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This is a simulation for a double stars system in circular motion.

Assume the mass for stars are M,m.
The density is assume to be the same for both stars, so the radius for M will change when M is changed.
The distance between star is d,

The radius for the circular motion are r and d-r,
where $M\, r=(d-r)\,m$ , and $\frac{GMm}{d^2}=M\,r w^2=m(d-r) w^2$

The black dot is the center of mass (CM) for the binary stars ststem.


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# How do you solve -5x - 3y = 5 and 5x - 5y = -5?

May 9, 2016

$x = - 1 , y = 0$

#### Explanation:

$- 5 x - 3 y = 5$
$5 x - 5 y = - 5$

$\implies - 8 y = 0$

$\implies y = 0$

Using the obtained value of $y$, obtain the value of $x$ using equation

$- 5 x - 3 y = 5$
$- 5 x - 3 \left(0\right) = 5$
$- 5 x = 5$
$x = - 1$

format: plain


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Here we are now at the middle of the fourth large part of this talk.get nowheremore quotes

# b: 1

In Silico Flurries: Computing a world of snow. Scientific American. 23 December 2017

# data visualization + art

If you like space, you'll love my the 12,000 billion light-year map of clusters, superclusters and voids. Find the biggest nothings in Boötes and Eridanus.

# IAU Sky Constellation Resources

null
from an undefined
place,
undefined
create (a place)
an account
of us
— Viorica Hrincu

Having recently drawn a few skycharts (Superclusters & Voids, Sanctuary Project), I was frustrated by the lack of parsable resources for the IAU Constellations.

Finding a plain-text parsable definition of the asterisms proved impossible. So I created my own.

## IAU Constellation Shapes

All the constellation shapes were derived by manually examining the IAU map and cross-referencing the stars to the Yale Catalogue of Bright Stars.

The list of IAU constellation shapes in the file linked to above conveniently includes the J2000 right ascention and declination for each stars in the pair, along with their HR index, magnitude, Greek letter designation and name. See the file header for all the details.

For example, Cassiopeia's familiar "W" appears as 4 lines that indicate the connections between HR stars 21-168-264-403-542.

$Cas 21 2.294583 59.149722 2.27 bet Caph|bet Cas|11 Cas 168 10.127083 56.537222 2.23 alf Schedar|alf Cas|18 Cas Cas 168 10.127083 56.537222 2.23 alf Schedar|alf Cas|18 Cas 264 14.177083 60.716667 2.47 gam BU 499A|BU 1028|gam Cas|27 Cas Cas 264 14.177083 60.716667 2.47 gam BU 499A|BU 1028|gam Cas|27 Cas 403 21.454167 60.235278 2.68 del Ruchbah|BUP 19A|del Cas|37 Cas Cas 403 21.454167 60.235278 2.68 del Ruchbah|BUP 19A|del Cas|37 Cas 542 28.598750 63.670000 3.38 eps Segin|eps Cas|45 Cas$

The names are obtained from IAU Catalog of Star Names (IAU-CSN) and Simbad's name fields "NAME", "*" and "**", in that order. You can conveniently browse the Simbad database by any star identifier. For example, for Sirius the URL is http://simbad.u-strasbg.fr/simbad/sim-id?Ident=sirius.

Please report any errors to me.

## Bitmap, SVG and PDF Constellation Shapes

The shapes of all the constellations and the stars that define the asterisms shown in the image below. I also include all the 110 Messier objects with common names in this map (hollow circles).

The map also shows the galactic equator and the ecliptic. The vernal equinox, summer solstice, autumn equinox and winter solstice occur along the ecliptic at right ascension 0/360 (Pices), 270 (Sagittarius), 180 (Vigo) and 90 (Gemini/Taurus).

Whole-sky star charts are traditionally drawn with 360 right ascention on the left in decreasing order towards 0 on the right.

All 88 IAU constellations and their stars. Also included are all 110 Messier objects.

If you're interested in more astronomical resources, check out my Universe Superclusters and Voids resource page.

## Detailed Sky Chart

I've also created detailed charts that include all the 9,110 stars in the Yale Catalogue of Bright Stars. These are labeled by their Greek designation with the constellation as well as with their IAU name.

A closeup of the detailed map in the region of Auriga and Perseus. The Pleiades can be seen at the bottom right.

The maps also show all 110 Messier objects, labeled by their index and, where available, a common name. All the labels in these maps have been lovingly adjusted manually to avoid ambiguity and overlap. Available are blue, black and white background versions.

Sky chart with all 9,110 BSC stars, 88 IAU constellations and all 110 Messier objects. Where available, objects are labled with their common name. Also shown are the galactic equator and the ecliptic. (BUY ARTWORK)
Sky chart with all 9,110 BSC stars, 88 IAU constellations and all 110 Messier objects. Where available, objects are labled with their common name. Also shown are the galactic equator and the ecliptic. (BUY ARTWORK)
Sky chart with all 9,110 BSC stars, 88 IAU constellations and all 110 Messier objects. Where available, objects are labled with their common name. Also shown are the galactic equator and the ecliptic. (BUY ARTWORK)
VIEW ALL

# Predicting with confidence and tolerance

Wed 07-11-2018
I abhor averages. I like the individual case. —J.D. Brandeis.

We focus on the important distinction between confidence intervals, typically used to express uncertainty of a sampling statistic such as the mean and, prediction and tolerance intervals, used to make statements about the next value to be drawn from the population.

Confidence intervals provide coverage of a single point—the population mean—with the assurance that the probability of non-coverage is some acceptable value (e.g. 0.05). On the other hand, prediction and tolerance intervals both give information about typical values from the population and the percentage of the population expected to be in the interval. For example, a tolerance interval can be configured to tell us what fraction of sampled values (e.g. 95%) will fall into an interval some fraction of the time (e.g. 95%).

Nature Methods Points of Significance column: Predicting with confidence and tolerance. (read)

Altman, N. & Krzywinski, M. (2018) Points of significance: Predicting with confidence and tolerance Nature Methods 15:843–844.

Krzywinski, M. & Altman, N. (2013) Points of significance: Importance of being uncertain. Nature Methods 10:809–810.

# 4-day Circos course

Wed 31-10-2018

A 4-day introductory course on genome data parsing and visualization using Circos. Prepared for the Bioinformatics and Genome Analysis course in Institut Pasteur Tunis, Tunis, Tunisia.

Composite of the kinds of images you will learn to make in this course.

# Oryza longistaminata genome cake

Mon 24-09-2018

Data visualization should be informative and, where possible, tasty.

Stefan Reuscher from Bioscience and Biotechnology Center at Nagoya University celebrates a publication with a Circos cake.

The cake shows an overview of a de-novo assembled genome of a wild rice species Oryza longistaminata.

Circos cake celebrating Reuscher et al. 2018 publication of the Oryza longistaminata genome.

# Optimal experimental design

Tue 31-07-2018
Customize the experiment for the setting instead of adjusting the setting to fit a classical design.

The presence of constraints in experiments, such as sample size restrictions, awkward blocking or disallowed treatment combinations may make using classical designs very difficult or impossible.

Optimal design is a powerful, general purpose alternative for high quality, statistically grounded designs under nonstandard conditions.

Nature Methods Points of Significance column: Optimal experimental design. (read)

We discuss two types of optimal designs (D-optimal and I-optimal) and show how it can be applied to a scenario with sample size and blocking constraints.

Smucker, B., Krzywinski, M. & Altman, N. (2018) Points of significance: Optimal experimental design Nature Methods 15:599–600.

Krzywinski, M., Altman, N. (2014) Points of significance: Two factor designs. Nature Methods 11:1187–1188.

Krzywinski, M. & Altman, N. (2014) Points of significance: Analysis of variance (ANOVA) and blocking. Nature Methods 11:699–700.

Krzywinski, M. & Altman, N. (2014) Points of significance: Designing comparative experiments. Nature Methods 11:597–598.

# The Whole Earth Cataloguer

Mon 30-07-2018
All the living things.

An illustration of the Tree of Life, showing some of the key branches.

The tree is drawn as a DNA double helix, with bases colored to encode ribosomal RNA genes from various organisms on the tree.

The circle of life. (read, zoom)

All living things on earth descended from a single organism called LUCA (last universal common ancestor) and inherited LUCA’s genetic code for basic biological functions, such as translating DNA and creating proteins. Constant genetic mutations shuffled and altered this inheritance and added new genetic material—a process that created the diversity of life we see today. The “tree of life” organizes all organisms based on the extent of shuffling and alteration between them. The full tree has millions of branches and every living organism has its own place at one of the leaves in the tree. The simplified tree shown here depicts all three kingdoms of life: bacteria, archaebacteria and eukaryota. For some organisms a grey bar shows when they first appeared in the tree in millions of years (Ma). The double helix winding around the tree encodes highly conserved ribosomal RNA genes from various organisms.

Johnson, H.L. (2018) The Whole Earth Cataloguer, Sactown, Jun/Jul, p. 89

# Why we can't give up this odd way of typing

Mon 30-07-2018
All fingers report to home row.

An article about keyboard layouts and the history and persistence of QWERTY.

My Carpalx keyboard optimization software is mentioned along with my World's Most Difficult Layout: TNWMLC. True typing hell.

TNWMLC requires seriously flexible digits. It’s 87% more difficult than using a standard Qwerty keyboard, according to Martin Krzywinski, who created it (Credit: Ben Nelms). (read)

McDonald, T. (2018) Why we can't give up this odd way of typing, BBC, 25 May 2018.


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Data Scientist

The Fast and the Furiously Frugal

Fast and frugal trees

I’ve praised the utility of decision trees in other scenarios especially where accountability and transparency of decision making is important. Here we explore why decision trees are a good introduction to Machine Learning and its ability to spot patterns in data providing insight. Decision trees are arguably easier to interpret and more inline with human thinking than some other ML methods, thus we write a post here to use a fast and frugal tree method, providing a quick solution to a classification problem.

