text:
# American Institute of Mathematical Sciences

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December  2015, 5(4): 807-826. doi: 10.3934/mcrf.2015.5.807

## Generalization on optimal multiple stopping with application to swing options with random exercise rights number

 1 Département de Mathématiques, Institut Supérieur d'Informatique et de Mathématiques de Monastir, Avenue de la Korniche, B.P. 223, 5000 Monastir, Tunisia, Tunisia

Received  December 2014 Revised  April 2015 Published  October 2015

This paper develops the theory of optimal multiple stopping times expected value problems by stating, proving, and applying a dynamic programming principle for the case in which both the reward process and the number of stopping times are stochastic. This case comes up in practice when valuing swing options, which are somewhat common in commodity trading. We believe our results significantly advance the study of option pricing.
Citation: Noureddine Jilani Ben Naouara, Faouzi Trabelsi. Generalization on optimal multiple stopping with application to swing options with random exercise rights number. Mathematical Control and Related Fields, 2015, 5 (4) : 807-826. doi: 10.3934/mcrf.2015.5.807
##### References:
 [1] C. Blanchet-Scalliet, N. El-Karoui, M. Jeanblanc and L. Martellini, Optimal investment and consumption decisions when time-horizon is uncertain, Journal of Mathematical Economics, 44 (2008), 1100-1113. doi: 10.1016/j.jmateco.2007.09.004. [2] R. Carmona and S. Dayanik, Optimal multiple stopping of linear diffusions, Mathematics of Operations Research, 33 (2008), 446-460. doi: 10.1287/moor.1070.0301. [3] R. Carmona and N. Touzi, Optimal multiple stopping and valuation of swing options, Mathematical Finance, 18 (2008), 239-268. doi: 10.1111/j.1467-9965.2007.00331.x. [4] N. Chaidee and K. Neammanee, Berry-Esseen bound for independent random sum via Stein's method, International Mathematical Forum, 3 (2008), 721-738. [5] N. Chaidee and M. Tuntapthai, Berry-Esseen bounds for random sums of non-i.i.d. random variables, International Mathematical Forum, 4 (2009), 1281-1288. [6] S. Christensen, A. Irle and S. Jürgens, Optimal multiple stopping with random waiting times, Sequential Analysis: Design Methods and Applications, 32 (2013), 297-318. doi: 10.1080/07474946.2013.803814. [7] S. Dayanik and I. Karatzas, On the optimal stopping times problem for one-dimensional diffusions, Stochastic Processes and their Applications, 9 (2003), 342-351. [8] E. B. Dynkin, Markov Processes: Theorems and Problems, 1st edition, Plenum Press, New York, 1969. [9] R. Elliott, M. Jeanblanc and M. Yor, On models of default risk, Mathematical Finance, 10 (2000), 179-195. doi: 10.1111/1467-9965.00088. [10] S. W. He, J. G. Wang and J. A. Yan, Semimartingale Theory and Stochastic Calculus, Science Press, Beijing, CRC Press, Boca Raton, 1992. [11] K. Itô and H. P. McKean, Diffusion Processes and Their Sample Paths, 1st edition, Springer-Verlag, Berlin, 1974. [12] N. Jilani Ben Naouara and F. Trabelsi, Biological application of optimal stopping, Int. J. of Mathematical Modelling and Numerical Optimisation, 5 (2014), 229-264. [13] N. Jilani Ben Naouara and F. Trabelsi, General undiscounted non-linear optimal multiple stopping of linear diffusions with boundary classification, to appear in Int. J. of Mathematics in Operational Research. [14] S. Karlin and H. Taylor, A Second Course in Stochastic Processes, Academic press, San Diego, 1981. [15] M. Kobylanski, M. C. Quenez and E. Rouy, Optimal multiple stopping time problem, The Annals of Applied Probability, 21 (2011), 1365-1399. doi: 10.1214/10-AAP727. [16] M. Pointier, Pricing Rules Under Asymmetric Information, http://www.math.univ-toulouse.fr/ pontier/squfirenze.pdf. [17] M. Tomomi and A. Katsunori, Lower bounds for Bruss' odds problem with multiple stopping, preprint, arXiv:1204.5537. [18] F. Trabelsi, Study of undiscounted non-linear optimal multiple stopping times problems on unbounded intervals, Int. J. Operational Research, 5 (2013), 225-254. doi: 10.1504/IJMOR.2013.052462. [19] F. Trabelsi and M. B. Zoghlami, On undiscounted non-linear optimal multiple stopping, Int. J. Operational Research, 14 (2012), 387-416. doi: 10.1504/IJOR.2012.047512. [20] A. B. Zeghal and M. Mnif, Optimal multiple stopping and valuation of swing options in Lévy models, Int. J. Theoretical and Applied Finance, 9 (2006), 1267-1297. doi: 10.1142/S0219024906004037.

show all references

##### References:
 [1] C. Blanchet-Scalliet, N. El-Karoui, M. Jeanblanc and L. Martellini, Optimal investment and consumption decisions when time-horizon is uncertain, Journal of Mathematical Economics, 44 (2008), 1100-1113. doi: 10.1016/j.jmateco.2007.09.004. [2] R. Carmona and S. Dayanik, Optimal multiple stopping of linear diffusions, Mathematics of Operations Research, 33 (2008), 446-460. doi: 10.1287/moor.1070.0301. [3] R. Carmona and N. Touzi, Optimal multiple stopping and valuation of swing options, Mathematical Finance, 18 (2008), 239-268. doi: 10.1111/j.1467-9965.2007.00331.x. [4] N. Chaidee and K. Neammanee, Berry-Esseen bound for independent random sum via Stein's method, International Mathematical Forum, 3 (2008), 721-738. [5] N. Chaidee and M. Tuntapthai, Berry-Esseen bounds for random sums of non-i.i.d. random variables, International Mathematical Forum, 4 (2009), 1281-1288. [6] S. Christensen, A. Irle and S. Jürgens, Optimal multiple stopping with random waiting times, Sequential Analysis: Design Methods and Applications, 32 (2013), 297-318. doi: 10.1080/07474946.2013.803814. [7] S. Dayanik and I. Karatzas, On the optimal stopping times problem for one-dimensional diffusions, Stochastic Processes and their Applications, 9 (2003), 342-351. [8] E. B. Dynkin, Markov Processes: Theorems and Problems, 1st edition, Plenum Press, New York, 1969. [9] R. Elliott, M. Jeanblanc and M. Yor, On models of default risk, Mathematical Finance, 10 (2000), 179-195. doi: 10.1111/1467-9965.00088. [10] S. W. He, J. G. Wang and J. A. Yan, Semimartingale Theory and Stochastic Calculus, Science Press, Beijing, CRC Press, Boca Raton, 1992. [11] K. Itô and H. P. McKean, Diffusion Processes and Their Sample Paths, 1st edition, Springer-Verlag, Berlin, 1974. [12] N. Jilani Ben Naouara and F. Trabelsi, Biological application of optimal stopping, Int. J. of Mathematical Modelling and Numerical Optimisation, 5 (2014), 229-264. [13] N. Jilani Ben Naouara and F. Trabelsi, General undiscounted non-linear optimal multiple stopping of linear diffusions with boundary classification, to appear in Int. J. of Mathematics in Operational Research. [14] S. Karlin and H. Taylor, A Second Course in Stochastic Processes, Academic press, San Diego, 1981. [15] M. Kobylanski, M. C. Quenez and E. Rouy, Optimal multiple stopping time problem, The Annals of Applied Probability, 21 (2011), 1365-1399. doi: 10.1214/10-AAP727. [16] M. Pointier, Pricing Rules Under Asymmetric Information, http://www.math.univ-toulouse.fr/ pontier/squfirenze.pdf. [17] M. Tomomi and A. Katsunori, Lower bounds for Bruss' odds problem with multiple stopping, preprint, arXiv:1204.5537. [18] F. Trabelsi, Study of undiscounted non-linear optimal multiple stopping times problems on unbounded intervals, Int. J. Operational Research, 5 (2013), 225-254. doi: 10.1504/IJMOR.2013.052462. [19] F. Trabelsi and M. B. Zoghlami, On undiscounted non-linear optimal multiple stopping, Int. J. Operational Research, 14 (2012), 387-416. doi: 10.1504/IJOR.2012.047512. [20] A. B. Zeghal and M. Mnif, Optimal multiple stopping and valuation of swing options in Lévy models, Int. J. Theoretical and Applied Finance, 9 (2006), 1267-1297. doi: 10.1142/S0219024906004037.
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text:
## anonymous 5 years ago radial of x =6

1. anonymous

$\sqrt{x}=6$ $\sqrt{x}^2=6^2$ $x=6$

2. anonymous

Sorry that last part should be: $x=36$

format: plain


text:
# How remove specified substring and everything after it to the end of a string?

If I have, say, the string,

txt = "now is the time for all (good) men"

then I want to remove the substring beginning with "for" and ending at the end of the string, namely, remove:

subtext = "for all (good) men"

I do realize that an expression of the form StringReplace[txt, regex ->""] or StringReplace[txt, pattern->""] will do it, but I don't know what regular expression or pattern to use.

The part of the substring to be omitted definitely begins with the specific text "for", but anything could follow it.

• StringReplace[txt, "for" ~~ ___ -> ""]
– paw
Oct 26 '15 at 0:21
• You don't have to use a regular expression; a string expression will also work: for example, "for" ~~ ___ ~~ EndOfString. You can see the corresponding regular expression using StringPatternPatternConvert. In this case it is (?ms)for.*\z. Oct 26 '15 at 0:22
• That works. Want to make it an Answer? And do I really need the part ~~~ EndOfSting? Oct 26 '15 at 0:24
• @paw: Please post your comment as an answer, as it seems to supply a simpler method than using a regular expression. Oct 26 '15 at 21:22

The desired replacement rule can be formulated using string expressions:

"for" ~~ ___-> ""

The pattern represents the concatenation of the string "for" with any sequence of zero or more characters.

Example:

txt = "now is the time for all (good) men";
StringReplace[txt, "for" ~~ ___ -> ""]

"now is the time "

This is how to do it without patterns:

StringDrop[
txt,
StringPosition[txt, "for"][[1, 1]] ;;
]
(* Out: "now is the time " *)

With patterns you have a few different options, as people have remarked in the comments. They gave you the string pattern version, this is the regular expression version:

StringReplace[txt, RegularExpression["for.*$"] -> ""] (* Out: "now is the time " *)  where . denotes any character, * denotes zero or more times, and $ denotes the end of the string.

(I see now that Oleksandr gave a regular expression as well, I didn't see that when I wrote the answer, but I'll let it remain.)

