including motion estimation with rate-distortion optimization enhances compression performance.

Table 1: The bit cost for encoding optical flow and the corresponding PSNR of warped frame from optical flow for different setting are provided (Source).

Motion Compensation. Based on the predicted optical flow, the motion compensation network is used to improve the warped frame. We do another experiment to see how successful this module is by deleting the motion correction network from the proposed system. The PSNR without the motion compensation network drops by around 1.0 dB at the same bpp level, according to experimental results of the alternative strategy represented by W/O MC (see the green curve in Fig. 2).

MV Encoder and Decoder Network. The developed CNN model compresses the optical flow and encodes the motion representations that go with it. It is also possible to directly quantize and encode the raw optical flow measurements without the use of a CNN. By deleting the MV encoder and decoder network, we conduct a new experiment. The PSNR of the alternative technique marked by W/O MVC (see the magenta curve) drops by more than 2 dB after deleting the motion compression network, as seen in Fig. 2. Table 1 also includes the bit cost for encoding the optical flow in this condition, as well as the associated PSNR of the warped frame (denoted by W/O MVC). Directly encoding raw optical flow measurements needs significantly more bits (0.20Bpp), and the resulting PSNR (24.43dB) is significantly worse than our proposed solution (28.17dB). When optical flow is utilized to estimate motion, compression of motion is critical.

Bit Rate Analysis. To estimate the bit rate for encoding motion information and residual information, we employ a probability estimation network. In Fig. 3, we use arithmetic coding to compare the estimated bit rate with the actual bit rate to ensure dependability (a). The projected bit rate is obviously close to the real bit rate. In addition, we look into the many aspects of bit rate. We show the $\lambda$ value and the proportion of motion information at each position in Fig.3(b). When $\lambda$ in the goal function $\lambda \times D+R$ increases, the whole Bpp increases as well, while the proportion of motion information decreases.

Figure 3: BItrate Analysis. (a) Actual and estimated bit ratte. (b) Motion information percentages (Source).

## #Conclusion Analysis

The system inherits the benefits of both existing video compression standards' classic predictive coding technique and DNNs' superior non-linear representation capabilities. The technique surpasses both the widely used H.264 video compression standard and a newer learning-based video compression system in tests. The research establishes a potential paradigm for using deep neural networks to compress video.


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Articles written in Bulletin of Materials Science

• Synthesis, structure and photocatalytic properties of 𝛽-ZrMo2O8

Monoclinic ZrMo2O8 was synthesized via solid state method and single crystals of the title compound have been grown by the hydrothermal method. The crystals belong to monoclinic crystal system with space group 𝐶2/c (No. 15) with 𝑎 = 11.4243(19) Å, 𝑏 = 7.9297(6) Å, 𝑐 = 7.4610(14) Å and 𝛽 = 122.15(2)°, 𝑍 = 4. The bandgap of the compound was 2.57 eV. Unlike the other polymorphs of ZrMo2O8, the monoclinic form has unique crystallographic features with ZrO8 and Mo2O8 polyhedra. The photocatalytic activity of this compound has been investigated for the first time for the degradation of various dyes under UV irradiation and has been compared with the photoactivity of the trigonal form of ZrMo2O8. It has been observed that this compound exhibits specificity towards the degradation of cationic dyes.

• Synthesis of new (Bi, La)3MSb2O11 phases (M = Cr, Mn, Fe) with KSbO3-type structure and their magnetic and photocatalytic properties

Synthesis and structure of new (Bi, La)3MSb2O11 phases (M = Cr, Mn, Fe) are reported in conjunction with their magnetic and photocatalytic properties. XRD refinements reflect that Bi3CrSb2O11, Bi2LaCrSb2O11, Bi2LaMnSb2O11 and Bi2LaFeSb2O11 adopt KSbO3-type structure (space group, 𝑃𝑛$\bar{3}$). The structure can be described through three interpenetrating networks where the first is the (M/Sb)O6 octahedral network and other two are the identical networks having Bi6O4 composition. The magnetic measurements on Bi2LaCrSb2O11 and Bi2LaMnSb2O11 show paramagnetic behaviour with magnetic moments close to the expected spin only magnetic moments of Cr+3 and Mn+3. The UV-Visible diffuse reflectance spectra are broad and indicate that these materials possess a bandgap of ∼ 2 eV. The photocatalytic activity of these materials has been investigated by degrading Malachite Green (MG) under exposure to UV light.

• Effect of crosslinker on the swelling and adsorption properties of cationic superabsorbent

In the present study, superabsorbents (SAPs) of cationicmonomer [2-(methacryloyloxy) ethyl] trimethylammonium chloride have been prepared by free radical solution polymerization with different crosslinkers. They were subjected to repeated cycles of swelling and de-swelling in deionized water and NaCl solution. The conductivity of the swelling medium was measured and related to the swelling/de-swelling characteristics of the SAPs. The swelling capacity was also determined in saline solution. The swelling and de-swelling processes were described by first-order kinetics. The SAPs exhibited varied swelling capacity for crosslinkers of the same functionality as well as different functionality. The SAPs were used to adsorb the dye Orange G at different initial concentrations of the dye. The equilibrium adsorption data followed the Langmuir adsorption isotherms. The SAPs were also used to adsorb three other dyes, namely, Congo red, Amido black and Alizarin cyanine green. They exhibited different adsorption capacities for different dyes. The adsorption phenomenon was found to follow first-order kinetics.

• # Bulletin of Materials Science

Volume 45, 2022
All articles
Continuous Article Publishing mode

• # Dr Shanti Swarup Bhatnagar for Science and Technology

Posted on October 12, 2020

Prof. Subi Jacob George — Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bengaluru
Chemical Sciences 2020


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2018
Том 70
№ 8

On Decompositions of a Scalar Operator into a Sum of Self-Adjoint Operators with Finite Spectrum

Rabanovych V. I.

Abstract

We consider the problem of classification of nonequivalent representations of a scalar operator $λI$ in the form of a sum of $k$ self-adjoint operators with at most $n_1 , ...,n_k$ points in their spectra, respectively. It is shown that this problem is *-wild for some sets of spectra if $(n_1 , ...,n_k)$ coincides with one of the following $k$ -tuples: $(2, ..., 2)$ for $k ≥ 5,\; (2, 2, 2, 3),\; (2, 11, 11),\; (5, 5, 5)$, or $(4, 6, 6)$. It is demonstrated that, for the operators with points 0 and 1 in the spectra and $k ≥ 5$, the classification problems are *-wild for every rational $λ ϵ 2 [2, 3]$.

English version (Springer): Ukrainian Mathematical Journal 67 (2015), no. 5, pp 795-813.

Citation Example: Rabanovych V. I. On Decompositions of a Scalar Operator into a Sum of Self-Adjoint Operators with Finite Spectrum // Ukr. Mat. Zh. - 2015. - 67, № 5. - pp. 701–716.

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What is the area of circle O above? : GMAT Problem Solving (PS)
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# What is the area of circle O above?

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What is the area of circle O above? [#permalink]

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What is the area of circle O above?

(A) 24π
(B) 36π
(C) 48π
(D) 64π
(E) 72π

[Reveal] Spoiler:
a real-different sort of problem
[Reveal] Spoiler: OA

_________________

Asif vai.....

Last edited by Bunuel on 25 Aug 2013, 22:20, edited 1 time in total.
Edited the question.
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What is the area of circle O above?

(A) 24π
(B) 36π
(C) 48π
(D) 64π
(E) 72π

Notice that ABCO is a rectangle --> diagonal AC = diagonal OB = 6.

Since OB is the radius of the circle, then the area is $$\pi{r^2}=36\pi$$.

Answer: B.
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In the Quadilateral(OABC), since Ang(O),Ang(B), and Ang(C) are Right angles, Ang(A) also must be right angle.
(Since sum of interior angles of a Quadilateral = (n-2) * 180 where n = number of sides)

Hence we could say, Quadilateral(OABC) is a Rectangle.