This post is based on another blog post, which makes a rather contrived comparison of a logistic regression and a decision tree. Arguable the difficulty of interpretation of logistic regression to the non-expert could be avoided by having a suitable front end, as many users will not need to see how the decision tool operates under the hood. However, the FFTrees package provides a user with everything they need in a few lines of code, and comes with excellent vignettes and documentation using some classic ML datasets.

We read, tidy and normalise the data as described in a previous post.

We then split the data into a training and testing set to assess how well the classifier performs on data it hasn’t seen.

Now that we’ve created the object, we can print it to the console to get basic information.

The printout tells us that the final fft object contains 6 different trees, and the largest tree only uses 4 of the original 12 cues. To see the best tree, we can simply plot the fft object:

There’s one of our fast and frugal trees! In the top section of the plot, we see that the data had 237 true Pass cases, and 113 true Fail cases. In the middle section, we see the actual tree. The tree then starts by looking at the cue G2. If the value is less than 0.53, the tree decides that the person is a Pass If the value is not less than 0.53, then the tree looks at G1. If the G1 > 0.44, the tree decides the patient is a Pass. If G1 <= 0.44, the tree decides that the person is a Fail.

That’s the whole decision algorithm! Quite simple to follow but not very powerful at extracting extra detail that would provide additional insight. I think all teachers would expect the previous two terms exam grades to be good predictors of the final exam pass or fail status. However, it does validate this assumption and provides a cut-off point for strategies targeting students who need extra help to get them up to a pass.

This fast and frugalness does miss some aspects as shown in the performance section of the plot where we see 43 misses. Where students passed but were predicted to fail, we probably prefer this to the alternative of Falsely predicting a pass.

Performace

As you can see, the tree performed exceptionally well: it made correct diagnoses in 301 (107+181+13) out of all 350 cases (86% correct). Additional performance statistics, including specificity (1 - false alarm rate), hit rate, d-prime, AUC (area under the curve) are also displayed. Finally, in the bottom right plot, you can see an ROC curve which compares the performance of the trees to CART (in red) and logistic regression (in blue).

Viewing other trees

Now, what if you want a tree that rarely misses true Fail cases, at the cost of additional false alarms (those who would pass anyway)? As Luan, Schooler, & Gigerenzer (2011) have shown, you can easily shift the balance of errors in a fast and frugal tree by adjusting the decisions it makes at each level of a tree. The FFTrees function automatically builds several versions of the same general tree that make different error trade-offs. We can see the performance of each of these trees in the bottom-right ROC curve. Looking at the ROC curve, we can see that tree number 3 has a very high specificity, but a smaller hit-rate compared to tree number 4. We can look at this tree by adding the tree = 3 argument to plot(). As teachers who are concerned that every-child matters we may prefer to err on the side of caution, depsite it adding expense (more teachers to help out) and wasting the time of some students.

Conclusion

FFTrees is a nice, easy to interpret gateway R package into classification techniques and Data Science and Machine Learning. I think it is at its most useful when challenging and testing assumptions held by experts in the field. This leads to an evidence based approach to decision making whereby you could show this FFT plot to parents to explain why there child needs to attend extra sessions during the School Holidays, for example.

Try it youself with the ‘mushrooms’ dataset (see ‘?mushrooms’ for details, it’s loaded in with the ‘FFT’ package). Or predict your seminal quality (if relevant) with the ‘fertility’ dataset.


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# zbMATH — the first resource for mathematics

Čech homology characterizations of infinite dimensional manifolds. (English) Zbl 0538.57006
Let $$X$$ denote a locally compact ANR. It is shown that $$X$$ is a $$Q$$-manifold if and only if $$X$$ has the disjoint disks property and the disjoint Čech carriers property. This result can be regarded as the infinite- dimensional version of the known fact that when $$n\geq 5$$, a generalized n-manifold Y is a topological $$n$$-manifold if and only if $$Y$$ has the disjoint disk property; the proof relies on the result of H. Torunczyk [Fundam. Math. 106, 31-40 (1980; Zbl 0346.57004)] that $$X$$ is a $$Q$$-manifold if and only if $$X$$ has the disjoint $$k$$-cells property for each $$k\geq 0$$. Several applications are given to the problems of recognizing $$Q$$-manifolds among products, proper cell-like images of $$Q$$-manifolds, and sums of $$Q$$-manifolds. An example is also given of a cellular decomposition of $$Q$$ whose nondegenerate elements form a null-sequence, but whose decomposition space is not a $$Q$$-manifold.
Reviewer: R.Sher

##### MSC:
 57N20 Topology of infinite-dimensional manifolds 57N60 Cellularity in topological manifolds 58B05 Homotopy and topological questions for infinite-dimensional manifolds 57P99 Generalized manifolds 55N07 Steenrod-Sitnikov homologies
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# Gaussian filter

Last updated

In electronics and signal processing, a Gaussian filter is a filter whose impulse response is a Gaussian function (or an approximation to it, since a true Gaussian response is physically unrealizable as it has infinite support). Gaussian filters have the properties of having no overshoot to a step function input while minimizing the rise and fall time. This behavior is closely connected to the fact that the Gaussian filter has the minimum possible group delay. It is considered the ideal time domain filter, just as the sinc is the ideal frequency domain filter. [1] These properties are important in areas such as oscilloscopes [2] and digital telecommunication systems. [3]

## Contents

Mathematically, a Gaussian filter modifies the input signal by convolution with a Gaussian function; this transformation is also known as the Weierstrass transform.

## Definition

The one-dimensional Gaussian filter has an impulse response given by

${\displaystyle g(x)={\sqrt {\frac {a}{\pi }}}e^{-ax^{2}}}$

and the frequency response is given by the Fourier transform

${\displaystyle {\hat {g}}(f)=e^{-{\frac {\pi ^{2}f^{2}}{a}}}}$

with ${\displaystyle f}$ the ordinary frequency. These equations can also be expressed with the standard deviation as parameter

${\displaystyle g(x)={\frac {1}{{\sqrt {2\pi }}\sigma }}e^{-{\frac {x^{2}}{2\sigma ^{2}}}}}$

and the frequency response is given by

${\displaystyle {\hat {g}}(f)=e^{-{\frac {f^{2}}{2\sigma _{f}^{2}}}}}$

By writing ${\displaystyle a}$ as a function of ${\displaystyle \sigma }$ with the two equations for ${\displaystyle g(x)}$ and as a function of ${\displaystyle \sigma _{f}}$ with the two equations for ${\displaystyle {\hat {g}}(f)}$ it can be shown that the product of the standard deviation and the standard deviation in the frequency domain is given by

${\displaystyle \sigma \sigma _{f}={\frac {1}{2\pi }}}$,

where the standard deviations are expressed in their physical units, e.g. in the case of time and frequency in seconds and hertz, respectively.

In two dimensions, it is the product of two such Gaussians, one per direction:

${\displaystyle g(x,y)={\frac {1}{2\pi \sigma ^{2}}}e^{-{\frac {x^{2}+y^{2}}{2\sigma ^{2}}}}}$ [4] [5] [6]

where x is the distance from the origin in the horizontal axis, y is the distance from the origin in the vertical axis, and σ is the standard deviation of the Gaussian distribution.

## Digital implementation

The Gaussian function is for ${\displaystyle x\in (-\infty ,\infty )}$ and would theoretically require an infinite window length. However, since it decays rapidly, it is often reasonable to truncate the filter window and implement the filter directly for narrow windows, in effect by using a simple rectangular window function. In other cases, the truncation may introduce significant errors. Better results can be achieved by instead using a different window function; see scale space implementation for details.

Filtering involves convolution. The filter function is said to be the kernel of an integral transform. The Gaussian kernel is continuous. Most commonly, the discrete equivalent is the sampled Gaussian kernel that is produced by sampling points from the continuous Gaussian. An alternate method is to use the discrete Gaussian kernel [7] which has superior characteristics for some purposes. Unlike the sampled Gaussian kernel, the discrete Gaussian kernel is the solution to the discrete diffusion equation.

Since the Fourier transform of the Gaussian function yields a Gaussian function, the signal (preferably after being divided into overlapping windowed blocks) can be transformed with a Fast Fourier transform, multiplied with a Gaussian function and transformed back. This is the standard procedure of applying an arbitrary finite impulse response filter, with the only difference that the Fourier transform of the filter window is explicitly known.

Due to the central limit theorem, the Gaussian can be approximated by several runs of a very simple filter such as the moving average. The simple moving average corresponds to convolution with the constant B-spline (a rectangular pulse), and, for example, four iterations of a moving average yields a cubic B-spline as filter window which approximates the Gaussian quite well. A moving average is quite cheap to compute, so levels can be cascaded quite easily.

In the discrete case the standard deviations are related by

${\displaystyle \sigma \cdot \sigma _{f}={\frac {N}{2\pi }}}$

where the standard deviations are expressed in number of samples and N is the total number of samples. Borrowing the terms from statistics, the standard deviation of a filter can be interpreted as a measure of its size. The cut-off frequency of a Gaussian filter might be defined by the standard deviation in the frequency domain yielding

${\displaystyle f_{c}=\sigma _{f}={\frac {1}{2\pi \sigma }}}$

where all quantities are expressed in their physical units. If ${\displaystyle \sigma }$ is measured in samples the cut-off frequency (in physical units) can be calculated with

${\displaystyle f_{c}={\frac {F_{s}}{2\pi \sigma }}}$

where ${\displaystyle F_{s}}$ is the sample rate. The response value of the Gaussian filter at this cut-off frequency equals exp(-0.5)≈0.607.

However, it is more common to define the cut-off frequency as the half power point: where the filter response is reduced to 0.5 (-3 dB) in the power spectrum, or 1/2 ≈ 0.707 in the amplitude spectrum (see e.g. Butterworth filter). For an arbitrary cut-off value 1/c for the response of the filter the cut-off frequency is given by

${\displaystyle f_{c}={\sqrt {2\ln(c)}}\cdot \sigma _{f}}$

For c=2 the constant before the standard deviation in the frequency domain in the last equation equals approximately 1.1774, which is half the Full Width at Half Maximum (FWHM) (see Gaussian function). For c=2 this constant equals approximately 0.8326. These values are quite close to 1.

