text:
# Labeled Booleans

Labeled booleans are used to record different related boolean flags.

## Recording API

### set

Sets one of the labels in a labeled boolean metric to a specific value.

import org.mozilla.yourApplication.GleanMetrics.Accessibility

Accessibility.features["high_contrast"].set(isHighContrastEnabled())

import org.mozilla.yourApplication.GleanMetrics.Accessibility;

Acessibility.INSTANCE.features["high_contrast"].set(isHighContrastEnabled());

Accessibility.features["screen_reader"].set(isScreenReaderEnabled())
Accessibility.features["high_contrast"].set(isHighContrastEnabled())

from glean import load_metrics

metrics.accessibility.features["high_contrast"].set(is_high_contrast_enabled())

#![allow(unused)]
fn main() {
use glean_metrics;

accessibility::features.get("high_contrast").set(is_high_contrast_enabled());
}

import * as acessibility from "./path/to/generated/files/acessibility.js";

acessibility.features["high_contrast"].set(this.isHighContrastEnabled());

C++

#include "mozilla/glean/GleanMetrics.h"

mozilla::glean::accessibility::features.Get("high_contrast"_ns).Set(false);

JavaScript

Glean.accessibility.features.screen_reader.set(true);
Glean.accessibility.features["high_contrast"].set(false);

#### Recorded Errors

• invalid_label:
• If the label contains invalid characters. Data is still recorded to the special label __other__.
• If the label exceeds the maximum number of allowed characters. Data is still recorded to the special label __other__.

## Testing API

### testGetValue

Gets the recorded value for a given label in a labeled boolean metric.

import org.mozilla.yourApplication.GleanMetrics.Accessibility

// Do the booleans have the expected values?
assertEquals(False, Accessibility.features["high_contrast"].testGetValue())

import org.mozilla.yourApplication.GleanMetrics.Accessibility;

// Do the booleans have the expected values?
assertEquals(False, Acessibility.INSTANCE.features["high_contrast"].testGetValue());

@testable import Glean

// Do the booleans have the expected values?
XCTAssertEqual(false, try Accessibility.features["high_contrast"].testGetValue())

from glean import load_metrics

# Do the booleans have the expected values?
assert not metrics.accessibility.features["high_contrast"].test_get_value()

#![allow(unused)]
fn main() {
use glean_metrics;

// Do the booleans have the expected values?
assert!(!accessibility::features.get("high_contrast").test_get_value(None).unwrap());
}

import * as accessibility from "./path/to/generated/files/acessibility.js";

assert(!(await accessibility.features["high_contrast"].testGetValue()));

C++

#include "mozilla/glean/GleanMetrics.h"

ASSERT_EQ(
true,
ASSERT_EQ(
false,
mozilla::glean::accessibility::features.Get("high_contrast"_ns).TestGetValue().unwrap().ref());

JavaScript

Assert.equal(true, Glean.accessibility.features["screen_reader"].testGetValue());
Assert.equal(false, Glean.accessibility.features.high_contrast.testGetValue());

### testHasValue

Whether or not any value was recorded for a given label in a labeled boolean metric.

import org.mozilla.yourApplication.GleanMetrics.Accessibility

// Was anything recorded?
assertTrue(Accessibility.features["high_contrast"].testHasValue())

import org.mozilla.yourApplication.GleanMetrics.Accessibility;

// Was anything recorded?
assertEquals(True, Acessibility.INSTANCE.features["high_contrast"].testHasValue());

@testable import Glean

// Was anything recorded?
XCTAssert(Accessibility.features["high_contrast"].testHasValue())

from glean import load_metrics

# Was anything recorded?
assert metrics.accessibility.features["high_contrast"].test_has_value()

### testGetNumRecordedErrors

Gets the number of errors recorded for a given labeled boolean metric in total.

import org.mozilla.yourApplication.GleanMetrics.Accessibility

// Did we record any invalid labels?
assertEquals(0, Accessibility.features.testGetNumRecordedErrors(ErrorType.InvalidLabel))

import org.mozilla.yourApplication.GleanMetrics.Accessibility;

// Did we record any invalid labels?
assertEquals(0, Acessibility.INSTANCE.features.testGetNumRecordedErrors());

@testable import Glean

// Were there any invalid labels?
XCTAssertEqual(0, Accessibility.features.testGetNumRecordedErrors(.invalidLabel))

from glean import load_metrics

# Did we record any invalid labels?
assert 0 == metrics.accessibility.features.test_get_num_recorded_errors(
ErrorType.INVALID_LABEL
)

#![allow(unused)]
fn main() {
use glean::ErrorType;

use glean_metrics;

// Did we record any invalid labels?
assert_eq!(
1,
accessibility::features.test_get_num_recorded_errors(
ErrorType::InvalidLabel
)
);
}

import * as accessibility from "./path/to/generated/files/acessibility.js";
import { ErrorType } from "@mozilla/glean/<platform>";

assert(
1,
await accessibility.features.testGetNumRecordedErrors(ErrorType.InvalidLabel)
);

## Metric parameters

Example labeled boolean metric definition:

accessibility:
features:
type: labeled_boolean
description: >
a11y features enabled on the device.
bugs:
- https://bugzilla.mozilla.org/000000
data_reviews:
- https://bugzilla.mozilla.org/show_bug.cgi?id=000000#c3
- me@mozilla.com
expires: 2020-10-01
labels:
- high_contrast
...

### Extra metric parameters

#### labels

Labeled metrics may have an optional labels parameter, containing a list of known labels. The labels in this list must match the following requirements:

##### Important

If the labels are specified in the metrics.yaml, using any label not listed in that file will be replaced with the special value __other__.

If the labels are not specified in the metrics.yaml, only 16 different dynamic labels may be used, after which the special value __other__ will be used.

Removing or changing labels, including their order in the registry file, is permitted. Avoid reusing labels that were removed in the past. It is best practice to add documentation about removed labels to the description field so that analysts will know of their existence and meaning in historical data. Special care must be taken when changing GeckoView metrics sent through the Glean SDK, as the index of the labels is used to report Gecko data through the Glean SDK.

## Data questions

• Which accessibility features are enabled?


text:
Dataset Open Access

# Binary black-hole surrogate waveform catalog

Scott E. Field; Chad R. Galley; Jan S. Hesthaven; Jason Kaye; Manuel Tiglio; Jonathan Blackman; Béla Szilágyi; Mark A. Scheel; Daniel A. Hemberger; Patricia Schmidt; Rory Smith; Christian D. Ott; Michael Boyle; Lawrence E. Kidder; Harald P. Pfeiffer; Vijay Varma

### DataCite XML Export

<?xml version='1.0' encoding='utf-8'?>
<identifier identifierType="DOI">10.5281/zenodo.6726994</identifier>
<creators>
<creator>
<creatorName>Scott E. Field</creatorName>
</creator>
<creator>
</creator>
<creator>
<creatorName>Jan S. Hesthaven</creatorName>
</creator>
<creator>
<creatorName>Jason Kaye</creatorName>
</creator>
<creator>
<creatorName>Manuel Tiglio</creatorName>
</creator>
<creator>
<creatorName>Jonathan Blackman</creatorName>
</creator>
<creator>
<creatorName>Béla Szilágyi</creatorName>
</creator>
<creator>
<creatorName>Mark A. Scheel</creatorName>
</creator>
<creator>
<creatorName>Daniel A. Hemberger</creatorName>
</creator>
<creator>
<creatorName>Patricia Schmidt</creatorName>
</creator>
<creator>
<creatorName>Rory Smith</creatorName>
</creator>
<creator>
<creatorName>Christian D. Ott</creatorName>
</creator>
<creator>
<creatorName>Michael Boyle</creatorName>
</creator>
<creator>
<creatorName>Lawrence E. Kidder</creatorName>
</creator>
<creator>
<creatorName>Harald P. Pfeiffer</creatorName>
</creator>
<creator>
<creatorName>Vijay Varma</creatorName>
</creator>
</creators>
<titles>
<title>Binary black-hole surrogate waveform catalog</title>
</titles>
<publisher>Zenodo</publisher>
<publicationYear>2022</publicationYear>
<dates>
<date dateType="Issued">2022-05-11</date>
</dates>
<resourceType resourceTypeGeneral="Dataset"/>
<alternateIdentifiers>
<alternateIdentifier alternateIdentifierType="url">https://zenodo.org/record/6726994</alternateIdentifier>
</alternateIdentifiers>
<relatedIdentifiers>
<relatedIdentifier relatedIdentifierType="DOI" relationType="IsSupplementTo">10.1103/PhysRevX.4.031006</relatedIdentifier>
<relatedIdentifier relatedIdentifierType="DOI" relationType="IsSupplementTo">10.1103/PhysRevLett.115.121102</relatedIdentifier>
<relatedIdentifier relatedIdentifierType="DOI" relationType="IsSupplementTo">10.1103/PhysRevD.95.104023</relatedIdentifier>
<relatedIdentifier relatedIdentifierType="DOI" relationType="IsSupplementTo">10.1103/PhysRevD.96.024058</relatedIdentifier>
<relatedIdentifier relatedIdentifierType="arXiv" relationType="IsSupplementTo">arXiv:1809.09125</relatedIdentifier>
<relatedIdentifier relatedIdentifierType="DOI" relationType="IsSupplementTo">10.1103/PhysRevD.99.064045</relatedIdentifier>
<relatedIdentifier relatedIdentifierType="arXiv" relationType="IsSupplementTo">arXiv:1905.09300</relatedIdentifier>
<relatedIdentifier relatedIdentifierType="DOI" relationType="IsVersionOf">10.5281/zenodo.1215752</relatedIdentifier>
<relatedIdentifier relatedIdentifierType="URL" relationType="IsPartOf">https://zenodo.org/communities/sxs</relatedIdentifier>
<relatedIdentifier relatedIdentifierType="URL" relationType="IsPartOf">https://zenodo.org/communities/zenodo</relatedIdentifier>
</relatedIdentifiers>
<rightsList>
<rights rightsURI="info:eu-repo/semantics/openAccess">Open Access</rights>
</rightsList>
<descriptions>
<description descriptionType="Abstract">
This repository contains all publicly available numerical relativity surrogate data for waveforms produced by the Spectral Einstein Code. The base method for building surrogate models can be found in Field et al., PRX 4, 031006 (2014).

Several numerical relativity surrogate models are currently available in this catalog:

Current models

NRHybSur2dq15.h5 &mdash; This is a surrogate model for binary black hole systems with a high mass ratio (up to 15), but restricted to nonprecessing spins and no secondary spin. Before constructing the surrogate, the NR waveforms are hybridized with SEOBNRv4HM to include the early inspiral. Therefore this model covers 9.5 solar mass or higher total mass system for ground-based detectors. A paper describing it can be found at Yoo et al., arxiv:2203.10109. It is evaluated with the gwsurrogate Python package, which can be found on PyPI.

NRSur7dq4.h5 &mdash; This is a surrogate model for binary black hole mergers with generic spins and mass ratios up to 4. A paper describing it can be found at Varma et al., arxiv:1905.09300. It is evaluated with the gwsurrogate Python package, which can be found on PyPI. Instructions for evaluating this surrogate can be found at this example IPython code.