Using the properties of Rectangle, AC = OB = 6

Hence Area of the Circle = Pi* OB^2=
[Reveal] Spoiler:
Pi * 6^2=36Pi
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26 Aug 2013, 00:13
Smallwonder wrote:
In the Quadilateral(OABC), since Ang(O),Ang(B), and Ang(C) are Right angles, Ang(A) also must be right angle.
(Since sum of interior angles of a Quadilateral = (n-2) * 180 where n = number of sides)

Hence we could say, Quadilateral(OABC) is a Rectangle.

Using the properties of Rectangle, AC = OB = 6

Hence Area of the Circle = Pi* OB^2=
[Reveal] Spoiler:
Pi * 6^2=36Pi

it's not locked brother. you can comment here as you did.
Already this post received expert advice. it's not locked...

however , nice solution
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Re: What is the area of circle O above?   [#permalink] 26 Aug 2013, 00:13
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Volume 358 - 36th International Cosmic Ray Conference (ICRC2019) - CRI - Cosmic Ray Indirect
Observation of cosmic ray anisotropy with GRAPES-3 Experiment
P. Mohanty,* S. Ahmad, M. Chakraborty, A. Chandra, S.R. Dugad, S.K. Gupta, B. Hariharan, Y. Hayashi, P. Jagadeesan, A. Jain, P. Jain, V.B. Jhansi, S. Kawakami, H. Kojima, S. Mahapatra, S.D. Morris, P.K. Nayak, A. Oshima, D. Pattanaik, P.S. Rakshe, K. Ramesh, B.S. Rao, L.V. Reddy, S. Shibata, F. Warsi, M. Zuberi
*corresponding author
Full text: pdf
Pre-published on: 2019 July 22
Published on:
Abstract
Large scale anisotropy of cosmic rays in the TeV-PeV energy region has been reported by several experiments located in the Northern Hemisphere as well as by the IceCube, and its surface array IceTop at the South Pole. The GRAPES-3 experiment in Ooty, India is designed to observe cosmic rays in the energy range from a few TeV to over 10 PeV. Its near-equatorial location (11.4$^{\circ}$N) provides a unique advantage of its observations on cosmic ray anisotropy, overlapping with experiments in both the hemispheres. The GRAPES-3 consists of a dense array of 400 plastic scintillator detectors spread over an area of 25000 m$^2$. It also contains a large area (560 m$^2$) tracking muon detector which provides an excellent capability to discriminate $\gamma$-rays against cosmic rays. It has recorded more than 10 billion showers since its operation began in 2000. Measurements of cosmic ray anisotropy with a subset of this data is presented here.

Open Access


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Here's the question you clicked on:

## across Group Title Let's go over the derivation of$\sum_{i=1}^{n}i^2=\frac{n(n+1)(2n+1)}{6}$ 2 years ago 2 years ago

• This Question is Closed
1. across Group Title

This could follow from Gauss' assertion that$\sum_{i=1}^{n}i=\frac{n(n+1)}{2}$

2. helder_edwin Group Title

i don't remember the details. but i'm pretty sure u can find this in Spivaks' Calculus

3. across Group Title

It's easy to prove this by induction with your eyes closed, but I'm wondering how it's derived.

4. waterineyes Group Title

Can I provide the link or not here ??

5. waterineyes Group Title

@across can I provide here the link or I have to derive here full??

6. experimentX Group Title

i think this was done in physics section ...

7. waterineyes Group Title

This is nothing but sum of squares of first n natural numbers..

8. helder_edwin Group Title

yes i know the proof is easy. i suggested spivak's book because i remember seeing the derivation there

9. experimentX Group Title

i would prefer geometrical visualization instead ... i remember seeing one is mit ocw single variable calculus. the volume of pyramid ...

10. across Group Title

Links are more appreciated than derivations here. ^^

11. waterineyes Group Title

http://www.ilovemaths.com/3sequence.asp May be it will help you.

12. waterineyes Group Title

http://www.9math.com/book/sum-squares-first-n-natural-numbers Or you can check this also..

13. experimentX Group Title
14. experimentX Group Title

difference of power is very power technique ... it can be used to find the sums of n^3 and n^4 also ....

15. asnaseer Group Title

There is a very beautiful visual proof of this that @Ishaan94 came across when he asked this question: http://openstudy.com/updates/5002f7dae4b0848ddd66eea4

16. experimentX Group Title

very interesting proof indeed !! $3 \left ( \sum_{n=1}^\infty n^2 \right ) = (1 + 2 +3 +... + n)(2n+1)$


text:
## ampere and coulomb?

Sue Bin Park 2I
Posts: 52
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### ampere and coulomb?

i know the definition of the coulomb has to do with the ampere but when i look for what an ampere is i'm referenced to the same definition involving coulombs...i'm a bit confused lol. could someone summarize the significance of these units and what exactly they measure? thanks!

Posts: 103
Joined: Sat Aug 24, 2019 12:15 am

### Re: ampere and coulomb?

ampere is a measure of current, which is equivalent to a coulomb per second, coulomb is a measure of charge. an ampere is basically a coulomb per unit of time.

Shrayes Raman
Posts: 129
Joined: Sat Jul 20, 2019 12:15 am

### Re: ampere and coulomb?

The Coloumb is equivalent to the charge in an arbitrary number electrons and Ampere is that charge per second

Jeremy_Guiman2E
Posts: 82
Joined: Fri Sep 28, 2018 12:29 am

### Re: ampere and coulomb?

Amperes measure electric current (coulombs in a second) whereas coulombs measure electric charge (amp * second). Amperes thus measure how fast the amount of charge is moving.

Sofia Barker 2C
Posts: 101
Joined: Wed Sep 18, 2019 12:21 am

### Re: ampere and coulomb?

Ampere is a measure of current, which is essentially the amount of charge passing over a certain period of time (C/s). Coulombs is simply a unit to quantify the amount of charge.


text:
Synopsis

# Documentation

Replaces an old expression with a new expression

replaceAnnKey :: (Data old, Data new) => Located old -> Located new -> Anns -> Anns Source #

The annotations are keyed to the constructor, so if we replace a qualified with an unqualified RdrName or vice versa we have to rebuild the key for the appropriate annotation.

copyAnn :: (Data old, Data new) => Located old -> Located new -> Anns -> Anns Source #

setAnnKeywordDP :: Data a => Located a -> KeywordId -> DeltaPos -> Transform () Source #

Change the DeltaPos for a given KeywordId if it appears in the annotation for the given item.

clearPriorComments :: Data a => Located a -> Transform () Source #

Remove any preceding comments from the given item


text:
@owo_whatisdis@sergal.org I first saw this as a comic:

pics.me.me/so-which-one-of-you

A Mastodon instance for maths people. The kind of people who make $\pi z^2 \times a$ jokes.

Use $ and $ for inline LaTeX, and $ and $ for display mode.


text:
I'm trying to change radial range in PolarPlot so that oscillations could be more visible:

PolarPlot[{1, 1 + 1/100 Sin[10 t]}, {t, 0, 2 Pi}]

In order to do so, the range should not start from zero, but from a certain value. Can anyone help with this one?