A simple moving average corresponds to a uniform probability distribution and thus its filter width of size ${\displaystyle n}$ has standard deviation ${\displaystyle {\sqrt {({n}^{2}-1)/12}}}$. Thus the application of successive ${\displaystyle m}$ moving averages with sizes ${\displaystyle {n}_{1},\dots ,{n}_{m}}$ yield a standard deviation of

${\displaystyle \sigma ={\sqrt {\frac {{n}_{1}^{2}+\cdots +{n}_{m}^{2}-m}{12}}}}$

(Note that standard deviations do not sum up, but variances do.)

A gaussian kernel requires ${\displaystyle 6{\sigma }-1}$ values, e.g. for a ${\displaystyle {\sigma }}$ of 3 it needs a kernel of length 17. A running mean filter of 5 points will have a sigma of ${\displaystyle {\sqrt {2}}}$. Running it three times will give a ${\displaystyle {\sigma }}$ of 2.42. It remains to be seen where the advantage is over using a gaussian rather than a poor approximation.

When applied in two dimensions, this formula produces a Gaussian surface that has a maximum at the origin, whose contours are concentric circles with the origin as center. A two dimensional convolution matrix is precomputed from the formula and convolved with two dimensional data. Each element in the resultant matrix new value is set to a weighted average of that elements neighborhood. The focal element receives the heaviest weight (having the highest Gaussian value) and neighboring elements receive smaller weights as their distance to the focal element increases. In Image processing, each element in the matrix represents a pixel attribute such as brightness or a color intensity, and the overall effect is called Gaussian blur.

The Gaussian filter is non-causal which means the filter window is symmetric about the origin in the time-domain. This makes the Gaussian filter physically unrealizable. This is usually of no consequence for applications where the filter bandwidth is much larger than the signal. In real-time systems, a delay is incurred because incoming samples need to fill the filter window before the filter can be applied to the signal. While no amount of delay can make a theoretical Gaussian filter causal (because the Gaussian function is non-zero everywhere), the Gaussian function converges to zero so rapidly that a causal approximation can achieve any required tolerance with a modest delay, even to the accuracy of floating point representation.

## Related Research Articles

In probability theory, a normaldistribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is

A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases, and then decreases back to zero. It can typically be visualized as a "brief oscillation" like one recorded by a seismograph or heart monitor. Generally, wavelets are intentionally crafted to have specific properties that make them useful for signal processing.

In probability and statistics, Student's t-distribution is any member of a family of continuous probability distributions that arise when estimating the mean of a normally-distributed population in situations where the sample size is small and the population's standard deviation is unknown. It was developed by English statistician William Sealy Gosset under the pseudonym "Student".

In signal processing and statistics, a window function is a mathematical function that is zero-valued outside of some chosen interval, normally symmetric around the middle of the interval, usually near a maximum in the middle, and usually tapering away from the middle. Mathematically, when another function or waveform/data-sequence is "multiplied" by a window function, the product is also zero-valued outside the interval: all that is left is the part where they overlap, the "view through the window". Equivalently, and in actual practice, the segment of data within the window is first isolated, and then only that data is multiplied by the window function values. Thus, tapering, not segmentation, is the main purpose of window functions.

In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the form

In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution for nonnegative-valued random variables. It is essentially a chi distribution with two degrees of freedom.

In signal processing, a finite impulse response (FIR) filter is a filter whose impulse response is of finite duration, because it settles to zero in finite time. This is in contrast to infinite impulse response (IIR) filters, which may have internal feedback and may continue to respond indefinitely.

In mathematics and in signal processing, the Hilbert transform is a specific linear operator that takes a function, u(t) of a real variable and produces another function of a real variable H(u)(t). This linear operator is given by convolution with the function . The Hilbert transform has a particularly simple representation in the frequency domain: It imparts a phase shift of ±90° to every frequency component of a function, the sign of the shift depending on the sign of the frequency. The Hilbert transform is important in signal processing, where it is a component of the analytic representation of a real-valued signal u(t). The Hilbert transform was first introduced by David Hilbert in this setting, to solve a special case of the Riemann–Hilbert problem for analytic functions.

The Voigt profile is a probability distribution given by a convolution of a Cauchy-Lorentz distribution and a Gaussian distribution. It is often used in analyzing data from spectroscopy or diffraction.

Gaussian noise, named after Carl Friedrich Gauss, is statistical noise having a probability density function (PDF) equal to that of the normal distribution, which is also known as the Gaussian distribution. In other words, the values that the noise can take on are Gaussian-distributed.

In image processing, a Gabor filter, named after Dennis Gabor, is a linear filter used for texture analysis, which essentially means that it analyzes whether there is any specific frequency content in the image in specific directions in a localized region around the point or region of analysis. Frequency and orientation representations of Gabor filters are claimed by many contemporary vision scientists to be similar to those of the human visual system. They have been found to be particularly appropriate for texture representation and discrimination. In the spatial domain, a 2D Gabor filter is a Gaussian kernel function modulated by a sinusoidal plane wave.

In image processing, a Gaussian blur is the result of blurring an image by a Gaussian function.

In probability theory, the Rice distribution or Rician distribution is the probability distribution of the magnitude of a circularly-symmetric bivariate normal random variable, possibly with non-zero mean (noncentral). It was named after Stephen O. Rice.

In probability theory, calculation of the sum of normally distributed random variables is an instance of the arithmetic of random variables, which can be quite complex based on the probability distributions of the random variables involved and their relationships.

In imaging science, difference of Gaussians (DoG) is a feature enhancement algorithm that involves the subtraction of one Gaussian blurred version of an original image from another, less blurred version of the original. In the simple case of grayscale images, the blurred images are obtained by convolving the original grayscale images with Gaussian kernels having differing width. Blurring an image using a Gaussian kernel suppresses only high-frequency spatial information. Subtracting one image from the other preserves spatial information that lies between the range of frequencies that are preserved in the two blurred images. Thus, the DoG is a spatial band-pass filter that attenuates frequencies in the original grayscale image that are far from the band center.

The Gabor transform, named after Dennis Gabor, is a special case of the short-time Fourier transform. It is used to determine the sinusoidal frequency and phase content of local sections of a signal as it changes over time. The function to be transformed is first multiplied by a Gaussian function, which can be regarded as a window function, and the resulting function is then transformed with a Fourier transform to derive the time-frequency analysis. The window function means that the signal near the time being analyzed will have higher weight. The Gabor transform of a signal x(t) is defined by this formula:

In probability theory, the inverse Gaussian distribution is a two-parameter family of continuous probability distributions with support on (0,∞).

In the areas of computer vision, image analysis and signal processing, the notion of scale-space representation is used for processing measurement data at multiple scales, and specifically enhance or suppress image features over different ranges of scale. A special type of scale-space representation is provided by the Gaussian scale space, where the image data in N dimensions is subjected to smoothing by Gaussian convolution. Most of the theory for Gaussian scale space deals with continuous images, whereas one when implementing this theory will have to face the fact that most measurement data are discrete. Hence, the theoretical problem arises concerning how to discretize the continuous theory while either preserving or well approximating the desirable theoretical properties that lead to the choice of the Gaussian kernel. This article describes basic appproaches for this that have been developed in the literature.

In mathematics, the Weierstrass transform of a function f : RR, named after Karl Weierstrass, is a "smoothed" version of f(x) obtained by averaging the values of f, weighted with a Gaussian centered at x.

In probability theory, the rectified Gaussian distribution is a modification of the Gaussian distribution when its negative elements are reset to 0. It is essentially a mixture of a discrete distribution and a continuous distribution as a result of censoring.

## References

1. Filtering in the Time and Frequency Domains by Herman J. Blinchikoff, Anatol I. Zverev
2. R.A. Haddad and A.N. Akansu, "A Class of Fast Gaussian Binomial Filters for Speech and Image Processing," IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 39, pp 723-727, March 1991.
3. Shapiro, L. G. & Stockman, G. C: "Computer Vision", page 137, 150. Prentence Hall, 2001
4. Mark S. Nixon and Alberto S. Aguado. Feature Extraction and Image Processing. Academic Press, 2008, p. 88.


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# How do black holes accrete mass?

Thanks to time dilation, a distant observer watching a man fall in to a black hole will only see him asymptotically approach the event horizon. So how do black holes ever get bigger?

• Just because the distant observer sees it, doesn't make it true. – chharvey Nov 10 '12 at 1:41

## 4 Answers

In the frame of reference of the black hole (if there is such a thing), the man is indeed absorbed by the black hole. But the information will never reach the observer since any photon emitted past the event horizon (which is not the black hole surface) will never reach the observer.

• I don't think this answers the question. By the way, the frame of reference "of the black hole" is not well-defined, and the even[t] horizon is not a bad definition for a black hole "surface"... – hwlin Nov 10 '12 at 4:15
• I thought your question implied that it would take an infinite amount of time for the man to fall in the black hole. Which is not true. Also, I always thought the size of the black hole was smaller or equal to the Schwarzschild radius, which is the event horizon. – FrenchKheldar Nov 10 '12 at 4:39

I don't think there's a paradox here: The infalling observer will observe himself passing the horizon. The observer at infinity does not see this happen, but she will notice, with time, that the region from which she never receives signals has expanded.

This question crops up a lot, and I think it illustrates an interesting aspect of GR. Specifically what exactly does the question "How do black holes accrete mass?" mean?

In GR most questions don't make sense unless you specify what observer you're asking about or whether you want an observer independant reply. In the case of objects falling into a black hole the observer independant reply is easy because the geodesic of an object falling into a black hole is easy to calculate. See any introductory GR textbook, or if you want to ask a new question about this I'll be happy to post the details.

But most people asking the question have an observer in mind, and typically it's the Schwarzschild observer i.e. the observer sitting stationary at infinity (or far enough away to be effectively infinitely far away). In this case you're quite correct that this observer will never see an object reach the event horizon, but then the Schwarzschild observer will never even see the black hole form.

