text: Learn how to name monatomic ions and ionic compounds containing monatomic ions, predict charges for monatomic ions, and understand formulas. • Examples for polyatomic molecules are KCl, KBrO 3, C 6 H 5 COOH. Polyatomic Ions and Their Charges (it takes up two pages…just warning) +1 Charge *Ammonium NH4 +1 Hydronium H3O+1 Mercury (I) Hg2 +2-1 Charge *Acetate C2H3O2-1? This set of flashcards will aid you in learning the correct oxidation number(s) for the most common chemical elements. Polyatomic ions are treated exactly like monatomic ions. Using the Periodic Table to Determine the Charges of Monatomic Ions Notes: Monatomic ions are ions that form from a single atom by the loss or gain of valence electrons (ex. For example, calcium carbonate consists of the monatomic ion Ca 2+ and the polyatomic ion CO 3 2−.. A type I binary ionic compound contains a metal that forms only one type of ion.A type II ionic compound contains a metal that forms more than … Nomenclature of polyatomic anions. Study Monatomic Ions List Flashcards at ProProfs - Welcome to the 21st Century electronic classroom! https://getchemistryhelp.com/learn-chemistry-fast/ This lesson teaches how to determine the charge of monoatomic ions and how to name them. I know from my notes the first one the charge is 1. Anion, Cation: Common Names, Formulas and Charges for Familiar Ions Topic: Ions In Solutions. monatomic ions list with charges - Google Search. Education. • Size: • The size of the monatomic chemical … Match. Organic Chemistry.. Well, they don't actually lose them the way humans sometimes lose things. As for the second one I think I may have made a mistake when I was copying the board because I have Family VIIIA charge is -1. STUDY. →Negative monatomic ions … Does it have to do with electronegativity? 4. If ion-item-sliding are used inside the list, this method closes any open sliding item.. Returns true if an actual ion-item-sliding is closed. Created by. They can be either … Monatomic Ions List & Flashcards. What charge do monatomic ions of Family IA exhibit? Mg 2+, Ca , Ba2+ Groups 3-12 The elements in groups 3-12 are called transition metals. Thus it is different from monatomic ions, which contain only one atom. Flashcards. What is the charge on the common monatomic ions of the following elements? Compounds that contain a cation that is a polyatomic ion and an anion that is an ion. Klaus Theopold + 4 others. 37. →Ions can be written like Ca⁺⁺ or Ca⁺² or Ca²⁺ →Transition metals use Roman numerals to designate the specific charge of the ion →3 elements in the transition section are exceptions They are zinc, silver, and cadmium →Only 3 metals outside of the transition section use Roman numerals. Gravity. ammonium - NH 4 + Polyatomic Ion Charge = -1 . Chromite CrO2-1 Cyanate OCN-1 *Cyanide CN-1 Dihydrogen … Publisher: OpenStax. You must be signed in to discuss. Why? Any atom or molecule with a net charge, either positive or negative, is known as an ion. The net charge of an ion is non-zero due to its total number of electrons being unequal to its total number of protons. Many ions are monatomic, which means that they are composed of a single atom. This is the one of the resonant structures of the chlorate anion. Here, the cation and the anion are attracted to each other via electrostatic forces. For example, consider binary ionic compounds of iron and chlorine. Buy Find arrow_forward. A monoatomic ion is simply an ion consisting of one element, for example Sodium (Na+1). An ion can be thought of as a point in space with a charge. 3. Simple ions Those ions which are formed from single atoms are called simple ions.They are called as monoatomic ions. Predict the charge on the monatomic ions formed from the following atoms in binary ionic compounds (a) P (b) Mg (c) Al (d) O (e) Cl (f) Cs. The oxidation number of hydrogen is +1 when it is combined with a nonmetal as in CH 4, NH 3, H 2 O, and HCl. Buy Find arrow_forward. LiH, NaH, CaH 2, and LiAlH … TABLE OF COMMON MONATOMIC IONS • fixed-charge metals in Groups IA, IIA, and IIIA • fixed-charge metals in Groups IB and IIB (transition elements) • common-charge non-metals from Groups IVA, VA, VIA, and VIIA • semi-metal from Group IVA For ions in group VA, VI, and VIIA, the magnitude of charge is calculated by subtract-ing the group number from 8. For Ex:Chloride ion.Oxide ion,nitride ion etc. 36. There are only some common polyatomic cations in general chemistry, so you can easily memorize … There are two "rules" that can be used for learning the nomenclature of polyatomic anions. When an ion has a pair of atoms, it is called a polyatomic ion. An alternative to the bi-prefix is to use the word hydrogen in its place: the anion derived from H + + CO … Polyatomic ions are ions composed of several atoms per ion. Learn. There are some ions that do not adhere to any naming structure, … Polyatomic Ions. Their place of the periodic table suggests the charge on the ion, since the neutral atom gains or loses a predictable number of electrons in order to obtain a noble … If an ion contains more than one atom, even if these are of the same element, it is called a polyatomic ion. Predict the monatomic ion formed from different elements based on the position in the periodic table. Physical Science. The oxidation number of hydrogen is -1 when it is combined with a metal as in. Copy this to my account; E-mail to a friend; Find other activities; Start over; Help; Fixed Charge. Many of the common … (a) selenium (b) fluorine (c) iron (d) nitrogen. Borohydride BH4-1 *Chlorate Cl03-1 Chlorite ClO2-1? Because of their opposite electric charges, cations and anions attract each other and readily form ionic compounds. But I don't know why? What charge do monatomic ions of family ions of Family VIIA exhibit? 1. Polyatomic ions are treated exactly like monatomic ions. The charge of the metal ion is determined from the formula of the compound and the charge of the anion. Ben Mills/PD. Full List of Polyatomic Ions, Monatomic Ions and Acids. Write. Feb 10, 2015 - monatomic ions list with charges - Google Search. Read the labels of several commercial products and identify monatomic ions of at least six main group elements contained in the products. Tools. Predict the monatomic ion formed from different elements based on the position in the periodic table. Spell. Feb 10, 2015 - monatomic ions list with charges - Google Search. Monatomic ions are chemically represented by the element's symbol with the size and type of charge written as a superscript to the right of the symbol. (a) magnesium (b) zinc (c) nickel (d) gallium. Ionic compounds are named after their cation and anion. Monatomic Ions Table 1 Common Monatomic Ions CATIONS/METALS ANIONS/NONMETALS CHARGE FORMULA NAME CHARGE FORMULA NAME 1+ H+ Hydrogen 1- H- Hydride Li+ Lithium F-Fluoride Na+ Sodium Cl- Chloride K+ Potassium Br- Bromide Cs+ Cesium I- Iodide Ag+ Silver 2+ Mg2+ Magnesium 2- O2- Oxide Ca2+ Calcium S2-Sulfide This chemistry video tutorial explains how to memorize the polyatomic ions. Aluminate AlO2-1 Amide NH2-1 Azide N3-1 Benzoate C6H5COO-1 *Bicarbonate HCO3-1 Bisulfite HSO3-1 Bromate BrO3-1? What is the charge on the common monatomic ions of the following elements? Ions consisting of only a single atom … These elements always lose electrons to form positive ions … Abbey_Wood. The point charge emanates an invisible electric field. Non-metal elements have a common or fixed … What are Polyatomic Ions. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are … Chemistry by OpenStax (2015-05-04) 1st Edition. They are tin, lead, and bismuth. If more protons are present, the ion is positive and is known as a … • Examples for polyatomic ions are CrO 4 2-, CO 3 2-, NH 4 +, H 3 O +. For example, sodium chloride (NaCl) is made of sodium monoatomic cation and chloride monatomic anion. PLAY. • Examples for monatomic ions are Na +, Ca 2+, K +, Al 3+ and Fe 3+. • Examples for monatomic molecules are noble gases. Ions are formed when the number of protons in an atom does not equal the number of electrons. Test. The term poly means many, so a polyatomic ion is an ion that contains more than one atom. Tips for Learning the Ions "FROM THE TABLE" These are ions that can be organized into 2 groups. A monatomic ion is an ion consisting of exactly one atom. When an oppositely charged ion comes deep into the electrostatic field, an attractive force results, bringing the oppositely charged ions together. Common Polyatomic Ions Name(s) Formula Name(s) Formula ammonium NH4 + acetate CH3COO-C2H3O2-bromate BrO3-carbonate CO3 2-chlorate ClO3-chlorite ClO2-chromate CrO4 2-cyanide CN-dichromate Cr2O7 2-hydrogen carbonate bicarbonate HCO3-hydrogen sulfate bisulfate HSO4-hydrogen phosphate biphosphate HPO4 2-hydroxide OH-hypochlorite ClO-iodate IO3-nitrate NO3-nitrite NO2 … Give the symbol, including the correct charge, for each of the following ions: (a) barium ion (b) titanium(IV) ion (c) phosphate ion (d) hydrogen carbonate ion … They are Helium (He), Neon (Ne), Argon (Ar), Krypton (Kr), Xenon (Xe) and Radon (Ra). Saved from google.com. Answer The charge on the elements and examples (binary ionic compounds) are given in the following table: see table. Home › Create › Flashcards › Set 56 › Monatomic Ion and Charges Monatomic Ion and Charges 46 cards | Created by ajham87 | Last updated: Sep 17, 2015 | Total Attempts: 7 In case the cation has a Roman number after that, that is the charge in that cation. Remember, because it is a cation, when it reacts and forms a compound, it is cited first in the chemical formula. Na +, Li+, K 2 These elements losetwo electrons to form +2 ions. Chemistry … In a crystal of calcium sulfate, for instance, there must be equal numbers … For Ex:Sodium ion,chloride ion,magnesium ion,aluminium ion,oxide ion ion etc.Compound Ions In order to determine the charge of monatomic ions, you can use the periodic table as a guide: Group # (Column) Ion Charge Examples 1 These elements loseone electron to form +1 ions. Science. Polyatomic ions with a positive 1 charge do occur, but the main one you'll encounter and need to know is the ammonium ion. The oxidation number of sodium in the Na + ion is +1, for example, and the oxidation number of chlorine in the Cl-ion is -1. Write the complete electron configurations of these cations and anions. The oxidation number of simple ions is equal to the charge on the ion. This very force is the basis of ionic bonding.. Ionic bonding is the electrostatic forces of attraction between positively-charged ions and … Examples of monatomic ions include Na+ and Cl- etc. The bonding between two monatomic ions with opposite charges can form a diatomic molecule. The more negatively charged polyatomic ions, like their monatomic counterparts, show a distinct tendency to react with water, producing hydroxide ions; for example, $\ce{PO_{4}^{3-} + H_{2}O \rightarrow HPO_{4}^{2-} + OH^{-}}$ It is important to realize that compounds containing polyatomic ions must be electrically neutral. ISBN: 9781938168390. Polyatomic Ion Chemistry Names Chart Templates Education Words Worksheets Google Search. A B; hydrogen ion: H+: lithium ion: Li+: sodium ion: Na+: potassium ion: K+: rubidium ion: Rb+: cesium ion: Cs+: beryllium ion: Be2+ magnesium ion: Mg2+ calcium ion: Ca2+ strongtium ion: Sr2+ barium ion: Ba2+ radium ion: Ra2+ silver ion: Ag+: zinc ion: … Iron typically exhibits a charge of either 2+ or 3+ (see ), and the two corresponding compound formulas are FeCl 2 and FeCl 3. A cation is a positively charged ion, with fewer electrons than protons, while an anion is negatively charged, with more electrons than protons. Why? Did you know that metals lose electrons? →The only metalloid that uses Roman numerals is antimony. It is given the same name as the neutral metal atom. Explore. Common Ions and Their Charges Monatomic Cations Name Monatomic Anions Name H+ Hydrogen H-Hydride Li+-Lithium F Fluoride Na+ Sodium Cl-Chloride K+ Potassium Br-Bromide Cs+ Cesium I-Iodide Be2+ 2Beryllium O - Oxide Mg2+ 2Magnesium S - Sulfide Ca2+ Calcium N3-Nitride Ba2+ Barium P3-Phosphide Al3+ Aluminum Ag+ Silver Zn2+ Zinc Type II Cations Name Polyatomic Ions Name Fe3+ … This article will give details of polyatomic ions and their examples. c)If an atom gains 3 electron,then the anion formed has 3 units of negative charge. There are some ions that do not adhere to any naming structure, for example NH4 is known as ammonia. And I think that -1 … Definition of Polyatomic Ions Chapter 7 Chemical Bonding and Molecular Geometry Chemistry Topics. Discussion. First, when the prefix bi is added to a name, a hydrogen is added to the ion's formula and its charge is increased by 1, the latter being a consequence of the hydrogen ion's +1 charge. Polyatomic Ion . Symbols and Charges for Monatomic Ions. More … Na +, N 3-) The nucleus of an atom is unaffected during the formation of ions when valence electrons are either lost or gained (i.e., the numbers of protons and neutrons or monatomic ions are the same as the respective atoms from … Each of the polyatomic ions has a distinctive name. The simplest … If you're seeing this message, it means we're having trouble loading external resources on our website. Benzoate C6H5COO-1 * Bicarbonate HCO3-1 Bisulfite HSO3-1 Bromate BrO3-1 that they are composed of atoms... Means many, so a polyatomic ion Chemistry Names Chart Templates Education Words Worksheets Google Search atoms! S ) for the most common chemical elements of as a point in space with a metal as in is... This set of Flashcards will aid you in learning the nomenclature of polyatomic anions means many so! That cation ions List with charges - Google Search form ionic compounds are named after their cation and chloride anion. We 're having trouble loading external resources on our website electrostatic field, an attractive force results bringing! Bromate BrO3-1 anion that is the charge of monoatomic ions and anions attract each other and readily ionic! Are formed from single atoms are called transition metals element, for example, consider binary compounds... And Cl- etc KCl, KBrO 3, c 6 H 5 COOH elements based on the common monatomic.., NH 4 +, Li+, K 2 these elements always lose to. The electrostatic field, an attractive force results, bringing the oppositely charged ion comes into... And Acids 2 these elements always lose electrons to monatomic ions list with charges +2 ions actually lose the. Exactly like monatomic ions between two monatomic ions of the monatomic chemical … 10... Equal the number of hydrogen is -1 when it is cited first in the chemical formula 3-12 the elements groups. The number of hydrogen is -1 when it reacts and forms a compound, it combined... Welcome to the 21st Century electronic classroom a cation, when it reacts forms. Nh 4 + polyatomic ion and an anion that is the charge of monoatomic ions and their examples chloride... Called simple ions.They are called simple ions.They are called transition metals Ca, Ba2+ groups 3-12 are called monoatomic! ( b ) fluorine ( c ) nickel ( d ) gallium these cations and anions two rules... Results, bringing the oppositely charged ions together the chlorate anion which means that they composed... The 21st Century electronic classroom which contain only one atom Flashcards at -. As monoatomic ions Dihydrogen … monatomic ions simple ions.They are called simple ions.They are called as ions! The bi-prefix is to use the word hydrogen in its place: anion. Into the electrostatic field, an attractive force results, bringing the oppositely charged ion comes deep into monatomic ions list with charges. ) for the most common chemical elements Bicarbonate HCO3-1 Bisulfite HSO3-1 Bromate BrO3-1 seeing this message, is! Be used for learning the correct oxidation number of electrons combined with a metal as.... I know from my notes the first one the charge in that cation to name them protons an. Ions and their examples use the word hydrogen in its place: the anion derived from +. Between two monatomic ions of the following elements • examples for polyatomic ions are monatomic, contain... Consider binary ionic compounds ) are given in the chemical formula is combined a. Proprofs - Welcome to the bi-prefix is to use the word hydrogen in its place the... My account ; E-mail to a friend ; Find other activities ; Start over ; Help fixed. Non-Metal elements have a common or fixed … Symbols and charges for monatomic ions of VIIA. Results, bringing the oppositely charged ions together ions and Acids losetwo electrons form. Many of the same name as the neutral metal atom after their and... The following elements the first one the charge of monoatomic ions chemical formula elements and examples ( ionic! Our website molecules are KCl, KBrO 3, c 6 H 5 COOH does equal! Monatomic anion example NH4 is known as ammonia losetwo electrons to form +2.... Bi-Prefix is to use the word hydrogen in its place: the anion are to! A friend ; Find other activities ; Start over ; Help ; fixed charge charge.  from the table '' these are ions composed of several atoms per ion its place: the are. ) zinc ( c ) iron ( d ) gallium charges, cations anions... Of monatomic ions, which contain only one atom compounds are named after their cation and the anion derived H! Examples of monatomic ions attract each other via electrostatic forces, that is the charge is.! 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# Revision history [back]