• How get the regular expression to work in case the substring to be deleted begins with, or otherwise includes, special characters such as ( that have special meaning in regexes? I tried the usual method of trying to escape, as StringReplace["now is the time (for) all good men", RegularExpression["$$for$$.*$"] -> ""] but that just returns the original string. Oct 26 '15 at 19:46 • @murray You can resort to this instead: RegularExpression["[(]for[)].*$"] - square brackets in regex means "any of these characters" and in this case we only list one character, so it only matches that one. Oct 26 '15 at 20:05
• Thanks for the fix to using RegularExpression`. I'm still happier using a Wolfram Language string expression. Oct 27 '15 at 2:40


text:
## Class AHURootedTreeIsomorphismInspector<V,E>

• java.lang.Object
• org.jgrapht.alg.isomorphism.AHURootedTreeIsomorphismInspector<V,E>
• Type Parameters:
V - the type of the vertices
E - the type of the edges
All Implemented Interfaces:
IsomorphismInspector<V,E>

public class AHURootedTreeIsomorphismInspector<V,E>
extends java.lang.Object
implements IsomorphismInspector<V,E>
This is an implementation of the AHU algorithm for detecting an (unweighted) isomorphism between two rooted trees. Please see mathworld.wolfram.com for a complete definition of the isomorphism problem for general graphs.

The original algorithm was first presented in "Alfred V. Aho and John E. Hopcroft. 1974. The Design and Analysis of Computer Algorithms (1st ed.). Addison-Wesley Longman Publishing Co., Inc., Boston, MA, USA."

This implementation runs in linear time (in the number of vertices of the input trees) while using a linear amount of memory.

Note: If the input graph is directed, it effectively considers only the subtree reachable from the specified root.

For an implementation that supports unrooted trees see AHUUnrootedTreeIsomorphismInspector.
For an implementation that supports rooted forests see AHUForestIsomorphismInspector.

Note: This inspector only returns a single mapping (chosen arbitrarily) rather than all possible mappings.

Author:
Alexandru Valeanu
• ### Constructor Summary

Constructors
Constructor Description
AHURootedTreeIsomorphismInspector(Graph<V,E> tree1, V root1, Graph<V,E> tree2, V root2)
Construct a new AHU rooted tree isomorphism inspector.
• ### Method Summary

All Methods
Modifier and Type Method Description
IsomorphicGraphMapping<V,E> getMapping()
Get an isomorphism between the input trees or null if none exists.
java.util.Iterator<GraphMapping<V,E>> getMappings()
Get an iterator over all calculated (isomorphic) mappings between two graphs.
boolean isomorphismExists()
Check if an isomorphism exists.
• ### Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
• ### Constructor Detail

• #### AHURootedTreeIsomorphismInspector

public AHURootedTreeIsomorphismInspector(Graph<V,E> tree1,
V root1,
Graph<V,E> tree2,
V root2)
Construct a new AHU rooted tree isomorphism inspector. Note: The constructor does NOT check if the input trees are valid.
Parameters:
tree1 - the first rooted tree
root1 - the root of the first tree
tree2 - the second rooted tree
root2 - the root of the second tree
Throws:
java.lang.NullPointerException - if tree1 or tree2 is null
java.lang.NullPointerException - if root1 or root2 is null
java.lang.IllegalArgumentException - if tree1 or tree2 is empty
java.lang.IllegalArgumentException - if root1 or root2 is an invalid vertex
• ### Method Detail

• #### getMappings

public java.util.Iterator<GraphMapping<V,E>> getMappings()
Get an iterator over all calculated (isomorphic) mappings between two graphs.
Specified by:
getMappings in interface IsomorphismInspector<V,E>
Returns:
an iterator over all calculated (isomorphic) mappings between two graphs
• #### isomorphismExists

public boolean isomorphismExists()
Check if an isomorphism exists.
Specified by:
isomorphismExists in interface IsomorphismInspector<V,E>
Returns:
true if there is an isomorphism, false if there is no isomorphism
• #### getMapping

public IsomorphicGraphMapping<V,E> getMapping()
Get an isomorphism between the input trees or null if none exists.
Returns:
isomorphic mapping, null is none exists


text:
# Video pipeline demonstrating applying labels based on timestamps or frame numbers¶

## Goal¶

In this example, we will go through the creation of a pipeline using the VideoReader operator to read videos along with their labels. We will apply labels to the video based on frame numbers as input, and read timestamp and frame number labels along with decoded video frames.

To make it clearer, let’s look at how we can obtain these sequences and how to use them!

First let’s import the libraries.

[1]:

import os
import numpy as np
import tempfile

[2]:

from nvidia.dali.pipeline import Pipeline
import nvidia.dali.ops as ops
import nvidia.dali.types as types

try:
from PIL import Image
has_PIL = True
except ImportError:
has_PIL = False

Then we define that in each iteration of the pipeline we want 4 batches and each batch has sequence length of 1 frame. We want to run 10 iterations for illustrative purpose.

[3]:

BATCH_SIZE=4
COUNT=1
ITER=10
frame_num_based_labels=True

The video we have used in this example is a 10 seconds long, 25fps, 512x512 resolution video with timestamp burned into the frame itself in the top half of the video and frame number in the bottom half. The burned text in the decoded frame can be easily used to verify that we are indeed returned the correct frame number and timestamp labels.

DALI_EXTRA_PATH environment variable should point to the place where data from DALI extra repository is downloaded. Please make sure that the proper release tag is checked out.

[4]:

dali_extra_path = os.environ['DALI_EXTRA_PATH']
file_list_txt = dali_extra_path +  "/db/video/test_label.mp4 0\n"
file_list_txt += dali_extra_path + "/db/video/test_label.mp4 1 0 100\n"
file_list_txt += dali_extra_path + "/db/video/test_label.mp4 2 100 200\n"
file_list_txt += dali_extra_path + "/db/video/test_label.mp4 3 200 250\n"
tf = tempfile.NamedTemporaryFile()
tf.write(str.encode(file_list_txt))
tf.flush()

## Defining the pipeline¶

We can then define a minimal Pipeline that will output directly the VideoReader outputs.

Notice that we set file_list_frame_num=True, this means that the labels we provided above using file_list will be interpreted as frame numbers. test_label.mp4 1 0 100 is interpreted as applying label 1 from frame number 0 to 100(excluding).

If however you wished to apply labels using timestamp, simply set file_list_frame_num=False (default behaviour), and provide timestamps you wish to apply the label to. For example, if you wished to apply a label 1 to frames from 3 seconds to 8.5 seconds: test_label.mp4 1 3 8.5

In this example we have set both enable_frame_num and enable_timestamps to True for illustrative purpose. You can enable or disable either of them depending on your particular use case.

[5]:

class VideoPipe(Pipeline):
def __init__(self, batch_size, num_threads, device_id, data):
self.input = ops.VideoReader(device="gpu", file_list = data, sequence_length=COUNT,
shard_id=0, num_shards=1, random_shuffle=True, initial_fill=10,
image_type=types.RGB, dtype=types.FLOAT,
file_list_frame_num=frame_num_based_labels, enable_frame_num=True, enable_timestamps=True)

def define_graph(self):
return output

## Visualizing the results¶

We will use matplotlib to display the frames we obtained and the associated labels.

[6]:

%matplotlib inline
from matplotlib import pyplot as plt
import matplotlib.gridspec as gridspec

[7]:

pipe = VideoPipe(batch_size=BATCH_SIZE, num_threads=2, device_id=0, data=tf.name)
pipe.build()
gs = gridspec.GridSpec(ITER, 1)

for i in range(ITER):
sequences_out, label, start_frame_num, timestamps = pipe.run()
sequences_out = sequences_out.as_cpu().as_array()
label  = label.as_cpu().as_array()
start_frame_num = start_frame_num.as_cpu().as_array()
timestamps = timestamps.as_cpu().as_array()
batch_sequences = sequences_out[0]
sample_frame = batch_sequences[0]
if has_PIL:
im = Image.fromarray(sample_frame.astype('uint8'))
fig = plt.figure(figsize = (16,128), facecolor="#76b900")
plt.subplot(gs[i])
plt.axis("off")
plt.title("label=" + str(label[0][0]) + '\n'
"frame number=" + str(start_frame_num[0][0]) + '\n'
"timestamp=" + str(round(timestamps[0][0], 2)), fontsize=20)
plt.imshow(im)
plt.close()
tf.close()


text:
# Osmotic concentration

(Redirected from Osmolar)

Osmotic concentration, formerly known as osmolarity,[1] is the measure of solute concentration, defined as the number of osmoles (Osm) of solute per litre (L) of solution (osmol/L or Osm/L). The osmolarity of a solution is usually expressed as Osm/L (pronounced "osmolar"), in the same way that the molarity of a solution is expressed as "M" (pronounced "molar"). Whereas molarity measures the number of moles of solute per unit volume of solution, osmolarity measures the number of osmoles of solute particles per unit volume of solution.[2] This value allows the measurement of the osmotic pressure of a solution and the determination of how the solvent will diffuse across a semipermeable membrane (osmosis) separating two solutions of different osmotic concentration.

## Unit

The unit of osmotic concentration is the osmole. This is a non-SI unit of measurement that defines the number of moles of solute that contribute to the osmotic pressure of a solution. A milliosmole (mOsm) is 1/1,000 of an osmole. A microosmole (μOsm) (also spelled micro-osmole) is 1/1,000,000 of an osmole.

## Types of solutes

Osmolarity is distinct from molarity because it measures osmoles of solute particles rather than moles of solute. The distinction arises because some compounds can dissociate in solution, whereas others cannot.[2]

Ionic compounds, such as salts, can dissociate in solution into their constituent ions, so there is not a one-to-one relationship between the molarity and the osmolarity of a solution. For example, sodium chloride (NaCl) dissociates into Na+ and Cl ions. Thus, for every 1 mole of NaCl in solution, there are 2 osmoles of solute particles (i.e., a 1 mol/L NaCl solution is a 2 osmol/L NaCl solution). Both sodium and chloride ions affect the osmotic pressure of the solution.[2]

Another example is magnesium chloride (MgCl2), which dissociates into Mg2+ and 2Cl ions. For every 1 mole of MgCl2 in the solution, there are 3 osmoles of solute particles.