What the Schwarzschild observer can do is measure the curvature at some non-zero distance from the event horizon, and from this infer that the black hole must exist. If the observer does this, then throws some mass into the black hole and remeasures the curvature they will find it has increased. This allows them to infer that the mass of the black hole has increased i.e. the black hole has accreted mass.

It's tempting to get excessively philosophical and ask what we mean by "exist", but I think this temptation should be resisted as long as the observer independant calculation gives a clear result. If I do measurements of some object and those measurements are consistent with the Schwarzschild metric of a black hole then I think I'm justified in concluding that the object is a black hole.

• Thanks so much, great answer. And yes I meant an observer at infinity. What do you mean by "the observer will never see the BH form?" The Schwarzschild solution describes an eternal BH, so no one will see it form, right? – hwlin Nov 17 '12 at 19:35
• Yes, the Schwarzschild metric strictly describes an eternal black hole, but it's a good approximation for black holes of a finite age. Well, arguably black holes of a finite age don't exist, but you know what I mean :-) – John Rennie Nov 18 '12 at 10:31
• I don't understand. We have observed black holes in binary systems and other various places, but we are observers near infinity. How can we observe these black holes if we can never have seen them form? – PyRulez Jul 13 '15 at 1:58
• @PyRulez: we have never observed a black hole. However we have observed objects so compact and massive that we now they would form a black hole given infinite time. Note that although a black hole takes an infinite time, a collapsing object collapses to an object very similar to a black hole very quickly. For example if you were to look at Sagittarius A$^*$ it would be indistinguishable from a black hole even though technically it isn't (yet). – John Rennie Jul 13 '15 at 5:37
• @JohnRennie That's what I thought. There aren't any singularties either, aren't there? – PyRulez Jul 13 '15 at 6:51

Firstly, just considering the behaviour of infalling matter in a Schwarzschild metric doesn't directly answer the "how can they possibly grow?" question, since this metric describes an unchanging "eternal" black hole with constant mass. The other problem is that the traditional event horizon, which is often described as the defining feature of the black hole, only makes sense in the context of the entire picture of the spacetime, i.e. including its full time evolution. This is because the EH is defined as the boundary of the region in which photons are unable to escape to future null infinity, and we don't know for a fact that a photon can make such an escape until we have a picture of the entire spacetime laid out before us.

For this reason, people have considered other, more local (i.e. not requiring the whole of space-time), black hole definitions based on trapping horizons, and dynamical horizons. Here is a reference describing a dynamical horizon. In this ref fig. 4 shows a Penrose diagram of the Vaidya spacetime, in which infalling radiation causes the dynamical horizon to grow. Surprisingly, the traditional event horizon, however, reaches back even into the flat (blue) region. In the right hand picture, infall of radiation stops at $\nu=\nu_0$ and from then on the black hole looks like a Schwarzschild black hole.

Additional information on the Vaidya-Schwarzschild scenario is available here and a general treatment of various options for horizon definition here.


text:
# Why do we need F to be Banach space instead of a normed space in this corollary of Uniform boundedness theorem?

Let $E$ and $F$ be two Banach spaces. Let $(T_n)$ be a sequence of continuous linear operators from $E$ into $F$ such that for every $x \in E, T_n(x)$ converges (as $n \to \infty$) to a limit denoted by $T(x)$. Then we have

(a) $\sup \Vert T_n\Vert < \infty$

(b)$T \in L(E,F)$,

(c) $\Vert T\Vert_{L(E,F)} \le \liminf_{n\to \infty} \Vert T_n\Vert_{L(E,F)}.$

Why do we need $F$ to be Banach space instead of a normed space in this corollary of the Uniform boundary theorem?

• I think it should be $\|T\|\leq\liminf_{n}\|T_{n}\|$? – user284331 Jan 10 '18 at 2:43
• Uniform boundedness principle, not "Uniform boundary theorem"? – Robert Israel Jan 10 '18 at 2:50

The following is taken from An Introduction to Banach Space, Robert E. Megginson, page 45, 46:

Let $\mathcal{F}$ be a nonempty family of bounded linear operators from a Banach space $X$ into a normed space $Y$. If $\sup\{\|Tx\|: x\in\mathcal{F}\}$ is finite for each $x\in X$, then $\sup\{\|T\|: T\in\mathcal{F}\}$ is finite.

And as a corollary:

Let $(T_{n})$ be a sequence of bounded linear operators from a Banach space $X$ into a normed space $Y$ such that $\lim_{n}T_{n}x$ exists for each $x\in X$. Define $T:X\rightarrow Y$ by the formula $Tx=\lim_{n}T_{n}x$. Then $T$ is a bounded linear operator from $X$ into $Y$.

So I think the completeness of $Y$ is not needed.

We do not need $F$ to be a Banach space. Given $E$, $F$, $T_n$ and $T$ as specified but with $F$ a normed space, you can simply replace $F$ by its completion $\widetilde{F}$, and get the same situation with $T_n$ and $T$ mapping $E$ into $\widetilde{F}$. The theorem for operators from $E$ into $\widetilde{F}$ then implies the theorem for operators from $E$ into $F$.

• Thanks! This point of view helps me see the connection much more clearly . – Xiao Jan 10 '18 at 5:09


text:
# Hierarchy of TeXnical Accomplishments

Pretty much the same thing as the above, except instead of taking a snapshot of Unix users, it takes a snapshot of TeX users.


text:
# [SOLVED]Problems in LaTeX

#### ZaidAlyafey

##### Well-known member
MHB Math Helper
Today I faced lots of problems in LaTeX it just drove me crazy ...

1- when I preview post a MATH tags just appears from no where in between some code ! ...

2-Some code just stops rendering but when I post it in another compiler it just works fine , but when preveiwing it here , it doesn't work

3-I faced this problem multiple times , what drives me crazy that if I copy and paste the same code to another compiler it works :

$$\displaystyle \int^{\infty}_0\left(e^{-bx}-\frac{1}{1+ax}\right)\,\frac{dx}{x}=\log\left(\frac{a}{b} \right)-\gamma$$

#### Chris L T521

##### Well-known member
Staff member
Today I faced lots of problems in LaTeX it just drove me crazy ...

1- when I preview post a MATH tags just appears from no where in between some code ! ...

2-Some code just stops rendering but when I post it in another compiler it just works fine , but when preveiwing it here , it doesn't work

3-I faced this problem multiple times , what drives me crazy that if I copy and paste the same code to another compiler it works :

$$\displaystyle \int^{\infty}_0\left(e^{-bx}-\frac{1}{1+ax}\right)\,\frac{dx}{x}=\log\left(\frac{a}{b} \right)-\gamma$$
I'm not sure what to say about (1), but for (2) you need to introduce more spacing into your LaTeX code, if I'm not mistaken. I think Ackbach pointed out somewhere that once your string of TeX commands exceeds a certain length without spacing, mathjax doesn't know how to compile it properly. So with enough spacing, I get your example to look like this:

$$\displaystyle \int^{\infty}_0 \left(e^{-bx}- \frac{1}{1+ax}\right) \,\frac{dx}{x} = \log\left(\frac{a}{b} \right)-\gamma$$

#### Klaas van Aarsen

##### MHB Seeker
Staff member
Another problem is that regular text formatting and latex do not work well together.
When you copy+paste latex, there is a tendency that some formatting commands come along, breaking latex.
When replying, click the quick button Switch Editor to Source Mode at the left top.
Then you can see the formatting commands and remove them if necessary.

#### ZaidAlyafey

##### Well-known member
MHB Math Helper
I'm not sure what to say about (1), but for (2) you need to introduce more spacing into your LaTeX code, if I'm not mistaken. I think Ackbach pointed out somewhere that once your string of TeX commands exceeds a certain length without spacing, mathjax doesn't know how to compile it properly. So with enough spacing, I get your example to look like this:

$$\displaystyle \int^{\infty}_0 \left(e^{-bx}- \frac{1}{1+ax}\right) \,\frac{dx}{x} = \log\left(\frac{a}{b} \right)-\gamma$$
I thought this is the problem , but what do you mean by enough spacing ?

#### Klaas van Aarsen

##### MHB Seeker
Staff member
I thought this is the problem , but what do you mean by enough spacing ?
After 50 characters without a space, vBulletin inserts an invisible space for you.
It breaks latex.

MHB Math Helper

#### ZaidAlyafey

##### Well-known member
MHB Math Helper
Another problem is that regular text formatting and latex do not work well together.
When you copy+paste latex, there is a tendency that some formatting commands come along, breaking latex.
When replying, click the quick button Switch Editor to Source Mode at the left top.
Then you can see the formatting commands and remove them if necessary.
I've long been searching for that Switch Editor to Source Mode button , could you please upload an image for that ...

#### MarkFL

##### Administrator
Staff member
On the toolbar, it's the button on the left, at the top.

#### Klaas van Aarsen

##### MHB Seeker
Staff member
I've long been searching for that
Switch Editor to Source Mode button , could you please upload an image for that ...

As you can see the quick button at the left top is pressed.
In the reply text you can see for instance [I] now.

#### Jameson

##### Administrator
Staff member
I recommend keeping the editor in the "standard editor" option. If you go to "Settings" -> "General settings" -> "Miscellaneous Options" -> "Message Editor Interface:", you'll see you have 3 choices. It seems like many users choose the "enhanced editor" option but you can do everything with the standard editor that you can with the enhanced and I never have any problems using it. So you can edit your default choice there or you can just push the button that I like Serena pointed out, but I'm not sure that this will stay as the default option. Either way, I definitely recommend the choice where you see each BB code tag.


text:
# BibLatex warning as below. Any clue? Thanks [closed]

Here is what I want to accomplish, i.e., apa-style citation

\usepackage{babel}
\usepackage{csquotes}
\usepackage[style=apa]{biblatex}
\DeclareLanguageMapping{american}{american-apa}

But I received the following error message:

Package biblatex warning: No "backend" specified, using Biber backend. To use BibTex, load biblatex with the "backend=bibtex" option.

!Latex Error: Command \bibhand already defined Or name \end... illegal, see p. 192 of the manual.