NRHybSur3dq8.h5 &mdash; This is a surrogate model for binary black hole systems with generic mass ratios but restricted to nonprecessing spins. Before constructing the surrogate, the NR waveforms are hybridized with post-Newtonian waveforms to include the early inspiral. Therefore this model covers the full stellar mass range for ground-based detectors. A paper describing it can be found at Varma et al., PRD 99, 064045 (2019).&nbsp; It is evaluated with the gwsurrogate Python package, which can be found on PyPI. Instructions for evaluating this surrogate can be found this example IPython code.

NRSur7dq4Remnant &mdash; This is a surrogate model for mass, spin, and recoil kick velocity of the remnant BH left behind in generically precessing binary black hole mergers, with mass ratios up to 4. A paper describing it can be found at Varma et al., arxiv:1905.09300. It is evaluated with the surfinBH Python package, which can be found on PyPI. Installation instructions and an ipython help notebook can be found in the same link.

Older models

SpEC_q1_10_NoSpin_nu5thDegPoly_exclude_2_0.h5 &mdash; A surrogate model for binary black hole mergers with non-spinning black holes. This is described in Blackman et al., PRL 115, 121102 (2015). It is evaluated with the gwsurrogate python package, which can be found on PyPI. Instructions for evaluating this surrogate can be found in tutorials included with the gwsurrogate package and in this example IPython code.

NRSur4d2s_FDROM_grid12.h5 and NRSur4d2s_TDROM_grid12.h5 &mdash; These are fast frequency-domain and time-domain (respectively) surrogate models for binary black hole mergers where the black holes may be spinning, but the spins are restricted to a parameter subspace which includes some but not all precessing configurations. NRSur4d2s_FDROM_grid12.h5 is the NRSur4d2s_FDROM model described in Blackman et al., PRD 95, 104023, (2017), and NRSur4d2s_TDROM_grid12.h5 is built from the underlying (slower) NRSur4d2s time-domain model in the same way but without the FFTs. These surrogates are also evaluated using gwsurrogate, and a tutorial can be found in this example IPython code.

NRSur7dq2.h5 &mdash; This is a surrogate model for binary black hole mergers with generic spins. A paper describing it can be found at Blackman et al., PRD 96, 024058 (2017). This surrogate is evaluated through a standalone python package contained in NRSur7dq2.tar.gz, which has simple installation instructions in its README file. A tutorial can be found for evaluating this surrogate in this example IPython code.

&nbsp;

&nbsp;

If you find these surrogate models useful in your own research please cite the Field et al., PRX (2014) paper as well as the relevant paper describing the specific numerical relativity surrogate model, if available (e.g., the Blackman et al. 2015 paper for non-spinning binary black hole coalescences).

Caveats:

Evaluating surrogate models outside of the ranges they were trained upon may give inaccurate results. Please use with caution when extrapolating.

The surrogate data available here for non-spinning binary black holes produced in Blackman et al. 2015 contains the (2,0) mode. However, this mode was not used in the paper. While this surrogate can predict a (2,0) mode, current numerical relativity simulations may not yet be able to accumulate (non-oscillatory) Christodoulou memory sufficiently. The surrogate (2,0) mode is founded upon basis SpEC waveforms that have been hybridized with leading order post-Newtonian waveforms. Therefore, the (2,0) mode can be included in the mode&rsquo;s output but should be used with caution. Currently, the default option to evaluate this surrogate (using GWSurrogate) is to exclude all m=0 modes.

</description>
</descriptions>
</resource>

22,437
32,589
views

format: plain


text:
# zbMATH — the first resource for mathematics

An elementary problem equivalent to the Riemann hypothesis. (English) Zbl 1098.11005
Let $$\sigma(n)$$ denote the sum-of-divisors function as usual, and write $$H_n = \sum_{j=1}^{n}{1\over j}$$. The author proves that the problem of showing $$\sigma(n)\leq H_n + e^{H_n}\log H_n$$, with equality only for $$n=1$$, is equivalent to the Riemann Hypothesis (RH). The proof depends on two bounds for $$\sigma(n)$$ given by G. Robin [J. Math. Pures Appl. (9) 63, 187–213 (1984; Zbl 0516.10036)].
The first bound is, assuming RH, $$\sigma(n) \leq e^{\gamma}n\log\log n$$ for $$n \geq 5041$$. Robin’s proof made use of delicate estimates involving prime-counting functions in terms of zeros of the Riemann zeta-function, and explicit error estimates for prime-counting functions. The second bound is, if RH is false, then there exist constants $$0<\beta < {1\over 2}$$ and $$C>0$$ such that $$\sigma(n) \geq e^{\gamma}n\log\log n + {Cn\log\log n\over (\log n)^{\beta}}$$ holds for infinitely many $$n$$.
The paper contains an exposition of highly composite numbers, superior highly composite numbers, superabundant numbers and collosally abundant numbers, and a brief account of the work of L. Alaoglu and P. Erdős [Trans. Am. Math. Soc. 56, 448–469 (1944; Zbl 0061.07903)] on these matters.

##### MSC:
 11A25 Arithmetic functions; related numbers; inversion formulas 11M26 Nonreal zeros of $$\zeta (s)$$ and $$L(s, \chi)$$; Riemann and other hypotheses
##### Keywords:
The sum-of-divisors function; Riemann Hypothesis
Full Text:


text:
Finding the transfer function of a opamp filter circuit

Well, I've the following third order filter circuit:

simulate this circuit – Schematic created using CircuitLab

And I know that the transfer function looks like:

$$\mathscr{H}\left(\text{s}\right)=\frac{1}{\alpha_1\cdot\text{s}^3+\alpha_2\cdot\text{s}^2+\alpha_3\cdot\text{s}+1}\tag1$$

Where:

• $$\alpha_1=\text{C}_1\cdot\text{C}_2\cdot\text{C}_3\cdot\text{R}_1\cdot\frac{\text{R}_2\cdot\text{R}_3}{\text{R}_2+\text{R}_3}\cdot\left(\text{R}_1+\text{R}_2\right)\tag2$$
• $$\alpha_2=\text{C}_1\cdot\text{C}_2\cdot\text{R}_1\cdot\left(\text{R}_1+\text{R}_2\right)+\text{C}_2\cdot\text{C}_3\cdot\text{R}_3\cdot\left(\text{R}_1+\text{R}_2\right)\tag3$$
• $$\alpha_3=\text{C}_1\cdot\text{R}_1+\text{C}_2\cdot\left(\text{R}_1+\text{R}_2+\text{R}_3\right)\tag4$$

But my question is how can I use current node analyses to find this transfer function (assuming an ideal opamp)?

My work:

I wrote the current node equations:

1. $$\text{I}_1=\text{I}_{\text{R}_1}+\text{I}_{\text{C}_1}+\text{I}_{\text{R}_2}\tag5$$
2. $$\text{I}_2=\text{I}_{\text{R}_2}+\text{I}_{\text{R}_3}+\text{I}_{\text{C}_3}\tag6$$
3. $$\text{I}_3=\text{I}_{\text{R}_3}+\text{I}_{\text{C}_2}\tag7$$
4. $$\text{I}_4=\text{I}_{\text{C}_3}\tag8$$
5. $$\text{I}_5=\text{I}_4=\text{I}_{\text{C}_3}\tag9$$
6. And of course I know that ideal opamp equation: $$\text{V}_+=\text{V}_-\tag{10}$$

But now I do not know how to continue using the voltages at those nodes:

• @TonyStewart.EEsince'75 I want to use this for frequency analysis so it is not DC (I think, or I misted the point you're trying to make. If that is so I'm sorry). – Looper Mar 14 '18 at 20:25
• Vdifferential(f)=0 is the point. or Vin+=Vin- – Tony Stewart Sunnyskyguy EE75 Mar 14 '18 at 20:26
• Why don't you try the fast analytical techniques described electronics.stackexchange.com/questions/338350/… rather going for a tedious KVL-KCL classical analysis? – Verbal Kint Mar 14 '18 at 20:44
• @VerbalKint Thanks for the link. I see your work, but I also want to get it with algebra and math instead of using a program. – Looper Mar 14 '18 at 20:49
• @Looper, there is no program, just determining the circuit time constants and combining them to form the denominator coefficients. You can beat the FACTs on these applications! : ) – Verbal Kint Mar 14 '18 at 21:05


text: Abstract—A reconfigurable mesh, R-mesh for short, is a two-dimensional array of processors connected by a grid-shaped reconfigurable bus system. Each processor has four I/O ports that can be locally connected during execution of algorithms. This paper considers the <it>d</it>-dimensional Euclidean Minimum Spanning Tree (EMST) and the All Nearest Neighbors (ANN) problem. Two results are reported. First, we show that a minimum spanning tree of <it>n</it> points in a fixed <it>d</it>-dimensional space can be constructed in <it>O</it>(1) time on a <tmath>$\sqrt {n^3}\times \sqrt {n^3}$</tmath> R-mesh. Second, all nearest neighbors of <it>n</it> points in a fixed <it>d</it>-dimensional space can be constructed in <it>O</it>(1) time on an <it>n</it>×<it>n</it> R-mesh. There is no previous <it>O</it>(1) time algorithm for the EMST problem; ours is the first such algorithm. A previous R-mesh algorithm exists for the two-dimensional ANN problem; we extend it to any <it>d</it>-dimensional space. Both of the proposed algorithms have a time complexity independent of <it>n</it> but growing with <it>d</it>. The time complexity is <it>O</it>(1) if <it>d</it> is a constant.


text:
### Home > CALC > Chapter 4 > Lesson 4.4.1 > Problem4-125

4-125.

Imagine a hollow rubber ball. The rubber is 1-inch thick and the radius is 3 inches.

Volume = big sphere − small sphere

$\frac{4}{3}\pi(4)^3-\frac{4}{3}\pi(3)^3=\frac{148}{3}\pi \text{ un}^{3}$


text:
ï»¿ Describe Watson and Crick model of DNA. from Biology Biomolecules Class 11 Assam Board

Book Store

Currently only available for.
CBSE Gujarat Board Haryana Board

Previous Year Papers

Download the PDF Question Papers Free for off line practice and view the Solutions online.
Currently only available for.
Class 10 Class 12
Describe Watson and Crick model of DNA.

Watson and Crick proposed the model of DNA in 1953.

According to Watson and Crick model of DNA :

1. The Double Helix consists of two polynucleotide chains, which are coiled like a rope in helical (spiral) fashion.

2. The two strands of polynucleotides are  run in the opposite direction and are antiparallel.

3. The backbone is formed by the sugar-phosphate-sugar chain.

4. The nitrogen bases lie perpendicular to this
backbone but face inside.

5. A always pairs with T and G pairs with C. A and T are joined by two hydrogen bonds while the G and C pair have three bonds.

6. A base pair represents each step of ascent of the spiral. At each step the strand turns 36$°$.

7. There are ten base pairs in a complete turn which has ten steps.

8. The pitch is 34Å. the rise per base pair is 3.4Å.

The Watson-Crick model of DNA.

450 Views

What is the percentage of nucleic acids in cells ?