Nonionic compounds do not dissociate, and form only 1 osmole of solute per 1 mole of solute. For example, a 1 mol/L solution of glucose is 1 osmol/L.[2]

Multiple compounds may contribute to the osmolarity of a solution. For example, a 3 Osm solution might consist of: 3 moles glucose, or 1.5 moles NaCl, or 1 mole glucose + 1 mole NaCl, or 2 moles glucose + 0.5 mole NaCl, or any other such combination.[2]

## Definition

The osmolarity of a solution, given in osmoles per liter (osmol/L) is calculated from the following expression:

${\displaystyle \mathrm {osmolarity} =\sum _{i}\varphi _{i}\,n_{i}C_{i}}$

where

• φ is the osmotic coefficient, which accounts for the degree of non-ideality of the solution. In the simplest case it is the degree of dissociation of the solute. Then, φ is between 0 and 1 where 1 indicates 100% dissociation. However, φ can also be larger than 1 (e.g. for sucrose). For salts, electrostatic effects cause φ to be smaller than 1 even if 100% dissociation occurs (see Debye–Hückel equation);
• n is the number of particles (e.g. ions) into which a molecule dissociates. For example: glucose has n of 1, while NaCl has n of 2;
• C is the molar concentration of the solute;
• the index i represents the identity of a particular solute.

Osmolarity can be measured using an osmometer which measures colligative properties, such as Freezing-point depression, Vapor pressure, or Boiling-point elevation.

## Osmolarity vs. tonicity

Osmolarity and tonicity are related but distinct concepts. Thus, the terms ending in -osmotic (isosmotic, hyperosmotic, hyposmotic) are not synonymous with the terms ending in -tonic (isotonic, hypertonic, hypotonic). The terms are related in that they both compare the solute concentrations of two solutions separated by a membrane. The terms are different because osmolarity takes into account the total concentration of penetrating solutes and non-penetrating solutes, whereas tonicity takes into account the total concentration of non-freely penetrating solutes only.[3][2]

Penetrating solutes can diffuse through the cell membrane, causing momentary changes in cell volume as the solutes "pull" water molecules with them. Non-penetrating solutes cannot cross the cell membrane; therefore, the movement of water across the cell membrane (i.e., osmosis) must occur for the solutions to reach equilibrium.

A solution can be both hyperosmotic and isotonic.[2] For example, the intracellular fluid and extracellular can be hyperosmotic, but isotonic – if the total concentration of solutes in one compartment is different from that of the other, but one of the ions can cross the membrane (in other words, a penetrating solute), drawing water with it, thus causing no net change in solution volume.

## Plasma osmolarity vs. osmolality

Plasma osmolarity can be calculated from plasma osmolality by the following equation:[4]

Osmolarity = osmolality * (ρsol − ca)

where:

• ρsol is the density of the solution in g/ml, which is 1.025 g/ml for blood plasma.[5]
• ca is the (anhydrous) solute concentration in g/ml – not to be confused with the density of dried plasma

According to IUPAC, osmolality is the quotient of the negative natural logarithm of the rational activity of water and the molar mass of water, whereas osmolarity is the product of the osmolality and the mass density of water (also known as osmotic concentration).

In simpler terms, osmolality is an expression of solute osmotic concentration per mass of solvent, whereas osmolarity is per volume of solution (thus the conversion by multiplying with the mass density of solvent in solution (kg solvent/kg solution).

${\displaystyle {\text{osmolarity}}=\sum _{i}\varphi _{i}\,n_{i}m_{i}}$

where mi is the molality of component i.

Plasma osmolarity/osmolality is important for keeping proper electrolytic balance in the blood stream. Improper balance can lead to dehydration, alkalosis, acidosis or other life-threatening changes. Antidiuretic hormone (vasopressin) is partly responsible for this process by controlling the amount of water the body retains from the kidney when filtering the blood stream.[6]


text:
# The Beta Prior, Likelihood, and Posterior

September 4, 2013
By

(This article was first published on Statistical Research » R, and kindly contributed to R-bloggers)

The Beta distribution (and more generally the Dirichlet) are probably my favorite distributions.  However, sometimes only limited information is available when trying set up the distribution.  For example maybe you only know the lowest likely value, the highest likely value and the median, as a measure of center.  That information is sufficient to construct a basic form of the distribution.  The idea of this post is not to elaborate in detail on Bayesian priors and posteriors but to give a real working example of using a prior with limited knowledge about the distribution, adding some collected data and arriving at a posterior distribution along with a measure of its uncertainty.

The example below is a simple demonstration on how a prior distribution and current data can be combined and form a posterior distribution.  Earlier this year I gave a presentation at a conference where I modified this simple version of my code to be substantially more complex and I used the Dirichlet distribution to make national predictions based on statewide and local samples.   So this approach has some very useful applied statistical properties and can be modified to handle some very complex distributions.

A Note on Posterior Intervals

The posterior interval (also called a credible interval or credible region) provides a very intuitive way to describe the measure of uncertainty.  Unlike a confidence interval (discussed in one of my previous posts), a credible interval does in fact provide the probability that a value exists within the interval.  With this interval it is based on calculating the probability of different values given the data.  The graph above shows the different values (identified as theta) as well as the simulated posterior interval limits (alpha=.05 in black and .01 in light gray).  In other words the probability that theta is a member of the 95% credible interval is 0.95 (written as $P\left( \theta \in CI \right)$ = 0.95).

The Prior and Posterior Distribution: An Example

The code to run the beta.select() function is found in the LearnBayes package.  This is a great function because by providing two quantiles one can determine the shape parameters of the Beta distribution.  This is useful to find the parameters (or a close approximation) of the prior distribution given only limited information.  If additional quantiles are known then they can be incorporated to better determine the shape parameters of the Beta distribution.

library(LearnBayes)
Q = data.frame(
quantile=c(
median=0.5,
maximum=0.99999,
minimum=0.00001),
prior=c(
median=0.85,
maximum=0.95,
minimum=0.60)
)

optimalBeta = function(Q) {
q1q = Q$quantile[1] q1p = Q$prior[1]
q2q = Q$quantile[2] q2p = Q$prior[2]
q3q = Q$quantile[3] q3p = Q$prior[3]

# find the beta prior using quantile1 and quantile2
q.med = list(p=q1q, x=q1p)
q.max = list(p=q2q, x=q2p)
q.min = list(p=q3q, x=q3p)

# prior parameters using median and max, and median and min
prior.A = beta.select(q.med,q.max)
prior.B = beta.select(q.med,q.min)

prior.Aa = prior.A[1]
prior.Ab = prior.A[2]

prior.Ba = prior.B[1]
prior.Bb = prior.B[2]

## find the best possible beta prior
## Set a start and stop point range to find the best parameters
if (prior.Aa < prior.Ba) {
start.a = prior.Aa
stop.a = prior.Ba
} else {
start.a = prior.Ba
stop.a = prior.Aa
}

if (prior.Ab < prior.Bb) {
start.b = prior.Ab
stop.b = prior.Bb
} else {
start.b = prior.Bb
stop.b = prior.Ab
}
seq.a = seq(from=start.a, to=stop.a, length.out=1000)
seq.b = seq(from=start.b, to=stop.b, length.out=1000)

seq.grid = expand.grid(seq.a, seq.b)

prior.C.q1 = qbeta(q1q, seq.grid[,1], seq.grid[,2])
prior.C.q2 = qbeta(q2q, seq.grid[,1], seq.grid[,2])
prior.C.q3 = qbeta(q3q, seq.grid[,1], seq.grid[,2])

## Different distance measurements, manhattan, euclidean, or otherwise.
## It would be interesting to run a simulation to measure a variety of distance measurements.
prior.C.delta = abs(prior.C.q1 - q1p) + abs(prior.C.q2 - q2p) + abs(prior.C.q3 - q3p)
## prior.C.delta = sqrt( (prior.C.q1 - q1p)^2 + (prior.C.q2 - q2p)^2 + (prior.C.q3 - q3p)^2 )

optimize.seq = cbind(seq.grid, prior.C.q1, prior.C.q2, prior.C.q3, prior.C.delta)

## Minimize the delta, if the min-delta is not unique then choose the first occurence
best.a = optimize.seq[,1][ optimize.seq[,6]==min(optimize.seq[,6])][1]
best.b = optimize.seq[,2][ optimize.seq[,6]==min(optimize.seq[,6])][1]

return(list(a=best.a,b=best.b))
}

prior.dist = optimalBeta(Q)

##########################################################
## Take a look at only the prior
##########################################################
curve(dbeta(x,prior.dist$a,prior.dist$b)) # plot the prior
abline(v=Q$prior[1]) ########################################################## ## Take a look at only the likelihood with given successes ########################################################## calcLikelihood = function(successes, total){ curve(dbinom(successes,total,x)) # plot the likelihood } calcLikelihood(45, 50) ## e.g. 45/50 sucesses ## calculate some properties of the Beta distribution calcBetaMode = function(aa, bb) { beta.mode = (aa - 1)/(aa + bb - 2) return(beta.mode) } calcBetaMean = function(aa, bb) { beta.mean = (aa)/(aa + bb) return(beta.mean) } calcBetaVar = function(aa, bb) { beta.var = (aa * bb)/(((aa + bb)^2) * (aa + bb + 1)) return(beta.var) } calcBetaMedian = function(aa, bb) { beta.med = (aa-1/3)/(aa+bb-2/3) return(beta.med) } calcBetaSkew = function(aa, bb) { beta.skew = ( 2*(bb-aa)*sqrt(aa+bb+1) ) /( (aa+bb+2)/sqrt(aa+bb) ) return(beta.skew) } ########################################################## ## Take a look at the prior, likelihood, and posterior ########################################################## priorToPosterior = function(successes, total, a, b) { ## Note the rule of succession likelihood.a = successes + 1 likelihood.b = total - successes + 1 ## Create posterior posterior.a = a + successes; posterior.b = b + total - successes theta = seq(0.005, 0.995, length = 500) ## Calc density prior = dbeta(theta, a, b) likelihood = dbeta(theta, likelihood.a, likelihood.b) posterior = dbeta(theta, posterior.a, posterior.b) ## Plot prior, likelihood, and posterior ## Can be used to scale down the graph if desired. ## However, the density is different for each prior, likelihood, posterior m.orig = apply( cbind(prior, likelihood, posterior), 2, max) m = max(c(prior, likelihood, posterior)) plot(theta, posterior, type = "l", ylab = "Density", lty = 2, lwd = 3, main = paste("Prior: beta(", round(a,2), ",", round(b,2), "); Data: B(", total, ",", successes, "); ", "Posterior: beta(", round(posterior.a,2), ",", round(posterior.b,2), ")", sep=""), ylim = c(0, m), col = 1) lines(theta, likelihood, lty = 1, lwd = 3, col = 2) lines(theta, prior, lty = 3, lwd = 3, col = 3) legend("topleft",y=m, c("Prior", "Likelihood", "Posterior"), lty = c(3, 1, 2), lwd = c(3, 3, 3), col = c(3, 2, 1)) prior.mode = calcBetaMode(a, b) likelihood.mode = calcBetaMode(likelihood.a, likelihood.b) posterior.mode = calcBetaMode(posterior.a, posterior.b) prior.mean = calcBetaMean(a, b) likelihood.mean = calcBetaMean(likelihood.a, likelihood.b) posterior.mean = calcBetaMean(posterior.a, posterior.b) prior.med = calcBetaMedian(a, b) likelihood.med = calcBetaMedian(likelihood.a, likelihood.b) posterior.med = calcBetaMedian(posterior.a, posterior.b) prior.var = calcBetaVar(a, b) likelihood.var = calcBetaVar(likelihood.a, likelihood.b) posterior.var = calcBetaVar(posterior.a, posterior.b) prior.skew = calcBetaSkew(a, b) likelihood.skew = calcBetaSkew(likelihood.a, likelihood.b) posterior.skew = calcBetaSkew(posterior.a, posterior.b) print(paste("Mode: prior=",prior.mode,"; Likelihood=",likelihood.mode,"; Posterior=",posterior.mode)) print(paste("Mean: prior=",prior.mean,"; Likelihood=",likelihood.mean,"; Posterior=",posterior.mean)) print(paste("~Approx Median (for a and b > 1): prior=",prior.med,"; Likelihood=",likelihood.med,", for Posterior=",posterior.med)) print(paste("Var: prior=",prior.var,"; Likelihood=", likelihood.var,"; Posterior=",posterior.var)) print(paste("Skewness: prior=",prior.skew,"; Likelihood=",likelihood.skew,"; Posterior=",posterior.skew)) return(list(a=posterior.a,b=posterior.b)) } posterior.out = priorToPosterior(25,50, prior.dist$a, prior.dist$b) # 25/50 is current data beta.sim = rbeta(1000000,posterior.out$a, posterior.out\$b)
abline(v=quantile(beta.sim, prob=c(.05/2, 1-.05/2)), col='#000000', lwd=2)
abline(v=quantile(beta.sim, prob=c(.01/2, 1-.01/2)), col='#EEEEEE', lwd=2)