I have no clue what this means.

## closed as unclear what you're asking by moewe, Adam Liter, user31729, Werner, ChrisSJun 2 '14 at 22:20

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• And can you please post a complete, minimal document allowing us to reproduce the problem? The first warning is harmless as it just informs you that biber will be the backend used; the error indicates that you are loading some other package that has defined \bibhang and we need the minimal document to see where the problem might be. – Gonzalo Medina May 29 '14 at 13:30


text:
Norm (mathematics)

Main Article
Discussion
Related Articles  [?]
Bibliography  [?]
Citable Version  [?]

This editable Main Article is under development and subject to a disclaimer.

In mathematics, a norm is a function on a vector space that generalizes to vector spaces the notion of the distance from a point of a Euclidean space to the origin.

Formal definition of norm

Let X be a vector space over some subfield F of the complex numbers. Then a norm on X is any function ${\displaystyle \|\cdot \|:X\rightarrow \mathbb {R} }$ having the following four properties:

1. ${\displaystyle \|x\|\geq 0}$ for all ${\displaystyle x\in X}$ (positivity)
2. ${\displaystyle \|x\|=0}$ if and only if x=0
3. ${\displaystyle \|x+y\|\leq \|x\|+\|y\|}$ for all ${\displaystyle x,y\in X}$ (triangular inequality)
4. ${\displaystyle \|cx\|=|c|\|x\|}$ for all ${\displaystyle c\in F}$

A norm on X also defines a metric ${\displaystyle d}$ on X as ${\displaystyle d(x,y)=\|x-y\|}$. Hence a normed space is also a metric space.


text:
Is 0 a stable equilibrium of $x''(t) + 2x^3(t) = 0$?
Is 0 an asymptotically stable equilibrium of $x''(t) + 2x^3(t) = 0$?


text:
# Each statement describes a transformation of the graph of f(x)=3/x. which statement correctly describes the graph of y= f(x- 7) +

###### Question:

Each statement describes a transformation of the graph of f(x)=3/x. which statement correctly describes the graph of y= f(x- 7) + 3? A It is the graph of f translated 3 units up and 7 units to the left.
B. It is the graph of f translated 7 units down and 3 units to the right.
C It is the graph of f translated 7 units up and 3 units to the
right!
D. It is the graph of f translated 3 units up and 7 units to the right.

### Solve for x Plz help

Solve for x Plz help $Solve for x Plz help$...

### Visit us again, but the most of us to take care, I will have the? ???

Visit us again, but the most of us to take care, I will have the? ???...

### Hello solve this algebra equation. not sure how to start it. 2x+7=-2(×-3)+5

Hello solve this algebra equation. not sure how to start it. 2x+7=-2(×-3)+5...

### An unbalanced force of 20 newtons is applied to a4.0-kilogram mass at rest. The acceleration of theobject is closest to

An unbalanced force of 20 newtons is applied to a 4.0-kilogram mass at rest. The acceleration of the object is closest to...

SOMEONE PLEASE HELP ASAP $SOMEONE PLEASE HELP ASAPPPPPPPP IF YOU ANSWER THIS CORRECTLY I WILL GIVE YOU A GREAT RATE$...

### The ratio of the weight of an object on Earth to an object on Planet X is 4 to 9. If a person weighs 230 pounds on Earth, find

The ratio of the weight of an object on Earth to an object on Planet X is 4 to 9. If a person weighs 230 pounds on Earth, find his weight on Planet X. Round to the nearest whole number....

### What will most likely change our ways of communicating in the future?

What will most likely change our ways of communicating in the future? ...

### Apiece of wood has a volume of 30 cm³. it has a length of 2 cm and a width of 3 cm. what is the height of the wood?

Apiece of wood has a volume of 30 cm³. it has a length of 2 cm and a width of 3 cm. what is the height of the wood?...

### Who formed the democratic-republican party? a. authors of the declaration of independence b. believers in the implied powers

Who formed the democratic-republican party? a. authors of the declaration of independence b. believers in the implied powers of the constitution c. followers of george washington d. followers of thomas jefferson...

### If Lauren reads for 30 minutes in 6 minutes in a 240 page novel, what is the constant of proportionally

If Lauren reads for 30 minutes in 6 minutes in a 240 page novel, what is the constant of proportionally...

### 1 The simile used at the beginning of paragraph 6, which comparesthe first sprinkles of rain to army scouts, conveys the idea

1 The simile used at the beginning of paragraph 6, which compares the first sprinkles of rain to army scouts, conveys the idea that A something powerful is about to happen B the author will be safe during the storm C the storm may be less severe than expected Dit is too late for the author to seek ...

### Please im bored can anyone join my remind

Please im bored can anyone join my remind...

### What are the 2 errors? VIOLATORS WILL BE TOWED AND FIND FIFTY DOLLAR

What are the 2 errors? VIOLATORS WILL BE TOWED AND FIND FIFTY DOLLAR...

### You should present a cogent argument based on your critical analysis of the questions posed, using appropriate psychological terminology. It is not enough

You should present a cogent argument based on your critical analysis of the questions posed, using appropriate psychological terminology. It is not enough to answer a question by merely listing facts. Jennifer is interested in training two of her new puppies to participate in agility competitions. ...

### In this lesson, INERTIA is compared to because 'things want to keep doing what they are already doing.'asteadfastnessblazinesscstubbornnessdfocusing

In this lesson, INERTIA is compared to because "things want to keep doing what they are already doing." a steadfastness b laziness c stubbornness d focusing...


text:
## The Geometry Of Least Squares

Here’s a simple and intuitive way of looking at the geometry of a least squares regression:

Take the bottom left point in the triangle below as the origin O. For the linear model:

$$Y=X\beta + \epsilon$$

Both $Y$ and $$X\beta$$ are vectors, and the residual vector $$\epsilon$$ is the difference. The standard least squares error technique uses $$\epsilon^2$$ or $$(Y-X\beta)^T(Y-X\beta)$$ as the error measure to be minimised, and this leads to the calculation of the $$\beta$$ coefficient vector.

Geometrically, the beta coefficients calculated by the least squares regression minimise the squared length of the error vector. This turns out to be the projection of $$Y$$ on to $$X\beta$$ – i.e. the perpendicular vector that turns (O, $$Y$$, $$X\beta$$) into a right-angled triangle.

The projection of $$Y$$ onto $$X\beta$$ is done using the projection matrix P, which is defined as

$P = X\left(X^{T}X\right)^{-1}X^{T}$

So $$X\beta = \hat{Y} = PY$$.

Using the Pythagorean theorem:

$$Y^TY = \hat{Y}^T\hat{Y} + (Y-X\beta)^T(Y-X\beta)$$

In other words, the total sum of squares = sum of squares due to regression + residual sum of squares. This is a fundamental part of analysis of variance techniques.


text:
# Mainnet Voting Guide¶

This guide was last updated on May 2, 2017.

This page is intended to give a brief introduction to how agenda voting works and details the process for setting your tickets to cast your preferred vote for any agenda.

## Introduction¶

There is a two-phase process for voting to implement consensus changes that would create a hard forking scenario.

First, it’s important to note that the Decred blockchain has specifically designated two different block intervals for the voting process. There is a Stake Version Interval (SVI) of 2016 blocks (~1 week) and a Rule Change Interval (RCI) of 8064 blocks (~4 weeks). 4 Stake Version Intervals fit within 1 Rule Change Interval.

The first step of the voting process is to meet the upgrade threshold on the network. After the hard fork code is released (such as the sdiff algorithm change in v1.0.0), a majority of the nodes on the network participating in PoW/PoS need to first upgrade before the voting can be scheduled to begin. For Proof-of-Work, at least 95% of the 1000 most recent blocks must have the latest block version. For Proof-of-Stake, 75% of the votes cast within a single SVI must have the latest vote version. Once miner and voter upgrade thresholds are met, the voting is scheduled to begin on the first block of the next RCI (due to there being 4 SVIs per RCI, it can take up to 6048 blocks [3 SVIs] for the next RCI to begin).

The second step of this process is the actual voting. A single RCI transpires while a maximum of 40320 votes are cast. The votes are tallied at the final block of the RCI, and outcomes are determined prior to the next block being mined.

There are a few possible outcomes of a vote:

1. If more than 90% of all votes within the RCI are “Abstain” votes, the agenda vote remains active for the next RCI.
2. If all non-abstaining votes within the RCI fail to meet the 75% Yes or No majority threshold, the agenda vote remains active for next RCI.
3. If 75% of all non-abstaining votes within the RCI are in support of the agenda (“Yes”), the agenda is considered locked in and the consensus changes will activate 8064 blocks after the vote passed.
4. If 75% of all non-abstaining votes within the RCI are in opposition of the agenda (“No”), the agenda fails and the consensus changes will never activate.
5. If an agenda reaches its expiration before ever reaching a 75% majority vote, the agenda expires and the consensus changes will never activate.

Below is a diagram of the entire cycle for a single agenda with consensus upgrades.

## How To Vote¶

To participate in voting, you first need a wallet. If you don’t have one already, visit the Beginner Guide, choose a wallet, and follow the installation and setup guides.

Next, you’ll need to learn the basics of Proof-of-Stake. You’ll need to be able to buy tickets with your application of choice.

Finally, you’ll need to learn how to set the votechoice for your tickets in order to cast a “Yes”, “No”, or “Abstain” vote for an agenda. By default, your tickets will cast “Abstain” votes. If you are solo-staking with dcrwallet, go here. If you are pool-staking with dcrwallet, Paymetheus, or Decrediton, go here.

## Solo-voting¶

Through the command-line, you’ll want to familiarize yourself with the dcrctl --wallet getvotechoices and dcrctl --wallet setvotechoice "agendaid" "choiceid" commands.