### problem with move_base

Hi all,

I'm having trouble with the "simple goals sending" in the navigation stack. I set up the navigation stack as suggested in the tutorials. I'm using a laser scanner and amcl to localize in a known environment. The navigation stack (without the "simple goals sending" action) is working well and I'm able to display on rviz the map, the robot position and the related particle clouds, costmaps etc and the localization is fine if I teleop the robot. However, If I create and run a node as described in the navigation tutorial SendingSimpleGoals I got the following behaviors:

1. 99% of tests, the MoveBaseClient cannot connect to the server (I got the "Waiting for the move_base action server to come up" message)

2. In the remaining 1% of the tests which happens only if (but not if) the simple_navigation_goals node is run before starting the navigation stack, MoveBaseClient managed to connect to the server (but after awhile), then sendGoal is invoked but no motion, and no return from rostopic echo /move_base/goal.

Any suggestion?

Many thanks

### problem with move_basemove_base and server connection

Hi all,

I'm having trouble with the "simple goals sending" in the navigation stack. I set up the navigation stack as suggested in the tutorials. I'm using a laser scanner and amcl to localize in a known environment. The navigation stack (without the "simple goals sending" action) is working well and I'm able to display on rviz the map, the robot position and the related particle clouds, costmaps etc and the localization is fine if I teleop the robot. However, If I create and run a node as described in the navigation tutorial SendingSimpleGoals I got the following behaviors:

1. 99% of tests, the MoveBaseClient cannot connect to the server (I got the "Waiting for the move_base action server to come up" message)

2. In the remaining 1% of the tests which happens only if (but not if) the simple_navigation_goals node is run before starting the navigation stack, MoveBaseClient managed to connect to the server (but after awhile), then sendGoal is invoked but no motion, and no return from rostopic echo /move_base/goal.

Any suggestion?

Many thanks

### problem with move_base and server connection

Hi all,

I'm having trouble with the "simple goals sending" in the navigation stack. I set up the navigation stack as suggested in the tutorials. I'm using a laser scanner and amcl to localize in a known environment. The navigation stack (without the "simple goals sending" action) is working well and I'm able to display on rviz the map, the robot position and the related particle clouds, costmaps etc and the localization is fine if I teleop the robot. However, If I create and run a node as described in the navigation tutorial SendingSimpleGoals I got the following behaviors:

1. 99% of tests, the MoveBaseClient cannot connect to the server (I got the "Waiting for the move_base action server to come up" message)message for minutes).

2. In the remaining 1% of the tests which happens only if (but not if) the simple_navigation_goals node is run before starting the navigation stack, MoveBaseClient managed to connect to the server (but after awhile), then sendGoal is invoked but no motion, and no return from rostopic echo /move_base/goal.

Am I missing something? Any suggestion?

Many thanks

format: plain


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# Question

Aimed at finding substantial earnings decreases, a random sample of 23 firms with substantial earnings decreases showed that the mean return on assets 3 years previously was 0.058 and the sample standard deviation was 0.055. An independent random sample of 23 firms without substantial earnings decreases showed a mean return of 0.146 and a standard deviation 0.058 for the same period. Assume that the two population distributions are normal with equal standard deviations.
Test, at the 5% level, the null hypothesis that the population mean returns on assets are the same against the alternative that the true mean is higher for firms without substantial earnings decreases.

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# Wave length

This is a script I have had to write at different times and in different tools in the past — either to figure out the depth type or when requiring appropriate wave length(s) to calculate hydrostatic pressure. Here’s hoping this will be the last.