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text:
# Differential Forms

## Initialization

Run the following code to initialize $\LaTeX$ output and load some macros, either by clicking on "Evaluate" or by typing Shift+Enter.

Now enter a line element in the box below, adapting the given code as needed. First declare any parameters or functions, then provide a list of (orthogonal) coordinates using the MakeC command as shown below (note the double parentheses). Finally, use the Makeg command to enter both the signature (number of $-$ signs) and the diagonal coefficients ($g_{ii}$) of the line element $ds^2=g_{ij}dx^idx^j$. The result should be the volume element $\omega$.

## Code

Now run the Sage code in the boxes below. (You can also enter your own code.)


text:
# P Natural Water Contains a Small Amount of Tritium ( 3 1 H ). this Isotope Beta-decays with a Half-life of 12.5 Years. a Mountaineer While Climbing Towards a Difficult Peak Finds Debris - Physics

Sum

Natural water contains a small amount of tritium (""_1^3H). This isotope beta-decays with a half-life of 12.5 years. A mountaineer while climbing towards a difficult peak finds debris of some earlier unsuccessful attempt. Among other things he finds a sealed bottled of whisky. On returning, he analyses the whisky and finds that it contains only 1.5 per cent of the ""_1^3H radioactivity as compared to a recently purchased bottle marked '8 years old'. Estimate the time of that unsuccessful attempt.

#### Solution

Given:
Half-life time of tritium, T_"1/2" = 12.5 years

Disintegration constant, lambda = 0.693/12.5 per year

Let A0 be the activity, when the bottle was manufactured.
Activity after 8 years (A) is given by

A = A_0e^((-0.693)/(12.5) xx 8)  .....(1)

Let us consider that the mountaineering had taken place years ago.
Then, activity of the bottle (A') on the mountain is given by

A' = A_0e^(-lambdat)

Here, A' = (Activity of the bottle manufactured 8 years ago) × 1.5 %

A' = A_0e^((-0.693)/(12.5) xx 8) xx 0.015 ...(2)

Comparing (1) and (2)

(-0.693)/12.5 t = (-0.6931 xx 8)/12.5 + "In"   [0.015]

⇒ (-0.693)/12.5 t = (-0.693)/12.5 xx 8 - 4.1997

⇒ 0.639  t = 58.040

⇒ t = 83.75 years

Is there an error in this question or solution?

#### APPEARS IN

HC Verma Class 11, Class 12 Concepts of Physics Vol. 2
Chapter 24 The Nucleus
Q 34 | Page 443


text:
ï»¿ CBSE Important Questions(Short questions) for Heredity and Evolution

# CBSE Important Questions(Short questions) for Heredity and Evolution

Q1. What is speciation?
Q2. What are genes and where they are located?
Q3. An organ like a wing in birds is an advantage to the organism. Did they appear in different stages or were formed due to single sudden change in them?
Q4. What happens when a dead insect is caught in hot mud? Give two points?
Q5. If it said that, ‘there is no real progress in the idea of evolution, then what evolution is’? Explain.
Q6. Who is the father of Genetics?
Q7. Write the scientific name of men and garden pea?
Q8. What conclusion was drawn by Mendel after obtaining F2 progeny in a dihybrid cross?
Q9. What is the study of fossils known as?

Q10. Identification of human races was done by a particular trait. Name the trait?
Q11. Explain what happen when in a group of organism, if organisms are not able to survive in changing environmental conditions?
Q12. Explain the two ways by which individuals with a particular trait may increase n a population?
Q13. What is the outcome of genetic drift between two sub. Populations.
Q14. Sometimes, accidently a dead body  or its part get buried under depositing sediments and are preserved. These are fossils. How can the estimation of the age of fossils be done?
Q15. Why are wings of butterfly ad wings of bat considered analogous? Give reason.
Q16. What is the significance of variation?
Q17. What is natural selection?
Q18. What are dominant and recessive traits?
Q19. Only variations that confer an advantage to an individual organism will survive in a population. Do you agree with this statement? Why or why not?
Q20. Why did Mendel select pea plant for conducting his experiment on inheritance?
Q21. The sex of the children is determined by what they inhent from their father and not their mother. Justify.
Q22. What are acquired characters? Give example.
Q23. Explain the terms analogous and homologous organs with examples?
Q24. How gene is responsible for controlling a trait? Explain by giving an example.
Q25. Define Genetics.

link to this page by copying the following text

Given below are the links of some of the reference books for class 10 Science.

You can use above books for extra knowledge and practicing different questions.

### Practice Question

Question 1 Which among the following is not a base?
A) NaOH
B) $NH_4OH$
C) $C_2H_5OH$
D) KOH
Question 2 What is the minimum resistance which can be made using five resistors each of 1/2 Ohm?
A) 1/10 Ohm
B) 1/25 ohm
C) 10 ohm
D) 2 ohm
Question 3 Which of the following statement is incorrect? ?
A) For every hormone there is a gene
B) For production of every enzyme there is a gene
C) For every molecule of fat there is a gene
D) For every protein there is a gene

Note to our visitors :-

Thanks for visiting our website.
DISCLOSURE: THIS PAGE MAY CONTAIN AFFILIATE LINKS, MEANING I GET A COMMISSION IF YOU DECIDE TO MAKE A PURCHASE THROUGH MY LINKS, AT NO COST TO YOU. PLEASE READ MY DISCLOSUREÂ FOR MORE INFO.


text:
# Gaussian Curvature

## Homework Statement

Is the gaussian curvature at a point on the surface $$g(x,y)=xy$$
$$\frac{1}{(x^2+y^2+1)^2}?$$

## Homework Equations

shape operator:
$$S(\textbf{x})=-D_\textbf{x}\hat{\textbf{n}}=\frac{\partial (n_x, n_y)}{\partial (x,y)}$$

Gaussian Curvature = $$|S(\textbf{x})|$$

$$\hat{\textbf{n}}=\frac{\nabla g}{\|\nabla g\|}$$

## The Attempt at a Solution

I basically plugged stuff into the above equations. I'm not sure if they're all correct.


text:
## 材料和方法

### 用深度学习的方法分割细胞核

Overview of the nuclei segmentation procedure. $DRAN_{BL}$ and $DRAN_{BD}$ are the models for nuclei blob detection and boundary detection, respectively.

#### 多尺度聚合

Multiscale Deep Residual Network (MDRAN) architecture. MDRAN composes of 3 DRANs at 3 scales (x0.5, x1.0, x2.0) and a decoder (in dash rectangle), aggregating 3 scales together and generating a segmentation map at x1.0 scale. In the decoder, the convolution block [128, 5 × 5, 256] denotes [128 input channels, 5 × 5 kernel, 256 output channels].

### 两步的自动化方法对整个载片图像进行分类

Overview of the NSCLC classification framework. (A) Workflow for training the neural network to classify input patches as either non-diagnostic (ND), lung adenocarcinoma (LUAD), or lung squamous cell carcinoma (LUSC). (B) Workflow for processing the WSIs within the test set to obtain probability maps for each class. (C) Workflow for the random forest regression model. Features are extracted from LUAD and LUSC probability maps and then fed as input into the random forest model. SN stands for stain normalization by method of Reinhard et al. (2001).

## 结果

### 分割效果评估方法

DICE系数衡量了实际上和算法输出之间的重叠部分，但是没有考虑到分离和融合的情况，“分离”是本是一个核，算法输出是多个核，“融合”是本是多个核，算法输出一个核。使用DICE系数，一个算法把两个接触或者重叠的核看作一个和正确的分割为两个，具有相同的DICE。DICE-2可以解决“分割”的问题。伪代码如下：

Q and P are the sets of segmented objects (nuclei). The two DICE coefficients were computed for each image tile in the test dataset. The score for the image tile was calculated as the average of the two dice coefficients. The score for the entire test dataset was computed as the average of the scores of all the image tiles.

### 分割细胞核的深度学习方法实验性评估

1.考虑到补丁的宽度和高度，在[-0.05,0.05]范围内的随机垂直和水平位移
2.在[-45°，45°]度范围内随机旋转
3.随机垂直和水平翻转，概率为0.5
4.随机剪切，强度范围为[-0.4π，0.4π]
5.随机调整大小，比率范围为[0.6,2.0]

Image patch generation. To avoid zero-padding in augmentation, a patch of size 200 × 200 is first provided. Subsequently, the center region of 102 × 102 is cropped and fed into the network as input. For an input of size 102 × 102, the network provides a segmentation map of size 54 × 54.

Head-to-head comparison between MDRANBL and DRANBL on the test set. Test images are ordered by the ascending order of MDRAN DICE_1. The shaded area indicates that the images were scanned at 20x magnification.

Examples of nuclei segmentation via the multiscale aggregation (MDRANBL+DRANBD) and single scale (DRANBL+DRANBD) approach. The images from top to bottom are the 1st, 11th, 20th, and 25th image tile in the test set.