The first command, dcrctl --wallet getvotechoices, returns JSON resembling this:

{
"version": 4,
"choices": [
{
"agendaid": "sdiffalgorithm",
"agendadescription": "Change stake difficulty algorithm as defined in DCP0001",
"choiceid": "abstain",
"choicedescription": "abstain voting for change"
},
{
"agendaid": "lnsupport",
"agendadescription": "Request developers begin work on Lightning Network (LN) integration",
"choiceid": "abstain",
"choicedescription": "abstain from voting"
}
]
}

The second command, dcrctl --wallet setvotechoice "agendaid" "choiceid", let’s you set your votechoice. "agendaid" is found via the getvotechoices command above, and "choiceid" can be yes, no, or abstain.

For example, issuing dcrctl --wallet setvotechoice sdiffalgorithm yes results in the following changes to dcrctl --wallet getvotechoices:

{
"version": 4,
"choices": [
{
"agendaid": "sdiffalgorithm",
"agendadescription": "Change stake difficulty algorithm as defined in DCP0001",
"choiceid": "yes",
"choicedescription": "change to the new algorithm"
},
{
"agendaid": "lnsupport",
"agendadescription": "Request developers begin work on Lightning Network (LN) integration",
"choiceid": "abstain",
"choicedescription": "abstain from voting"
}
]
}

## Stakepool-voting¶

If your Stakepool has updated to the latest stakepool software, you will find a “Voting” page in the navigation menu with dropdown options for each agenda. After you’ve chosen how you want your tickets to vote, simply press the “Update Voting Preferences” to save your votechoices. Below you’ll find an image of the votechoices for vote version 4.

The easiest method to track your how your tickets actually voted is to use the block explorer to view each of your votes.

The block explorer has been updated to display YES, NO, and ABSTAIN votes for each agenda with each ticket. The first and second tickets in the following example image voted “ABSTAIN” for both agendas while the third ticket voted “YES” for both agendas.

You can also manually verify your vote by inputting the txid for the ticket purchase transaction into the block explorer or clicking the link for a Voted Ticket on your stakepool’s website. Below is a quick walkthrough, starting from a stakepool’s website. The red numbers of each image corresponds to the step number.

1. Go to your stakepool’s Tickets page, open the Voted Tickets section and click the txid for the ticket you want to inspect. This will open the block explorer to the ticket purchase transaction. (Figure 3)
2. Press the ‘+’ button to view the transaction details. (Figure 4)
3. Press the ‘>’ on the 1st output of the transaction (the ‘stakesubsmission’ output) to open the block explorer to the actual voting transaction. (Figure 4)
4. The vote transaction should already be expanded, if not, press the ‘+’ button. (Figure 5)
5. Locate the second output of the transaction. Viewing the OP_RETURN of this output, the first 4 digits represent the individual votebits and the last 8 represent the vote version in hexadecimal. In the images below, a version “04” vote voted “01”, both numbers are in hexadecimal (aka 0x04 and 0x01). (Figure 5)
6. Compare the block explorer votebits with the chart below and you’ll see that the ticket voted to abstain from both lnsupport and sdiffalgo agendas while verifying the previous block as valid. (Figure 6)

figure 3:

figure 4:

figure 5:

figure 6:

## Tracking Vote Progress¶

voting.decred.org is an official website set up to track the progress of upgrading and voting.


text:
Computer Science

Indexing with modulus

Let’s generate a list of numbers from 1 to 6:

numbers = list(range(1, 7))
print(numbers)

This will print:

[1, 2, 3, 4, 5, 6]

Now let’s generate a new list of numbers that has a length of 25 and gets its numbers from the list above:

more_numbers = []

for i in range(25):
next_number = numbers[i % len(numbers)]
more_numbers.append(next_number)

print(more_numbers)

This will print:

[1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1]

Notice that we use range(25) to append to more_numbers 25 times. Then we use i % len(numbers) to get an index that will loop through the numbers and wrap back to the beginning as needed. Because numbers has a length of 6, The variable i will take the values 0 1 2 3 4 5 and then 0 1 2 3 4 5 and so forth.


text:
• Mean Value Estimates of the Error Terms of Lehmer Problem

• # Fulltext

https://www.ias.ac.in/article/fulltext/pmsc/120/04/0507-0513

• # Keywords

Lehmer problem; Kloosterman sum; Gauss sum.

• # Abstract

Let 𝑝 be an odd prime and 𝑎 be an integer coprime with 𝑝. Denote by $N(a,p)$ the number of pairs of integers $b, c$ with $bc\equiv a(\mathrm{mod} p),1\leq b, c < p$ and with $b, c$ having different parity. The main purpose of this paper is to study the mean square value problem of $N(a,p)-\frac{1}{2}(p-1)$ over interval $(N, N+M]$ with $M,N$ positive integers by using the analytic methods, and finally by obtaining a sharp asymptotic formula.

• # Author Affiliations

1. School of Mathematics and Statistics, Southwest University, Chongqing 400715, People’s Republic of China
2. Research Center for Basic Science, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, People’s Republic of China

• # Proceedings – Mathematical Sciences

Volume 131, 2021
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019


text:
Requests for technical support from the VASP group should be posted in the VASP-forum.

# Self-consistent screening in Tkatchenko-Scheffler method

A computationally efficient way to account for electrodynamic response effects, in particular the interaction of atoms with the dynamic electric field due to the surrounding polarizable atoms was proposed by Tkatchenko et al[1]. In this method, termed TS+SCS, the frequency-dependent screened polarizabilities ${\displaystyle \alpha ^{{SCS}}(\omega )}$ are obtained by solving the self-consistent screening equation:

${\displaystyle \alpha _{{i}}^{{SCS}}(\omega )=\alpha _{{i}}(\omega )-\alpha _{{i}}(\omega )\sum _{{i\neq j}}\tau _{{ij}}\alpha _{{j}}^{{SCS}}(\omega )}$

where ${\displaystyle \tau _{{ij}}}$ is the dipole-dipole interaction tensor and ${\displaystyle \alpha _{{i}}(\omega )}$ is the effective frequency-dependent polarizability, approximated by

${\displaystyle \alpha _{{i}}(\omega )={\frac {\alpha _{{i}}}{1+\left(\omega /\omega _{i}\right)^{2}}}}$

with the characteristic mean excitation frequency ${\displaystyle \omega _{i}={\frac {4}{3}}{\frac {C_{{6ii}}}{(\alpha _{{i}})^{2}}}}$. The dispersion coefficients are computed from the Casimir-Polder integral:

${\displaystyle C_{{6ii}}={\frac {3}{\pi }}\int _{0}^{{\infty }}\alpha _{{i}}^{{SCS}}(\omega )\alpha _{{i}}^{{SCS}}(\omega )\,d\omega .}$

The van der Waals radii of atoms are obtained by rescaling the radii computed using DFT-TS:

${\displaystyle R_{{0i}}^{{SCS}}=\left({\frac {\alpha _{{i}}^{{SCS}}}{\alpha _{{i}}}}\right)^{{1/3}}R_{{0i}}.}$

The dispersion energy is computed using the same equation as in the original Tkatchenko-Scheffler method but with corrected parameters ${\displaystyle C_{{6ii}}^{{SCS}}}$, ${\displaystyle \alpha _{{i}}^{{SCS}}}$, and ${\displaystyle R_{{0i}}^{{SCS}}}$. The TS+SCS method is invoked by setting IVDW=2|20 and LVDWSCS=.TRUE.. In addition to parameters controlling the Tkatchenko-Scheffler method, the following optional parameters can set by the user:

• VDW_SR=0.97 scaling factor ${\displaystyle s_{{R}}}$
• SCSRAD=120 cutoff radius (in ${\displaystyle \AA }$) used in the calculation of ${\displaystyle \tau _{{ij}}}$
• LSCALER0=.TRUE. decider whether to use the equation above for ${\displaystyle R_{{0i}}^{{SCS}}}$ to re-scale the parameter ${\displaystyle R_{{0}}}$ (LSCALER0=.TRUE.) or not

Details of the implementation of the TS+SCS method in VASP and the performance tests made on various crystalline systems are presented in reference [1].

## IMPORTANT NOTES

• This method requires the use of POTCAR files from the PAW dataset version 52 or later.
• This method is incompatible with the setting ADDGRID=.TRUE..
• This type of calculation may be time-consuming for large systems. Note that the SCS contribution to gradients and stress tensor is only modest (but non-negligible) in many cases. In the initial stages of relaxation of large systems, or if only energy is of interest, the calculation can be accelerated by setting LSCSGRAD=.FALSE..
• The default value for the parameter VDW_SR (which is, in general, different from that used in the unscreened Tkatchenko-Scheffler method method) is available only for the PBE functional. If a functional other than PBE is used, the value for VDW_SR must be specified in the INCAR file.


text:
# How would you deal with categorical data in a naive Bayesian classifier?

I've built a little naive Bayesian classifier that works with Boolean and real values. Boolean distributions are dealt with via Bernoulli distributions, while real valued data are dealt with kernel mixture estimators. I'm currently in the process of adding count data in.

How would one deal with categorical data though, e.g. Monday, Tuesday, Wednesday, or Toyota, Honda, Ford?

My initial thoughts are to assign a number to each category, treat it as a normal real value and round to the nearest integer category on prediction. That seems very wrong to me though.

• Not sure about the correct answer, but your initial thought is definitely incorrect (as you suggest), as that implies that there is an ordering to the categories. Dec 17 '12 at 20:51
• I'm not sure I follow what you mean "treat it as a normal real value and round to the nearest integer category on prediction", but it sounds like an incorrect approach. Categorical valued data is treated similar to boolean data. It is a discrete variable and should be treated as discrete, not real value. Just assign integers $1,...,K$ as labels to your $K$ categories, then use the counts to estimate parameters of your categorical distribution just as you used the counts to estimate the Bernoulli's. Dec 17 '12 at 21:21
• Correct, it is the incorrect approach. I wasn't sure what the correct technique was. Also, oddly enough the platform I'm using, Mathematica, doesn't seem to have a categorical/discrete distribution available to it. Jarad, thanks for the pointer I'll get working on this.
– user17880
Dec 17 '12 at 22:36


text:
# Python API Reference¶

## Chi2¶

mitsuba.python.chi2.BSDFAdapter(bsdf_type, extra, wi=[0, 0, 1], ctx=None)

Adapter to test BSDF sampling using the Chi^2 test.