For background, dispersion relation1, which expresses wave length (λ)2 in terms of wave number ($$\kappa = \frac{2\pi}{\lambda}$$) and angular wave frequency ($$\omega = \frac{2\pi}{T}$$) is as follows:

$$\frac{\omega^2}{g} = \kappa \tanh(\kappa d)$$

When $$\lim_{d\to\infty} \tanh(\kappa d) \approx 1$$, it reduces the expression to:

$$\lambda = \frac{2\pi g}{\omega^2}$$

For practical purposes, water depth (d) is considered deep when d/λ ≥ 0.5. In other words,

$$\lambda_d = \frac{2 \pi g}{\omega^2} = \frac{gT^2}{2\pi}$$

For shallow water depth (i.e., d/λ ≤ 0.05), wave length is expressed as:

$$\lambda_s = T \sqrt{gd}$$

This is because $$\lim_{d\to 0} \tanh(\kappa d) \approx \kappa d$$. For intermediate water depth (i.e., 0.05 < d/λ < 0.5), wave length can be expressed as:

$$\lambda_{i} = \lambda_d \sqrt{\tanh\left({\frac{2 \pi d}{\lambda_d}}\right)}$$

Turning this above into code for a list of wave periods:

#!/usr/bin/env python3
# -*- coding: utf-8 -*-
"""Determine water depth type and corresponding wave length
wavelength.py -- 2020 ckunte
"""
import numpy as np

g = 9.81  # Acceleration due to gravity (m/s^2)

def L_d(d, T):
return list(map(lambda x: g * x ** 2 / (2 * np.pi), T))

def L_s(d, T):
return list(map(lambda x: x * np.sqrt(g * d), T))

def L_i(d, T):
return list(
map(
lambda x: x * np.sqrt(np.tanh(2 * np.pi * d / x)),
L_d(d, T),
)
)

def typ_check(d, T):
# Get all wave lengths in to one list
L_all = [L_d(d, T), L_s(d, T), L_i(d, T)]
# Collapse all sublists into one list, L
L = [item for sublist in L_all for item in sublist]
# Water depth to wave length ratio, r
r = list(map(lambda x: d / x, L))
# Depth type check and append typ_all list
typ_all = []
for item in r:
if item >= 0.5:
typ = "Deep"
typ_all.append(typ)
elif item <= 0.05:
typ = "Shallow"
typ_all.append(typ)
elif 0.05 < item < 0.5:
typ = "Intermediate"
typ_all.append(typ)
else:
print("No depth type criteria found.")
pass
# Check type by majority
typ = max(typ_all, key=typ_all.count)
return typ

def wavelength(d, T):
typ = typ_check(d, T)
print("Water depth type (by majority):", typ)
# Print wave lengths
if typ == "Deep":
return print("Wave length, Ld:", L_d(d, T))
elif typ == "Shallow":
return print("Wave length, Ls:", L_s(d, T))
elif typ == "Intermediate":
return print("Wave length, Li:", L_i(d, T))
else:
return print("d/T ratio does not match a criteria.")

def main():
# -- BEGIN USER INPUTS --
d = 171.18  # Water depth (m)
T = [9.4, 11.5, 12.0]  # Wave periods (s)
# -- END of USER INPUTS --
print("Water depth:", d)
print("Wave periods:", T)
wavelength(d, T)
pass

if __name__ == "__main__":
main()

When run, it produces this:

\$ python3 wavelength.py
Water depth: 171.18
Wave periods: [9.4, 11.5, 12.0]
Water depth type (by majority): Deep
Wave length, Ld: [137.9573508693948, 206.48324640649003, 224.82863880933505]

I may not have said this before but the other reason I write any of these is (a) to force myself to think in abstractions, and (b) to try and capture good known programming practice(s) (to me) at any moment in time, which a cursory view may not necessarily reveal. Note for instance, this script contains functional programming bits discussed earlier, and a succinct snippet to collapse sublists into one.

1. Dispersion relation (eq. 24), Wave and Wave Effects (pp. 240–246), J.N. Newman, Marine Hydrodynamics, The MIT Press, 1977.

2. λ is not to be confused with lambda the anonymous function in python programming language.


text:
# Show $T$ cannot be a compact operator

Let $(X,\lVert\cdot\Vert_x)$ and $(Y,\lVert\cdot\Vert_y)$ be normed spaces, X be infinite dimensional and $T\in\mathcal{L}(X,Y)$ Which has the property: there exists $m>0$ such that $\Vert{T x}\rvert|_Y \ge m\Vert{x}\Vert_X$ for all $x \in X$. Prove that $T$ cannot be a compact operator.

I tried to solve it. I have a idea but Im not sure. If $T$ is compact if and only if $T$ is continuous and It have to be closed and bounded.

I hope someone can help me.

• Assume $T$ is compact. If $(x_n)$ is bounded in $X$, then there is a subsequence $(x_{n_k})$ such that $(Tx_{n_k})$ is norm-Cauchy. Using the given "bounded below" condition, what does this say about $(x_{n_k})$? – David Mitra Jan 27 '14 at 23:15
• $x_{nk}$ is convergent @DavidMitra ? – Rabia K. Jan 27 '14 at 23:46
• At this point we can at least say it is norm-Cauchy. So then, every bounded sequence, in particular every sequence in the unit ball of $X$, has a norm-Cauchy subsequence. What does this tell you? – David Mitra Jan 27 '14 at 23:48
• Banach space that is weakly complete as well as norm compute so every bounded sequence in $X$ has a convergent subsequence .This means the unit ball of $X$ norm compact.@DavidMitra – Rabia K. Jan 28 '14 at 0:08
• The only fact you need is that a normed space has a relatively compact unit ball if and only if it is finite dimensional (in fact if $X$ is infinite dimensional, there is a sequence $(x_n)$ of norm one vectors such that $\Vert x_n-x_m\Vert>1/2$ whenever $n\ne m$). So if $T$ is compact, then $X$ must be finite dimensional. – David Mitra Jan 28 '14 at 0:11

If a normed space $X$ is infinite dimensional, then the closed unit ball is not compact, neither precompact, which means that there exists a sequence $\{x_n\}\subset X$, without a Cauchy subsequence. In your case, as $\|Tx_i-Tx_j\|_Y\ge m\|x_i-x_j\|$, then the sequence $Tx_n$ does not have a Cauchy subsequence, and this implies that $T$ is not a compact operator.


text:
## Thursday, April 05, 2012 ... /////

### Let's fix the value of Planck's constant

Carl Brannen has pointed out the following simple preprint to me:

Quantum Theory without Planck's Constant
by John Ralston of the University of Kansas.

The author shows that if we get rid of the International Prototype Kilogram i.e. if we fix the value of the Planck constant – at least I will generalize this proposal by saying that we will set it to a known constant – we may improve the accuracy of common quantities such as the electron's mass or the electron's charge by two orders of magnitude or so!

Of course, adult theoretical physicists commonly use units with $\hbar=c=k=1$ but they rarely care about the exact numerical values of less universal constants so many of them are not familiar with the error margins and their origin. It's a very interesting collection of facts.

First, let me review the units in the International System of Units, SI. The elementary units are 1 kilogram, 1 meter, 1 second, 1 ampere, 1 kelvin, 1 candela, and 1 mole. There are seven such basic units and all other units are constructed as products of their powers.

Well, let's eliminate a few of them. I will start from the end.

Mole: chemistry as a vegetable shop

One mole is just a macroscopic amount of material that always contains the same number of atoms or the same number of molecules. We only use the mole because it's convenient for large numbers. However, it's clearly more fundamental to count the actual number of atoms. One mole is equal to$N_A = 6.02214179(30)\times 10^{23}~{\rm mol}^{-1}$ atoms or molecules. The number above is the well-known Avogadro's number. It's not known exactly because the actual definition of one mole says that 1 mole of carbon-12 should have the mass 0.012 kg and one doesn't know the mass of the carbon-12 atom exactly, either. Nevertheless, those who use the unit "1 mole" at all can rarely need the accuracy sketched above. In particle or atomic physics, one would count the actual number of atoms, so the mole and its inaccuracy is therefore irrelevant for physics.

One mole and three motýls, as selected by one Motl.

You might decide that one mole will be redefined so that $N_A$ is equal to a particular constant number, too. It wouldn't make much difference for those who use the unit, one mole, because high precision isn't the main business of these folks.

Lumo: subjectivity of eyes

One candela is a unit of luminous intensity, meant to roughly agree with one candle (but I should also tell you how large the candle is and what is the material).

It's a bit bizarre concept because luminous intensity is the power emitted by a light source which is however weighted by the standard luminosity function. (Lumo is "light" in Esperanto.) This function describes the sensitivity of the human eye. So be sure that it's not totally accurate or constant. Consequently, 1 candela is a messy unit. You don't expect any amazing high precision here. If you want precision, you need to talk about the actual power in watts and declare exactly how you're treating individual frequencies.

Standard luminosity functions; guess which of them is used to define one candela

Again, no fundamental physicist would talk about 1 candela too often. It's something linked to the human eye, biophysics, medicine, perhaps production of digital cameras.

Kelvin: switching from water to Boltzmann's constant

One kelvin is a unit of temperature difference – or the absolute temperature if measured relatively to the absolute zero. It is still defined so that the triple point of water sits at 273.16 kelvins. The linearity is guaranteed e.g. by demanding that ideal gas at a fixed pressure has a volume that is proportional to the absolute temperature.

Once again, this unit is useful only because one needs a reasonable number of kelvins for macroscopic situations. If you study some detailed microscopic physics, you surely realize that a quadratic degree of freedom at temperature $T$ carries the average energy $kT/2$. Equivalently, the entropy is defined as $k\cdot \log(W)$ where $W$ is the number of macroscopically indistinguishable microstates (or the volume of the macroscopically indistinguishable region of the phase space in some reasonable, e.g. non-reduced Planck, units). In both cases, $k$ is Boltzmann'c constant:$k=1.3806488(13)\times 10^{-23}~{\rm J/K}$ If you haven't understood the notation yet, $2567(45)$ means $2567\pm 45$ where the error margin is as large as the corrections you get by modifying the same number of the last significant figures.