Examples of correct and incorrect nuclei segmentation. Our method (Bottom) is able to distinguish the boundary of the non-highly overlapping nuclei fairly well but (Top) fails on the highly overlapping nuclei with disproportionate stains.

### 整个载玻片图像的分类

Test WSIs with overlaid probability maps. Blue/purple indicates a region classified as diagnostic LUAD, green indicates a region classified as diagnostic LUSC, yellow/orange refers to a region classified as non-diagnostic.


text:
## Tutorial: Javascript Promises: There and Back Again in Half the Time

If you are familiar with javascript, the chances are that you have found this post. I found it to be a great introduction to promises in javascript, but I also found the post to be rather long and not good as a reference. This post, however is an attempt to condense the information found in the tutorial from html5rocks (Update! HTML5 Rocks has been turned into the Google Developers Web Fundamentals) into a nice neat package that is not only useful as a tutorial, but also useful as a reference.

### The Problem and the Solution

When performing asynchronous tasks in javascript, you usually have to use callbacks and event listeners. This can quickly get unwieldy and difficult to understand. Promises were invented to rectify this problem. They allow you to chain function calls asynchronously - which generally cleans up your code and makes it easier to read:

function get(url)
{
return new Promise(function(resolve, reject) { //important!
var ajax = new XMLHttpRequest();
if(ajax.status >= 200 && ajax.status < 300)
resolve(ajax.response); //return the response
else
reject(ajax.response);
}
});
}

The important bit is the returning of a promise instead of actually performing the action requested. When creating a promise, a function is taken as the only argument. This function is then called with 2 arguments: a function that should be called upon success (usually called resolve()), and a function that should be called upon failure (usually called reject()).

Any number of asynchronous actions can be performed before either resolve() or reject() are called, but I would adivse that you put one async call in each promisified function.

The resolve() and reject() functions take a single argument: the value that should be passed onto the next function in the chain.

### and then....

Once a promise has been returned, an API is exposed that allows you to chain functions together in a long string through the function then():

//                         then() adds a function to the chain
//                                        |--------v
get("https://starbeamrainbowlabs.com/humans.txt").then(function(response) {
//we got a response :D
console.info("Got response", response);
}, function(error) {
//something went wrong :(
console.error("Something went wrong:", error);
});

then() takes a pair of functions as it's arguments. The first function is called upon success (and passed the value that was passed to resolve()), and the second (optional) function is called upon failure (and passed the value that was passed to reject()).

### Repetition

The best part of javascript promises is the ability to chain then() as many times as you like:

get("https://example.com/urls.txt").then(function(text) {
return text.split("\n");
}).then(function(lines) {
lines.forEach(function(url) {
get(url).then(function(response) {
console.log("got", url, "with text", response);
});
});
});

### Infinite chaining

The problem with the above is that lots of ajax calls a being made at once. If you used this in your next big web app, it could bring your server down having all the requests being sent out at once.

There is, however, a solution to this too. We can process each value in an array one at a time using Array.reduce():

get("https://example.com/urls.txt").then(function(text) {
return text.split("\n");
}).then(function(lines) {
lines.reduce(function(sequence, url) {
return squence.then(function() {
return get(url);
}).then(function(response) {
console.info("got", response);
});
}, function() { return new Promise(function(resolve) { resolve(); }); });
});

In this way, a new promise is created that resolves straight away (you can also use the Promise.resolve function is a shortcut here, but some implementations - especially Node.JS based ones - don't support this), and a promise for each url are then tacked onto the end of it. These are then called in a row - once th eprevious ajax request has completed, the next one will be sent. This can also be helpful when you are fetching content from a list of urls and you want them to be kept in order.

### Conclusion

Promises are cool - they let you simplify complex asynchronous code. You can chain them with then() as many times as you like - and you can use Array.reduce() to chain calls that iterate over each value in an array.


text:
# No bollocks, just rings

Today (one of) the goals of this blog will be neglected, and we’ll focus solely on its URL. No $\mathbb{F}_{1}$ or Riemann hypothesis, just some old-fashioned ring theory.

Misplaced Poetics

I’d like write up one of (Jacobson-) Herstein’s commutativity theorems in noncommutative ring theory. It’s a beautiful testament to the power of subdividing rings into nontrivial classes, and is filled to the brim with small, simple, but extremely elegant ideas. Part of the theorem’s appeal stems from the fact that once confronted with its statement, you’re bound to ask: who cares?! It seems so useless, and that somehow makes it all the more fun. Of course, there was a good reason for proving the theorem, but I’ll just leave you to ponder some applications. I’m in a “l’art pour l’art”-mood, hoping Poe doesn’t turn in his grave.

Trivial?

Theorem. A ring $R$ is commutative iff for any $a,b \in R$, there exists an integer $n(a,b)>1$ such that $(ab-ba)^{n(a,b)}=ab-ba$.

Of course, your first idea, like everyone else’s, is that this can be proven by (possible) extreme amounts of formula bashing. I encourage you to try this, as I have, and get very tired (and a tad bit frustrated) after wasting the better part of a Friday morning (and wacking half a tree). If you do manage to find a proof by pure calculation, let me know! The way we’ll tackle the theorem is by a reduction argument along the following lines:

• If $R$ is a division ring: see the next paragraph
• If $R$ is a left primitive ring: reduce to the previous case
• If $R$ is a semiprimitive ring: reduce to the previous case
• If $R$ is a (general) ring: reduce to the previous case

Just lovely, isn’t it.

Thé lemma

Crucial for proving the first part of the theorem is the following lemma, which will be applied in the next paragraph.

Lemma. Let $D$ be a division ring with nonzero characteristic. If $a$ is a non-central, periodic element in $D^{*}$, then there exists an additive commutator $b$ in $D^{*}$ such that $bab^{-1}=a^{i} \neq a$, for some $i>0$.

Taking $F_{p}$ to be the prime subfield of $D$, the $F_{p}$-algebra generated by $a$ is a finite subfield of $D$, of dimension $n$ over $F_{p}$, which we’ll denote $K$. The order of this field is $p^{n}$, and $a^{p^{n}}=a$. Since $a$ isn’t central, the inner derivation $\delta$, defined by $\delta(x)=ax-xa$, is non-zero. You can easily check that it is however a $K$-linear mapping of the $K$-vector space $D$. Using the Frobenius, one sees that $\delta^{p^{n}}=\delta$. Since any element $b \in K$ satisfies $b^{p^{n}}=b$, the following equation holds true $$t^{p^{n}} -t= \prod_{b \in K}(t-b) \in K[t].$$ Substituting $\delta$ into this equation, and remembering that $\delta \neq 0$ and all mono’s are left cancellable, there has to exist a $b_{0} \in K^{*}$ such that $\delta-b_{0}$ is not a mono. This means $\delta$ has an eigenvector $d$ with eigenvalue $b_{0}$. Using the definition of $\delta$, it follows that $dad^{-1}=a-b_{0} \in K \ \backslash \{a\}$. Since $a$ and $dad^{-1}$ have the same order in the cyclic group $K^{*}$, they generate the same subgroup, and $dad^{-1}=a^{i} \neq a$. To make sure $d$ is an additive commutator, replace it by $\delta(d)$, and you’ll see that the equation still holds up.

Division rings

Supposing $R$ is a division ring which doesn’t equal its centre $F$, you can easily see that there must exist an additive commutator which isn’t central, say $a=bb’-b’b$. Taking an element $c$ in $F^{*}$, a quick calculation shows that $ca$ is an additive commutator which can’t be central. The assumptions then imply that there is a number $k$ such that $1=a^{k}=(ca)^{k}=c^{k}a^{k}$, so any element in $F^{*}$ is periodic, and $D$ has non-zero characteristic. Using the theorem in the previous paragraph on $a$, an additive commutator $y$ exists, such that the group generated by $a$ and $y$ is a finite, periodic subgroup of $D^{*}$. This means the group has to be cyclic, which contradicts the statement $yay^{-1} \neq a$, and $R$ is commutative.

The proof of the pudding …

Let’s finish the proof. Suppose $R$ is a left primitive ring. The structure theorem for left primitive rings (basically a revamped version of Jacobson & Chevalley’s density theorem), says $R$ is isomorphic to a matrix ring over a skew field $D$, or $R$ has an infinite number of subrings $R_{m}$, each one having $M_{m}(D)$ as a quotient. Supposing $m>1$, and denoting by $E_{ij}$ the matrix with a $1$ in the $ij$-place and zeroes everywhere else, the obvious identity $$E_{11} E_{12} – E_{12} E_{11}=E_{12}$$ gives us a contradiction (just raise it to a power greater than $1$). This means $R \cong D$, and we reduced it to a previous case.

If our ring is semiprimitive, the Jacobson radical, denoted $J(R)$, is zero. Given a left primitive ideal $M_{i}$, take the quotient ring $R/M_{i}$, which is obviously left primitive, and inherits the property in the theorem because it’s a surjective image of $R$. This means each one of these rings is commutative. Now $R$ can be embedded in the product of all these rings, since the Jacobson radical, which is equal to the intersection of all left primitive ideals, is zero. Thus $R$ is commutative.

For a random ring, $R/J(R)$ is semiprimitive, and thus commutative. For $a, b$ in the ring, we know there exists an $n$ such that $(ab-ba)(1-(ab-ba)^{n-1})=0$. Since $ab-ba$ sits in $J(R)$, we know $(1-(ab-ba)^{n-1})$ is invertible, and $ab-ba=0$. Voilà!

More of this fun stuff can be found in Lam’s A first course in noncommutative rings and Lectures on modules and rings.

## 2 thoughts on “No bollocks, just rings”

1. I have a proof of this for n=1. Using a formula rather than, l think, this way u have done it.


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# Is the energy of an orbital dependent on temperature?

In the Schrodinger Equation's solution for electron orbital energy levels of the hydrogen atom there is no temperature dependency. $$E_n = - \frac{m_{\text{e}} \, e^4}{8 \, \epsilon_0^2 \, h^2 \, n^2}$$

Maybe this is due to ignoring the effects of temperature when deriving the Hamiltonian. No sources I have seen mention any assumptions about temperature. Are electron orbital energies dependent on temperature in spite of this common equation?

If you have hydrogen gas heated to near ionization then it should take less than 13.6 eV to remove an electron.

For some reason Im having trouble confirming any of this thru google. Is it 13.6 eV no matter what temperature it is or is there a temperature dependency?

• Dec 3, 2020 at 4:37
• Temperature is undefined for a single atom. Dec 3, 2020 at 9:55
• @my2cts not for a classical atom, but single atom quantum states with well-defined temperatures do exist. Of course, these are not eigenfunctions of the hamiltonian Dec 3, 2020 at 13:54
• @Tristan What is a classical atom? How can a single atom state exist that is not an eigenfunction? Do you mean an incoherent state described by a density matrix? Dec 3, 2020 at 20:21
• true, classical atom doesn't really make sense. My point was mostly that whilst classically the temperature of a single object doesn't make sense, in a quantum system it can Dec 4, 2020 at 9:41

## 4 Answers

Maybe this is due to ignoring the effects of temperature when deriving the Hamiltonian.