5 - 7 percent
634 Views

What are major elements of a cell ?

Carbon, hydrogen, oxygen and nitrogen are the major elements of a cell.
911 Views

Name one monosaccharide which provides energy ?

Glucose
698 Views

What is the general formula of carbohydrates ?

The carbohydrate has a general formula - CnH2nOn Or (CH2O)n.
596 Views

Give the names of two monosaccharide sugars.

Fructose and glucose are two monosaccharides
670 Views


text:
## Zach has $8.15 in his pocket. The money is in quarters and dimes. There are a total of 53 coins. How many quarters and dimes does Zach have Question Zach has$8.15 in his pocket. The money is in quarters and dimes. There are a total of 53 coins. How many quarters and dimes does Zach have in his pocket?

in progress 0
7 months 2021-07-23T11:42:26+00:00 1 Answers 5 views 0

34 dimes

19 quarters

Step-by-step explanation:

815=25x+10y

-1325=-25x+-25y

-510=-15y

510=15y

510÷15=y

34=y

53-34=x

19=x


text:
# Complex integrals and the possibility of avoiding parametric equations

I've been playing around with this equation:

$\displaystyle\int_{-\pi}^\pi{\displaystyle\frac{1-e^{3it}}{1-e^{it}}dt}$

Now it seems to me that we can (possibly) split the integral into four seperate integrals; one for each quadrant of the complex plane. Doing this, we can determine the complex part if we know the real part, and vice versa. So I therefore ponder that it might be possible to simplify the exponential expressions into expressions of variables. In other words, I wonder if we can rewrite the $e^{it}$s as $u$s.

I'm extremely interested in the potential of a variable substitution, but maybe I'm missing the obvious. I can't tell you exactly what I'm looking for, except for information concerning the possibility of this method. What is known? If there is a known way, I'd really enjoy seeing it. It would make many series expansions that I'm thinking about much easier, I'm almost certain. If no methods are known, I'd appreciate it very much if someone could take the time to elaborate on all of the reasons why this method is not known.

I'm very interested in going into detail on this matter, because it seems to decide issues concerning a great amount of potential.

-
I'm wondering if we could get a list of potential options to avoid parametrizations. Feel free to construct your own example(s); you don't have to use mine. –  Matt Groff Nov 24 '10 at 1:26
Did you try to simplify the integrand? $1-t^3$ has a zero at $t=1$, hence you can write $1-t^3=(1-t)\cdot(....)$. –  AD. Nov 24 '10 at 8:29
No, but that's definitely an option; thanks! –  Matt Groff Nov 24 '10 at 13:04

## 1 Answer

I don't really see why you don't want to just divide and then integrate the three terms... but if you insist on not using parameterizations: you can rewrite your integral as $$-i\int_{-\pi}^{\pi} {1 - e^{3it} \over 1 - e^{it}} ie^{it} {1 \over e^{it}}\,dt$$ Then you do a complex variables version of the $u$-substitution you are trying. You let $z = e^{it}$ and $dz = i e^{it} dt$. Your integral becomes $$-i\int_{|z| = 1} {1 - z^3 \over z(1 - z)}\,dz$$ Dividing $(1 - z)$ into $(1 - z^3)$ turns this into $$-i \int_{|z| = 1} ({1 \over z} + 1 + z) \,dz$$ Using the residue theorem this immediately evaluates into $-i 2\pi i = 2\pi$.

-
Good answer. I don't really think we need the residue theorem for this (though it's always fun to shoot sparrows with cannons), as the original integrand can be directly calculated to be $1 + e^{it} + e^{2it}$. This expression can be integraded by hand. –  Gunnar Magnusson Nov 23 '10 at 23:41
I know... he just asked for a way of doing it without using parameterizations. –  Zarrax Nov 24 '10 at 0:06
Well, I 'fess up to be the one who made him look at formulating his integrals of ratios of exponentials as contour integrals... but this one is yet another nice way of sidestepping parametrizations. –  Ｊ. Ｍ. Nov 24 '10 at 0:38
I should have mentioned my motivations...My mathematical background is a disjointed mess, and I'm currently focused on very complicated integrals (a simpler version being the one above). So I'm excited to learn as much as I can as fast as I can, and this seems to have potential. The power series expansions I've been working with are turning into a mess, and I was hoping for a quick and easy fix. As for J.M., he's been full of suggestions and I've been eager to try to learn about them all. I guess I'm more than a little disorganized, but I'm attempting to get a basic idea of things. –  Matt Groff Nov 24 '10 at 0:50
Thinking about more difficult integrals, I'm having trouble picking out terms for the residue theorom. Are there a lot of options besides looking at residues? The reason I'm even integrating is to find the constant term. But again, I'd like to get a general feel for the basics. –  Matt Groff Nov 24 '10 at 1:10


text:
# Difference between revisions of "Enable large arrays: Build octave such that it can use arrays larger than 2Gb."

The following only applies to systems that have 64-bit pointers (64bit architecture).

Starting with Octave 4.4.0, 64-bit indexing is the default for targets with 64-bit pointers. You can override that default by specifying --disable-64 when configuring Octave.

However, if the configure script determines that the BLAS library uses 32-bit integers, then operations using the following libraries are limited to arrays with dimensions that are smaller than 2^31 elements:

• BLAS
• LAPACK
• QRUPDATE
• SuiteSparse
• ARPACK

Additionally, the following libraries use "int" internally, so maximum problem sizes are always limited:

• glpk
• Qhull

To determine the integer size of the BLAS library used by Octave, the following code can be executed:

clear all;
N = 2^31;
## The following line requires about 8 GB of RAM!
a = b = ones (N, 1, "single");
c = a' * b

If the BLAS library uses 32-bit integers, an error will be thrown:

error: integer dimension or index out of range for Fortran INTEGER type

Otherwise, if the BLAS library uses 64-bit integers, the result is:

c = 2^31 = 2147483648

Note that the test case above usually requires twice the memory, if a and b are not assigned by a = b = .... Note further, that the data type "single" has a precision of about 23 binary bits. In this particular example no rounding errors occur.

### Versions prior to Octave 4.4

On previous versions of Octave, the default is that the size of a single Octave array cannot have more than approximately 2^31 elements, even on systems that use 64-bit pointers. This is because array indices were limited to 32-bit signed integers by default. Trying to create one will produce the following error:

>> a = zeros (1024*1024*1024*3, 1, 'int8');
error: out of memory or dimension too large for Octave's index type

You will obtain this error even if your system has enough RAM to create this array (3 GB in the above case).

To use arrays with more than (approximately) ${\displaystyle 2^{31}}$ elements, Octave has to be configured with the option --enable-64. This option is experimental and you are (as always) encouraged to submit bug reports if you find a problem. With this option, Octave will use internally 64-bit integers for array dimensions and indexing. However, all numerical libraries used by Octave will need to use also 64-bit integers for array dimensions and indexing, and in most cases they need to be compiled from source.

For details about how to compile these libraries please read the GNU Octave manual, or alternatively you can use MXE (M Cross Environment) which takes care of the different packages automatically (especially the configure flags --enable-64 and --enable-fortran-int64).


text:
# What real numbers do algebraic numbers cover?

Hardy and Wright mention ( though don't give a proof ) that any finite combination of real quadratic surds is an algebraic number. For example $\sqrt{11+2\sqrt{7}}$. Are all finite combinations of cube root, fourth root ... $n^{th}$ root also algebraic ? such as $\sqrt[3]{2+3\sqrt[7]{5+3\sqrt{6}}}+\sqrt[9]{2}$.

• Yes, they are all algebraic. Feb 2, 2017 at 17:06
• Yep. You are simply using elementary operators and functions, and the field of elementary functions is closed under (finitely many of) those operations. Feb 2, 2017 at 17:07
• They are algebraic, but there are algebraic numbers which can't be expressed only with nested radicals. Feb 2, 2017 at 17:08
• @Watson throw up a concrete example and that would be an answer in and of itself. Feb 2, 2017 at 17:10
• Note that changing your example can invalidate answer. In particular, I was working on mine and had almost posted it when you changed the target. Feb 2, 2017 at 17:29

The usual ways to prove this is to use some basic linear algebra.

Assume $\alpha$ and $\beta$ are algebraic numbers.

The field $\mathbb Q[\alpha][\beta]$ is a finite-dimensional vector space over $\mathbb Q$. Let $d$ be the dimension of the vector space.

Then $1,\alpha+\beta,(\alpha+\beta)^2,\cdots,(\alpha+\beta)^d$ are $d+1$ elements of that vector space, and thus must be linearly dependent. But that means that $\alpha+\beta$ is the root of a rational polynomial of degree at most $d$.

The same for $\alpha\beta$.

More specifically, if $\alpha$ is the root of a rational polynomial of degree $d_1$ and $\beta$ is the root of a rational polynomial of degree $d_2$ then $\alpha+\beta$ and $\alpha\beta$ are both roots of a polynomial of degree at most $d_1d_2$.

Finally, if $\alpha$ is the root of the rational polynomial $p(x)$ then $\sqrt[n]\alpha$ is a root of $p(x^n)$.

So this means that $5+3\sqrt6$ is the root of a polynomial of degree $2$, and $3\sqrt[7]{5+3\sqrt6}$ is a root of a polynomial of degree $14$ and $\sqrt[3]{2+3\sqrt[7]{5+3\sqrt6}}$ is the root of a polynomial of degree $42$ and finally $\sqrt[3]{2+3\sqrt[7]{5+3\sqrt{6}}}+\sqrt[9]{2}$ is a root of a polynomial of degree $42\cdot 9=378$. There might be a polynomial of smaller degree, but we get this upper bound.

• Shouldn't the lowest degree that you have shown be $42 \cdot 9=378?$ As you say, it could be lower. Feb 2, 2017 at 17:28
• Yep, brain freeze. @RossMillikan Feb 2, 2017 at 17:44
• But you stated without proof the key part of the demonstration: that $\Bbb Q[\alpha][\beta]$ is finite dimensional over $\Bbb Q$. That is the important part. The rest is just interpreting what this means. Feb 3, 2017 at 1:12
• Yeah, I didn't prove everything. @PaulSinclair Feb 3, 2017 at 2:02

Yes, you can unpack them into a polynomial that they satisfy. For your example, we can write $$x=\sqrt[3]{2+3\sqrt[7]{5+3\sqrt{6}}}\\x^3=2+3\sqrt[7]{5+3\sqrt{6}}\\\frac 13(x^3-2)=\sqrt[7]{5+3\sqrt{6}}\\\left(\frac 13(x^3-2)\right)^7=5+3\sqrt 6\\\frac 19\left(\left(\frac 13(x^3-2)\right)^7-5\right)^2-6=0$$

Note: this uses a previous example in the question. The approach is the same. Note that the algebraics are closed under the field operations, so the sum of algebraics is again algebraic.

• The OP edited the example, for $\sqrt[3]{2+3\sqrt[7]{5+3\sqrt{6}}}+\sqrt[9]{2}$ this is more difficult. Feb 2, 2017 at 17:12
• @Watson: That is why I appealed to closure under addition. Feb 2, 2017 at 17:26

Hint: Let $x = \sqrt{11+2\sqrt{7}}$, square this expression $x^2=11+2\sqrt{7} \implies x^2-11=2\sqrt{7}$. Now square again to get a polynomial with rational coefficients. You can use a similar approach for the second expression.