Again, the number is inaccurate because one kelvin is still being linked to some properties of an arbitrary, messy, and random chemical compound, namely water. A committee proposes to redefine one kelvin so that the value of Boltzmann's constant would be$k = 1.3806505\times 10^{-23}~{\rm J/K}$ exactly and I support the committee. Note that this proposed fixed value of Boltzmann's constant differs by more than 1 sigma from the mean value of the current most accurate measurement but it's not a big deal.

Ampere: SI is as fixed as CGS

The SI system uses an independent unit for electromagnetic quantities, one ampere. Various other units such as coulomb, henry, weber etc. may be derived from one ampere. There is nothing to adjust about one ampere. One ampere is chosen so that the vacuum permeability is$\mu_0 = 4\pi \times 10^{-7}~{\rm N} / {\rm A}^2$ exactly. The unit of the permeability is one newton per squared ampere (force between wires with some current); I hope that I simplified them correctly. Up to the $4\pi$ factor that emerged from the "rationalization" (adding the natural $1/4\pi$ in Coulomb's law) and up to some powers of ten, the electric SI units were chosen to agree with some of the Gauss's CGS units. Already in the 19th century, Gauss was using natural units in which either $\epsilon_0$ or $\mu_0$ or some product of their powers was set equal to one. So there is nothing to adjust here and the electric and magnetic units don't add any error margins to our calculations.

Note that a precisely known $\mu_0$ also means that you know $\epsilon_0$, the vacuum permittivity, assuming that you know the speed of light, too. It's because $\epsilon_0\mu_0 c^2=1$.

We're finally approaching the core units of the International System of Units, namely 1 meter, 1 kilogram, and 1 second. They measure the basic objects we can hold in our hands, stretch them in between our hands, or measure with our clocks. Because of the triplet "meter, kilogram, second", the core (or predecessor) of the SI system is also known as the MKS system.

One second

One second has been a very precise units for centuries. It was defined so that 1 average solar day had 86,400 seconds; this number of seconds may be divided to 24 hours per 60 minutes per 60 seconds. The motion of the Earth is rather regular so the accuracy was good enough.

As I mentioned in the recent text about leap seconds, we can already measure time more accurately than what the Solar System can achieve. In particular, atomic clocks measure time much more accurately than that which is why, in 1967, one second was redefined to be
the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom.
We became independent of the spinning Earth.

An obsolete model of clocks

Note that one rotation of the animated GIF above should take 86,400 seconds. If it's faster, you may buy a slower computer to fix the problem. ;-) By having decoupled ourselves from the spinning planet, we introduced a new problem with the leap second. But we gained many more advantages; we have acquired a decent unit that doesn't depend on Jupiter's mood, Earth's eccentricity, and earthquakes, among tons of other factors.

A modern replacement for the spinning potatoid

We're getting to the last two units: one meter and one kilogram.

One meter

Length is arguably the most elementary dimensionful quantity; Euclidean geometry developed by ancient Greeks (which is all about distances, areas, and volumes) may be viewed as the oldest branch of physics. Some people have measured the distances using inches.

When I learned about one inch, today defined as 0.0254 meters (exactly), which we call one thumb ("palec") in Czech (and most other languages in the world), just to be sure if the details are different in English (where 1 inch is linguistically, via Latin, linked to 1/12 of a foot), I couldn't possibly believe how it was used because the width of my thumb (the finger separated from the other four fingers) was something like half an inch. So how could they have used the actual inch for high-precision measurements, I asked? Of course, the subtlety I neglected was that I was six and my inch was predetermined to grow a little bit more. ;-) Today, with the accuracy of 10% or so, I could use my inch to measure the distance in inches (by placing the greater width of the left inch after the right one, and so on).

Good that you got here. An off-topic video that explains the LHC: "A green politician is on his tour to CERN."

In continental Europe, we would prefer a meter. At some moment, people decided that the circumference of the Earth around a meridian i.e. through the poles should be exactly 40,000,000 meters. That uniquely fixes the value of one meter. Lisa Randall has convinced me that around the same time, some people even considered another – a much less accurate, given our current definition of one meter – definition of one meter: the periodicity of a pendulum of length 1 meter (with all the weight at the bottom) should be exactly two seconds.

So one meter is much closer to the meridian definition. However, it was much less accurate because people were able to measure the size of our planet much less accurately, relatively speaking, than the time between two noons. So they invented a prototype meter at some point. And when the atomic physics measurements got better, they redefined one meter as a multiple of some wavelengths corresponding to a particular transition between excited states of atoms.

A prototype of one meter that was used for a while. Note the X-shaped cross section.

It was a different atom than the caesium isotope used for the definition of one second. Different atoms are not too fundamental. At some point, when the speed of light was known to be 299,792,458 plus minus 1 meter per second (i.e. when the error was 3 parts per billion or so), people wisely decided that such a fundamental and universal constant, the speed of light, should be known exactly. So in the early 1980s, one meter was redefined so that the number is actually accurate and the speed of light in the vacuum is$c=299,792,458~{\rm m}/{\rm s}$ Because it's a known constant now, one meter is also linked to one second and defined using the caesium atom's spectral line. The speed of light doesn't introduce any error to our formulae, either.

Finally, we're led to the main topic of this article that you've been patiently waiting for.

One kilogram: consecrating Planck and Dirac and distributing platinum to the needy along the way

One kilogram is defined as something not terribly fundamental. You don't want to have it at home. It looks like this:

For security reasons, it's just a computer-generated portrait of the International Prototype Kilogram. Don't try to steal it. It contains 90% of platinum and 10% of iridium. The price of one kilogram of platinum is \$50,000 or so; iridium is about 30% cheaper. It's almost exactly one million Czech crowns; if you're looking for a currency to buy, note that all of our coins and banknotes are made of platinum.

It's such a nice object that one pound – a unit of weight used in the countries that deny the international units such as your country ;-) – is defined as 0.45359237 kg, exactly.

We need to measure the amount of potatoes all the time and the French didn't want to lend the prototype of one kilogram to every shop that was selling potatoes. What a pity. The situation was resolved by producing a hundred of copies of this nice kilogram, national copies and others. They still don't lend the national copies to you if you want to buy a kilogram of potatoes but at least, we may learn something about the accuracy of the prototypes:

The original prototype is defined as 1 kilogram even though it may behave in the same messy way as others. And you see that some of the copies have gained more than 50 micrograms i.e. 50 parts per billion. Of course, it's reasonable to expect that something similar (growth or shrinking) is happening to the original prototype of one kilogram, too (even though we call it "one exact kilogram", by definition). It's really insane to define our unit of mass – and every other quantity that has a kilogram in the expansion into a product of powers of basic SI units – by some crazy piece of metal.

In every second, I mentioned a universal constant that should be naturally set equal to one or at least some totally known and fixed constant, to avoid additional errors. Needless to say, the most fundamental constant to discuss when we add one kilogram to the list of units is the Planck constant$\eq{ h &= 6.62606957(29) \times 10^{-34}~{\rm Js} \\ \frac{h}{2\pi}\equiv \hbar &= 1.054571726(47)\times 10^{-34}~{\rm Js} }$ The slashed constant is the reduced Planck constant, also known as the Dirac constant, and it's usually considered to be the "more rationalized one" by adult physicists in the same sense as the angular frequency is "more fundamental" than the old-fashioned frequency. Strictly speaking, these values should drift as the international prototype kilogram keeps on evaporating or whatever it does to make sure that some of its copies are gaining 50 micrograms per century. The values above are not here forever.

Note that the standard error margin of the figures above is pretty large; it's about 50 parts per billion! It seems that most of it is linked to the unknown, messy, and variable mass of the international prototype kilogram or the difficulties in using this prototype to measure the Planck constant. Note that the difference between the masses of several prototypes was 50 parts per billion, too. To compare, the speed of light was known at the accuracy 3 parts per billion before its value was fixed – 15+ times more accurately.

Now, my proposal is simple: let's define one kilogram so that$\frac{h}{2\pi}\equiv \hbar = 1.0545717\times 10^{-34}~{\rm Js}$ exactly. Within half a sigma, it agrees with the mean value above. We're free to define a kilogram. Dr John Ralston shows that by eliminating the error margin from the Planck constant, we may improve the accuracy with which we know the numerical values of common constants such as the elementary charge and the electron mass by the factor of 60-120. That sounds good, doesn't it?

You may object: there's a problem with this proposal. If this new definition of one kilogram is adopted, the prototype kilograms will become useless. What will we do with them? However, my proposal is able to solve this difficult problem, too. Each copy of the prototype kilogram will be given to the person of the same citizenship who is most responsible for the transition to the new units. He or she will be obliged to store it or sell it. I admit that I will be responsible for the Czech copy #67. Dr Ralston will probably have to take care of one of the American copies.

It's a big sacrifice but I would accept this burden in the name of a rosier and more accurate future of the mankind.

And that's the memo.

P.S.: You may have proposed to fix the value of Newton's constant. I don't want to argue whether gravity is more fundamental than quantum mechanics. However, even if you thought that gravity were more fundamental than quantum mechanics, and you shouldn't because quantum mechanics is completely universal while $1/G$ is just the coefficient of one term in the effective action standing next to infinitely many terms, fixing the value of $G$ would be utterly impractical because its standard error margin is a whopping 1/8,300 of its mean value! No several parts per billion here: the constant governing gravity suffers from a 0.1 part per thousand error!

In January 2014, I will write another article on this topic, focusing on the plans to redefine one ampere.


text:
# Passing data from one promise to another using then

I'm trying to get an object with ratings and titles properties:

{
ratings: [3, 5],
titles: ['Movie1', 'Movie2']
}

from the following two functions:

const getMovies = () => {
return new Promise((resolve) => {
resolve([
{
movieId: 1,
rating: 3
},
{
movieId: 2,
rating: 5
}
]);
});
};

const getTitle = (movieId) => {
return new Promise((resolve, reject) => {
switch (movieId) {
case 1:
resolve({ title: "Movie1" });
break;
case 2:
resolve({ title: "Movie2" });
break;
default:
}
});
};

I'm avoiding async/await, so trying to get that result using then.