How are you defining heat at the quantum level? The Schrodinger equation describes how objects behave at the quantum level and heat describes a form energy which is transferred between objects of different temperatures. Temperature is a macroscopic quantity and not microscopic. The equation above describes the energy of electrons in a specific shell and these electrons can change energy by the absorption or emission of photons, and photons do not possess temperature.

No sources I have seen mention any assumptions about temperature.

For exactly those reasons.

Are electron orbital energies dependent on temperature in spite of this common equation?

No they are not.

If you have hydrogen gas heated to near ionization then it should take less than 13.6 eV to remove an electron.

No. Heating hydrogen will not cause the absorption of photons which is needed to ionise hydrogen (other methods for ionisation of atoms exist, but I’m speaking in the context of this question). Moreover, the hydrogen will be ionised upon the absorption of photon with this energy and not any less. This is the crux of the term energy and other quantities are quantised at a microscopic level which gave rise to quantum mechanics.

Is it 13.6 eV no matter what temperature it is or is there a temperature dependency?

Once again temperature is not relevant here. For ionisation to occur, a photon must be absorbed (there are other ways to ionise atoms too). So to answer your question, there is no such dependence.

• Maybe it would be instructive to mention that the stability of spectra for a given atom is what allows us to measure a doppler shift from stars moving away or towards us. astro.ucla.edu/~wright/doppler.htm Dec 3, 2020 at 8:42
• It is because the spectrum of a given atom, is independent of temperature that the doppler shift of the spectrum can be used to measure the velocity of the star.. If the spectrum could be variable then it could not be used for such a measurement. Dec 3, 2020 at 9:03
• If you heat up hydrogen such that kT >> ground state energy, you can in fact effect the spectra (you will of course will ionize the atoms into a plasma). Dec 4, 2020 at 9:07
• It might help to also add that temperature is a property of an ensemble, reflecting the translational energy of particles, not the "internal" energy of electron states. Dec 4, 2020 at 22:39

In addition to @Dr jh's answer, I would like to add that the effect of temperature on the spectra of atoms is part of what is known as as Doppler broadening of the lines.

In atomic physics, Doppler broadening is the broadening of spectral lines due to the Doppler effect caused by a distribution of velocities of atoms or molecules. Different velocities of the emitting particles result in different Doppler shifts, the cumulative effect of which is the line broadening. This resulting line profile is known as a Doppler profile. A particular case is the thermal Doppler broadening due to the thermal motion of the particles. Then, the broadening depends only on the frequency of the spectral line, the mass of the emitting particles, and their temperature, and therefore can be used for inferring the temperature of an emitting body.

Italics mine

• Btw, this was indeed instructive and a great response showing that temperature is independent of temperature via an empirical example which is independent of any opinions. Plus I didnt know of this very cool phenomenon. Dec 13, 2020 at 17:34

I think this question belies a misunderstanding of the nature of temperature.

Temperature is not an input to fundamental physical laws, it is something that comes out of the physical laws when applied to a large number of objects. There is no explicit temperature in the Schrödinger equation for a hydrogen atom. The concept of temperature only emerges when you consider the Schrödinger equation for a large number of hydrogen atoms that can exchange energy with each other.

The Schrödinger equation (or Newton's laws or Maxwell's equations) establish the ground rules for how atoms and molecules (or any other objects) behave. However, it's usually difficult to work out what these rules predict for anything more than simple systems. The point of statistical mechanics is to predict the average behavior of a large number of such simple systems under the constraints provided by the ground rules (fundamental physical laws). Temperature is a concept that emerges from the statistics of large number of degrees of freedom that can exchange energy. Again, it is not an input to the Schrödinger equation or Newton's laws, but a consequence of them!

Temperature is a macroscopic property. Individual atoms don't have temperature. An analogy would be income inequality: it makes sense to ask how much income inequality a country or state or city has, but it doesn't make sense to ask how much income inequality a single person has. A simplified version of what temperature is is that it is a measure of how much difference between velocities there is between atoms. Just as it doesn't make sense to ask how much difference between incomes there is when you're talking about a single person, it doesn't make sense to ask how much difference in velocities there is if you're talking about a single atom.

If anything, heating a gas will increase the ionization energy. The ionization energy is given for the frame of reference of the atom. If an atom is moving at a high speed relative to us, then the ionization energy in our frame energy is increased.

It's not clear what you mean by "heated to near ionization". According to this, 13.6 eV corresponds to 158 thousand degrees Kelvin. So if you were to heat hydrogen to that temperature, you would see ionization from the collisions. But that doesn't mean the ionization energy is decreased, it means the energy is met.


text:
# Ideals in a multiplier algebra on the ball

@article{Clouatre2015IdealsIA,
title={Ideals in a multiplier algebra on the ball},
author={Raphael Clouatre and Kenneth R. Davidson},
journal={arXiv: Operator Algebras},
year={2015}
}
• Published 10 November 2015
• Mathematics
• arXiv: Operator Algebras
We study the ideals of the closure of the polynomial multipliers on the Drury-Arveson space. Structural results are obtained by investigating the relation between an ideal and its weak-$*$ closure, much in the spirit of the corresponding classical facts for the disc algebra. Zero sets for multipliers are also considered and are deeply intertwined with the structure of ideals. Our approach is primarily based on duality arguments.
10 Citations
• Mathematics
• 2023
. Let H be a space of analytic functions on the unit ball B d in C d with multiplier algebra Mult( H ) . A function f ∈ H is called cyclic if the set [ f ] , the closure of { ϕf : ϕ ∈ Mult( H ) } ,
• Mathematics
Transactions of the American Mathematical Society
• 2020
The main objects under study are quotients of multiplier algebras of certain complete Nevanlinna--Pick spaces, examples of which include the Drury--Arveson space on the ball and the Dirichlet space
• Mathematics
Journal of the London Mathematical Society
• 2022
In the context of Dirichlet type spaces on the unit ball of Cd$\mathbb {C}^d$ , also known as Hardy–Sobolev or Besov–Sobolev spaces, we compare two notions of smallness for compact subsets of the
• Mathematics
Mathematische Annalen
• 2022
Henkin functionals on non-commutative C-algebras have recently emerged as a pivotal link between operator theory and complex function theory in several variables. Our aim in this paper is
• Quanlei Fang
• Mathematics
Interpolation and Realization Theory with Applications to Control Theory
• 2019
The Drury–Arveson Space, as a Hilbert function space, plays an important role in multivariable operator theory. We give a brief survey of some aspects of the multipliers on the space.
. We prove a noncommutative version of Bishop’s peak interpolation-set theorem.
• Mathematics
Handbook of Analytic Operator Theory
• 2019
We give a brief review of the Drury-Arveson space and its associated operator theory. We then survey those recent results on that space where analytical methods play a predominant role. A number of
• Mathematics
• 2020
We study the multiplier algebras $A(\mathcal{H})$ obtained as the closure of the polynomials on certain reproducing kernel Hilbert spaces $\mathcal{H}$ on the ball $\mathbb{B}_d$ of $\mathbb{C}^d$.

## References

SHOWING 1-10 OF 27 REFERENCES

• W. Rudin
• Mathematics
• 1957
Let K and C be the closure and boundary, respectively, of the open unit disc U in the complex plane. Let be the Banach algebra whose elements are those continuous complex functions on K which are
• Mathematics
• 1999
A free semigroup algebra is the weak operator topology closed algebra generated by a set of isometries with pairwise orthogonal ranges. The most important example is the left regular free semigroup
• Mathematics
• 2015
We consider a number of examples of multiplier algebras on Hilbert spaces associated to discs embedded into a complex ball in order to examine the isomorphism problem for multiplier algebras on
• Mathematics
• 2013
We study the structure of bounded linear functionals on a class of non-self-adjoint operator algebras that includes the multiplier algebra of every complete Nevanlinna-Pick space, and in particular
Preliminaries.- The Automorphisms of B.- Integral Representations.- The Invariant Laplacian.- Boundary Behavior of Poisson Integrals.- Boundary Behavior of Cauchy Integrals.- Some Lp-Topics.-
Let Bn be the open unit ball of C , and let H(B^) be the algebra of bounded analytic functions on Bn, endowed with the uniform norm on Bn. The ball algebra is the space A(Bn) = C(Bn) n //°°(Z?ra),
Given a variety of algebras V, we study categories of algebras in V with a compatible structure of uniform space. The lattice of compatible uniformities of an algebra, Unif A, can be considered a
In this paper, we explore certain Banach spaces of analytic functions. In particular, we study the space A -I, demonstrating some of its basic properties including non-separability. We ask the


text:
## Geometry: Common Core (15th Edition)

3y = 120$^{\circ}$ (y - 15) = 25$^{\circ}$
All angles in a triangle must add up to 180∘. We can therefore add the three given angle measures together, set them equal to 180∘, and then solve for y. 35 + 3y + (y-15) = 180 35 - 35 + 3y + (y-15) = 180 - 35 3y + (y-15) = 145 (We can re-write (y-15) as y - 15.) 3y + y - 15 = 145 3y + y - 15 + 15 = 145 + 15 3y + y = 160 We can now combine like-terms and then solve for y. 4y = 160 4y = 160 y = 40$^{\circ}$ We can now solve for each individual angle measure. The angle labelled 3y really equals (3 x 40) = 120$^{\circ}$. The angle labelled (y - 15) really equals (40 - 15) = 25$^{\circ}$.


text:
## Asterisk-dagger footnote in the title page only (footmisc package)

Hi everybody,

I have a problem in using the footmisc package. What I need to do is to use a asterisk-dagger footnote for the title of the paper, and then the normal numerated footnotes. But by using the footmisc package, it changes all the footnotes with symbols..

is there another way to proceed or I am just using the package wrongly?

Thanks,

Stefano Lombardi

### The footmisc package is, I

The footmisc package is, I believe, designed to modify the footnotes in the body of the document.  The front matter footnotes are handled separately by the typesetting specification.  The typesetting specification you are using and the class options can change the typeset numbering of footnotes.  For example, if you use the standard LaTeX article with footnotes in the front matter, the footnotes are marked using symbols, unless you select to use a title page.  Then the front matter footnotes are treated the same as the footnotes in the body of the document.

Overriding the typesetting specification to change the appearance of the footnote numbering is still possible, but gets a little tricky.  Post a sample document and it should be possible to modify it to get what you want.

### Hi George, thanks for the

Hi George,

thanks for the immediate reply. I attached the sample file.