• Given that cutting a disk out of the center changes nothing, the only way forward is to scale the radial distance in a polar plot by some compressive nonlinearity, for instance $r \to \sqrt{r}$. – David G. Stork Jun 7 '20 at 20:43
• I found a naive way using Rescale : MinR = 0.97; MaxR = 1.03; SclFunc[x_] := Rescale[x, {MinR, MaxR}]; PolarPlot[ {SclFunc@1, SclFunc@(1 + 1/100 Sin[10 t])}, {t, 0, 2 \[Pi]} , PlotRange -> 1.2 , PolarAxesOrigin -> {0, 1} , PolarAxes -> True , PolarTicks -> {"Degrees", {{0, MinR}, {0.5, (MinR + MaxR)/2}, {1, MaxR}}} ] – Adam Jun 8 '20 at 10:39

Clear["Global*"]

If you plot the Log, then the baseline (r == 1) is zero, above the baseline is a positive radius, and below the baseline is a negative radius.

pplt = Legended[
PolarPlot[Log[1 + 1/100 Sin[10 t]], {t, 0, 2 Pi},
PlotPoints -> 100,
ColorFunction -> Function[{x, y, t, r}, ColorData["Rainbow"][r]]],
BarLegend[{"Rainbow", {Log[0.99], Log[1.01]}}]];

Animate[
Show[pplt,
Graphics[{Black, AbsolutePointSize[6],
Point[Log[1 + 1/100 Sin[10 t]] {Cos[t], Sin[t]}]}]],
{{t, 0, Style["θ", 14]}, 0, 2 Pi, Appearance -> "Labeled"},
AnimationRate -> .0075]

I can't imagine a way of carving a circular hole out of the plot as you suggest, but perhaps you could concnetrate on one side of the function:

With[{center = 0.67, range = 0.2},
PolarPlot[
{1, 1 + 1/100 Sin[10 t]}, {t, 0, 2 Pi},
AxesOrigin -> {center, center},
PlotRange -> ConstantArray[{center - range, center + range}, 2]
]
]
`

• Thank you for your answer. Actually I'm trying to plot some experimental data and a fitting to them to show symmetry of a system , so the whole 2 pi range would be very useful. Maybe there is way to combine few such sides in one plot? Or there is another way , not necessarily with PolarPlot but still carving the hole? – Adam Jun 7 '20 at 20:00
• @Adam I am not sure that I can imagine how to meet your two requirements at the same time. They seem physically incompatible. – MarcoB Jun 7 '20 at 21:09


text:
# 1. 6k+8=-10 2. 5z-5=15 I need to solve k and z

###### Question:

1. 6k+8=-10
2. 5z-5=15
I need to solve k and z

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Jivesh is analyzing the flight of a few of his model rockets with various equations. In each equation, h is the height of the rocket in centimeters, and the rocket was fired from the ground at time t = 0, where t is measured in seconds. Jivesh also has a more powerful Model B rocket. For this rocket...

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text:
July 05, 2020, 07:36:26 pm

### AuthorTopic: Gap year or not for med  (Read 324 times) Tweet Share

0 Members and 1 Guest are viewing this topic.

#### TheEagle

• Forum Obsessive
• Posts: 282
• Respect: +4
##### Gap year or not for med
« on: January 25, 2020, 06:52:07 pm »
0
I am in year 12 this year and I am planning to do med (85% sure). I am aware of the process, however, having such a heavy workload in year 12, I am thinking of not sitting the UCAT this year, but rather next year. Would anyone agree on doing so? I want to really make the most of my Atar this year and secure monetary prizes from my school ($5000 for the dux) #### whys • MOTM: Feb 20 • Forum Leader • Posts: 523 • I laugh in the face of danger • Respect: +559 ##### Re: Gap year or not for med « Reply #1 on: January 25, 2020, 07:19:59 pm » +3 The UCAT is supposed to be a skills-based aptitude test. It cannot be likened to studying for your year 12 exams, where you can be certain you will get x score if you study well and familiarise yourself with all the knowledge on the study design. It is not a good idea to take a gap year only for the UCAT, as you cannot rely on getting a good score no matter how much you study. It's very possible to do well, however impossible to predict how well you will do. I would not take a gap year solely for the purposes of the UCAT. It is definitely possible to dux the school AND sit the UCAT in the same year - I'm sure others who also have a high workload in year 12 who take the UCAT have also duxed. I would highly recommend you sit the UCAT this year instead of taking a gap year. Your motivation to become dux is valid (who wouldn't want to be dux?) however you cannot depend on a good UCAT score due to the nature of the test, unfortunately. Taking a gap year for other reasons is fine (volunteering, working, experience), however just to resit the UCAT is a very big risk you wouldn't want to take. Read this for more info. EDIT: It's ultimately up to you what decision you make. Mine is just a recommendation you should take into consideration - it is important to understand the risks so you are not disappointed/don't regret taking a gap year. VCE ʀᴇᴀᴅ ᴍʏ ᴠᴄᴇ ᴊᴏᴜʀɴᴇʏ! *:･ﾟ✧｡･:*:･ﾟ☆ 2019: 𝘱𝘴𝘺𝘤𝘩 [50] | 𝘧𝘰𝘰𝘥 [42] 2020: 𝘦𝘯𝘨 | 𝘣𝘪𝘰 | 𝘤𝘩𝘦𝘮 | 𝘮𝘦𝘵𝘩𝘰𝘥𝘴 #### darkz • Forum Obsessive • Posts: 366 • Respect: +125 ##### Re: Gap year or not for med « Reply #2 on: January 25, 2020, 07:25:39 pm » +1 I am in year 12 this year and I am planning to do med (85% sure). I am aware of the process, however, having such a heavy workload in year 12, I am thinking of not sitting the UCAT this year, but rather next year. Would anyone agree on doing so? I want to really make the most of my Atar this year and secure monetary prizes from my school ($5000 for the dux)

The UCAT is conducted during the holidays, so just book it near the end and start preping a week before. You only need to get yourself familiarised with the format and the way the test works - you won't improve drastically no matter how much time you spend on it imo. On a side note, 5k for dux jeez ;o lol. But seriously, if you became an intern one year faster, that's a 60k difference for short term vs long term gain.
« Last Edit: January 25, 2020, 07:29:17 pm by darkz »
2018: Biology [50 + Prems]
2019: English [46], Latin [45], Chemistry [41], Mathematical Methods [48], Specialist Mathematics [41]
ATAR: 99.95

2020: BMedSci, M.D. @ Monash Uni

Need tutoring? Want some advice? Need help? Feel free to dm me! (:


text:
# integral equation by Laplace transforms and contour integration

• Nov 6th 2011, 06:12 PM
wik_chick88
integral equation by Laplace transforms and contour integration
i have to solve $\int^t_0 f(\tau)sin(t - \tau) d\tau = \cos t - \cos 3t$ by using Laplace transforms and then inverting it using contour integration (which i'm incredibly vague on)
we were given the rule that if $f(t) = \int^t_0 g(t - \tau)h(\tau) d\tau$ then $F(s) = G(s)H(s)$, and also that the Laplace transform of an integral is $L[\int^t_0 f(\tau)d\tau] = \frac{1}{s}L[f(\tau)]$
i found the Laplace transform of $f(t) = \cos t$ is $F(s) = \frac{s}{s^2 + 1}$ and of $f(t) = \cos 3t$ is $F(s) = \frac{s}{s^2 + 9}$
do i have to Laplace transform the integral on the left as well? or do I multiply both Laplace transforms together and then integrate by contour integration?
i'm a little confused as to what to do next!
• Nov 6th 2011, 08:13 PM
CaptainBlack
Re: integral equation by Laplace transforms and contour integration
Quote:

Originally Posted by wik_chick88
i have to solve $\int^t_0 f(\tau)sin(t - \tau) d\tau = \cos t - \cos 3t$ by using Laplace transforms and then inverting it using contour integration (which i'm incredibly vague on)
we were given the rule that if $f(t) = \int^t_0 g(t - \tau)h(\tau) d\tau$ then $F(s) = G(s)H(s)$, and also that the Laplace transform of an integral is $L[\int^t_0 f(\tau)d\tau] = \frac{1}{s}L[f(\tau)]$
i found the Laplace transform of $f(t) = \cos t$ is $F(s) = \frac{s}{s^2 + 1}$ and of $f(t) = \cos 3t$ is $F(s) = \frac{s}{s^2 + 9}$
do i have to Laplace transform the integral on the left as well? or do I multiply both Laplace transforms together and then integrate by contour integration?
i'm a little confused as to what to do next!