• true, yes, but this isn't super easy to generalize. A bit more care must be taken in proving this for all nested, $n$th degree surds Feb 2, 2017 at 17:09
• @MrYouMath I updated my example. Feb 2, 2017 at 17:10

I think you're asking if a sum of two algebraic numbers is algebraic, where "algebraic" is defined as "the root of a polynomial with rational coefficients".

While there are linear algebra reasons for this to be true, there is actually a nice direct calculation in terms of resultants.

If $\alpha_i$ are the roots of a polynomial $f$ and $\beta_j$ are the roots of a polynomial $g$, both with rational coefficients, then

$$\prod_{i} \prod_j (x - (\alpha_i + \beta_j))$$

when expanded, is a polynomial with rational coefficients. This polynomial can be computed by a resultant:

$$\operatorname{Res}_t(f(t), g(x-t))$$

and other formulations of the resultant make it obvious the result is a polynomial with rational coefficients.

Alternatively, Galois theory nearly immediately proves the product polynomial defined above has rational coefficients, because the polynomial is preserved by any permutation of the $\alpha$'s and of the $\beta$'s.

All the numbers you list are algebraic.

Lemma. If $\alpha$ and $\beta$ are algebraic, then $\alpha+\beta$ and $\alpha\beta$ are algebraic.

Proof. Since $\beta$ is algebraic (over $\mathbb{Q}$), it is also algebraic over $\mathbb{Q}(\alpha)$. Hence $\mathbb{Q}(\alpha,\beta)$ is a finite extension of $\mathbb{Q}(\alpha)$, which in turn is finite over $\mathbb{Q}$. Therefore $\mathbb{Q}(\alpha,\beta)$ is finite by the dimension formula: $$[\mathbb{Q}(\alpha,\beta):\mathbb{Q}]= [\mathbb{Q}(\alpha,\beta):\mathbb{Q}(\alpha)] [\mathbb{Q}(\alpha):\mathbb{Q}]$$ Any finite extension of $\mathbb{Q}$, say $n$-dimensional, consists of algebraic elements, because if $\gamma$ is an element, $\{1,\gamma,\gamma^2,\dots,\gamma^n\}$ is linearly dependent, so we find a polynomial with coefficients in $\mathbb{Q}$ that has $\gamma$ as root. In particular $\alpha+\beta$ and $\alpha\beta$ belong to $\mathbb{Q}(\alpha,\beta)$, so they're algebraic. QED

Note that this doesn't say that an algebraic extension (that is, all of whose elements are algebraic) is finite. Just that a finite extension is algebraic.

Lemma. If $\alpha$ is algebraic, then $\sqrt[n]{\alpha}$ is algebraic.

Proof. $\sqrt[n]{\alpha}$ is the root of $X^n-\alpha$, so it is algebraic over $\mathbb{Q}(\alpha)$, hence algebraic over $\mathbb{Q}$ by the same argument on dimensions as before. QED

A nested radical can be defined inductively as

1. A rational number is a nested radical;
2. If $\alpha_1,\dots,\alpha_k$ are nested radical, $q_1,\dots,q_k$ are rational numbers and $n$ is an integer, then $$\sqrt[n]{q_1\alpha_1+\dots+q_k\alpha_k}$$ is a nested radical;
3. nothing else is a nested radical.

Proposition. Any rational linear combination of nested radicals is algebraic.

Proof. Dismantle the number using the two lemma above. QED

However, there are algebraic numbers that cannot be expressed as sums of nested radicals. If this were true, then every equation with rational coefficients would be solvable by radicals; this is false as proved by Galois: radical extensions have solvable Galois group, but not all Galois groups of polynomials are solvable.

The complex algebraic numbers are what is called an algebraically closed field:

this implies that if $a_0,a_1,a_2,\dots, a_n$ are real algebraic numbers then the real roots of the polynomial $a_nx^n+a_{n-1}x^{n-1}+\dots+a_0$ are all algebraic numbers. This is way stronger than what you need.

Yes, this holds in general. Recall that an extension $F/\Bbb Q$ is algebraic iff for each $\alpha\in F$ we have

$$[\Bbb Q(\alpha):\Bbb Q]=\dim_{\Bbb Q} \Bbb Q(\alpha)< \infty$$

that is, if every element in it generates a finite extension. Similarly any finite extension is algebraic since if $\beta\in F$ then $\Bbb Q(\beta)\subseteq F\implies [\Bbb Q(\beta):\Bbb Q]\le [F:\Bbb Q]$. For a nested radical expression, this is easy then, take the most nested radical, adjoin the needed roots and proceed by induction.

In your first example $\alpha=\sqrt{11+2\sqrt 7}$, then we see that the field $\Bbb Q(\sqrt 7)$ has a polynomial for this which is ${1\over 2}(x^2-11)-\sqrt 7 = 0$ so the splitting field of this polynomial, which has degree at most $2$ over $\Bbb Q(\sqrt 7)$, and therefore degree at most $4$ over $\Bbb Q$, is algebraic, hence $\alpha$ is algebraic.

For your other example and more generally any example, you do similarly, since there are only finitely many nested radicals each time, you get splitting fields of finite degree, implying all such expressions are algebraic.

Just to work out the more complicated example, $\alpha = \sqrt[3]{2+3\sqrt[7]{5+3\sqrt{6}}}+\sqrt[9]{2}$ first adjoin $\sqrt 6$, and let $F_1=\Bbb Q(\sqrt 6)$. Then look at the splitting field, $F_2$, of $(x/3)^7-5-3\sqrt 6$ over $\Bbb Q(\sqrt 6)$. Then the splitting field, $F_3$ of $x^3-2-3\sqrt[7]{5+3\sqrt 6}$ over $F_2$, finally let $F_4=F_3(\sqrt[9]{2})$. Because each step had finite degree, $[F_4:\Bbb Q]<\infty$ showing $\alpha$ is algebraic.

One more since we're on a roll, look at $\sqrt{\sqrt{23}+\sqrt 5}$ this gives you a slight problem at first it seems since $\sqrt{23}$ and $\sqrt{5}$ are both the most nested radical, but as usual $[F(\alpha,\beta): E]\le [F(\alpha):E][F(\beta):E]$ and we can reduce to the simpler case by the classical result that sums and products of algebraic numbers are algebraic (proved in the same way with finite degrees).

The question in the title itself:

What real numbers do algebraic numbers cover?

has a rather amusing answer:

Practically none.

There are only countably many algebraic numbers, since each one is a root of some polynomial over $\mathbb{Q}$, and there are countably many such polynomials each of which has finitely many roots in $\mathbb{C}$. But there are uncountably many real numbers.

• Feb 3, 2017 at 8:28
• @CountIblis: We can be agnostic about it, if you don't believe in full semantics of power-sets or higher-order logic. A concrete analogous statement would be that given every computable enumeration of computable reals (represented by a program that always halts on an input $n$ and outputs the $n$ digit), there is a computable real that is computably distinguishable from every member of that enumeration. I view uncountability as indicating complexity, not necessarily size. Feb 3, 2017 at 8:41


text:
Performance Introduction (Solutions) Haskell Performance Introduction Step by Step Examples Graph reduction Laziness Time and space profilingStrictness Algorithm complexity Data structuresParallelism edit this chapter

Execution Model

Introducing Lazy Evaluation

Programming is not only about writing programs that work but also about programs that require little memory and time to execute on a computer. While both time and memory use are relatively straightforward to predict in an imperative programming language or in strict functional languages like LISP or ML, things are different here. In Haskell, expressions are evaluated on demand. For instance, the expression

```head (map (2 *) [1 .. 10])
```

will be evaluated as follows

``` ⇒ head (map (2 *) (1 : [2 .. 10]))      ([1 .. 10])
⇒ head (2 * 1 : map (2 *) [2 .. 10])    (map)
⇒ 2                                     (*)
```

The function `head` demands only the first element of the list and consequently, the remaining part `map (2 *) [2 .. 10]` of the list is never evaluated. Since this strategy only performs as much evaluation as necessary, it's called lazy evaluation. The chapter Graph reduction will present it in detail.

While lazy evaluation is the commonly employed implementation technique for Haskell, the language standard only specifies that Haskell has non-strict denotational semantics without fixing a particular execution model. A function `f` with one argument is said to be strict if it doesn't terminate or yields an error whenever the evaluation of its argument will loop forever or is otherwise undefined. Writing ⊥ for the "result" of an infinite loop, the definition for strictness is

A function `f` is called strict if `f ⊥ = ⊥` .

For example, trying to add `1` to a number that loops forever will still loop forever, so `⊥+1 = ⊥` and the addition function `(+1)` is strict. The function `head` is also strict since the first element of a list won't be available if the whole list is undefined. But it may well be that the first element is well-defined while the remaining list is not. In this case, we have

``` head (x : ⊥) = x
```

This equation means that `head` does not evaluate the remaining list, otherwise it would loop as well. Thus, such purely algebraic strictness properties are a great help for reasoning about execution time and memory. In strict languages like LISP or ML, we would always have `head (x:⊥) = ⊥`, whereas Haskell being "non-strict" means that we can write functions which are not strict, like the above property of `head` or the simplest example `(const 1) ⊥ = 1`. With the help of the constant `undefined` from the Prelude, we can even explore strictness interactively

```> head (1 : undefined)
1
*** Exception: Prelude.undefined
```

Strictness and ⊥ will be used through these chapters. Graph reduction presents further introductory examples and the denotational point of view is elaborated in Denotational semantics.

Example: fold

The best way to get a first feeling for lazy evaluation is to study an example. Therefore, we will consider some prototypical use case of `foldr` and its variants. The needed basics for studying the time and space complexity of programs are recapped in chapter Algorithm complexity.

Time

Consider the following function `isPrime` that examines whether a number is a prime number or not

``` isPrime n     = not \$ any (`divides` n) [2..n-1]
d `divides` n = n `mod` d == 0
any p         = or . map p
or            = foldr (||) False
```

The helper function `any` from the Haskell Prelude checks whether there is at least one element in the list that satisfies some property `p`. For `isPrime` , the property is "being a divisor of `n`".

The amount of time it takes to evaluate an expression is of course measured by the number of reduction steps. If `n` is a prime number, the above algorithm will examine the full list of numbers from 2 to `n-1` and thus has a worst-case running time of $O(n)$ reductions. However, if `n` is not a prime number, we do not need to loop through every one of these numbers, we can stop as soon as we found one divisor and report `n` as being composite. The joy of lazy evaluation is that this behavior is already built-in into the logical disjunction `||`!

```isPrime 42
⇒ not \$ any (`divides` 42) [2..41]
⇒ not ( or (map (`divides` 42) [2..41]                         ) )
⇒ not ( or ((42 `mod` 2 == 0) :      map (`divides` 42) [3..41]) )
⇒ not (     (42 `mod` 2 == 0) || or (map (`divides` 42) [3..41]) )
⇒ not (                 True  || or (map (`divides` 42) [3..41]) )
⇒ not True
⇒ False
```

The function returns `False` after seeing that 42 is even, `||` does not look at its second argument when the first one determines the result to be `True`. In other words, we have the following strictness property

``` True || ⊥ = True
```

Of course, the above algorithm can be implemented with a custom loop. But the crux of lazy evaluation is that we could formulate the algorithm in a transparent way by reusing the standard `foldr` and still get early bail-out. Put differently, lazy evaluation is about formulating fast algorithms in a modular way. An extreme example is to use infinite data structures to efficiently modularize generate & prune - algorithms. This and many other neat techniques with lazy evaluation will be detailed in the chapter Laziness.