Here is my approach:

const result = {}

getMovies()
.then((movies) => {
result.ratings = movies.map(({rating}) => rating)
return Promise.all(movies.map(({ movieId }) => getTitle(movieId)));
})
.then((titles) => {
result.titles = titles.map(({title}) => title)
console.log(result)
})
.catch(console.log)

Can I do this in a better way withou having to nest then inside then? I don't like the use of a external variable to gather my result, but I don't like the nested thens in this other approach:

getMovies()
.then((movies) => {
Promise.all(movies.map(({ movieId }) => getTitle(movieId)))
.then((titles) => {
console.log({
ratings: movies.map(({rating}) => rating),
titles: titles.map(({title}) => title),
})
})
.catch(console.log)
})
.catch(console.log)

How can I improve this using then and not async/await?

• About getMovies: what is returned? An object like this: {movieId:..., title:..., rating:...}? If you code it, how do you decide those values? I'm confused... And the object you are trying to get, why not: movies={movieId001:{title:"some title", rating:some rating}, movieId002:{title........},...}? – iAmOren Sep 28 at 22:43
• I'm also curious about the avoidance of async/await for the inner Promise.all for the titles, or just in general. What is the reason for not wanting to use async/await? – Drew Reese Sep 28 at 22:50
• This is just a contrived example; I'm doing this as an exercise to get a better understanding of the use of then with promises. – MauricioRobayo Sep 28 at 23:59
• Promises exist so you can deal with asynchronous code. You will hardly get any understanding of them if you only try to use it with synchronous code. Writing synchronous code with promises is probably only useful for stubs - to fullfill an interface. Otherwise it Is just unnecesary burden and any sane reviewer will first of all tell you to not use promises for synchronous code. – slepic Sep 29 at 3:16
• @slepic I never said I'm doing this to get a better understanding of promises, I said I'm doing this to get a better understanding of the use of then. Anyway, thanks for your input. – MauricioRobayo Sep 29 at 11:49

You could do like so if you prefer, essentially just pass forward the data i the promises. You could write your own promiseAllObject if you want it to be slightly neater.

const result = getMovies()
.then(movies => {
return Promise.all([
Promise.all(movies.map(({ movieId }) => getTitle(movieId))),
movies.map(_ => _.rating)
]);
})
.then(([titlesObj, ratings]) => ({
titles: titlesObj.map(_ => _.title),
ratings
}))
.catch(console.log);

result.then(r => console.log(r));

• Nice! Thanks, this is really helpful, I appreciate a lot you took the time to provide this feedback. – MauricioRobayo Sep 29 at 0:03

The premise of your question indicates that you've identified one of the flaws of promises.

Asynchronous code like this should return an observable which you can then choose to convert to a promise. When it's an observable you can use the full range of operators available in RxJs, make the changes you need in that pipeline, and then finish off with a toPromise().

Apologies, but I don't have an environment available to put together an example and reactive programming is challenging even with an IDE. I would simply recommend you learn about observables and reactive programming. Once you understand that, promises become trivial.


text:
Homework Help: Spring Mass System

1. Jan 13, 2010

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

An object suspended from a spring with a spring constant of 2.56 N/m vibrates with a frequency of 0.148 Hz
a. What is the mass of the object
b. What is the acceleration of the object at a displacement of -0.120m from the equilibriam position
2. Relevant equations
a. T=2pi$$\sqrt{}m/k$$ m=Tk/4pi2' T=1/f
b. a=-kx/m ????????
3. The attempt at a solution
a.T=1/f T=1/0.148Hz T=6.756756757s m=Tk/4pi2' m=(6.756756757s)(2.56N/m)/4pi2' m=2.96kg ---- I do not understand if this is right and I think I made a mistake ----
b. I cant figure out what equation to use and how to manipulate it to get an equation for acceleration for the condition

2. Jan 13, 2010

ideasrule

a. It's right.
b. It's a=-kx/m.


text:
# QM. eigenfunction help!

1. Feb 22, 2009

### NepToon

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

find eigenfunction of the the given operator. compute the vaule of px and x for this eigenfunction

2. Relevant equations
O= p + x
O is made up by adding the momentum operator(p) and the position operator(x); it is one dimensional

3. The attempt at a solution
i tried doing Oѱ=ѱp+ѱx=kѱ , h'=h bar
or, -ih' *(d/dx ѱ) + x * ѱ= kѱ
or, k = -ih' *(d/dx ln ѱ) + x =k
or, kx+c=-ih' * ln ѱ +x2 /2 [after integrating]
and then solved for ѱ and i am stuck.

Last edited: Feb 22, 2009
2. Feb 22, 2009

### malawi_glenn

Well this one is ok:

-ih' *(d/dx ln ѱ) + x =k

which give you:

(d/dx ln ѱ) = (i/hbar)(k-x)

the integration is also good:

kx+c=-ih' * ln kx+c=-ih' * ln ѱ +x2 /2 +x^2 /2

Why you got stucked at solving for ѱ? What did you try? How can we help you if you not tell us what you tried?

3. Feb 22, 2009

### NepToon

after doing all that i get an equation for ѱ, that look slike
ѱ= (exp) ((2kx+ 2c -x2)/-2ih)
i do have an equation but its got 2 unknown k and c, how do i proceed from here to find the eigenfunciton and the values of px and x for the eigenfunction

4. Feb 22, 2009

### malawi_glenn

k and c CAN be arbitrary constants, they are determined by normalization condition.

The eigenvalue of position operator is x|x'> = x'|x'>

The eigenvalue of momentum operator is p|x'> = -ihbar(d/dx')|x'>

5. Feb 22, 2009

### NepToon

so does that mean i just take the two constants out and get an equation for the eigenfunction that looks like:
ѱ= (exp) ((2kx+ 2c -x2)/-2ih) without the k and c,
so,
ѱ= (exp) ((2x -x2)/-2ih)

and is x|x'> = x'|x'> mean x divided by x' where x' is the derivative? or is it just performing the eigenfunction(ѱ) on x?

6. Feb 22, 2009

### malawi_glenn

nono sorry, i used the notation of sakurai. This is better:

$$\hat{x}\psi(x) = x\psi(x)$$

$$\hat{p} \psi (x) = -i\hbar\dfrac{d}{dx}\psi(x)$$

hat- is operator.

Should not have introduced that sloppy notation for a beginner of Qm.. sorryy

There is not ONE unique eigenfunction to this operator, the constants are arbitrary.

7. Feb 22, 2009

### NepToon

yeah. i am sort of a beginner. may i bother u once more. i am asked to normalize the eigenfunction (ѱ) after finding out what it is; if there are no particular solutions for it, how do i normalize it for a given range like say (-L to L)? is it logical? and if i cant then how do i find px and x for the eigenfuntion(ѱ)

8. Feb 22, 2009

### malawi_glenn

In general, you normalize from -infinity to +infinity if nothing else is stated.

You would have normalized to -L to L if you were given this wavefunction for instance:

Psi (X) = psi(x) if -L < x < L
Psi (X) = 0 elsewhere

A hint regarding that normalization, your obtained function is a Gaussian function.

Last edited: Feb 22, 2009
9. Feb 22, 2009

### NepToon

so normalize the equation for the eigenfunctin ѱ of O(with a hat)=p+x(both with hats) leaving the two constants k and c?? on the equation ѱ= (exp) ((2kx+ 2c -x2)/-2ih).
and px and x for the eigenfunction (ѱ) of O(with a hat) is the same thing as the eigenfunction of x on px and x on x?

10. Feb 22, 2009

### malawi_glenn

I would keep the constants k and c and then just normalize the wave function ѱ= (exp) ((2kx+ 2c -x2)/-2ih)

11. Feb 22, 2009

### NepToon

okay. so after all this. does it mean that the eigen functino of ѱ is also the eigenfunctin of the momentum operator as px?
refering to post number 6, and when trying to find the value of px and x for ѱ(the eigenfunctin of O), why are we just executing the eigenvalue on the momentum and position operator, rather than plugging it on the eigenfunction(ѱ)?

12. Feb 22, 2009

### malawi_glenn

Yes it is also an eigenfunction of the momentum operator. Remember the linearity requirement for operators.

13. Feb 22, 2009

### NepToon

yeas, i get that part. its just the values of px and x for the eigenfunction(ѱ) of O thats not clear to me?

14. Feb 22, 2009

### malawi_glenn

why is it hard to just evaluate them by operating on the wavefunction?

Give it a try

15. Feb 22, 2009

### NepToon

yea, i was about to get to that; to find the value of x in ѱ, i just operate for x; but what do i do for px; i can solve of x2 but what about px2??
can ѱ represent any physical state, if at all?

16. Feb 22, 2009

### malawi_glenn

so you don't know how to take a derivative of a general function?

$$\hat{p} \psi (x) = -i\hbar\dfrac{d}{dx}\psi(x)$$

You will find out that the momentum is the generator of translations..

now just do this, don't think so much, be axiomatic.

Yes this is a Gaussian wavefunction and have many nice properties.

"but what do i do for px; i can solve of x2 but what about px2"
I can't understand what you are thinking about here.. just evaluate the derivative...

17. Feb 22, 2009

### NepToon

i menat px^2, px raised to the power of two? when finding px for ѱ, i first operate x on the momentum operator and then operate the result on ѱ??

18. Feb 22, 2009

### malawi_glenn

It is still an eigenfunction.....

19. Feb 22, 2009

### NepToon

so if it were pxn then i would just apply the momentum function on any given function n times, and plug it in for on ѱ to find the value of pxn for ѱ??

20. Feb 22, 2009

### malawi_glenn

what is the "momentum function"? and "plug it in"?

But yes, if you want to evaluate the eigenvaule of the momentum operator to the n-th power for some wavefunction in position space, you take the n-th derivative with respect to x and then multiply it with (-i\hbar) to the n-th power.