Thank you very much,

Stefano

### The attached seems to work.

The attached seems to work.  I added a command in the document preamble to change footnote numbering to use symbles, then in the body of the document I added in a TeX field the command to use arabic numbering.

See http://www.mackichan.com/techtalk/how/fnsymbol.html for some information about footnote numbering.

### Hi George, I follow the

Hi George,

I follow the sample.tex you attached: the first footnote has an asterisk; but the second footnote begins with number 2 instead of 1. I am just wondering how I can make it possible that the rest of footnotes start with 1.
I checked the sample.tex, the rest footnotes begin with 1; and it is very strange that I can only see one in the sample.tex file, but I have two pages for the PDF file converted from it. I am just wondering where the first page comes from? It is not in the sample.tex file, right?

Thanks,
Wu

### Dear George, If you add

Dear George,
If you add another footnote into the document you sent and try to make the second one arabic numberred, you get the attached file. So, the second footnote has a marking of 2 but it is shown with a symbol in the footnote.

Is there any way to have the first footnote to be an asteriks and the rest numberred beginning from 1. It seems to me that if one wants to have two footnotes on the title page, your method does not work.

### Dear George, How can I have a

Dear George,

How can I have a marked footnote along with a numberred footnote on the title page? Moreover, the first footnote in the body should continue the numbering on the title page (abstract must be on the title page, not on a separate page).

In sum, I am trying to have the following:

Title page: (footnote *, footnote 1)
Body: (footnote 2, footnote 3, ... and so on)

I attached my (imperfect) sample SWP file.

Thank you,
Eren

### I can't find a way to do what

I can't find a way to do what you want.  I was able to get an arabic footnote mark in the title page, but the footnote itself was still indicated using symbols.  This is controlled inside the macros that format the title page and I wasn't able to find a suitable replacement to get the effect you are looking for.

I have two suggestions.  One is to locate a typesetting specification that does what you want.  If this is a requirement of the journal you are submitting to, find out if there is a LaTeX typesetting specification for that journal, or find a sample document from someone else that has successfully used this format with LaTeX for the journal.  The other suggestion is to not use the front matter and instead create your own title page using the titlepage environment.  You would need to visually format the title page, but you would be able to add TeX fields to change the footnote symbols and footnote counter to meet your needs.

### Thank you very much for the

Thank you very much for the help, it perfectly works.

Regards,

Stefano Lombardi


text:
Myung, McDonnell, Kazinets, Seo, and Moskowitz: Relationships between Household Smoking Restrictions and Intention to Quit Smoking among Korean American Male Smokers in California

Abstract

This cross-sectional study examined the relationships between household smoking restrictions and intentions to quit smoking among Korean American male smokers in California. We used data from the California Korean American Tobacco Use Survey (CKATUS), which was conducted in 2004 using computer-assisted telephone interviewing. Among the 2,545 respondents who participated in the CKATUS, the 387 male smokers who answered a question assessing their intention to quit smoking were included in the final analyses. In univariable analyses, smokers who reported having household smoking restrictions were more likely to intend to quit smoking as compared with those who did not (P<0.01). Other independent correlates of having an intention to quit smoking were being less than 50 yr of age, having spent more than 50% of one's life in the US, being assimilated, and having other smokers in the household. In a multiple logistic regression analysis, the significant correlates of having an intention to quit smoking were household smoking restrictions (complete or partial restriction vs. no restriction on smoking; odds ratio, 2.54; 95% confidence interval, 1.22-5.28) and absence of other smokers in the household. In conclusion, smoking restrictions in the household are associated with an intention to quit smoking among Korean American male smokers in California.

INTRODUCTION

As secondhand smoke has been found to be a health risk to non-smokers, household smoking bans as well workplace smoking bans have become common in the United States. The prevalence of household smoking bans has increased from 43% in 1993 to 61% in 1999 (1). Concerning household smoking bans, a variety of topics such as secondhand smoke, childhood asthma, health status, and tobacco use behavior have been examined (2-6). Further, several studies have reported the relationship between household smoking bans and intention to quit smoking or actual smoking cessation. A prospective study found that a complete household ban at baseline was associated with twice the odds of a subsequent quit attempt (7). An international panel survey also reported that intention to quit at baseline was associated with implementing household bans between baseline and follow-up, an average of seven months later (8).
According to the 2005 California Health Interview Survey (CHIS), 31% of Korean American (KA) men were current smokers (9). This rate is higher than current smoking among other Asian American men (18%) or the average of all men in California (19%) (9).
Given this higher smoking rate among KA men, household smoking restrictions could be of particular relevance to this community. The aim of this study was to estimate the prevalence of household smoking restrictions and to examine the relationship between them and the intention to quit smoking among KA male smokers in California, using data from the California Korean American Tobacco Use Survey (CKATUS), which was conducted to identify tobacco use behavior among adults of Korean descent residing in California (Korean Californians) (10).

MATERIALS AND METHODS

The CKATUS was conducted between 2003 and 2004 to assess tobacco knowledge, attitudes, and behavior among 2,545 adults of Korean descent residing in California, using computer-assisted telephone interviewing (CATI) (10). The survey was sponsored by the Tobacco Control Section, California Department of Health Services, USA.

Methodology of the CKATUS

The methodology for this survey has been presented elsewhere (9). The sample was obtained from a list of telephone numbers registered to individuals with Korean surnames residing in California. To ensure a representative sample, the original sample was stratified based on generation, region, and gender.
The sample was modified to reflect gender proportions by region only, because the number of second generation Koreans residing in California was too small to locate enough individuals to achieve the original sampling design. To compensate for selection bias (which is the unequal probability of one person being selected into the sample over another person), and provide representative population estimates, we used a weighting technique. Data were weighted by number of adults per household, number of residential telephone lines, and region to represent the population characteristics of KAs in California.
The survey included questions from the California Tobacco Survey (CTS) (10), as well as additional items specific to this population, including a measure of acculturation. The CTS assesses cigarette smoking behavior and other tobacco use, attitudes toward smoking, and media exposure to smoking and health-related information. Interviews were conducted in English and Korean by Strategic Research Group (SRG, Columbus, OH, USA). The overall response rate for the survey was 48%.

Methodology of this study

We used the CKATUS data for the current study and received an exemption from the Institutional Review Board at the University of California, Berkeley because these data are publicly available and individual data are not identifiable.

Samples

Among the 2,545 total respondents (i.e., men and women) who participated in the CKATUS, 15.3% were current smokers. Of the current smokers, 387 adult men were included in the final analysis based on smoking at least 100 cigarettes in their lifetime, smoking everyday or some days at the time of the interview, and having responded to a question about their quitting intentions.

Measures

We evaluated participants' intention to quit smoking with the question, "What best describes your intentions regarding quitting? Would you say you...," "Are planning to quit in the next 30 days," "Contemplating quitting in the next 6 months," "May quit in the future, but not in the next 6 months," or "Never expect to quit." For the analysis, this was coded as a binary variable: any of the first three response categories was defined as having an "intention to quit," and the last response category was defined as "not having an intention to quit."
Household smoking restrictions were evaluated with the question, "What are the smoking rules or restrictions in your household, if any?" The four possible responses were, "Smoking is completely prohibited," "Smoking is generally banned for everyone with few exceptions," "Smoking is allowed in some rooms only," or "There is no restriction on smoking." This was coded as a binary variable: the first three response categories were defined as having "complete or partial ban on smoking," and the last response category was defined as "no restrictions on smoking."
Acculturation can be broadly defined as the cultural modification of an individual, group, or people by adapting to, or borrowing, traits from another culture (11). In the CKATUS it was defined as the process by which a subculture is absorbed into the dominant culture. Based on this operationalization, language proficiency is likely to be an appropriate indicator. Therefore, in the CKATUS, acculturation was defined using questions about language proficiency. Categories included traditional (not acculturated), bilingual (somewhat acculturated), or assimilated (acculturated). Respondents who spoke both English and Korean at least "pretty well" were categorized as bilingual and, therefore, somewhat acculturated. Those who only spoke Korean were categorized as traditional or not acculturated, and those who spoke only English were categorized as assimilated or acculturated.

Statistical analyses

We used weighted data to compensate for non-response bias and to provide representative population estimates. A Pearson chi-square test for categorical variables and a two-sample t-test for continuous variables were used in the univariable analyses to evaluate the association between smoking restrictions (complete or partial restriction vs. no restriction on smoking) or sociodemographic variables and intention to quit. We also conducted a multiple logistic regression to adjust for potential confounding including age (>45 yr vs. 18-44 yr), annual household income (>US$50,000 vs. ≤US$ 50,000), education (≥college graduate vs. <college graduate), proportion of life spent in the US (>50% vs. ≤50%), and number of smokers in the household (1 vs. 0, 2 or more vs. 0). Considering the multicollinearity among proportion of life spent in the US, generation, and acculturation, only the variable of proportion of life spent in the US, which was considered to be more objective and significant, were included in the final model. Odds ratios (OR) and 95% confidence intervals (CIs) were calculated using Stata SE 10.0 svy commands to adjust standard errors for the complex survey sample design (StataCorp, College Station, TX, USA). All statistical tests were two-sided, and we considered P<0.05 to be statistically significant.

Sample characteristics

Table 1 summarizes the characteristics of the adult male smokers included in the final analyses. About 60% had at least a college education. Most were first generation KAs (89.7%), and about 10% were assimilated. Almost 25% had spent more than half of their lives in the US.
Forty-one percent reported that they might quit smoking in the future, but not in the next 6 months, while 19.8% never expected to quit. Regarding smoking restrictions, 17.4% had households with no restriction on smoking, while 82.6% lived in households where smoking was completely or partially prohibited (40.2% and 42.4%, respectively; data not shown in Table 1). Almost half (48%) lived with other smokers.

Comparison of characteristics by intention to quit

Table 1 compares characteristics of respondents who had no intention to quit smoking with those who did. Compared with those who had no intention to quit, those who intended to quit smoking were significantly more likely to be younger than 50 yr (P=0.02), have spent more than 50% of their lives in the US (P=0.03), be assimilated (P=0.01), have restrictions on smoking in their household (P<0.01), and have other household members who smoked (P<0.01). Higher annual household income (P=0.05), higher education level (P=0.06), and second generation (P=0.06) had marginally significant associations with intentions to quit smoking.

Independent correlates of having an intention to quit smoking

As shown in Table 2, the significant independent correlates of having an intention to quit smoking in the multiple logistic regression analysis were household smoking restrictions (complete or partial restriction vs. no restriction on smoking; OR, 2.54; 95% CI, 1.22-5.28) and absence of other smokers in the household (2 or more vs. no other smokers in the household; OR, 0.19; 95% CI, 0.08-0.49). Higher education (≥college graduate vs. <college graduate; OR, 1.93; 95% CI, 0.98-3.82) and younger age (>45 yr vs. 18-44 yr; OR, 0.50; 95% CI, 0.25-1.00) had marginally significant associations with intention to quit smoking.