It's neither feasible nor necessary to perform a detailed graph reduction to analyze execution time. Shortcuts are available, like non-strictness or the fact that lazy evaluation will always take fewer reduction steps than eager evaluation. They will be presented in Graph reduction.

Space

While execution time is modeled by the number of reduction steps, memory usage is modeled by the size of an expression during evaluation. Unfortunately, it's harder to predict and deviating from the normal course lazy evaluation by more strictness can ameliorate it. More details in Graph reduction. Here, we will present the prototypical example for unexpected space behavior.

Naively summing a huge number of integers

```> foldr (+) 0 [1..1000000]
*** Exception: stack overflow
```

produces a stack overflow. What happens? The evaluation proceeds as follows

```foldr (+) 0 [1..1000000]
⇒ 1+(foldr (+) 0 [2..1000000])
⇒ 1+(2+(foldr (+) 0 [3..1000000]))
⇒ 1+(2+(3+(foldr (+) 0 [4..1000000]))
```

and so on. We see that the expression grows larger and larger, needing more and more memory. In this case, the memory is allocated on the stack for performing the pending additions after the recursive calls to `foldr` return. At some point, the memory needed exceeds the maximum possible stack size raising the "stack overflow" error. Don't worry whether it's stack or heap, the thing to keep in mind is that the size of the expression corresponds to the memory used and we see that in general, evaluating `foldr (+) 0 [1..n]` needs $O(n)$ memory.

But it should be possible to do it in $O(1)$ space by keeping an accumulated sum of numbers seen so far, exploiting that `+` is associative. (Some readers may notice that this means to make the function tail recursive.) This is what foldl does:

```foldl (+) 0 [1..n]
⇒ foldl (+) (0+1) [2..n]
⇒ foldl (+) ((0+1)+2) [3..n]
⇒ foldl (+) (((0+1)+2)+3) [4..n]
```

But much to our horror, the accumulated sum will not be reduced any further! It will grow on the heap until the end of the list is reached

``` ⇒ ... ⇒ foldl (+) ((((0+1)+2)+3)+...) []
⇒ ((((0+1)+2)+3)+...)
```

and subsequent reduction of this huge unevaluated sum will fail with a stack overflow, too. (So, just introducing an accumulating parameter doesn't make it tail recursive.)

The problem is that the unevaluated sum is an overly large representation for a single integer and it's cheaper to evaluate it eagerly. This is what `foldl'` does:

```foldl' f z []     = z
foldl' f z (x:xs) = z `seq` foldl' f (f z x) xs
```

Here, evaluating `a `seq` b` will reduce `a` to weak head normal form before proceeding with the reduction of `b`. With this, the sum proceeds as

```foldl' (+) 0 [1..n]
⇒ foldl' (+) (0+1) [2..n]
⇒ foldl' (+) (1+2) [3..n]
⇒ foldl' (+) (3+3) [4..n]
⇒ foldl' (+) (6+4) [5..n]
⇒ ...
```

in constant space.

For more details and examples, read the chapter Graph reduction.

Compared to eager evaluation, lazy evaluation adds a considerable overhead, even integers or characters have to be stored as pointers since they might be ⊥. An array of 1-byte characters is several times more compact than a `String = [Char]` of similar length. With strictness annotations, unboxed types and automatic strictness analysis, the overhead can be reduced. The Haskell wiki is a good resource [1] concerning these low-level details, the wikibook currently doesn't cover them.

↑Jump back a section

Algorithms & Data Structures

Algorithms

While this wikibook is not a general book on algorithms, there are many techniques of writing efficient programs unique to functional programming. The chapter Algorithm complexity recaps the big-O notation and presents a few examples from practice.

In Laziness, the focus is on exploiting lazy evaluation to write efficient algorithms in a modular way.

A common theme of Program derivation and Equational reasoning is to derive an efficient program from a specification by applying and proving equations like

```map f . reverse  = reverse . map f
filter p . map f = map f . filter (p . f)
```

This quest has given rise to a gemstone, namely a purely algebraic approach to dynamic programming which will be introduced in some chapter with a good name.

Another equational technique known as fusion or deforestation aims to remove intermediate data structures in function compositions. For instance, the composition on the left hand side of

``` map f . map g = map (f . g)
```

constructs and deconstructs an intermediate list whereas the right hand side is a single pass over the list. The general theme here is to fuse constructor-deconstructor pairs like

``` case (Just y) of { Just x -> f x; Nothing -> ...; } = f y
```

This will be detailed in some chapter with a good name.

Data structures

Choosing the right data structure is key to success. While lists are common and can lead surprisingly far, they are more a kind of materialized loops than a classical data structure with fast access and updates. Many languages have special support for particular data structures of that kind like arrays in C, hash tables in Perl or lists in LISP. Natively, Haskell favors any kind of tree. But thanks to parametric polymorphism and type classes, any abstract type like balanced binary trees is easy to use and reuse! The chapter Data structures details the natural choices of data structures for common problems.

Because Haskell is purely functional, data structures share the common trait of being persistent. This means that older version of a data structure are still available like in

``` foo set = (newset, set)
where newset = insert "bar" set
```

Compared to that, the default in imperative languages is ephemeral, i.e. data is updated in place and old versions are overridden. For example, arrays are ephemeral. That's why they are either immutable or require monads to use in Haskell. Fortunately, many ephemeral structures like queues or heaps have persistent counterparts and the chapter Data structures will also gently introduce some design principles in that direction, for instance concerning amortization.

↑Jump back a section

Parallelism

The goal of parallelism is to run an algorithm on multiple cores / computers in parallel for faster results. Because Haskell doesn't impose an execution order thanks to its purity, it is well-suited for formulating parallel algorithms.

Currently, parallelism in Haskell is still a research topic and subject to experimentation. There are combinators for controlling the execution order Control.Parallel.Strategies, but they are rather fine-grained. Research on a less ambitious but more practical alternative Data Parallel Haskell is ongoing.

The chapter Parallelism is not yet written but is intended to be a gentle introduction to parallel algorithms and current possibilities in Haskell.

 Performance Introduction Solutions to exercises Haskell Performance Introduction  >> Step by Step Examples  >> Graph reduction  >> Laziness  >> Time and space profiling  >> Strictness  >> Algorithm complexity  >> Data structures  >> Parallelism edit this chapter Haskell edit book structure
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text:
# Weierstrass M-test

Last updated

In mathematics, the Weierstrass M-test is a test for determining whether an infinite series of functions converges uniformly and absolutely. It applies to series whose terms are bounded functions with real or complex values, and is analogous to the comparison test for determining the convergence of series of real or complex numbers. It is named after the German mathematician Karl Weierstrass (1815-1897).

## Statement

Weierstrass M-test. Suppose that (fn) is a sequence of real- or complex-valued functions defined on a set A, and that there is a sequence of non-negative numbers (Mn) satisfying the conditions

• ${\displaystyle |f_{n}(x)|\leq M_{n}}$ for all ${\displaystyle n\geq 1}$ and all ${\displaystyle x\in A}$, and
• ${\displaystyle \sum _{n=1}^{\infty }M_{n}}$ converges.

Then the series

${\displaystyle \sum _{n=1}^{\infty }f_{n}(x)}$

converges absolutely and uniformly on A.

The result is often used in combination with the uniform limit theorem. Together they say that if, in addition to the above conditions, the set A is a topological space and the functions fn are continuous on A, then the series converges to a continuous function.

## Proof

Consider the sequence of functions

${\displaystyle S_{n}(x)=\sum _{k=1}^{n}f_{k}(x).}$

Since the series ${\displaystyle \sum _{n=1}^{\infty }M_{n}}$ converges and Mn 0 for every n, then by the Cauchy criterion,

${\displaystyle \forall \varepsilon >0:\exists N:\forall m>n>N:\sum _{k=n+1}^{m}M_{k}<\varepsilon .}$

For the chosen N,

${\displaystyle \forall x\in A:\forall m>n>N}$
${\displaystyle \left|S_{m}(x)-S_{n}(x)\right|=\left|\sum _{k=n+1}^{m}f_{k}(x)\right|{\overset {(1)}{\leq }}\sum _{k=n+1}^{m}|f_{k}(x)|\leq \sum _{k=n+1}^{m}M_{k}<\varepsilon .}$

(Inequality (1) follows from the triangle inequality.)

The sequence Sn(x) is thus a Cauchy sequence in R or C, and by completeness, it converges to some number S(x) that depends on x. For n>N we can write

${\displaystyle \left|S(x)-S_{n}(x)\right|=\left|\lim _{m\to \infty }S_{m}(x)-S_{n}(x)\right|=\lim _{m\to \infty }\left|S_{m}(x)-S_{n}(x)\right|\leq \varepsilon .}$

Since N does not depend on x, this means that the sequence Sn of partial sums converges uniformly to the function S. Hence, by definition, the series ${\displaystyle \sum _{k=1}^{\infty }f_{k}(x)}$ converges uniformly.

Analogously, one can prove that ${\displaystyle \sum _{k=1}^{\infty }|f_{k}(x)|}$ converges uniformly.

## Generalization

A more general version of the Weierstrass M-test holds if the common codomain of the functions (fn) is a Banach space, in which case the premise

${\displaystyle |f_{n}(x)|\leq M_{n}}$

is to be replaced by

${\displaystyle \|f_{n}(x)\|\leq M_{n}}$,

where ${\displaystyle \|\cdot \|}$ is the norm on the Banach space. For an example of the use of this test on a Banach space, see the article Fréchet derivative.

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In mathematics, Fatou's lemma establishes an inequality relating the Lebesgue integral of the limit inferior of a sequence of functions to the limit inferior of integrals of these functions. The lemma is named after Pierre Fatou.

In calculus, the squeeze theorem, also known as the pinching theorem, the sandwich theorem, the sandwich rule, the police theorem, the between theorem and sometimes the squeeze lemma, is a theorem regarding the limit of a function. In Italy, the theorem is also known as theorem of carabinieri.

In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which almost everywhere convergence of a sequence of functions implies convergence in the L1 norm. Its power and utility are two of the primary theoretical advantages of Lebesgue integration over Riemann integration.

In mathematics, the integral test for convergence is a method used to test infinite series of non-negative terms for convergence. It was developed by Colin Maclaurin and Augustin-Louis Cauchy and is sometimes known as the Maclaurin–Cauchy test.