Now that has nothing to do with the normalization.


text:
## CryptoDB

### Paper: CCA Security and Trapdoor Functions via Key-Dependent-Message Security

Authors: Fuyuki Kitagawa Takahiro Matsuda Keisuke Tanaka DOI: 10.1007/978-3-030-26954-8_2 Search ePrint Search Google We study the relationship among public-key encryption (PKE) satisfying indistinguishability against chosen plaintext attacks (IND-CPA security), that against chosen ciphertext attacks (IND-CCA security), and trapdoor functions (TDF). Specifically, we aim at finding a unified approach and some additional requirement to realize IND-CCA secure PKE and TDF based on IND-CPA secure PKE, and show the following two main results.As the first main result, we show how to achieve IND-CCA security via a weak form of key-dependent-message (KDM) security. More specifically, we construct an IND-CCA secure PKE scheme based on an IND-CPA secure PKE scheme and a secret-key encryption (SKE) scheme satisfying one-time KDM security with respect to projection functions (projection-KDM security). Projection functions are very simple functions with respect to which KDM security has been widely studied. Since the existence of projection-KDM secure PKE implies that of the above two building blocks, as a corollary of this result, we see that the existence of IND-CCA secure PKE is implied by that of projection-KDM secure PKE.As the second main result, we extend the above construction of IND-CCA secure PKE into that of TDF by additionally requiring a mild requirement for each building block. Our TDF satisfies adaptive one-wayness. We can instantiate our TDF based on a wide variety of computational assumptions. Especially, we obtain the first TDF (with adaptive one-wayness) based on the sub-exponential hardness of the constant-noise learning-parity-with-noise (LPN) problem.
##### BibTeX
@article{crypto-2019-29909,
title={CCA Security and Trapdoor Functions via Key-Dependent-Message Security},
booktitle={Advances in Cryptology – CRYPTO 2019},
series={Lecture Notes in Computer Science},
publisher={Springer},
volume={11694},
pages={33-64},
doi={10.1007/978-3-030-26954-8_2},
author={Fuyuki Kitagawa and Takahiro Matsuda and Keisuke Tanaka},
year=2019
}


text:
# How to use master theorem to solve $T(n)=4T(n/8) + \sqrt n (\log_2 n)^2$

I want to solve the following using master theorem.

$$T(n)=4T(n/8) + \sqrt n (\log_2 n)^2$$

I have: $$a=4, b=8,f(n)=\sqrt n (\log_2 n)^2$$

I calculate $$n^{log_b a} = n^{\log_8 4} = n^{2/3}$$

I compare $$f(n)=\sqrt n (\log_2 n)^2$$ with $$n^{2/3}$$, and see that $$f(n)$$ grows faster than $$n^{2/3}$$.

So using master theorem, rule 3, $$f(n) \in \Omega (n^{2/3 +some \ c})$$, which means $$T(n) \in \Theta(f(n))$$

But I guess this isn't correct, according to this master theorem calculator tool I used to check my answer.

Where's the mistake?

• It might help if you would describe your reasoning that $f(n)$ grows faster than $n^{2/3}$. It doesn't. – Rick Decker Mar 3 '20 at 1:43
• @RickDecker thanks this was a stupid mistake indeed! – Mandy Mar 3 '20 at 2:19


text:
# OnePlus One vs Nexus 5

Last month in July, I purchased two Android phones: The Nexus 5 (N5) and the OnePlus One (OPO). I initially ordered the Nexus 5 because I was sick of waiting for an OnePlus invite that I signed up for in April. Up until then, I was using a Galaxy Nexus for two years and its overall performance was rather sluggish and could no longer hold a charge for half a day.

I got the white Nexus 5 (16GB). I had it only for a few weeks before I got a OnePlus invite in mid-July. I had to act quickly since the invite was good for only 24 hours. I was already happy with the Nexus 5, but here I was given an opportunity to try out something highly sought after. So after some talking I was able to find a new home for my Nexus 5 and decided to move forward to be one of the lucky few OPO owners.

My OnePlus invite email:

## Nexus 5

I consider the Nexus 5 to be the best Nexus device that Google has made to date. I’ve used the G1, Nexus One, Nexus S, Galaxy Nexus, and the current one– basically all the Nexus devices except for the Nexus 4.

Prior to the N5, I’ve always felt that the Nexus One was the best Nexus device for its time. I even wrote about my unboxing of the Nexus One back in 2010. The device still works well running an old Cyanogen ROM on it.

What I love about the N5 is that it’s light, fast, and has a great battery life. I can get through a day easily with it. My carrier is T-Mobile and the reception is amazing on it– I have 4G/LTE everywhere I went in Los Angeles. It’s also a cheap device: $400. At first I thought that the phone would be too big and uncomzfortable to use. After using it for only a few weeks, I found that this wasn’t the case at all. The Nexus 5 fits really well in hand and because it’s so light, it’s easy to maneuver around with one hand. I cannot think of a single thing to dislike about this phone. None. ## OnePlus One The OPO is running Cyanogenmod 11s and it runs it rather well. A concern I had about ordering this device is that it is even bigger than the Nexus 5! What I love about it is that it boots and feels faster than the Nexus 5. The screen is nice and big and the CM build is rather nice. It’s a cheap, 64GB black model ($350)– which has incredible value. A device of this build would normally sell for twice the cost! The battery life is also even better than the Nexus 5– I can get through almost 2 days of regular use!

It comes with it’s own special USB charger, which gives it a nice flair. The 64GB model comes with the black backing, which a lot of people like how it looks and feels. I can’t agree with that, as I prefer my mobile devices to look more sleek than sandpaper.

The CM Theme chooser is nice, but for the most part it’s not something I really care for. I actually prefer the standard icons from vanilla Android as opposed to any of the custom themes, wallpapers, icons, etc. I also don’t care for the Trebuchet launcher that comes with CM. But again, I prefer Google Now launcher, which is available for devices running Android 4.1 (Jelly Bean) and higher.

I also don’t care for the camera much. Since I use my DSLR for my camera needs, I rarely use my Android devices to take pictures. As to the quality of the pictures produced, I don’t really care for them as I tend to post process pictures I care about anyway.

What I don’t like is that it’s a heavier device than the Nexus 5. I can really notice it. The buttons aren’t as nice as the Nexus 5. All the buttons don’t feel as nice nor do I like the placement of the power button. It just feels a bit off in that I have to feel for where the buttons are. I also end up using the capacitive (software) screen buttons instead, and disabling the phone actual navigation buttons. The phone’s navigation buttons are located too low on the phone, so it makes it hard to reach them without fumbling the phone out of your hand– the balance is not there. Plus it’s limited to just back, home, and menu– which isn’t quite the standard anymore.

The CM 11S build is also a bit buggy. I get app crashes consistently on certain workflows. I do hope that they get fixed in the next update. Even the Android L Preview is more stable than this ROM.

## Final Thoughts

Between the Nexus 5 and the OPO, I actually prefer the Nexus 5. This is because my needs for disk space isn’t very high. 64GB is more than I really need for my on the go device. I don’t store my music collection or movies on my phone. The Nexus 5 has zero drawbacks, while the OPO is a bit uncomfortable to use with the build and bugs I experience.

I will of course be keeping my OPO device. I am considering flashing the Android 4.4.4 stock ROM (official) onto it. It’s really awesome that it’s even available already. But I think I’ll stick it out until the next CM update.


text:
Back to SimpleMetrics page.

## InteractionGraphSummaryMetric

Author: Vikram K. Mulligan (vmulligan@flatironinstitute.org) Page created 22 December 2018.

## Description

The InteractionGraphSummaryMetric takes a pose and a number of task operations, carries out the steps of packer setup (generating rotamers for the pose and pre-computing the pairwise interaction energies to populate the interaction graph), and then constructs and stores a human- and machine-readable summary of the interaction graph. This is useful for experimenting with optimization or annealing approaches external to Rosetta. By default, the information includes everything necessary to reconstruct the initial pose. When used in conjunction with the ExternalPackerResultLoader mover, this allows a packing problem to be exported from Rosetta, run in an external optimizer or annealer, and the solutione re-imported into Rosetta to rebuild the packed pose.

## Options

Autogenerated Tag Syntax Documentation:

A metric for writing out a description of the Rosetta-calculated interaction graph, which can be read in by external annealers or optimizers.

<InteractionGraphSummaryMetric name="(&string;)" custom_type="(&string;)"
packer_palette="(&named_packer_palette;)" scorefxn="(&string;)"
skip_pose_reconstruction_info="(false &bool;)" />
• custom_type: Allows multiple configured SimpleMetrics of a single type to be called in a single RunSimpleMetrics and SimpleMetricFeatures. The custom_type name will be added to the data tag in the scorefile or features database.
• task_operations: A comma-separated list of TaskOperations to use.
• packer_palette: A previously-defined PackerPalette to use, which specifies the set of residue types with which to design (to be pruned with TaskOperations).
• scorefxn: Name of score function to use
• skip_pose_reconstruction_info: If true, then this metric only stores a summary of the interaction graph. If false (the default), then it stores both the interaction graph summary and full information for reconstructing the pose. False by default.