DISCUSSION

In this cross-sectional survey of KA adult male smokers in California, we found that household smoking restrictions were associated with having an intention to quit smoking. This finding is similar to those of previous studies (7, 8, 12). A prospective study reported that a complete ban at baseline, compared with no ban or a partial ban, was significantly associated with an increase in subsequent quit attempts (OR, 2.0) (7). The International Tobacco Control (ITC) Four Country Survey-a cohort study conducted in 2002 with an average of seven months follow-up-reported that intentions to quit at baseline were associated with implementing bans between baseline and follow-up (8). Bans at baseline were also associated with more quit attempts and success at follow-up (8). Another longitudinal study showed that smokers in the preparation stage (i.e., planning to quit seriously within the next 30 days and having made at least one-day quit attempt in the past year) were more likely to adopt a full ban than those either in precontemplation (having no intention to quit within the next 6 month) or contemplation (considering quitting seriously within the next 6 months) stages (12).
We also found that those who lived with 2 or more other smokers were less likely to intend to quit smoking, and those who were more educated were more likely to intend to quit smoking. Other significant factors associated with an intention to quit from univariable analyses included age and proportion of life spent in the US. Also, higher education and younger age had marginally significant associations with intentions to quit smoking. Several previous studies have reported that the intention to quit smoking was more prevalent among those who were younger, had higher income, and smoked fewer cigarettes per day (13-15). Our findings were consistent with earlier studies except for daily cigarette consumption, which may be due to missing data on this variable in the CKATUS.
Smokers who were linguistically assimilated and spent more than half of their life in the US were more likely to have an intention to quit smoking. These findings may be due to different social norms about smoking in the US and their country of origin (in this case, Korea). According to the Organisation for Economic Cooperation and Development (OECD) 2001 Health Data, current smoking prevalence of adult men in Korea (52.9%) was much higher than in the US (20.3%) (16). The low smoking prevalence in the US may, therefore, facilitate intentions to quit smoking, especially among more assimilated men.
Also, we performed univariable analyses and a multiple logistic regression analysis based on a different operationalization for the outcome variable (i.e., intention to quit in the next 6 months-stages of contemplation and preparation vs. no intention to quit in the next 6 months-stage of precontemplation). However, there was no significant association between household smoking restrictions and having an intention to quit smoking (P=0.40). Therefore, in the current study, we could conclude that those who had household smoking restrictions were significantly more likely to have an intention to quit smoking (vs. never expect to quit smoking), even though they were not likely to contemplate quitting in the near future (i.e., in the next 6 months).
Our study had several limitations. First, due to the cross-sectional nature of the survey, we are unable to assess whether the smoking restrictions in the household led to having an intention to quit smoking or vice versa. We were neither able to evaluate the causal direction between household smoking restrictions and intention to quit smoking, nor test the relationship between household smoking restrictions and successful smoking cessation. Although two longitudinal studies (13, 17) mentioned above indicated that stages of change at baseline predicted the adoption of smoking restrictions in the household, the direction in our study is unclear. Further large cohort studies are needed to confirm the directional relationship between these factors. Second, we were unable to evaluate these relationships in women because of the small number of women smokers in the CKATUS data. Larger studies are needed that focus on recruiting Korean American women who smoke. Last, the overall response rate for the survey was only 48%. The response rate varied by region, with the lowest response rate being 43% and the highest, 55%. Although interviewers attempted to complete an interview by calling at least ten times at different times of day and on different days of the week, the response rate was low, possibly leading to selection bias. However, we believe that we compensated for this by using a weighting technique.
In conclusion, we found that household smoking restrictions are associated with an intention to quit smoking among KA adult male smokers in California. The results suggest that the adoption of smoking bans in the household could encourage smokers to quit.

Figures and Tables

Table 1
Sample characteristics and comparison by intention to quit*(n=387)

*Participants were adult men current smokers among the 2,545 respondents in the California Korean American Tobacco Use Survey; The relationship between this variable and 'intention to quit' variable was evaluated for only 298 subjects due to missing data.

Percentages (%) and P values were calculated from analyses of complex survey design.

Table 2
Independent correlates of having an intention to quit smoking from the multiple logistic regression analysis (n=387)*

*OR and 95%CI were calculated from analyses of complex survey design and were adjusted for age, annual income, education, proportion of life spent in the US, smoking restrictions in the household, and number of smokers in the household, using weighted data.

OR, odds ratio; CI, confidence interval.

ACKNOWLEDGEMENT

The present article has been revised of its references based upon the erratum record in JKMS 2010;25(6):984.

Notes

This project was funded by the Centers for Disease Control and Prevention through Cooperative Agreement #U48/DP000033. Its contents are solely the responsibility of the authors and do not necessarily represent the official views of the Centers for Disease Control and Prevention.

References

1. Levy DT, Romano E, Mumford EA. Recent trends in home and work smoking bans. Tob Control. 2004. 13:258–263.
2. Pizacani BA, Martin DP, Stark MJ, Koepsell TD, Thompson B, Diehr P. Household smoking bans: which households have them and do they work? Prev Med. 2003. 36:99–107.
3. Blackburn C, Spencer N, Bonas S, Coe C, Dolan A, Moy R. Effect of strategies to reduce exposure of infants to environmental tobacco smoke in the home: cross sectional survey. BMJ. 2003. 327:257.
4. Halterman JS, Fagnano M, Conn KM, Szilagyi PG. Do parents of urban children with persistent asthma ban smoking in their homes and cars? Ambul Pediatr. 2006. 6:115–119.
5. Wamboldt FS, Balkissoon RC, Rankin AE, Szefler SJ, Hammond SK, Glasgow RE, Dickinson WP. Correlates of household smoking bans in low-income families of children with and without asthma. Fam Process. 2008. 47:81–94.
6. Shelley D, Yerneni R, Hung D, Das D, Fahs M. The relative effect of household and workplace smoking restriction on health status among Chinese Americans living in New York City. J Urban Health. 2007. 84:360–371.
7. Pizacani BA, Martin DP, Stark MJ, Koepsell TD, Thompson B, Diehr P. A prospective study of household smoking bans and subsequent cessation related behaviour: the role of stage of change. Tob Control. 2004. 13:23–28.
8. Borland R, Yong HH, Cummings KM, Hyland A, Anderson S, Fong GT. Determinants and consequences of smoke-free homes: findings from the International Tobacco Control (ITC) Four Country Survey. Tob Control. 2006. 15:Suppl 3. iii42–iii50.
9. California Health Interview Survey. CHIS 2005 Adult Public Use File, Release1 [computer file]. 2007. Los Angeles, CA: UCLA Center for Health Policy Research.
10. Carr K, Beers M, Kassebaum T, Chen MS Jr. California Korean American Tobacco Use Survey-2004. 2005. Sacramento, CA: Department of Health Services.
11. Merriam-Webster's Online Dictionary. Available at: http://www.merriam-webster.com/dictionary/acculturation.
12. Pizacani BA, Martin DP, Stark MJ, Koepsell TD, Thompson B, Diehr P. Longitudinal study of household smoking ban adoption among households with at least one smoker: Associated factors, barriers, and smoker support. Nicotine Tob Res. 2008. 10:533–540.
13. Siahpush M, McNeill A, Borland R, Fong GT. Socioeconomic variations in nicotine dependence, self-efficacy, and intention to quit across four countries: findings from the International Tobacco Control (ITC) Four Country Survey. Tob Control. 2006. 15:Suppl 3. iii71–iii75.
14. Abdullah AS, Ho LM, Kwan YH, Cheung WL, McGhee SM, Chan WH. Promoting smoking cessation among the elderly: what are the predictors of intention to quit and successful quitting? J Aging Health. 2006. 18:552–564.
15. Abdullah AS, Yam HK. Intention to quit smoking, attempts to quit, and successful quitting among Hong Kong Chinese smokers: population prevalence and predictors. Am J Health Promot. 2005. 19:346–354.
16. OECD Health Data 2001. OECD. Available at: http://stats.oecd.org/wbos/Index.aspx?DatasetCode=HEALTH.
17. Fava JL, Velicer WF, Prochaska JO. Applying the transtheoretical model to a representative sample of smokers. Addict Behav. 1995. 20:189–203.
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Seung-Kwon Myung
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Is using medior to describe mid-level expertise is right or wrong?


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# A question concerning convex functions

Consider points $a,b,c$ (not on a line) and $x_1,...,x_n$ in $\Bbb{R}^2$. I am looking for a necessary and sufficient condition in terms of the geometric configuration of these points such that for any convex function $f : \Bbb{R}^2 \longrightarrow \Bbb{R}$, $\frac{f(x_1)+\cdots+f(x_n)}{n} \leq \frac{f(a)+f(b)+f(c)}{3}$. (As a guess, I think the points $x_i$ must be in the triangle determined by $a,b,c$). If you know a reference or an idea please let me know.

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The necessary and sufficient conditions are: $(1/n)\sum x_j=(1/3)(a+b+c)$, and all $x_j$ lie inside the closed triangle $(a,b,c)$.

Proof of sufficiency. For every affine function, we have equality. Let $f$ be convex. Then there exists a unique affine function $g$ which matches $f$ at $a$, $b$ and $c$. And we have $f\leq g$ in the triangle $(a,b,c)$. Thus $$\sum f(x_j)/n\leq\sum g(x_j)/n=(g(a)+g(b)+g(c))/3=(f(a)+f(b)+f(c))/3.$$

Necessity. If some point is outside, it is easy to construct a piecewise affine function which is zero on the triangle and positive outside, so the inequality is violated. If $A=(x_1+...+x_n)/n$, does not coincide with $B=(a+b+c)/3$, then it is easy to see that there exists an affine function which violates the inequality: just take any affine $g$ such that $g(A)>g(B)$.

This has a straightforward generalization to any dimension.

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First, By the barycenter of $x_j$, you mean $\sum x_j$ or $\frac{\sum x_j}{n}$ ? Second, would you give a tip on how do you construct the affine function claimed in the last line of your answer. –  user30300 Jul 28 '13 at 9:16
About barycenter: second expression. It is also called the center of gravity. About affine function: take any one which is 0 at A and 1 at B. –  Alexandre Eremenko Jul 28 '13 at 11:34
@Alexandre: Reading the necessity part of you nice answer tells me you should edit the first line of your answer. I think you meant $\frac{\sum x_j}{n} = \frac{a+b+c}3$. –  user37834 Jul 28 '13 at 15:13
Silvi: thanks, I corrected the error. –  Alexandre Eremenko Jul 28 '13 at 16:05


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