The Arzelà–Ascoli theorem is a fundamental result of mathematical analysis giving necessary and sufficient conditions to decide whether every sequence of a given family of real-valued continuous functions defined on a closed and bounded interval has a uniformly convergent subsequence. The main condition is the equicontinuity of the family of functions. The theorem is the basis of many proofs in mathematics, including that of the Peano existence theorem in the theory of ordinary differential equations, Montel's theorem in complex analysis, and the Peter–Weyl theorem in harmonic analysis and various results concerning compactness of integral operators.

In mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence defines a series S that is denoted

In measure theory, an area of mathematics, Egorov's theorem establishes a condition for the uniform convergence of a pointwise convergent sequence of measurable functions. It is also named Severini–Egoroff theorem or Severini–Egorov theorem, after Carlo Severini, an Italian mathematician, and Dmitri Egorov, a Russian physicist and geometer, who published independent proofs respectively in 1910 and 1911.

In mathematics, the Riesz–Fischer theorem in real analysis is any of a number of closely related results concerning the properties of the space L2 of square integrable functions. The theorem was proven independently in 1907 by Frigyes Riesz and Ernst Sigismund Fischer.

A bounded real sequence is said to be almost convergent to if each Banach limit assigns the same value to the sequence .

In mathematics, uniform integrability is an important concept in real analysis, functional analysis and measure theory, and plays a vital role in the theory of martingales. The definition used in measure theory is closely related to, but not identical to, the definition typically used in probability.

In mathematics, the Fraňková–Helly selection theorem is a generalisation of Helly's selection theorem for functions of bounded variation to the case of regulated functions. It was proved in 1991 by the Czech mathematician Dana Fraňková.

In real analysis and measure theory, the Vitali convergence theorem, named after the Italian mathematician Giuseppe Vitali, is a generalization of the better-known dominated convergence theorem of Henri Lebesgue. It is a characterization of the convergence in Lp in terms of convergence in measure and a condition related to uniform integrability.

## References

• Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN   978-0-07-054236-5. OCLC   21163277.
• Rudin, Walter (May 1986). Real and Complex Analysis. McGraw-Hill Science/Engineering/Math. ISBN   0-07-054234-1.
• Rudin, Walter (1976). . McGraw-Hill Science/Engineering/Math.
• Whittaker, E.T.; Watson, G.N. (1927). A Course in Modern Analysis (Fourth ed.). Cambridge University Press. p. 49.


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# SSC CGL Tier 1 :: Quantitative Aptitude QA Test 66

## Home SSC CGL Tier 1 / Quantitative Aptitude QA Test 66 Questions and Answers

1 .
A pipe of diameter d can drain a certain water tank in 40 minutes. The time taken by a pipe of diameter 2d for doing the same job in
5 minutes
10 minutes
20 minutes
80 minutes
2 .
Two train of equal length are running on parallel lines in the same direction at 46 km/h and 36 km/h. The faster train passes, the slower train in 36 seconds. The length of each train is:
82 m
50 m
80 m
72 m
3 .
Two boats A and B start towards each other from two places, 108 km apart. Speeds of the boats A and B in still water are 12 km/hr and 15 km/hr respectively. If A proceeds down and B up the stream, they will meet after:
4.5 hours
4 hours
5.4 hours
6 hours
4 .
If $x ^{x\sqrt x}$ = $(x\sqrt x)^x$. then x equal to
$4 \over 9$
$2 \over 3$
$9\over 4$
$3\over 2$
5 .
If a$\neq$ b, then which of the following statements is true?
$a + b \over 2$ = $\sqrt {ab}$
$a + b \over 2$ < $\sqrt {ab}$
$a + b \over 2$ > $\sqrt {ab}$
All of the above
6 .
If a*b= 2a + 3b - ab, then the value of (3*5+ 5*3) is:
10
6
4
2
7 .
If x = 2 - $2^{1\over 3}$ + $2^{2\over 3}$ then the value of $x^3$ - 6$x^2$ + 18x = 18 is
22
33
40
45
8 .
If p+q = 10 and pq=5 then the numerical value of $p\over q$ + $q\over p$ will be:
16
20
22
18
9 .
If x and y are positive real numbers and xy=8, then the minimum value of 2x+y is:
9
17
10
8
10 .
If tan 2$\theta$ tan 4$\theta$ =1, then the value of tan 3$\theta$ is:
$\sqrt 3$
0
1
$1 \over \sqrt 3$


text:
# zbMATH — the first resource for mathematics

A note on the equality of the BLUPs for new observations under two linear models. (English) Zbl 1229.15023
Summary: We consider two linear models, $${\mathcal L}_1$$ and $${\mathcal L}_2$$, say, with new unobserved future observations. We give necessary and sufficient conditions in the general situation, without any rank assumptions, that the best linear unbiased predictor (BLUP) of the new observation under the model $${\mathcal L}_1$$ continues to be BLUP also under the model $${\mathcal L}_2$$.

##### MSC:
 15A42 Inequalities involving eigenvalues and eigenvectors 62J05 Linear regression; mixed models 62H12 Estimation in multivariate analysis 62H20 Measures of association (correlation, canonical correlation, etc.)


text:
# Tag Info

1

Before going to field theory, it seems instructive to first ask the same questions in point mechanics: Can the Lagrangian $L(q,v,t)$ depend on time explicitly? Yes. The Lagrangian $L(q,v,t)$ can depend explicitly on time. E.g. there could be external sources. On the other hand, if the Lagrangian does not depend on time explicitly, then the ...

1

We have such equation: $$H = \frac{\partial L}{\partial \dot{q}} \dot{q} - L$$ You can show by calculation, that it holds in your special case, too. Now we use chain rule, and Euler-Lagrange equation: \frac{dL}{dt} = \frac{\partial L}{\partial q} \dot{q} + \frac{\partial L}{\partial \dot{q}} \ddot{q} =\frac{d}{dt}\left(\frac{\partial L}{\partial ...

0

The mathematical concept that I was searching for in this question is the following: http://en.wikipedia.org/wiki/Hodge_dual I will not elaborate more, but except to say that, the Hodge dual allows you to define a conserved current corresponding to any choice of the "time axis". (Sorry for the vagueness).

0

Comments to the question (v2): It is often is possible to formulate (discrete) time-evolution equations/equations of motion (eoms) in a discretized theory. This is of course useful in computational physics. However OP asks for a variational action principle for a fully discretized theory. Hence we will not discuss further the case where eoms (without a ...

0

Not all conserved charges are obtained by integrating the time component of some conserved current. For example, momentum and angular momentum are conserved charges and are obtained by integrating a spatial component of a conserved current. So the equations and interpretation for conserved charges in a Euclidean theory are the same as in the ...

2

Assuming no quantum gravity, $\eta^{\mu\nu}$ is a constant and can be pulled out of the derivative and what remains looks like a $\delta^k_{\mu}$ or $\delta^k_{\nu}$-type expression (in the sense of a Kronecker $\delta$), pulling the $k$ into the $\partial^\mu$ or $\partial^\nu$ respectively. If you are confused about where the minus sign comes from, I ...

Top 50 recent answers are included


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# What is the energy scale of a Hamiltonian?

On the second page of this paper a term 'fundamental energy scale' is used while talking about a Hamiltonian. The context is implementing Deutsch's algorithm using Adiabatic Quantum Computation.

What is the fundamental energy scale of a Hamiltonian and why it is important?

-
As defined in (1), (2) or (11), the "hamiltonian" are dimensionless. But a true hamiltonian, representing energy must have a energy dimension. So you need a energy scale $E$,depending on the physical problem, to define the true hamiltonian, for instance $\tilde H_0 = E ~ H_0$. Maybe the author use these dimensionless hamiltonian to simplify the calculus. – Trimok Jul 16 '13 at 18:11
@Trimok, your comment partially answers my question. Is this kind of dimensionless Hamiltonian show up often in quantum mechanics or quantum information? – Omar Shehab Jul 16 '13 at 19:08
I don't know exactly. If $E$ is a constant, working with dimensionless hamiltonians is the same thing as working in units $E=1$. For instance, in Quantum field Theory, one very often work with units $c = \hbar = 1$. This simplifyes notations and calculus. As long as you know what $E$ exactly is (and it depends on the particular physical system your are studying), this is not a problem. – Trimok Jul 17 '13 at 7:16
@Trimok, so in QFT you do the calculation for $c = \hbar = 1$ and rescale the final result, right? – Omar Shehab Jul 17 '13 at 12:54
If you need to do that, yes. For instance, in QED, a lot of results (scattering amplitudes, etcc.) depends only on the coupling constant $\alpha = \frac{e^2}{\hbar c}$. Moreover, $c=1$ allows to put at the same level time, the $x_0$ coordinate and space, $x_1,x_2, x_3$ coordinates - This is Minkowski space-time. Note, in fact, that putting $c=1, \hbar=1, G=1$ is a natural way of thinking, because it corresponds to Planck units, and this simplifyes equations,see Wikipedia. – Trimok Jul 17 '13 at 16:09


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Tutorial on optimal framed clustering

Updated: 2020-12-12; Created: 2020-12-03

The FramedClust function finds a frame of consecutive points belonging to a large set of univariate data points to obtain the best clustering. The frame.size parameter indicates the number of points to be included in each frame. The actual width of a frame is the distance between the smallest and largest data points inside the frame.

Here we illustrate how to run the FramedClust function and visualize the results to explain what framed clustering is.

Data preparation

Any linear dataset can be used with the FramedClust function. The input vector X contains some points from a linear data. X does not have to be sorted. We will find the best K=7 clusters among all possible frames containing 50 points (frame.size = 50).

library(OptCirClust)
X = rgamma(70, 6)
K = 7
frame.size = 50

Performing framed clustering on the data

Now we demonstrate how to cluster the data using three different algorithms: the recommended fast and optimal "linear.polylog" algorithm, the brute-force and optimal algorithm by repeatedly calling "Ckmeans.1d.dp", and the slow and heuristic algorithm by repeatedly calling "kmeans".

# Our recommended method is the fast and optimal linear.polylog:
result_linear.polylog <- FramedClust(X, K, frame.size,  method = "linear.polylog")

# The slow and optimal via repeatedly calling Ckmeans.1d.dp:
result_Ckmeans.1d.dp <- FramedClust(X, K, frame.size,  method = "Ckmeans.1d.dp")

# The slow and heuristic via repeatedly calling kmeans:
result_kmeans <- FramedClust(X, K, frame.size, method = "kmeans")

Visualizing framed clusters

The clustering outcomes obtained from the FramedClust function can be visualized using the plot function.

plot(result_linear.polylog, main = "linear.polylog: optimal\n***Recommended***")

plot(result_Ckmeans.1d.dp, main = "Repeated Ckmeans.1d.dp: quadratic time\nalways optimal")

plot(result_kmeans, main = "Repeated kmeans: heuristic\nnot always optimal")

The points are colored according to their cluster. The black dashed line represents an optimal frame with the minimum sum of squared within-cluster distances. All points outside the frame are colored in gray. The "kmeans" option of the FramedClust function does not guarantee an optimal solution, while the other two method options do.


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## Tuesday, November 04, 2014

### Multiverse and the measure problem: no progress

Off-topic, CERN: former ATLAS spokeslady Fabiola Gianotti was chosen to be the next CERN Director General. Congrats but it's since 2016 and she may very well miss the first big post-Higgs discoveries.