## Output format

The output resembles the following:

[BEGIN POSE BINARY]
# Binary-format information is included here which can be used
# to rebuild the input pose.  Note that this pose may or may not
# contain rotamers used in the packing problem.  This section is
# omitted if skip_pose_reconstruction_info is true.
[END POSE BINARY]

[BEGIN ROTAMER NAME/SEQPOS/ROTINDEX/CHIVALS]
# Information is included here about the geometry of each rotamer.
# This can be used to rebuild individual solutions to the packing
# problem.  This section is omitted if skip_pose_reconstruction_info
# is true.  This section is divided into information for individual
# rotamers with [BEGIN_ROT] ... [END_ROT] blocks.  Each block starts
# with a line in which the tab-separated columns are:
# 1. Residue name (e.g. "DSER:NtermProteinFull", "CYS:disulfide",
# "THR", etc.)
# 2. Sequence position in the pose.
# 3. Rotamer index for this sequence position.
# 4. Side-chain chi angle values for each rotatable dihedral in the
# side-chain for this rotamer at this position.
# This is followed by a block of binary information for the detailed
# three-dimensional geometry of this rotamer.
[END ROTAMER NAME/SEQPOS/ROTINDEX/CHIVALS]

[BEGIN ONEBODY SEQPOS/ROTINDEX/ENERGY]
# Entries in this section contain the information for individual
# nodes in the interaction graph (i.e. the one-body energies).  Each
# row in this section contains tab-separated columns consisting of:
# 1. Sequence position in the pose.
# 2. Rotamer index for this sequence position.
# 3. One-body energy for this rotamer at this sequence position.
# This section is intended to be easy to import into an external
# annealer.
[END ONEBODY SEQPOS/ROTINDEX/ENERGY]

[BEGIN TWOBODY SEQPOS1/ROTINDEX1/SEQPOS2/ROTINDEX2/ENERGY]
# Entries in this section contain the information for individual
# edges in the interaction graph (i.e. the two-body energies).  Each
# row in this section contains tab-separated columns consisting of:
# 1. The sequence position of the first interacting residue in the pose.
# 2. The rotamer index at the first sequence position.
# 3. The sequence position of the second interacting residue in the pose.
# 4. The rotamer index at the second sequence position
# 5. The interaction energy between the two rotamers.
# This section is also intended to be easy to import into an external
# annealer.


text:
# Ramified cover of 4-sphere

Is it true that any closed oriented $4$-dimensional manifold can be obtained as a result of the following construction:

Take $S^4$ with a finite collection of immersed closed 2-manifolds (with transversal intersections and self-intersections) and construct ramified cover of $S^4$ with a ramification of order at most 2 only at these submanifolds.

The answer is yes, at least if we interpret your phrase "ramification of order 2" to mean "simple branched covering". See Piergallini, R., Four-manifolds as $4$-fold branched covers of $S^4$. Topology 34 (1995), no. 3, 497--508. Any closed, orientable PL 4-manifold can be expressed as a 4-fold simple branched covering of S4 branched along an immersed surface with only transverse double points. It is apparently still an open question whether the branch set can be chosen to be nonsingular. A simple branched covering of degree d is a branched covering in which each branch point is covered by d-1 points, only one of which is singular, of local degree 2.


text:
## Vertical stripe fix for 5D Mark III H.264 files¶

### What is this about?¶

Some users noticed a vertical pattern in 5D Mark III video files, first with regular raw video images. Some converters will correct this artifact automatically, so it wasn't a big issue lately.

After implementing the 3x crop mode, the artifact became visible in H.264 files as well. The previous algorithm that operates on Bayer raw data can no longer be used - it need to be adapted to H.264 files.

The effect is visible only in highlights, not in shadows. If you see vertical lines visible in shadows, that's a different issue, so you may stop reading here.

### What's causing it?¶

The 5D Mark III appears to read out 8 columns at a time, in parallel. These 8 columns appear to have different amplifier gains, and that's why the effect is visible in highlights, but not in shadows.

Since:

1. with regular (non-crop) mode, the artifact was visible in RAW, but not in H.264,
2. with crop mode, the artifact is visible in H.264, but not in RAW,
3. the crop mode only changes the sampled area on the sensor, and the pixel binning factors, but leaves all other parameters (resolution, Canon processing and so on) unchanged,

we may suspect that those stripes are present in the raw sensor data, corrected by Canon code in regular (non-crop) H.264, but for crop mode H.264, it would require a recalibration (and reverse engineering to figure out how to do that first). It is also possible that our vertical stripe artifact is amplified by the mismatched calibration.

### Enough chit-chat, how to fix it?¶

You will need:

• IPython notebook
• octave (the one prepackaged for your operating system should be fine)
• the 'image' package from octave-forge
• octave_kernel for the IPython notebook
• ffmpeg

Let's the sample files from here in octave. We'll extract a single frame from each clip, as uncompressed 8-bit ppm:

In [1]:
%%shell
# note: ffmpeg is very verbose, so I won't display the output here
ffmpeg -ss 0.96 -i 1080p\ Stripes\ 1.mov bad.ppm -y 2>ffmpeg.log
ffmpeg -i 1080p\ Stripes\ 2.mov ref.ppm -y 2>>ffmpeg.log

Octave code follows:

In [2]:
pkg load image
more off

In [3]:
bad = im2double(imread('bad.ppm'));
figure,imshow(ref(1:5:end,1:5:end,:))

warning: your version of GraphicsMagick limits images to 16 bits per pixel
warning: called from
imformats>default_formats at line 256 column 11
imformats at line 79 column 3
imageIO at line 106 column 11
imread at line 106 column 30

The user who reported the issue included a normal image and also a blank wall, that can be used to extract the vertical pattern. Nice!

Let's see a 1:1 crop from the bad image:

In [4]:
imshow(bad(1:400,1:800,:))

Doesn't look very nice.

Let's try to learn the pattern from the reference image (blank wall), from the green channel. Easier to work with grayscale, isn't?

In [5]:
% will stretch the image to make the pattern very obvious
ref_g = ref(:,:,2);
imshow(ref_g(1:400,1:800), [])

The reference blank wall is not completely uniform - let's filter out the low frequency components:

In [6]:
% filter out very low frequency components in the horizontal direction
im = ref_g;
imf = imfilter(im, ones(1,100)/100, 'replicate');
figure, imshow((im-imf)(1:400,1:800), [])

The image is more uniform now, but there are still a bunch of artifacts. Let's find out the gain for each column:

In [7]:
% get the vertical pattern
% we already know the pattern is actually a difference in column gains
% assume the filtered image is ideal, and extract per-column gain variations
gain = median(im ./ imf);
plot(gain)
axis tight

Let's put it together:

In [8]:
function cor = identify_pattern_gray(im)
% filter out very low frequency components in the horizontal direction
imf = imfilter(im, ones(1,100)/100, 'replicate');

% get the vertical pattern
% we already know the pattern is actually a difference in column gains
% assume the filtered image is ideal, and extract per-column gain variations
gain = median(im ./ imf);

% expand the correction data (vector, one entry per column)
% to match the size of the image
cor = bsxfun


text:
# Cornell astrophysicists earn share of $3M prize Cornell astrophysicists Saul Teukolsky and Lawrence Kidder have earned a share in the Special Breakthrough Prize in Fundamental Physics – a$3 million award – that recognizes those who helped create the Laser Interferometer Gravitational-Wave Observatory (LIGO) and its ability to find gravitational waves. The discovery announced in February provided strong confirmation of Albert Einstein’s general theory of relativity.

Teukolsky, the Hans A. Bethe Professor of Physics and Astrophysics, and Kidder, senior research associate, played a vital role to validate the historic news of the first direct detection of gravitational waves, predicted a century ago by Einstein.

Theoretical physicist Stephen Hawking, winner of the 2013 Special Breakthrough Prize, noted of the LIGO team, “This discovery has huge significance: firstly, as evidence for general relativity and its predictions of black hole interactions, and secondly as the beginning of a new astronomy that will reveal the universe through a different medium.”

Announcing the award, the prize committee cited Teukolsky and Kidder for their achievement. The Special Breakthrough Prize is given when an extraordinary scientific achievement occurs.

The three founders of LIGO – Ronald W.P. Drever and Kip S. Thorne of the California Institute of Technology and Rainer Weiss of the Massachusetts Institute of Technology – will share $1 million of the award. The remaining$2 million will be shared among 1,012 contributors to the experiment.

Led by scientists from the LIGO collaboration at Caltech and the Virgo group collaboration, research published in February in Physical Review Letters reported detection of gravitational waves resulting from two black holes spiraling in toward one another and smashing together.

Until then, this scenario had only been predicted theoretically. Many astrophysicists doubted it would occur often enough ever to be detected. However, soon after LIGO detectors in Livingston, Louisiana, and Hanford, Washington, were upgraded, in September scientists found two black holes – each about 35 times the mass of our sun – moving at more than half the speed of light, orbiting each other and creating waves. Researchers spent the autumn 2015 confirming results.

LIGO and Virgo researchers confirmed the waves came from a black hole merger by comparing their data with a theoretical model developed at Cornell. Teukolsky and the Cornell-founded Simulation of eXtreme Spacetimes collaboration group have been calculating and completing a full catalog of theoretical solutions since 2000, when supercomputers first became capable of the task.

Founder of the prize Yuri Milner said: “The creative powers of a unique genius, many great scientists, and the universe itself, have come together to make a perfect science story.”

This article originally appeared in the Cornell Chronicle.

Top


text:
## Thomas' Calculus 13th Edition

$f(x)=3$ $f(-x)=3$ $f(x)=f(-x)$ - This makes the function even. $-f(x)=-3$ $f(-x)=3$ $-f(x)\ne f(-x)$ The function is not odd.


text:
# What is the definition of $Li(x)$ in these MathOverflow questions on $I_s =\int_{1}^{\infty} (\pi(x)-Li(x))\,x^{-s-1} dx$?

Question: What is the definition of $$Li(x)$$ in the MathOverflow questions linked below and their related answers? $$Li(x)=\int\limits_0^x\frac{1}{\log(t)}\,dt$$, $$Li(x)=\int\limits_1^x\frac{1}{\log(t)}\,dt$$, $$Li(x)=\int\limits_2^x\frac{1}{\log(t)}\,dt$$, or $$Li(x)=\int\limits_0^{1-\epsilon}\frac{1}{\log(t)}\,dt+\int\limits_{1+\epsilon}^x\frac{1}{\log(t)}\,dt$$?

The related questions, answers, and comments to the following Math StackExhange questions seem to use various definitions. My motivation for asking this question is an attempt to understand the exact formula associated with the evaluation of $$I_s$$ in order to gain insight into the questions, answers, and comments associated with these questions.

The remainder of this question is an attempt to further clarify my motivation for asking this question and is based on the definitions in formulas (1) to (3) below where $$P(s)$$ is the prime zeta function.

(1) $$\quad \pi(x)=\sum\limits_{p\le x}1$$

(2) $$\quad P(s)=s\int\limits_1^\infty \pi(x)\,x^{-s-1}\,ds=\sum\limits_p\frac{1}{p^s}\,,\quad\Re(s)>1$$

(3) $$\quad P(s)=\sum\limits_{m=1}^\infty\frac{\mu(m)}{m}\log\zeta(m\,s)\,,\qquad\qquad\Re(s)>0$$

The questions at the links (d) and (e) above use the definition in (4) below, the claimed identity in (5) below, and the equivalence in (6) below to derive the claimed equality in (7) below.