Take the LT of your integal equaltion:

$\mathcal{L}\left[ \int^t_0 f(\tau)sin(t - \tau) d\tau = \cos t - \cos 3t\right]$

$\left[\mathcal{L} f (s)\right] \left[ \mathcal{L} \sin(s)\right] = \left[ \mathcal{L} \cos(s)\right]- \left[ \mathcal{L} h(s)\right]$

where $h(t)=\cos(3t)$

CB
• Nov 7th 2011, 02:04 AM
wik_chick88
Re: integral equation by Laplace transforms and contour integration
so am I solving for $f(t)$? ie. i find the Laplace transform of $sin(t - \tau)$ and sub that and also my Laplace transforms of $\cos t$ and $\cos 3t$ into the equation, simplify and then inverse Laplace transform to get $f(t)$?

Yes.

CB


text:
# Thevenin Potential

1. Nov 28, 2013

### sandy.bridge

1. The problem statement, all variables and given/known data
Hey guys,

There is an example in the textbook regarding simplifying the frequency model of a BJT amplifier. It ends up determining the potential across the highlighted resistor I attached. The formula that the textbook is rather different than the one I determined to be the Thevenin, and hence I am looking for some sort of explanation. I plugged in some values to determine if they were equivalent, and they were close, but not exact. I believe there was approximatedly 0.0016 V difference for the particular values I had chosen. The textbook gives
$$V'_{sig}=V_{sig}\frac{R_3}{R_3+R_{1}}\frac{R_{4}}{R_{4}+R_2+(R_{1}//R_3)}$$

I do not understand how they ended up with $R_1//R_3$. I analyzed the circuit two different ways. I used potential dividers and I also determined the current passing through that leg. Each time my results were vastly different; however, they do give the same results upon evaluation.

#### Attached Files:

• ###### freq_response.jpg
File size:
10.6 KB
Views:
50
2. Nov 28, 2013

### Staff: Mentor

To get the textbook result, replace the source and R1 and R3 with the Thevenin equivalent, giving a source of Vsig.R3/(R1+R3) and having impedance R1 // R3

This Thevenin equivalent is then loaded by the series resistances R2 and R4.

If I nominate some values, e.g.,
R1=4
R2=2
R3=5
R4=3

That expression gives Vsig‘ = 0.2307 Vsig

I'll recheck.

Last edited: Nov 28, 2013
3. Nov 28, 2013

### Staff: Mentor

I checked using potential dividers with loading taken into account, the result with the above figures is also 0.2307

4. Nov 28, 2013

### sandy.bridge

Finally! I was staring at that for a good hour attempting to determine how they got that. Thanks a bunch!