Parameter bsdf_type (string):

Name of the BSDF plugin to instantiate.

Parameter extra (string):

Additional XML used to specify the BSDF’s parameters.

Parameter wi (array(3,)):

Incoming direction, in local coordinates.

class mitsuba.python.chi2.ChiSquareTest(domain, sample_func, pdf_func, sample_dim=2, sample_count=1000000, res=101, ires=4)

Implements Pearson’s chi-square test for goodness of fit of a distribution to a known reference distribution.

The implementation here specifically compares a Monte Carlo sampling strategy on a 2D (or lower dimensional) space against a reference distribution obtained by numerically integrating a probability density function over grid in the distribution’s parameter domain.

Parameter domain (object):

An implementation of the domain interface (SphericalDomain, etc.), which transforms between the parameter and target domain of the distribution

Parameter sample_func (function):

An importance sampling function which maps an array of uniform variates of size [sample_dim, sample_count] to an array of sample_count samples on the target domain.

Parameter pdf_func (function):

Function that is expected to specify the probability density of the samples produced by sample_func. The test will try to collect sufficient statistical evidence to reject this hypothesis.

Parameter sample_dim (int):

Number of random dimensions consumed by sample_func per sample. The default value is 2.

Parameter sample_count (int):

Total number of samples to be generated. The test will have more evidence as this number tends to infinity. The default value is 1000000.

Parameter res (int):

Vertical resolution of the generated histograms. The horizontal resolution will be calculated as res * domain.aspect(). The default value of 101 is intentionally an odd number to prevent issues with floating point precision at sharp boundaries that may separate the domain into two parts (e.g. top hemisphere of a sphere parameterization).

Parameter ires (int):

Number of horizontal/vertical subintervals used to numerically integrate the probability density over each histogram cell (using the trapezoid rule). The default value is 4.

Notes:

The following attributes are part of the public API:

messages: string

The implementation may generate a number of messages while running the test, which can be retrieved via this attribute.

histogram: array

The histogram array is populated by the tabulate_histogram() method and stored in this attribute.

pdf: array

The probability density function array is populated by the tabulate_pdf() method and stored in this attribute.

p_value: float

The p-value of the test is computed in the run() method and stored in this attribute.

tabulate_histogram()

Invoke the provided sampling strategy many times and generate a histogram in the parameter domain. If sample_func returns a tuple (positions, weights) instead of just positions, the samples are considered to be weighted.

tabulate_pdf()

Numerically integrate the provided probability density function over each cell to generate an array resembling the histogram computed by tabulate_histogram(). The function uses the trapezoid rule over intervals discretized into self.ires separate function evaluations.

run(significance_level=0.01, test_count=1, quiet=False)

Run the Chi^2 test

Parameter significance_level (float):

Denotes the desired significance level (e.g. 0.01 for a test at the 1% significance level)

Parameter test_count (int):

Specifies the total number of statistical tests run by the user. This value will be used to adjust the provided significance level so that the combination of the entire set of tests has the provided significance level.

Returns → bool:

True upon success, False if the null hypothesis was rejected.

class mitsuba.python.chi2.LineDomain(bounds=[- 1.0, 1.0])

The identity map on the line.

mitsuba.python.chi2.MicrofacetAdapter(md_type, alpha, sample_visible=False)

Adapter for testing microfacet distribution sampling techniques (separately from BSDF models, which are also tested)

mitsuba.python.chi2.PhaseFunctionAdapter(phase_type, extra, wi=[0, 0, 1])

Adapter to test phase function sampling using the Chi^2 test.

Parameter phase_type (string):

Name of the phase function plugin to instantiate.

Parameter extra (string):

Additional XML used to specify the phase function’s parameters.

Parameter wi (array(3,)):

Incoming direction, in local coordinates.

class mitsuba.python.chi2.PlanarDomain(bounds=None)

The identity map on the plane

mitsuba.python.chi2.SpectrumAdapter(value)

Adapter which permits testing 1D spectral power distributions using the Chi^2 test.

class mitsuba.python.chi2.SphericalDomain

Maps between the unit sphere and a [cos(theta), phi] parameterization.

## Autodiff¶

class mitsuba.python.autodiff.Adam(params, lr, beta_1=0.9, beta_2=0.999, epsilon=1e-08)

Base class: mitsuba.python.autodiff.Optimizer

Implements the Adam optimizer presented in the paper Adam: A Method for Stochastic Optimization by Kingman and Ba, ICLR 2015.

__init__(params, lr, beta_1=0.9, beta_2=0.999, epsilon=1e-08)
Parameter lr:

learning rate

Parameter beta_1:

controls the exponential averaging of first order gradient moments

Parameter beta_2:

controls the exponential averaging of second order gradient moments

step()

class mitsuba.python.autodiff.Optimizer(params, lr)

__init__(params, lr)
Parameter params:

dictionary (name: variable) of differentiable parameters to be optimized.

Parameter lr:

learning rate

set_learning_rate(lr)

Set the learning rate.

disable_gradients()

Temporarily disable the generation of gradients.

class mitsuba.python.autodiff.SGD(params, lr, momentum=0)

Base class: mitsuba.python.autodiff.Optimizer

Implements basic stochastic gradient descent with a fixed learning rate and, optionally, momentum [SMDH13] (0.9 is a typical parameter value for the momentum parameter).

The momentum-based SGD uses the update equation

$v_{i+1} = \mu \cdot v_i + g_{i+1}$
$p_{i+1} = p_i + \varepsilon \cdot v_{i+1},$

where $$v$$ is the velocity, $$p$$ are the positions, $$\varepsilon$$ is the learning rate, and $$\mu$$ is the momentum parameter.

__init__(params, lr, momentum=0)
Parameter lr:

learning rate

Parameter momentum:

momentum factor

step()

mitsuba.python.autodiff._render_helper(scene, spp=None, sensor_index=0)

Internally used function: render the specified Mitsuba scene and return a floating point array containing RGB values and AOVs, if applicable

mitsuba.python.autodiff.render(scene, spp: Union[None, int, Tuple[int, int]] = None, unbiased=False, optimizer: mitsuba.python.autodiff.Optimizer = None, sensor_index=0)

Perform a differentiable of the scene scene, returning a floating point array containing RGB values and AOVs, if applicable.

Parameter spp (None, int, or a 2-tuple (int, int)):

Specifies the number of samples per pixel to be used for rendering, overriding the value that is specified in the scene. If spp=None, the original value takes precedence. If spp is a 2-tuple (spp_primal: int, spp_deriv: int), the first element specifies the number of samples for the primal pass, and the second specifies the number of samples for the derivative pass. See the explanation of the unbiased parameter for further detail on what these mean.

Memory usage is roughly proportional to the spp, value, hence this parameter should be reduced if you encounter out-of-memory errors.

Parameter unbiased (bool):

One potential issue when naively differentiating a rendering algorithm is that the same set of Monte Carlo sample is used to generate both the primal output (i.e. the image) along with derivative output. When the rendering algorithm and objective are jointly differentiated, we end up with expectations of products that do not satisfy the equality $$\mathbb{E}[X Y]=\mathbb{E}[X]\, \mathbb{E}[Y]$$ due to correlations between $$X$$ and $$Y$$ that result from this sample re-use.

When unbiased=True, the render() function will generate an unbiased estimate that de-correlates primal and derivative components, which boils down to rendering the image twice and naturally comes at some cost in performance $$(\sim 1.6 imes\!)$$. Often, biased gradients are good enough, in which case unbiased=False should be specified instead.

The number of samples per pixel per pass can be specified separately for both passes by passing a tuple to the spp parameter.

Note that unbiased mode is only relevant for reverse-mode differentiation. It is not needed when visualizing parameter gradients in image space using forward-mode differentiation.

Parameter optimizer (mitsuba.python.autodiff.Optimizer):

The optimizer referencing relevant scene parameters must be specified when unbiased=True. Otherwise, there is no need to provide this parameter.

Parameter sensor_index (int):

When the scene contains more than one sensor/camera, this parameter can be specified to select the desired sensor.

mitsuba.python.autodiff.render_torch(scene, params=None, **kwargs)

mitsuba.python.autodiff.write_bitmap(filename, data, resolution, write_async=True)

Write the linearized RGB image in data to a PNG/EXR/.. file with resolution resolution.

## Other¶

class mitsuba.python.util.ParameterMap(properties, hierarchy)

Dictionary-like object that references various parameters used in a Mitsuba scene graph. Parameters can be read and written using standard syntax (parameter_map[key]). The class exposes several non-standard functions, specifically torch(), update(), and keep().