Interstellar: the reviewers seem totally captivated by the black hole blockbuster.
Yesterday, Natalie Wolchover and Peter Byrne wrote an article for the Quanta Magazine:
In a Multiverse, What Are the Odds?
It's the first part of a series. The article has received no comments so far and no other media reacted. My reactions to this silence is mixed:
1. On one hand, this article refers to many actual big shots in physics research (and yes, I know all the physicists mentioned there in person), unlike most of the "news about science" that inform about the work of assorted cranks and 3rd class physicists
2. Despite these physicists' being very good and at the top, this whole "measure problem" research program is fundamentally flawed, so it's good news that at least one set of flawed ideas is being ignored by most of the media
Let me summarize the topic by saying that the multiverse – a very large universe or "conglomerate of connected universes" – naturally follows from eternal inflation. If inflation is right, it's likely to be eternal. And inflation is likely to be right, too. And this whole "anthropic" attitude is an additional speculative floor built on top of eternal inflation that wants to calculate the probabilities that we observe something "here and now" from looking at the other, mostly very distant and separated, parts of the whole multiverse. The multiverse is probably right; but this whole "anthropic" level is fallacious.

The article mentions various people who defend the anthropic reasoning, including Weinberg, Guth, Vilenkin, Bousso, Kleban, and others. Nathan Seiberg is quoted as a neutral person who would like to be anti-anthropic but he won't deny the evidence. Paul Steinhardt plays the role of the guy who denies the cosmic inflation itself.

Alan Guth is a key co-father of cosmic inflation and he pays lip service to the anthropic reasoning, too. In a video attached in the article, Guth says that a big problem of cosmology is how to calculate the percentage $$\infty/\infty$$ of the cows across the Universe that are violet (the Milka cow could be just one example). Steven Weinberg gave us the argument estimating the cosmological constant in the late 1970s and this argument is being promoted as a success of the anthropic reasoning, which is misleading, as I will point out.

Various random methods to calculate $$\infty/\infty$$ due to Vilenkin and others are said to lead to highly variable and unsatisfactory results. Raphael Bousso is promoted as the author of a better approach to the "measure problem" based on the causal diamonds.

However, the whole paradigm that one should look for a solution of the "measure problem" is fundamentally flawed. It's based on the assumption that the probabilities of various observations done "here and now" are affected by statistical properties of objects in the multiverse – either in the spacetime or in a spatial slice – because "we are typical" in some sense. But this very general assumption is wrong not just for one reason but for many reasons:
1. Calculating properties of the Universe "now and here" from the distribution of observers including the future ones is acausal, and therefore logically inconsistent, so the "spacetime counting" is just plain wrong (how many observers of some type will exist also depends on our current and future decisions, so these numbers can't influence these decisions in the backward direction)
2. The "spatial slice counting" is also wrong because there are no preferred "slices" of the Universe, due to relativity
3. All this "anthropic" counting requires one to be able to define what an observer is and what is not. However, science obviously can't give us an answer to this "bureaucratic" question who gets the "paperwork" – the difference between "non-observers" and "observers" is always gradual and quantitative
4. There is absolutely no justification for the assumption that "every observer whose immediate vicinity is looking the same is equally likely". It's just one possible probability distribution among infinitely many – and distributions that actually try to "discourage" highly populous representatives are at least equally justified (some of this discouragement has to exist to make the distributions normalizable, see the next item)
5. The uniform distribution doesn't even exist because the sets of observers in the multiverse are infinite and the normalized uniform distribution would have to have $$p(X) = (1/\infty)\cdot 1 = 0$$
6. Any attempt to regulate this problem with infinities leaders to a strong dependence on the cutoffs
The promoters of the anthropic principle only acknowledge the last problem. They see it as a minor technicality that just hasn't been solved yet. But it's not a technicality – it is just a small corollary of the previous, much more serious problems with the whole paradigm. And they're problems that the defenders of the anthropic principle completely ignore.

The basic issue that they still seem to misunderstand, after these decades (!), is that every probability that is uniquely and unambiguously calculable by the laws of physics has the form$P(\text{property at later moment} | \text{prop. at earlier }t).$ In other words, the laws of Nature allow you to calculate the probability that a particular initial state – known either "maximally" (as a pure state) or "probabilistically" (as a density matrix) – will evolve into a final state with certain properties. That's how dynamical laws of physics always work. In particular, I want to emphasize that the opposite probabilities – probabilities of something at an earlier moment given some assumptions about the later moment – are never uniquely calculable because they may only result from Bayesian inference and this inference is a "reverse problem" that inevitably depends on some arbitrary, subjective priors! So the results of the "opposite" (past assuming future) probabilities can never be fully dictated by the laws of physics!

The only "new twist" that cosmology may add into this template is that it may provide us with a definition of the "ultimate initial conditions of the Universe", perhaps by a formula similar to the Hartle-Hawking wave function. With this addition, the probabilities$P(\text{property at any moment}|\text{right initial cond.})$ may become meaningful, too.

In the future, it may be possible to calculate the probability that a particular compactification of string theory with certain extra properties has evolved from the right initial conditions. Other things could have evolved, too. They may even be "realized" in very distant parts of the Universe or the multiverse. But they don't affect what we observe here and now. Instead, they are consequences of the initial conditions, much like our observations here and now are consequences.

It's also misleading to say that Weinberg's estimate of the cosmological constant is a success of the anthropic reasoning. In reality, Weinberg has simply observed the existence of the stars and he has proven that the cosmological constant can't be too high and positive; and can't be too high and negative, otherwise the theory would imply that no stars exist. This is nothing else than a particular procedure to falsify a class of the hypotheses by the empirical data. It's the most ordinary scientific method – and Weinberg has played the role of an experimenter.

However, there are usually many very different ways to falsify a wrong theory by finding its predictions that contradict certain empirical observations. And it is never true (at least not demonstrably true) that a particular method to falsify the theories is the unique one. The anthropic approach (the anthropic lack of principles) is the belief that a procedure similar to Weinberg should be the only relevant argument or observations that constrains or determines some property of Nature. But this belief has clearly been wrong almost everywhere in the history of science and it is very likely that it will be wrong in most scientific advances in the future, too.

There simply exist – and will exist – much sharper, more concrete and well-defined ways to eliminate or strengthen hypotheses than just the observations that "some intelligent life exists"! Nature gives us many hints (and clear evidence), we see some of them, fail to see or overlook the others, but the task for science is to find as many useful hints as possible. And the anthropic lack of principles is telling us nothing whatsoever where we should look. On the contrary, it is telling us that we shouldn't look anywhere, perhaps with one exception of a stupid process of counting the observers.

Another part of Weinberg's success is that the inequality for the cosmological constant that he derived from the existence of stars is obeyed by the Universe around us – it has to be – but "barely so". The observed cosmological constant is "just" 100 times smaller than the maximal one that is allowed according to Weinberg's argument. The number 100 is much smaller than 10123 but it is still pretty large. The opinion that we have to eliminate the values that contradict the existence of stars is a fact. But there are also other things that affect the justifiable opinions about the value of the cosmological constant and it will never be possible to show that these other factors – e.g. conventional physics as we have known it for centuries – play no role. Whether the number 100 above is large or small is a matter of subjective taste, not science. This taste depends on some probabilistic distribution for the surviving viable values of the cosmological constant but without a law dictating the initial conditions, every such distribution is just a premature, subjective, idiosyncratic guess that simply cannot be "qualitatively more true" than others.

Now, the causal diamonds and light-like sheets and slices are very natural in physics. This importance is also linked to the fact that the light-cone gauge is such a natural gauge-fixed formulation in string/M-theory (Matrix theory naturally arises in the light-cone gauge, too); the light sheets may always be viewed as generalized black hole horizons, so they're relevant for the generalizations of the black hole entropy to general spacetimes; and they play many other roles. But I wouldn't say that the discussions about the initial conditions for the Universe are among them. At most, Bousso's focus on the light sheet is yet another observation showing a fundamental defect of the "measure problem" research as we knew it for many years. According to the horizon complementarity, the degrees of freedom behind the cosmic horizon shouldn't even be viewed as independent ones, so they can't simultaneously have well-defined values with the actual observables within our causal patch! That's why we shouldn't be talking about the number of observers behind the horizon if we also want to talk about more tangible questions that are observable within our patch: to speak about both is just like to assume that both $$x$$ and $$p$$ are well-defined in quantum mechanics.

But this observation weakens the efforts to calculate something about "now and here" from counting observers; it doesn't strengthen it in any way.

Incidentally, in a recent paper by Bena, Graña, Kuperstein, and Massai, it is being argued that in the KKLT construction of the "many stabilized flux vacua", the added anti-D3-brane actually makes the whole vacuum unstable, in the tachyonic sense, so the "uplifted vacua" are no good. Once supersymmetry is broken, calculations become significantly less controllable. So whether this criticism of KKLT is valid will remain an open question for some time, I guess.

Let me remind you that the KKLT is a semi-specific construction of large classes of type IIB string theory compactifications that starts from showing that many supersymmetric AdS vacua are possible solutions because of many values of the fluxes over cycles; and in the second part, KKLT argue that one may also "uplift" these SUSY AdS vacua to non-SUSY dS vacua by adding an ingredient that breaks supersymmetry, namely some anti-D3-branes (branes with the opposite charge than the D3-branes that have been there already before that final step).

KKLT are generally more achieved physicists than the critics and I think that they're generally slightly better "string mathematicians" able to calculate these matters, too. But otherwise my expectation is that the critics – these critics or similar ones – are likely to be correct. Why do I think so?

Because unbroken supersymmetry is the key feature that tends to produce moduli spaces, stable vacua, positivity of the energy (which is also linked to the stability at the stationary points), and large configuration spaces. When SUSY is broken, it just seems reasonable to expect that many "steep slopes" in the configuration spaces are created, ruining the stability of almost all conceivable stationary points, and the stationary points are likely to have some unstable directions. So quite generally, I do tend to think that the number of stable non-SUSY vacua is smaller than the number of stable SUSY vacua, in some sensible way to count them. I know that some people have the opposite expectation but it even seems plausible to me that – just like we would sometimes believe – if the analysis is done really perfectly, SUSY breaking leads to a single (!) realistic non-SUSY vacuum in four dimensions, and this vacuum has to be ours. Of course that this possibility is a wishful thinking to a large extent and the uniqueness isn't necessary for any established principle. But it may be a possibility, anyway.

#### 23 comments:

1. Hello Lubos,

but in another parts of string field theory a major progress was made, I think: http://phys.org/news/2014-11-field-theory-foundation-quantum-mechanics.html

Best regards and thank you for your very interesting postings,
Klaus Lange

2. Dear Klaus, first, this article (in Quanta Magazine, or this blog post) has nothing to do with string *field* theory. String field theory is just a tiny fraction of the research in string theory - a particular way to write down the formulae for perturbative amplitudes.

Second, the progress you reported is not new. The preprint was posted in July

http://arxiv.org/abs/arXiv:1407.6833

which is when I first responded to it in a comment. So far, the paper has 0 citations.