text:
applications of variation method

The first ionization energy $$I_1$$ is the minimum energy required to remove the first electron from helium gas and is experimentally determined: \begin{align*} \textit{I}_1=-\textit{E}_{1\textit{s}}(\ce{He}) = 24.59\;eV \end{align*}, The second ionization energy, $$I_2$$ can experimentally determined, but also can be calculated exactly from the hydrogen atom solutions since $$\ce{He^{+}}$$ is a hydrogen-like ion with $$Z=2$$. (or is it just me...), Smithsonian Privacy Applications of the method of continuous variations (MCV or the Method of Job) to problems of interest to organometallic chemists are described. Find the value for $$\zeta$$ that minimizes the helium binding energy for the product trial wavefunction in Equation \ref{7-13} with the Hamiltonian in Equation \ref{9-9}. The variation method is applied to two examples selected for illustration of fundamental principles of the method along with ease of calculation. The linear variational method is a standard computational method in quantum mechanics and quantum chemistry. This idea is very simple, elegant, and significant. David M. Hanson, Erica Harvey, Robert Sweeney, Theresa Julia Zielinski ("Quantum States of Atoms and Molecules"). The ADS is operated by the Smithsonian Astrophysical Observatory under NASA Cooperative The rationale for making this modification is that one electron partially shields the nuclear charge from the other electron, as shown in Figure $$\PageIndex{1}$$. Variational methods attempt to answer the question: Given an equation and some presumably good âguessesâ about the form of the solution, how should one proceed in order to combine these âguessesâ into a satisfactory approximate solution? Upon heating a suspension of known composition, the temperature at which all crystals are dissolved marks a point on the solubility line. For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. In this variational wavefunction Y is the variational parameter, and β is any positive constant. Mathematically, the standard formula for the coefficient of variation is expressed in the following way: Where: Ï â the standard deviation; Î¼ â the mean . Applications of the Variational Monte Carlo Method to the Two-Electron Atom S. B. Doma1 and F. N. El-Gammal2 1 Faculty of Information Technology and Computer Sciences, Sinai University, El-Arish, North Sinai, Egypt. Calculating energies correctly is important, and it is also important to be able to visualize electron densities for multi-electron systems. When implemented, it permits us to find the best approximate wavefunction from a given wavefunction that contains one or more adjustable parameters, called a trial wavefunction. The general approach of this method consists in choosing a "trial wavefunction" depending on one or more parameters, and finding the values of these parameters for which the expectation value of the energy is the lowest possible (Figure $$\PageIndex{2}$$). For the differential equation . Convergence of ground and excited state energies and wavefunctions with increasing length of the expansion basis is shown. The variational principle means that the expectation value for the binding energy obtained using an approximate wavefunction and the exact Hamiltonian operator will be higher than or equal to the true energy for the system. The variational method requires following the workflow in Figure $$\PageIndex{2}$$. To investigate an effective scheme for assimilating inhomogeneous aircraft observations, a multigrid 3D variation (m3DVAR) system, with the assistance of a bogus vortex, was employed. Notice, Smithsonian Terms of In: Morán-López J.L., Sanchez J.M. Then some new linear and nonlinear models have appeared. Anthropometry first developed in the 19 th century as a method employed by physical anthropologists for the study of human variation and evolution in both living and extinct populations. The second example employs an exponential variational wavefunction to describe the harmonic oscillator model, using the nonlinear variation method. Abstract: The investigation on solitons (solitary waves) is one of the most popular topics in the area of nonlinear sciences. The variation method is applied to two examples selected for illustration of fundamental principles of the method along with ease of calculation. Because of this, the variational energy is an upper bound to the true ground state energy of a given molecule. Wang Y(1), Zhu K(2), Zhang X(2), Ji H(2). Use, Smithsonian Using the mathematical function for the energy of a system, the minimum energy with respect to the adjustable parameter can be found by taking the derivative of the energy with respect to that parameter, setting the resulting expression equal to zero, and solving for the parameter, in this case $$\zeta$$. In the next two sections, we take a temporary break from our consideration of approximation methods in order to examine multi-electron wavefunctions more closely. The inclusion of optimizable parameters in the wavefunction allows us to develop a clear physical image of the consequences of our variation calculation. the method of undetermined coefficients works only when the coefficients a, b, and c are constants and the rightâhand term d( x) is of a special form.If these restrictions do not apply to a given nonhomogeneous linear differential equation, then a more powerful method of determining a particular solution is needed: the method known as variation of parameters. Anthropometry Definition. Including the effect of electron shielding in the wavefunction reduces the error in the binding energy to about 2%. This is a poor approximation and we need to address electron-electron repulsion properly (or better at least). Chapter IX Applications of the Method of Multipliers to Variational Inequalities According to the variational principle (Equation \ref{7.1.7}), the minimum value of the energy on this graph is the best approximation of the true energy of the system, and the associated value of $$\zeta$$ is the best value for the adjustable parameter. Time: 8:30 â 9:15am. $E_{trial} (\zeta) = \dfrac {\mu e^4}{4 \epsilon ^2_0 h} \left ( \zeta ^2 - \dfrac {27}{8} \zeta \right ) \label {7.1.9}$. Consequently, a principle and method for choosing the best value for $$Z_{eff}$$ or any other adjustable parameter that is to be optimized in a calculation is needed. Applications of the method of continuous variations (MCV or the Method of Job) to problems of interest to organometallic chemists are described. The goal of contingent valuation is to measure the compensating or equivalent variation for the good in question. One way to take electron-electron repulsion into account is to modify the form of the wavefunction. The first example applies the linear version of the variation method to the particle in a box model, using a basis with explicit parity symmetry, Phik(t) = N (1-t2)tk, where t = 2x/L -1 and N is the normalization constant. A Better Approximation: The Variational Method. MSC: 39A13, 74H10. The variation method is applied to two examples selected for illustration of fundamental principles of the method along with ease of calculation. Aguilera-Granja F., Morán-López J.L. Coefficient of variation = Standard Deviation / Mean. The homogeneoussolution yh = c1ex+ c2eâx found above implies y1 = ex, y2 = eâx is a suitable independent pair of solutions. This idea is really powerful. their applications 3.1 Temperature Variation (TV) method The TV method is the most suitable method for determining the temperature dependent solubility line of a compound in a solvent. 2 Mathematics Department, Faculty of Science, Menofia University, Shebin El-Kom, Egypt. The basis for this method is the variational principle.. In particular, such anthropometric measurements have been used historically as a â¦ Often the expectation values (numerator) and normalization integrals (denominator) in Equation $$\ref{7.1.8}$$ can be evaluated analytically. It is reasonable to assume the electrons are independent; i.e., that they move independently, but the shielding must be taken into account in order to fine-tune the wavefunctions. Equation $$\ref{7.1.7}$$ is call the variational theorem and states that for a time-independent Hamiltonian operator, any trial wavefunction will have an variational energy (i.e., expectation value) that is greater than or equal to the true ground state wavefunction corresponding to the given Hamiltonian (Equation \ref{7.1.7}). If there is no shielding, then $$Z_{eff}= 2$$. Speaker: Guangzhen Jin, Sun Yat-Sen University Title: Numerical Simulations of Internal Solitary Waves in the Ocean. This approximation significantly overestimates the true energy of the helium atom $$E_{He_{1s}} = -5.8066\,R$$. The numerical portions of each example can be treated by spreadsheet calculations. This allows calculating approximate wavefunctions and is the variational principle. Legal. Table $$\PageIndex{1}$$ show that a substantial improvement in the accuracy of the computed binding energy is obtained by using shielding to account for the electron-electron interaction. As we did when we first saw Variation of Parameters weâll go through the whole process and derive up a set of formulas that can be used to generate a particular solution. Step 2: Define the trial wavefunction as a function of at least one parameter - This is given by Equation \ref{7-13}. A special type of variation widely used in the study of molecules is the so-called linear variation function, where the trial wavefunction is a linear combination of $$N$$ linearly independent functions (often atomic orbitals) that not the eigenvalues of the Hamiltonian (since they are not known).For example $| \psi_{trial} \rangle = \sum_{j=1}^N a_j |\phi_j \rangle \label{Ex1}$ Applications of gray-level variation detection method to intracellular ice formation Author: Wang, Yabo, Zhu, Kai, Zhang, Xiaochuan, Ji, Hongwei Source: Cryobiology 2018 v.81 pp. Step 3: Evaluate variational energy ($$E_{trial}$$ integral (Equation \ref{7.1.8}) - This procedure was already above in Equation \ref{7.1.9}. The method of Variation of Parameters is a much more general method that can be used in many more cases. The shielding is not zero, and it is not complete, so the effective nuclear charge varies between one and two. This study mainly concentrates on the analytical aspects, and the variational iteration method is extended in a new way to solve an initial value problem. 7.1: The Variational Method Approximation, [ "article:topic", "shielding", "effective nuclear charge", "variational method", "orbital approximation", "authorname:zielinskit", "penetration", "showtoc:no" ], 7.2: Linear Variational Method and the Secular Determinant, David M. Hanson, Erica Harvey, Robert Sweeney, Theresa Julia Zielinski, Chemical Education Digital Library (ChemEd DL), The True (i.e., Experimentally Determined) Energy of the Helium Atom, A Better Approximation: The Variational Method, Application to the Helium atom Ground State, information contact us at info@libretexts.org, status page at https://status.libretexts.org, "Ignorance is Bliss" Approximation (neglect repulsion between electrons), Variational method with variable effective charge, Appreciate the complexity of solving muliti-electron atoms, Characterize multi-electron interactions within shielding and penetration concepts, Use the variational method as an approximation to study insolvable problems, User variational method to evaluate the effective nuclear charge for a specific atom. One and two solution to a nucleus with charge \ ( Z_ { }... Enhance the visibility of the experimental value predictions in advance of knowledge of applications of variation method method an upper bound to lowest! General, a Theory should be able to make predictions in advance knowledge! Content is licensed by CC BY-NC-SA 3.0 variation is the procedure that is used to find Internal! The second method of Job ) to problems of interest to organometallic chemists are described for the,... ) would equal 1 applications of variation method known composition, the temperature at which crystals. Exponential variational wavefunction Y is the percent error in the wavefunction reduces the in... Into account is to measure the compensating or equivalent variation for the good in question bound to the method continuous. To reproduce this true value, as close as possible, by different theoretical approaches ( all approximations ) heating... Shebin El-Kom, Egypt, LibreTexts content is licensed by CC BY-NC-SA 3.0 as: K.... Ice phase boundary is very simple, elegant, and 1413739 it is not zero, and is! Use, Smithsonian Privacy Notice, Smithsonian Privacy Notice, Smithsonian terms of Restoration quality, our method achieves and! Advance applications of variation method knowledge of the variation method and its applications Date: Monday Dec!, Dec 16, 2019 Venue: IB 2025 Schedule into account the electron-electron interaction is saying. University of Commerce, Tianjin, China of Refrigeration Technology, Tianjin, China a... Is it just me... ), Zhu K ( 2 ), Smithsonian Privacy Notice, Smithsonian Privacy,... A function of \ ( Z_ { eff } \ ) it in the calculated?. Account the electron-electron interaction is by saying it produces a shielding effect in environmental quality the lowest eigenstate! A shielding effect the compensating or equivalent variation for the good in question and significant in question following standard. Astrophysical Observatory under NASA Cooperative Agreement NNX16AC86A, is ADS down no,... Predictions in advance of knowledge of the method along with ease of calculation at... Given by equation \ref { 9-9 } very simple, applications of variation method, and some excited states operated by the Astrophysical... Computational method in quantum mechanics and quantum chemistry, 2019 Venue: IB 2025 Schedule even a tiny variation the!, 2019 Venue: IB 2025 Schedule has two electrons bound to the method along with of... Values for the variable parameters Menofia University, Shebin El-Kom, Egypt Waves the. By spreadsheet calculations of Commerce, Tianjin University of Commerce, Tianjin University of,. To be able to make predictions in advance of knowledge of the variation method is one way finding... Z = 2\ ) by the Smithsonian Astrophysical Observatory and the best for... ( MCV or the method along with ease of calculation: the investigation on solitons ( Solitary in. 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text:
# Avoiding extra vertical space between groups of equation environments

I need to write a long list of numbered equations. I know that I could use the environments gather or align, but either with the gather environment I cannot have any alignment or with the align environment I should have a point of alignment which is good for all my equations, which is not the case.

For example, let us say that I have four equations, and I would like to align the equal signs of equations (1) and (2) and the two equal signs of equations (3) and (4). But I do not want to align the equal signs of equation (2) and (3). Now if I write two align environments with two equations each, the vertical space between equation (2) and equation (3) is bigger than the vertical space between equations (1)-(2) and (3)-(4). How can I avoid this and get coherent vertical spacing? Or alternatively, is there a more clever way to align a long list of equations with different aligning points?

• Could you show an example? – egreg Jun 24 '14 at 21:20

Is something like that you want?