__init__(properties, hierarchy)

Private constructor (use mitsuba.python.util.traverse() instead)

torch() → dict

Converts all Enoki arrays into PyTorch arrays and return them as a dictionary. This is mainly useful when using PyTorch to optimize a Mitsuba scene.

set_dirty(key: str)

Marks a specific parameter and its parent objects as dirty. A subsequent call to update() will refresh their internal state. This function is automatically called when overwriting a parameter using __setitem__().

update() → None

This function should be called at the end of a sequence of writes to the dictionary. It automatically notifies all modified Mitsuba objects and their parent objects that they should refresh their internal state. For instance, the scene may rebuild the kd-tree when a shape was modified, etc.

keep(keys: list) → None

Reduce the size of the dictionary by only keeping elements, whose keys are part of the provided list ‘keys’.

mitsuba.python.util.is_differentiable(p)

mitsuba.python.util.traverse(node: mitsuba.core.Object)mitsuba.python.util.ParameterMap

Traverse a node of Mitsuba’s scene graph and return a dictionary-like object that can be used to read and write associated scene parameters.

mitsuba.python.math.rlgamma(a, x)

Regularized lower incomplete gamma function based on CEPHES

class mitsuba.python.xml.Files

Enum for different files or dicts containing specific info

class mitsuba.python.xml.WriteXML(path, split_files=False)

File Writing API Populates a dictionary with scene data, then writes it to XML.

data_add(key, value, file=0)

Add an entry to a given subdict.

key: dict key value: entry file: the subdict to which to add the data

add_comment(comment, file=0)

Add a comment to the scene dict

comment: text of the comment file: the subdict to which to add the comment

add_include(file)

Add an include tag to the main file. This is used when splitting the XML scene file in multiple fragments.

file: the file to include

wf(ind, st, tabs=0)

Write a string to file index ind. Optionally indent the string by a number of tabs

ind: index of the file to write to st: text to write tabs: optional number of tabs to add

set_filename(name)

Open the files for output, using filenames based on the given base name. Create the necessary folders to create the file at the specified path.

name: path to the scene.xml file to write.

set_output_file(file)

Switch next output to the given file index

file: index of the file to start writing to

write_comment(comment, file=None)

Write an XML comment to file.

comment: The text of the comment to write file: Index of the file to write to

write_header(file, comment=None)

Write an XML header to a specified file. Optionally add a comment to describe the file.

file: The file to write to comment: Optional comment to add (e.g. “# Geometry file”)

open_element(name, attributes={}, file=None)

Open an XML tag (e.g. emitter, bsdf…)

name: Name of the tag (emitter, bsdf, shape…) attributes: Additional fileds to add to the opening tag (e.g. name, type…) file: File to write to

close_element(file=None)

Close the last tag we opened in a given file.

file: The file to write to

element(name, attributes={}, file=None)

Write a single-line XML element.

name: Name of the element (e.g. integer, string, rotate…) attributes: Additional fields to add to the element (e.g. name, value…) file: The file to write to

get_plugin_tag(plugin_type)

Get the corresponding tag of a given plugin (e.g. ‘bsdf’ for ‘diffuse’) If the given type (e.g. ‘transform’) is not a plugin, returns None.

plugin_type: Name of the type (e.g. ‘diffuse’, ‘ply’…)

current_tag()

Get the tag in which we are currently writing

configure_defaults(scene_dict)

Traverse the scene graph and look for properties in the defaults dict. For such properties, store their value in a default tag and replace the value by \$name in the prop.

scene_dict: The dictionary containing the scene info

preprocess_scene(scene_dict)
Preprocess the scene dictionary before writing it to file:

• Reorder the scene dict before writing it to file.

• Separate the dict into different category-specific subdicts.

• If not splitting files, merge them in the end.

scene_dict: The dictionary containing the scene data

format_spectrum(entry, entry_type)

Format rgb or spectrum tags to the proper XML output. The entry should contain the name and value of the spectrum entry. The type is passed separately, since it is popped from the dict in write_dict

entry: the dict containing the spectrum entry_type: either ‘spectrum’ or ‘rgb’

format_path(filepath, tag)

Given a filepath, either copy it in the scene folder (in the corresponding directory) or convert it to a relative path.

filepath: the path to the given file tag: the tag this path property belongs to in (shape, texture, spectrum)

write_dict(data)

Main XML writing routine. Given a dictionary, iterate over its entries and write them to file. Calls itself for nested dictionaries.

data: The dictionary to write to file.

process(scene_dict)

Preprocess then write the input dict to XML file format

scene_dict: The dictionary containing all the scene info.

transform_matrix(transform)

Converts a mitsuba Transform4f into a dict entry. This dict entry won’t have a ‘type’ because it’s handled in a specific case.

transform: the given transform matrix

decompose_transform(transform, export_scale=False)

Export a transform as a combination of rotation, scale and translation. This helps manually modifying the transform after export (for cameras for instance)

transform: The Transform4f transform matrix to decompose export_scale: Whether to add a scale property or not. (e.g. don’t do it for cameras to avoid clutter)

mitsuba.python.xml.copy2(src, dst, *, follow_symlinks=True)

Copy data and metadata. Return the file’s destination.

The destination may be a directory.

If follow_symlinks is false, symlinks won’t be followed. This resembles GNU’s “cp -P src dst”.

mitsuba.python.xml.dict_to_xml(scene_dict, filename, split_files=False)

mitsuba.python.test.util.fresolver_append_path(func)

Function decorator that adds the mitsuba project root to the FileResolver’s search path. This is useful in particular for tests that e.g. load scenes, and need to specify paths to resources.

The file resolver is restored to its previous state once the test’s execution has finished.

mitsuba.python.test.util.getframeinfo(frame, context=1)

Get information about a frame or traceback object.

A tuple of five things is returned: the filename, the line number of the current line, the function name, a list of lines of context from the source code, and the index of the current line within that list. The optional second argument specifies the number of lines of context to return, which are centered around the current line.

mitsuba.python.test.util.make_tmpfile(request, tmpdir_factory)

mitsuba.python.test.util.stack(context=1)

Return a list of records for the stack above the caller’s frame.

mitsuba.python.test.util.tmpfile(request, tmpdir_factory)

Fixture to create a temporary file

mitsuba.python.test.util.wraps`(wrapped, assigned=('__module__', '__name__', '__qualname__', '__doc__', '__annotations__'), updated=('__dict__'))

Decorator factory to apply update_wrapper() to a wrapper function

Returns a decorator that invokes update_wrapper() with the decorated function as the wrapper argument and the arguments to wraps() as the remaining arguments. Default arguments are as for update_wrapper(). This is a convenience function to simplify applying partial() to update_wrapper().


text:
Physics Forums (http://www.physicsforums.com/index.php)
-   Calculus & Beyond Homework (http://www.physicsforums.com/forumdisplay.php?f=156)
-   -   f''(x) of 2xy - y^3 = 5 (http://www.physicsforums.com/showthread.php?t=585162)

 LearninDaMath Mar8-12 09:48 PM

f''(x) of 2xy - y^3 = 5

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

Here is my work/attempt. In blue is my answer. In green is the supposed correct answer.

The first derivative is surely correct, as it matches the answer I was supposed to get. However, if my second derivative answer is incorrect, can you determine where I I am making a mistake?

http://i1094.photobucket.com/albums/...plicit-1-1.jpg

 tjackson3 Mar8-12 10:06 PM

Re: f''(x) of 2xy - y^3 = 5

Here's a hint: your mistake comes on the third to last step (the line beginning f''(x) = -2(-2y)

 LearninDaMath Mar8-12 10:55 PM

Re: f''(x) of 2xy - y^3 = 5

Quote:
 Quote by tjackson3 (Post 3805625) Here's a hint: your mistake comes on the third to last step (the line beginning f''(x) = -2(-2y)
Would it be correct to approach this step like this:

http://i1094.photobucket.com/albums/...mplicit2-1.jpg

EDIT: On the first line, please regard the 12y^2 as just 12y. The second line is written as though the 12y is what I started off with.

 pylauzier Mar8-12 11:12 PM

Re: f''(x) of 2xy - y^3 = 5

Your mistake is in the 3rd to last step, where you have one term with a 2x-3y^2 denominator. In the following step you simplified the equation by multiplying (2x-3y^2) with the (2x-3y^2)^2 term. However the two 4y terms on the numerator were not being divided by 2x-3y^2.

 LearninDaMath Mar8-12 11:50 PM

Re: f''(x) of 2xy - y^3 = 5

Would this be the correct way to do that step?

http://i1094.photobucket.com/albums/...mplicit3-1.jpg

 LearninDaMath Mar9-12 12:59 AM

Re: f''(x) of 2xy - y^3 = 5

Okay, I was able to get the answer in the right form as the provided answer. However, it is off numerically by 2. The provided answer has 18xy and my answer has 16xy. I have always known 8 + 8 to equal 16...what gives?

http://i1094.photobucket.com/albums/...eimplicit4.jpg

 Mark44 Mar9-12 10:24 AM

Re: f''(x) of 2xy - y^3 = 5

1) You don't take "f' " of an expression. f'(x) is the derivative of f. The operator is d/dx. f' and f'' should not appear in your work at all.
2) You wrote f(x) = 2xy - y3 = 0. This is incorrect because a) the expression 2xy - y3 involves both x and y, not just x alone, and b) the equation 2xy - y3 = 0 defines an implicit relationship between x and y.

 LearninDaMath Mar11-12 04:47 AM

Re: f''(x) of 2xy - y^3 = 5

Ah, thats right. It's not a function is it. It's an equation, but not a function. Appreciate the input.

 stripes Mar11-12 07:18 AM

Re: f''(x) of 2xy - y^3 = 5

As Mark44 said, it's an implicit relationship. Correct me if I'm wrong, but doesn't the question seem a little strange? How can

$f(x) = 2xy - y^{3} = 5$?

That being the case, wouldn't we need to write

$f(x,y) = 2xy - y^{3} = 5$, or $f(x,y) = 2xy - y^{3} - 5$

since it is a function of two variables? Then we would just need to find $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$. If this isn't the case, we would just have

$2xy - y^{3} = 5$,

and we would need to find $\frac{dy}{dx}$ implicitly, not f'(x).

So guys, which is it? Or am I completely off on this one?

 Mark44 Mar11-12 02:15 PM

Re: f''(x) of 2xy - y^3 = 5

Quote:
 Quote by stripes (Post 3809320) As Mark44 said, it's an implicit relationship. Correct me if I'm wrong, but doesn't the question seem a little strange? How can $f(x) = 2xy - y^{3} = 5$? That being the case, wouldn't we need to write $f(x,y) = 2xy - y^{3} = 5$, or $f(x,y) = 2xy - y^{3} - 5$ since it is a function of two variables? Then we would just need to find $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$. If this isn't the case, we would just have $2xy - y^{3} = 5$, and we would need to find $\frac{dy}{dx}$ implicitly, not f'(x). So guys, which is it? Or am I completely off on this one?
No, I think you are on track here. f'(x) is meaningless here, since the left side of the equation 2xy - y3 - 5 = 0 represents a function of two variables.

The tacit assumption here, I believe, is that y is an implicit function of x, and goal is to find dy/dx using implicit differentiation. Note that this should be written as dy/dx = ..., not f'(x) = ..., so as to not cause confusion between the function (of two variables) in the first equation, and the function (of one variable) that relates x and y.

After dy/dx has been found, differentiate again to find d2y/dx2.

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