Third, the claim is really a fundamental misunderstanding of the interpretation of the string fields. The mathematical analogies between the operators in quantum mechanics and the non-commutative star product on one side - and the star-product merging the strings on the other side - has been known since the first paper by Witten about the cubic string field theory. So that's nothing new.

But this mathematical similarities doesn't mean that they're the same thing. A string field is a "matrix" on the space of string configurations, but it's also an operator on the Hilbert space. These are two completely different levels of the fields' being "operator-like". To some extent, the claim that the foundations of QM are "explained" by the string fields is as wrong as the claims that the foundations of QM are described by the droplets.

http://motls.blogspot.com/2014/07/droplets-and-pilot-waves-vs-quantum.html?m=1

And the latter is seriously wrong.

3. Hi Lubos,
Please tell me your views(if any) on this paper. I had recently come across it and it looked very curious. http://arxiv.org/abs/1312.6523

4. Hi! I think it's a nice paper, some of the mathematical observations may be new but some of them are older. The relationship to the division algebra is heuristically great but much of the structure evaporates for the larger algebras, especially the octonions.

5. They say:

"If the properties of this universe still seem atypical even in the habitable subset, then the multiverse explanation fails."

Why is that? I think it should be that the anthropic explantion fails.

6. Agreed. They simply adopt the dogma that the anthropic principle ("typicality") must be *automatically* associated with the multiverse. Of course that I disagree with that entirely.

7. But do we even need to expain QM? QM is fundamental it can't be explained by a deeper level. Not to mention that if this deeper level is not Quantum mechanical it has to be classical which is just stupid.

8. (This comment is to be taken as one coming from a mathematical mosquito with its neck sticking out!)

Dear Lumo,
Soon enough into reading this to me beautifully efficient and swiftly encompassing overview of and revisit to this topic, I was somehow deriving a slight satisfaction from sensing that your raging against, and your own sense of room to move away from, The AP seems to have slightly shriveled and shrunk, respectively. :->

9. Thanks for a nice discussion. I will have to study the earlier points more carefully in order to try to understand them better, but the second half of the essay makes points that are much clearer to a layman like me, and which help to clarify the issues in this very important question -- important because is likely to color humanity's "religious" views of the nature of the universe for eons to come. I must say I share your wishful thinking, but I am ready for anything.

10. "They simply adopt the dogma that the anthropic principle ("typicality") must be *automatically* associated with the multiverse. . ."

Theoretical cosmologists assume typicality but they are themselves highly atypical human beings. I find a certain irony in this.

11. I agree with you, Giotis. SFT is just an application (and much more specific type) of quantum mechanics, and like any field theory, it's "rather interesting" already before quantization, as a classical theory, too.

12. What an irony! And it's not just some similarity between the two problems. The relevant definition of "intelligent life" would almost certainly assume some "on Earth atypical" elevated IQ, because those who can't really get these cosmological theories may very well be clumped with the apes and maybe other organisms etc.

Even on Earth, one may get estimates of relative probabilities obtained from "typicality" that differ by many orders of magnitude depending on "who is allowed to be counted". And that's just the Earth where all "good enough" candidates are rather similar. In the multiverse, the "candidates to be called intelligent observers" are almost certainly much more diverse in number and in intensive quantities, so a more demanding version of "intelligent life" may very well shrink the pool not by the factor of 1 million like on Earth but by the factor of 10^{123} or higher.

13. landscapedoubterNov 4, 2014, 5:08:00 PM

Dear Lubos,

Just a comment on the following statement "KKLT are generally more achieved physicists than the critics and I think
that they're generally slightly better "string mathematicians" able to
calculate these matters, too.".

I would like you to inform you of the recent history of this issue. The first real criticism on anti-brane uplifting started in 2009, 6 years after KKLT. It took that long since computing the details of string theory vacua with broken SUSY is very dirty and the techniques were not available. At least 10 papers have been written since that are highly technical and demonstrate that anti-brane uplifting is likely to be problematic. The "pro-KKLT" camp has not written any paper in response, apart from a paper by Dymarsky (http://arxiv.org/abs/1102.1734) that contained many claims that have been shown to be wrong. The people that are working on this should not be considered as an "anti-KKLT camp". Instead they are working out the details with an open mind and the details seem to contain problems.

I am disappointed in the lack of scientific attitude of some members of the string pheno community. Instead of an honest scientific debate, the KKLT scrutiny is dismissed as politically motivated. A good scientist can defend his/her model by solid mathematics and not by authority arguments. Not a single computation has been done by the KKLT camp to disprove the critics. If they are the better "string mathematicians" then they should be able to show it.

In my opinion (for what it is worth) I would not bet my money on KKLT, but I would not say either it is shown to be inconsistent for sure (I would give it a 95% failure rate). It seems that the only way to tell for sure would be to do a computation at small length scales/or strong coupling.

14. Lightspeed is a robust observable (including exponents larger than 1) throughout the visible universe. Lightspeed survives both relativity and quantum mechanics. Lightspeed is impossibly small given the scale of the universe, across the solar system, even at circuit board dimensions.

One suspects there is a universal very small scale determinant rather than a large scale effect. The multiverse is then inversely wrong. Entanglement began intensely locally and resisted dissipative decay - a pervasive soliton background?

15. The paper on KKLT pulls my layman triggers for "this could be groundbreaking". A way to show that almost all of the landscape is unstable?

"Another feature that is important to understand is what is the endpoint of the tachyonic instability."

"Hence, such a construction will not give a long-lived de Sitter vacuum, but will either give an unstable one or one whose cosmological constant will jump down whenever the anti-D3 branes are shot out."

Tied to the cosmological constant? Could this force a de Sitter vacuum with a CC below some very low treshold? Absolutely amazing if so!

16. Are there any prominent researchers in the "pro-KKLT" camp? Specifically, KKLT themselves could reasonably think the papers pointing out problems are simply good work and have no interest in, or way to, find fault with them.

Catfighting aside, congrats on the "featured comment"; I had no idea such a feature existed.

17. Discussing of wonderful photos, examine out the NASA Astronomy Image of the Day Database. One pic per day with a brief medical summary.

Spybubble gratuit

18. The equivalence principle is more apropos. Unless there are a lot of small people running around that I cannot see the anthropic principle leaves much to be desired. While I am more than happy to entertain the thought that physics breaks down at small scales, where are all the little tiny people at? Isn't that more likely? Multiverse? Riiiiiiight..... http://youtu.be/UUARU7iawSA

19. Since vacuum energy is close to matter energy or if we assume that Einstein did not make a blunder, has there been a conjecture to tie both energies directly by some process. Maybe one is responsible for the other.

20. A bit of humor, the extreme badassism of top-level physicists. I saw it in the presentation “Eternal Inflation and the Measure Problem" by Alan Guth in Madrid passed month.

http://33.media.tumblr.com/a9e9e85c783a1ca0a5e50b658b6967a4/tumblr_nd67ljG81m1qaityko1_1280.png

21. kashyap vasavadaNov 5, 2014, 3:57:00 PM

Is it possible that there is a repulsive quantum field which gave rise to inflation and it decreased to a small value when inflation stopped and we interpret it as a small cosmological constant today? I know, this is a completely naive speculation on my part!! I suspect it is probably not possible to write a mathematical model for such a field otherwise some one would have done it. But a discussion will be interesting.

22. Lubos, you might enjoy http://arxiv.org/abs/1401.2938

23. I think this paper comes close to answering my question

http://philsci-archive.pitt.edu/398/1/cosconstant.pdf

page 20

The idea that the field concept has to be used with great care lies also
close at hand when we remember that all field effects in the last resort can
only be observed through their effects on matter. Thus, as Rosenfeld and
I showed, it is quite impossible to decide whether the field fluctuations
are already present in empty space or only created by the test bodies.52


text:
# Right-angled triangle

Geometry Level 4

$$\triangle ABC$$ is a right-angled triangle, where $$AC=3$$ and $$BC=4$$. Let $$P$$ be a point on the hypotenuse of the triangle. We drop perpendiculars from $$P$$ to $$AC$$ and $$BC$$, the footpoints of which are $$X$$ and $$Y$$.

If the minimum value of $$\overline{XY}$$ is $$\dfrac{m}{n}$$, where $$m$$ and $$n$$ are coprime integers, then find the value of $$m+n$$.

×


text:
# If a Bus Travelling at 20 M/S is Subjected to a Steady Deceleration of 5 M/S2, How Long Will It Take to Come to Rest ? - Science

Short Note
Sum

If a bus travelling at 20 m/s is subjected to a steady deceleration of 5 m/s2, how long will it take to come to rest ?

#### Solution

We have to find the time taken to reach the given final velocity.
We have:
Initial velocity (u) = 20 m/s
Final velocity  (v) = 0 m/s
Acceleration for the entire journey (a) = –5 ms2.
Let the time taken be (t).
We can calculate the time taken using the first equation of motion.
t = (v-u)/a
Time taken by the bus to come to rest:
= (0-20)/-5
= 4 s

Concept: Measuring the Rate of Motion - Speed with Direction
Is there an error in this question or solution?

#### APPEARS IN

Lakhmir Singh Science Class 9 Physics
Chapter 1 Motion
Short Answers | Q 33 | Page 21


text:
# 问题 T: Cashier Employment

#### 题目描述

A supermarket in Tehran is open 24 hours a day every day and needs a number of cashiers to fit its need. The supermarket manager has hired you to help him, solve his problem. The problem is that the supermarket needs different number of cashiers at different times of each day (for example, a few cashiers after midnight, and many in the afternoon) to provide good service to its customers, and he wants to hire the least number of cashiers for this job. The manager has provided you with the least number of cashiers needed for every one-hour slot of the day. This data is given as R(0), R(1), ..., R(23): R(0) represents the least number of cashiers needed from midnight to 1:00 A.M., R(1) shows this number for duration of 1:00 A.M. to 2:00 A.M., and so on. Note that these numbers are the same every day. There are N qualified applicants for this job. Each applicant i works non-stop once each 24 hours in a shift of exactly 8 hours starting from a specified hour, say ti (0 <= ti <= 23), exactly from the start of the hour mentioned. That is, if the ith applicant is hired, he/she will work starting from ti o‘clock sharp for 8 hours. Cashiers do not replace one another and work exactly as scheduled, and there are enough cash registers and counters for those who are hired. You are to write a program to read the R(i) ‘s for i=0..23 and ti ‘s for i=1..N that are all, non-negative integer numbers and compute the least number of cashiers needed to be employed to meet the mentioned constraints. Note that there can be more cashiers than the least number needed for a specific slot.

#### 输入

The first line of input is the number of test cases for this problem (at most 20). Each test case starts with 24 integer numbers representing the R(0), R(1), ..., R(23) in one line (R(i) can be at most 1000). Then there is N, number of applicants in another line (0 <= N <= 1000), after which come N lines each containing one ti (0 <= ti <= 23). There are no blank lines between test cases.

#### 输出

For each test case, the output should be written in one line, which is the least number of cashiers needed. If there is no solution for the test case, you should write No Solution for that case.

#### 样例输入 复制

1
1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
5
0
23
22
1
10

#### 样例输出 复制

1