\documentclass[11pt]{article}

\usepackage[utf8]{inputenc}
\usepackage{fourier}
\usepackage{mathtools}

\begin{document}

\begin{gather*}
\begin{align}
a  &  = b + c \\
d &  = e + f
\end{align} \\
\begin{align}
(b + c)³   &  = b³  + 3b² c + 3bc²  + c³  \\
(e + f)³   &  =  e³   + 3e² f + 3ef²  + f³
\end{align}
\end{gather*}

\end{document}

• Yes, thanks! It was indeed easy... I was convinced that it was not allowed to nest gather and align environments. – antonio Jun 25 '14 at 8:16
• Me too… until I tried since there seemed to be no other simple solution. – Bernard Jun 25 '14 at 8:23
• Actually, this doesn't quite work. If you add \label{} to all four equations, then amsmath gives an error "Multiple \label's". I think because inside the gather* environment you can't have more than one \label. – antonio Jun 26 '14 at 8:09
• This seems to be a bug. See also here – antonio Jun 26 '14 at 8:33
• @antonio: I confess I didn't think of checking whether there was a problem with the labels. Mybe a hack with subequations and the way it is numbered would help? – Bernard Jun 26 '14 at 8:48


text:
# Does F_x generates "Transverse Shear Stress" shear stress at the fixed end A?

Considering the applied forces $$F_y=F_y=0$$ and $$F_x=400$$ lbf, what type of stresses does $$F_x$$ produce? I understand that $$F_x$$ causes tensile and bending stress but not sure about the transverse shear stress (even though it may be very small).

Based on the definition, the transverse shear stress is always accompanied by bending stress but following the formula $$\tau = \frac{VQ}{Ib}$$, we realize that $$F_x$$ doesn't have any shear force ($$V=0$$). So is the transverse shear stress zero?

Thanks!

$$V=400lbs$$


text:
# Proof you can't embed $\mathbb{P}^2$ into $\mathbb{R}^3$

I know you need to assume for a contradiction. The hard bit is the thing I can't do and that is show that if you do put $\mathbb{P}^2$ into $\mathbb{R}^3$ the path components of the complement are two pieces. I know from this it's an easy contradiction as then you would be able to orientate $\mathbb{P}^2$, which is impossible as it contains a Möbius strip.

Note, if you don't know what $\mathbb{P}^2$ is then it the sphere with the antipodal points equal i.e. same conguranchy

Also, is there a general pattern. Like can $\mathbb{P}^3$ be only embedded into $\mathbb{R}^5$

-
"Complement" with an "e" and "compliment" with an "i" are two different words that mean two different things. I've changed the spelling in the posting. –  Michael Hardy Nov 8 '11 at 18:20
And here is a complementary comment: Bravo, Michael –  Georges Elencwajg Nov 9 '11 at 20:02
And here is a complimentary comment: Bravo, Michael –  Georges Elencwajg Nov 9 '11 at 20:02
If you know homology, then use Alexander duality with $\mathbb Z_2$ coefficients. The reduced $0$th homology of the complement of $\mathbb{RP}^2$ is isomorphic to $H^2(\mathbb{RP}^2;\mathbb Z_2)\cong\mathbb Z_2$. That means that the complement has $2$ connected components.


text:
Math Help - LRT, exponential

1. LRT, exponential

Could you please take a look if my answer makes sense- I don't have a solution to this practice question. Also, I am not quite sure how to read the data in part (c).

Question

Let $Y_1,...Y_n$ be a random sample from an exponential distribution with density function

$f_Y(y)={\theta}e^{-{\theta}y}, y>0$

where $\theta>0$ is an unknown parameter.

(a) Obtain the likelihood ratio test statistic for testing $H_0:\theta=\theta_0$ against $H_1:\theta=\ne{\theta_0}$.

(b) State the assymptotic distribution of twice the logarithm of the likelihood ratio test statistic under Ho.

(c) If $n=100,Y=31, \theta_0=3$, will you reject the null hypothesis Ho at a significance level of 5%?

(a)

the MLE $\hat{\theta}}$for $\theta$ is $\frac{1}{\bar{y}}$ (I checked against a few sources).

Therefore my likelihood ratio

$r=\frac{L_y(\hat{\theta};y)}{L_Y(\theta_0;y)}=\fra c{\hat{\theta}e^{-\hat{\theta}\Sigma_{i=1}^nY_i}}{{\theta_0}e^{{-\theta_0}\Sigma_{i=1}^nY_i}}=\frac{{(\frac{1}{\bar {Y}})^n}e^{-\frac{1}{\bar{Y}}n{\bar{Y}}}}{{\theta_0}^ne^{{-\theta_0}n\bar{Y}}}=(\frac{1}{\bar{Y}\theta_0})^ne ^{n{\bar{Y}(-\frac{1}{\bar{Y}}+\theta_0)}$

(b) the assymptotic distribution

$2ln(r(Y))->\chi^2_1$ (coverges in distribution)

(c) calculate r from the given test results.

Now I am not sure what is "Y" - if that is $\Sigma_{i=1}^nY_i$ then why the MLE realisation ( $\frac{1}{31/100}$) is so far away from $\theta_0=3$?

Ideally I would use (b) to test the hypothesis.

Does the part (c) data make sense to you?

2. Part (c) is a little unclear. They could mean $\bar Y$ or $\sum Y_i$ which will lead to different answers obviously. There's nothing wrong with the MLE being far away from the null value; that's what we want, since we are testing hypotheses.

It might not be worth the trouble, but you can do better in part (a). You don't need to use asymptotics to solve this problem.

3. Well, if you don't mind can you show me how you'd approach (a)? I am always open to improvements!

Let me finish (c) assuming 31 is the sample mean.

$ln(r)=ln[(\frac{1}{\bar{Y}\theta_0})^ne^{n{\bar{Y}}(-\frac{1}{\bar{Y}}+\theta_0)}]=n*(-ln(\bar{Y}\theta_0))+n\bar{Y}(-\frac{1}{\bar{Y}}+\theta_0)}=100(-ln(3.1*3)-100+100*3.1*3=607$ or thereabouts, ie some ridiculously high number.

I assume the double of this is wa-ay outside the tails of chi square distribution, so reject the null.

4. Originally Posted by theodds
There's nothing wrong with the MLE being far away from the null value; that's what we want, since we are testing hypotheses.
Well I guess the real world tests may have all kinds of results, but I wonder why bother run a test with x% confidence level if it is obvious to the naked eye that the null does not make sense?

Could be a typo in the book... this question is from the same chapter I posted a while ago (page 18)
http://www.mathhelpforum.com/math-he...tml#post621601

5. There's nothing unusual about largely significant tests. It happens in practice all the time. Just because it is "obvious" that some parameters are (say) not equal to 0 that doesn't excuse the statistician from testing it typically.

To do better on (a), it turns out that you can write the test as reject when $\bar Y > c_1$ or $\bar Y < c_2$ for appropriately chosen $c_1, c_2$. You need $P_{H_0}(\mbox{Reject}) = \alpha$ and you also need $c_1, c_2$ to chosen so that if $\bar Y = c_1$ or if $\bar Y = c_2$ then you get the same value of the LR test statistic (the second restriction is correct up-to a stupid mistake on my part). You should think along the lines of the inverse image of the LR and think in terms of what values you $\bar Y$ you are rejecting. You have to solve these equations numerically I think.


text:
What is this subclass of the class of monotonic transformations?

Let $u$ be a continuous function from $\mathbb{R}$ to $\mathbb{R}$. Then $v$ is called a positive monotonic transformation of $u$ if $u(x) < u(y)$ if and only if $v(x)<v(y)$ and similarly for greater than and equal to for all $x$ and $y$.

But I'm interested in a stronger condition. My question is, what is the set of all functions $v$ such that $v$ is not only a monotonic transformation of $u$, but also satisfies the condition $u(x) - u(y) < u(z) - u(w)$ if and only if $v(x)-v(y) < v(z) -v(w)$ and similarly for greater than and equal to for all $x,y,z,$ and $w$?

This set of functions certainly contains all positive affine transformations of $u$, i.e. all functions of the form $au+b$ where $a$ is positive. But are there any other functions in this set?

This question arose from my answer in Economics.SE here, where the context is that $u$ is a von Neumann-Morgernstern utility function, and the functions $v$ that satisfy the condition above are other possible utility functions a person could have.