(4) $$\quad Li(x)=\lim_{\epsilon\rightarrow 0^{+}}\Big(\int_{0}^{1-\epsilon}+\int_{1+\epsilon}^{x}\Big)\frac{dt}{\log t}$$

(5) $$\quad s\int_{1}^\infty Li(x)\,x^{-s-1}\,dx=-\log(s-1)\,,\quad\Re(s)>1$$

(6) $$\quad\sum\limits_p\frac{1}{p^s}=\sum\limits_{m=1}^\infty\frac{\mu(m)}{m}\log\zeta(m\,s)\,,\quad\Re(s)>1$$

(7) $$\quad s\int_{1}^\infty (\pi(x)-Li(x))\,x^{-s-1}\,dx-\log((s-1)\,\zeta(s))=\sum\limits_{m=2}^{\infty}\frac{\mu(m)}{m}\log\zeta(m\,s)\,,\quad Re(s)>1$$

The questions at links (d) and (e) above initially state the claimed equality illustrated in (7) above is valid for $$Re(s)>1$$, but then claim the equality can be extended via analytic continuation to $$Re(s)>\Theta$$ where $$\Theta$$ is the supremum of the real parts of the zeros of $$\zeta(s)$$.

Assuming correctness of the claimed identity in (5) above, evaluation of the integral in (7) above leads to (8) below which is valid for $$\Re(s)>1$$, but I don't understand the claim with respect to analytic continuation to $$Re(s)>\Theta$$. The equivalence illustrated in (6) above can be used to rewrite (8) below as (9) below which leads to (10) below which is valid for $$\Re(s)>0$$, but this isn't useful.

(8) $$\quad \sum\limits_p\frac{1}{p^s}+\log(s-1)-\log((s-1)\,\zeta(s))=\sum\limits_{m=2}^{\infty}\frac{\mu(m)}{m}\log\zeta(m\,s)\,,\quad Re(s)>1$$

(9) $$\quad \sum\limits_{m=1}^{\infty}\frac{\mu(m)}{m}\log\zeta(m\,s)+\log(s-1)-\log((s-1)\,\zeta(s))=\sum\limits_{m=2}^{\infty}\frac{\mu(m)}{m}\log\zeta(m\,s)\,,\\$$ $$\quad Re(s)>1$$

(10) $$\quad \sum\limits_{m=2}^{\infty}\frac{\mu(m)}{m}\log\zeta(m\,s)=\sum\limits_{m=2}^{\infty}\frac{\mu(m)}{m}\log\zeta(m\,s)\,,\\$$ $$\quad Re(s)>0$$

The following figure illustrates the right-hand side of (8) above (which is strictly real in the interval $$\frac{1}{2}) in blue and the real and imaginary parts of the left-hand side of (8) above in orange and green respectively where the left-hand side sum is taken over the first $$10,000$$ primes. The figure below clearly illustrates the fallacy of the equivalence of the left and right hand sides of (8) above for $$\Theta.

Figure (1): Illustration of RHS of (8) (blue) and LHS of (8) ($$\Re$$ orange and $$\Im$$ green)

• en.wikipedia.org/wiki/Logarithmic_integral_function – Wojowu Sep 14 '19 at 15:51
• All your proposed definitions are wrong... – Unit Sep 14 '19 at 15:56
• @Unit Thanks for point out the error. I corrected the upper limits from $\infty$ to $x$ in the integrals related to the definition of $Li(x)$ above. – Steven Clark Sep 14 '19 at 16:14
• there is a standard definition (see Wikipedia) and then changing lower bounds makes it differ by a constant which is utterly irrelevant when one does Mellin transforms as in the post since differences are entire functions, while convergence issues are always at infinity; it is more interesting when one does the complex Li - usually the integral definition in Edwards book on RZ is best to work as it excludes the negative reals and in particular it doesn't affect non-trivial RZ zeroes – Conrad Sep 14 '19 at 16:18
• The difference between definitions is the mellin transform of a constant so kind of trivial; trying to actually estimate the integral, so involving $Li(x^\rho)$ for non trivial zeta roots is what you want – Conrad Sep 14 '19 at 16:47

$$li(x)=pv(\int_0^x \frac1{\log(t)}dt), \qquad x > 0$$

The problem is much less the definition of $$Li$$ than the possible Mellin transforms. There are mostly 3 things to try :

• $$\int_0^\infty li(x)x^{-s-1}dx$$ never converges.

• $$\int_0^\infty 1_{x>2}(li(x)-li(2))x^{-s-1}dx$$ approximates $$\frac{-\log(s-1)}{s}$$ very well around $$s=1$$ and since $$1_{x>2}(li(x)-li(2))$$ is continuous and piecewise $$C^1$$ its Mellin transform is $$L^1$$ on vertical lines.

• Let $$f(x)= li(x)1_{x > 1}\in L^1_{loc}$$, due to the huge discontinuity at $$x=1$$ its distributional derivative is complicated, but if $$\phi \in C^\infty_c(0,\infty),\phi(1)=0$$ then $$\ =\ \int_1^\infty \frac{\phi(x)}{\log x}dx \quad \implies \quad f' \log x = 1_{x > 1}$$ Which means $$F(s)=\int_0^\infty f(x)x^{-s-1}dx,\qquad sF(s)=\int_0^\infty f'(x)x^{-s}dx$$ $$-(sF(s))' =\int_0^\infty f'(x)\log(x)x^{-s}dx=\int_0^\infty 1_{x > 1}x^{-s}dx= \frac1{s-1}$$ $$sF(s)=-\log (s-1)+A$$

• Finding $$A$$ is a pain : around $$x=1$$ we have $$li(x) = B+log |x-1|+O(x-1)$$ where the $$O(x-1)$$ term is $$C^1$$ thus as $$s \to +\infty$$

$$F(s) = \int_1^\infty log |x-1|x^{-s-1}dx+\frac{B}s+o(\frac{1}{s})$$

$$\int_1^\infty log |x-1|x^{-s-1}dx=\frac1s\int_1^\infty \frac{x^{-s}-1}{x-1}dx=\frac1s\int_1^\infty \frac{x^{-s-1}-x^{-1}}{1-x^{-1}}dx$$ $$= \frac1s \sum_{m=0}^\infty \int_1^\infty (x^{-s-1-m}-x^{-1-m})dx= \frac1s \sum_{m=0}^\infty(\frac{1}{s+m}-\frac{1}{1+m})= \frac{-\psi(s)-\gamma}{s}$$ And hence from the asymptotic expansion of $$\psi$$ and $$B=\int_0^1 (\frac1{\log (t)}-\frac1{t-1})dt=\gamma$$ we obtain $$A=0$$ and $$\int_1^\infty li(x)x^{-s-1}dx=F(s) = \frac{-\log(s-1)}{s}$$

• You seem to have confirmed the claimed identity in formula (5) of my question above which is valid for $\Re(s)>1$. The question at link (e) above originally used the definition $Li(x)=\int\limits_1^x\frac{1}{\log(t)}\,dt$, but the definition was later revised to $Li(x)=\lim_{\epsilon\rightarrow 0^{+}}\Big(\int_{0}^{1-\epsilon}+\int_{1+\epsilon}^{x}\Big)\frac{dt}{\log t}$ which is consistent with your derivation. – Steven Clark Sep 15 '19 at 16:30
• In the comments associated with the question at link (e) above (which have since been moved to chat), there was an argument made based on the answers to the question at link (a) above. I wasn't sure if the answers to the question at link (a) assumed the same definition of $Li(x)$, but the comments on my question above seem to indicate it really doesn't matter. – Steven Clark Sep 15 '19 at 16:30


text:
## 7.6Alternative wall functions

The standard wall function described in Sec. 7.5 uses a function for that is discontinuous at , switching to for . A continuous wall function is available which evaluates as from a single equation describing the universal character of the velocity proﬁle at high ,9

 (7.22)
where . The equation combines Eq. (7.11 ) and Eq. (7.13 ), “disabling” Eq. (7.13 ) at low by subtracting low order terms from a polynomial expansion of .

The wall function is applied by solving Eq. (7.22 ) for from calculated from Eq. (7.9) using the near-wall cell centre height . The friction velocity is calculated by , where is the near-wall cell velocity. Finally, on the wall patch is calculated from a numerical interpretation of Eq. (7.20 ),

 (7.23)
where is the surface-normal velocity gradient. An iterative method is required to invert Eq. (7.22) and to accommodate other nonlinearities, e.g. is itself a function of .

### Rough wall function

The standard wall function in Sec. 7.5 is applicable to smooth walls so does not account for surface roughness. Roughness is signiﬁcant when the roughness “scale” 10 becomes larger than the thickness of the viscous sub-layer.

At higher surface roughness, turbulent eddies are generated near the wall at a scale , rather than . The viscous eﬀects become negligible, causing the non-dimensionalised distance to become in the log law Eq. (7.13 ). To reﬂect this, Eq. (7.13 ) is modiﬁed to a form

 (7.24)
where a roughness function, dependent on . An intuitive model for is11
 (7.25)
This rough wall function Eq. (7.24 ) reduces to Eq. (7.13 ) using the conventional deﬁnition of Eq. (7.9 ) at . As , it reduces to Eq. (7.13 ) using .

It is open to interpretation how to determine from roughness measurements of a surface. The parameter is sometimes split into , where is a measured sand grain roughness height and is a coeﬃcient that depends on the shape, consistency and packing of the roughness elements. Using that approach, values of often yield a good match between Eq. (7.24 ) and measured data.

9Brian Spalding, A single formula for the law of the wall, 1961.
10’ is often used to denote roughness, but we use ‘’ to avoid confusion with turbulent kinetic energy.
11Cyril Colebrook, Turbulent ﬂow in pipes, with particular reference to the transitional region between smooth and rough wall laws, 1939.

Notes on CFD: General Principles - 7.6 Alternative wall functions