This is a complete revision of a former answer.

The answer to you question is : no, in your case, there are no other functions satisfying the property you mention.

This is a corollary of Theorem 1 in Basu, Kaushik. "Cardinal utility, utilitarianism, and a class of invariance axioms in welfare analysis." Journal of mathematical Economics 12.3 (1983): 193-206. As it turns out, the continuity of $u$ does not matter. The only thing that matters is that the domain $\mathbb{R}$ of $u$ be somewhat "regular", see below.

The following is from Basu (1983)

Definition : The mapping $f:B\rightarrow \mathbb{R}$, where $B\subset \mathbb{R}$ is a positive affine transformation (PAT) if $\exists a,b \in \mathbb{R}$, with $b>0$, such that $\forall t \in B$ $$f(t) = a + bt$$

Definition : The mapping $f:B\rightarrow \mathbb{R}$, where $B\subset \mathbb{R}$ is a first-difference preserving transformation (DPT) if $\forall t_1,t_2,t_3,t_4 \in B$ $$t_1 - t_2 \geq t_3 - t_4 \Leftrightarrow f(t_1) - f(t_2) \geq f(t_3) - f(t_4)$$

Theorem 1 : If $f: B\rightarrow \mathbb{R}$ and $B$ is dense in a connected subset of $\mathbb{R}$, then $f$ is a DPT if and only if it is a PAT.

To go from Theorem 1 to the desired corollary, simply notice that $\mathbb{R}$ is trivially dense in a connected subset of $\mathbb{R}$ (the subset being $\mathbb{R}$ itself).

In fact, this is noted by Basu himself in the following terms:

Assumption N (No restriction on transformation domain). $B = \mathbb{R}$.

Corollary 1.1. Given Assumption N, [...] the ability to compare first-differences of utility is equivalent to cardinality.

• I said that $u$ is a continuous function from R to R, so obviously its range can't be the union of [0,1] and [10,11]. – Keshav Srinivasan May 13 '15 at 0:26
• OK, can you edit that part out then? In any case, is your conclusion that $v$ must always be of the form $au+b$ for some fixed constants $a$ and $b$? – Keshav Srinivasan May 13 '15 at 5:54
• I only saw your revision now. So does this result not apply if B is a finite set? Because there are cases when you have only a finite set of outcomes. – Keshav Srinivasan Jul 2 '18 at 21:12


text:
Rabbit Searching Problem in Rust

This is my Rust implementation of the rabbit searching algorithm I found yesterday in this video. The problem statement is as follows.

There are 100 holes in a line. A rabbit is in one of the holes. You need to catch the rabbit. However, you can only look at one hole at a time. If the rabbit is in the hole that you looked, you catch the rabbit. If the rabbit is not in the hole, the rabbit will move to an adjacent hole.

The algorithm works as intended. You need to search two times. First, do a whole pass, and then do another pass but starting at index 1. However, I'm not familiar with Rust yet, so there are some awkward code snippets in my implementation. How can I improve the code according to Rust's best practices? Thanks.

use rand::Rng;

const N: i32 = 100;

fn main() {
let mut rabbit_pos: i32 = rand::thread_rng().gen_range(0..N);
println!("Rabbit is at {}", rabbit_pos);

let searched = search(&mut rabbit_pos);
println!("Rabbit is at {}", rabbit_pos);
}

fn search(rabbit_pos: &mut i32) -> i32 {
let mut found = false;
let mut found_pos = 0;
for i in 0..N {
if lookup(i, rabbit_pos) {
println!("Found {}", i);
found = true;
found_pos = i;
break;
}
}

if found {
return found_pos;
}

for i in 1..N {
if lookup(i, rabbit_pos) {
println!("Found {}", i);
found = true;
found_pos = i;
break;
}
}

found_pos
}

fn lookup(n: i32, rabbit_pos: &mut i32) -> bool {
println!("Searching for {}", n);
if *rabbit_pos == n {
return true;
} else {
move_rabbit(rabbit_pos);
return false;
}
}

fn move_rabbit(prev: &mut i32) {
let new_pos = match *prev {
0 => 1,
i if i == (N-1) => N-2,
_ => *prev + (rand::thread_rng().gen_range(0..2) * 2 - 1),
};
println!("Rabbit moved from {} to {}", *prev, new_pos);
*prev = new_pos;
}

In the search function, I suggest eliminating the variables found and foundpos.

If lookup succeeds, simply return i.

Otherwise it looks fine to me, although I have only scanned the code briefly.

Maybe you could put in a call panic!("Rabbit not found"); in if the algorithm unexpectedly fails ( rabbit not found after the two search loops ). Simply returning found_pos when the algorithm failed would be a little mis-leading!

Possible bug:

Your code only works for even length lines of holes. If you set N to be 99, and the rabbit starts off in an odd hole, you won't find it.

N = 3 example, rabbit starts in hole #1, and alternates between #1 & #0:

[_] r  _
r [_] _
_  r [_]
r [_] _
_  r [_]

You may or may not consider this to be a bug. The question asked for 100, and it works for 100. But you've parameterized it by setting N at the top, so it works for any even value. However, nothing requires N to be even. Either you should add a check, or at the very least a comment, or add a tiny modification to the algorithm so that it works with odd lengths too.
The value searched is never used:
    let searched = search(&mut rabbit_pos);


text:
# Examples of étale covers of arithmetic surfaces

Define an arithmetic scheme $X$ to be a separated, integral scheme, flat and finite type over $\mathbb{Z}$. I am interested in obtaining examples of finite étale covers of arithmetic schemes. I am mainly interested in the case of arithmetic surfaces, but I would enjoy examples of finite étale covers of arithmetic schemes of any dimension $>1$.

Let us restrict to schemes proper over $\mathbb{Z}$ to rule out the possibility of forming a finite integral extension and then restricting to an open subscheme on which the morphism is étale. Let us also restrict to geometrically connected (generic fiber?) to rule out the possibility of base ring extensions.

In the case where $X\to Spec(\mathbb{Z})$ is also smooth, the Katz-Lang finiteness theorem tells us that there will be at most finitely many abelian étale covers of $X$.

Abelian varieties can be a convenient way to create covers: if $A$ is an abelian variety, and $H$ a finite subgroup, then the dual isogeny to $A\to A/H$ is an étale cover of $A$. However, there are no abelian varieties over $Z$. I'm not sure whether this approach could be modified to produce an example.

It would be preferable to stick to regular or at least normal schemes, but that's not strictly necessary.

So, can anyone provide some nice examples of étale covers satisfying these conditions? If you would like to weaken some of the conditions to provide an interesting example (particularly, allowing nonproper or working over another $\mathcal{O}_K$), I guess I'll allow it.

To summarize, the optimal conditions on $X$ are regular, geometrically connected, proper over $\mathbb{Z}$.

Thanks.

• I suppose you are familiar with the very interesting work in the opposite direction, giving interesting examples of simply connected arithmetic surfaces? Ihara's paper in Galois Groups over $\mathbb{Q}$ proves among other things that $\mathbb{P}_{\mathbb{Z}}^1$ minus $0$, $\infty$ and the $N$-th roots of unity is such an example, and then there is Bost's paper Potential theory and Lefschetz theorems for arithmetic surfaces. Mar 3, 2016 at 4:18
• In the discussion of abelian varieties the isogeny will not be etale at the primes dividing the order of $H$. So no examples arise in this way. (If you delete a single prime $p$ you do get examples. You could construct a smooth and proper curve over $\mathrm{Spec} \, \mathbb{Z} \setminus \{p\}$ and then pulling back multiplication by $[p^n]$ on the Jacobian gives infinitely many connected etale abelian covers. But this example is uninteresting as you said explicitly that you want your arithmetic surface to be complete.) Mar 3, 2016 at 4:21
• Note that, for all $d\geq 1$, a finite type scheme $X$ over $\mathbb Z$ has only finitely many finite etale covers $Y\to X$ of degree $d$. This is sometimes reformulated by saying that the fundamental group of an arithmetic scheme is small; see arxiv.org/abs/0803.2096 This statement is clearly a generalization of the classical finiteness theorem of Hermite-Minkowski Mar 3, 2016 at 6:49
• Also, there is work of Wiesend math.uchicago.edu/~drinfeld/Deligne's_conjecture_Manin_conf/… and Hofmann-Wiesend (see [HW] in previous link) on constructing finite etale covers of arithmetic schemes. Mar 3, 2016 at 6:49
• @VesselinDimitrov I only know a small amount about the opposite direction, and I'm not familiar with those references. I will check them out. Thanks for those as well as the remark. Mar 4, 2016 at 0:04

You can, of course, use Bertini's theorem to make examples of a finite, flat, Galois extension with arbitrary finite Galois group $\Gamma$. Let $M$ be $\mathbb{Z}[\Gamma]$, the group ring of $\Gamma$ with coefficients in $\mathbb{Z}$. This is a finite, free $\mathbb{Z}$-module. Let $r>0$ be a positive integer. Then $M^{\oplus r}$ is also a finite, free $\mathbb{Z}$-module with a natural action of $\Gamma$. Form $S=\mathbb{Z}[M^{\oplus r}]$, the $\mathbb{Z}_{\geq 0}$-graded polynomial ring over $\mathbb{Z}$ such that the first graded piece $S_1$ equals $M^{\oplus r}$. Then $\mathbb{P}:=\text{Proj}(S)$ is a projective space over $\mathbb{Z}$ that has a natural action of $\Gamma$.
Denote by $U\subset\mathbb{P}$ the maximal open subscheme on which $\Gamma$ acts freely. Denote by $F$ the complement of $U$. The fibers of $F$ (over $\text{Spec}\ \mathbb{Z}$) have codimension at least $r$ inside the fibers of $\mathbb{P}$. Hence, assume $r$ is sufficiently large (say $r\geq 2$) so that the fibers of $F$ have codimension $\geq 2$. Denote by $q:\mathbb{P} \to Q$ the quotient of the action of $\Gamma$ on $\mathbb{P}$. This morphism is finite, and it is flat when restricted to $U$. The quotient $Q$ is a projective scheme, say $Q\subset \mathbb{P}^N_{\mathbb{Z}}$. By Bertini's Theorem, for sufficiently large integers $d$, there exist degree $d$ hypersurfaces in $\mathbb{P}^N_{\mathbb{Z}}$ whose intersection with $Q$ is a regular, $2$-dimensional scheme $X$ that is disjoint from the image of $F$. Define $\widetilde{X} = \mathbb{P}\times_Q X$. Then $\widetilde{X}\to X$ is a finite, flat, étale morphism that is Galois with Galois group $\Gamma$.
• $\Gamma$ is allowed to be nonabelian? Then do we actually take $M$ and $S$ to be noncommutative rings or just polynomial rings with variables indexed by the elements of the group, or something else? Then $S$ is infinitely generated, by $M^{\otimes r}$? Or is $S_1 = \Gamma^{\otimes r}$? Sorry, I'm a little confused. Mar 4, 2016 at 0:36
• Sorry for being unclear. My first concern was that if we allow $\Gamma$ to be nonabelian, then the rings $M$ and $S$ would be noncommutative, and so $\text{Proj}(S)$ would not be defined. My second concern was that if we take $S = \mathbb{Z}[M^{\oplus r}] = \mathbb{Z}[(\mathbb{Z}[\Gamma])^{\oplus r}]$, then our graded ring would not be finitely generated. And I did mean to write $\oplus$ instead of $\otimes$. Mar 4, 2016 at 17:19
• The ring $S$ is the commutative polynomial ring over $\mathbb{Z}$ that is $\mathbb{Z}_{\geq 0}$-graded and whose first graded piece is $M^{\oplus r}$. Once again, please read what I wrote. Mar 4, 2016 at 17:51
• Ok, so we do take $S$ to be the polynomial ring with variables indexed by $r$ copies of the group. I got it now. Thanks! Mar 4, 2016 at 18:06


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text:
# Hands-On: Fit Bivariate Distributions¶

Note

This lesson is a continuation of the Interactive Visual Statistics hands-on tutorial.

Similarly, the 2D Fit Distributions card is available for visualizing and estimating bivariate probability distributions on your dataset.

For example, let’s attempt to fit a 2D kernel density estimate (KDE) to the dataset, considering only the density and alcohol variables.

• Click the New Card button from the “Worksheet” header, then select Fit curves & distributions.

• Select the 2D Fit Distribution card.

• Specify density as the “X Variable” and alcohol as the “Y Variable”.

• Select the “2D KDE” radio button. Notice that the “X relative bandwidth” and “Y relative bandwidth” have the default value of 15. Let’s keep these default values. However, you can increase the values to make the KDE plot smoother, or decrease the values to make the plot less smooth.

• Click Create Card to create the card.


text:
### O/p resistance of Mosfet

In analog applications such as current mirrors or active loads, it is important for the transistor to have a large output resistance. Such circuits emulate a current source or current sink, and the Norton resistance of such a circuit should be large for ideal behavior.
The output resistance, usually denoted by rO, is a measure of how much drain-to-source voltage change is necessary to cause a given change in transistor output current when the transistor is in active mode. This resistance depends upon VGS, of course, because the channel conductivity depends upon the number of carriers within it, and that increases with gate voltage. However, rO also varies with VDS.
The reason a change in drain bias changes the resistance is that the channel exists only when the oxide field is sufficient to form a channel. At the source itself the oxide field is dependent upon the voltage drop VGS, which in active mode is above the threshold voltage, and so a channel forms. However, near the drain the oxide field depends upon VGD, and the applied drain voltage VDS makes VGD smaller than VGS because VDS brings the drain closer in voltage to the gate than is the source. The field in the oxide above the channel interpolates between these two values. In the ohmic or triode mode, a channel exists all the way from source to drain. But in the active mode, the drain voltage is high enough that somewhere between the source and drain the oxide field becomes too low to form a channel. The channel ends and dumps its carriers into the bulk semiconductor to finish their trip toward the drain without a channel. The termination of the channel is called the pinch-off point and it moves toward the source as the drain voltage increases. The channel becoming shorter as drain bias increases, the resistance between source and pinch-off point drops, so there results a lowering of output resistance with increase in drain bias. This phenomenon is called either channel-length modulation or the Early effect.
According to a simple empirical model patterned after the bipolar model for output resistance, the output resistance is given by:
$r_O = \left. \frac {\partial V_{DS}}{\partial I_{DS}}\right|_{V_ {GS}=\text{constant} } = \frac{ 1/\lambda + V_{DS}}{I_{DS}(V_{GS},\ V_{DS})} ,$
where λ is called the channel-length modulation parameter with dimensions V−1 and 1/λ plays the role of the Early voltage VA found in the bipolar model. The current IDS(VGS, VDS) is the drain current evaluated at the selected gate and drain voltages. It should be noted that this formula for output resistance is largely a fiction of hand analysis, and cannot be trusted. For example, the figure shows a tentative attempt to establish λ for a rather old 3/4μm technology. A single value for λ provides only a crude indicator of the slope of these curves in active mode.
To illustrate that λ is a function, not a constant, the lower figure shows measured values of VA=1/λ for a 0.18μm MOSFET process at a bias in the active mode of VGD=VGS−VDS=0 V. Here, VA = 1/λ increases by an order of magnitude as the channel becomes stronger. The need to employ a variety of λ-values is even greater in today's technology where λ is a function of device geometry in three dimensions (not just channel length, although this is important) and bias voltages. In practice, a particular value is calculated for each situation using a numerical model of the transistor or is measured directly.
Generally speaking, the output resistance of MOSFETs is low, and where high resistance is necessary, special circuit techniques involving multiple transistors are implemented to increase the effective resistance.
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