text stringlengths 14 5.77M | meta dict | __index_level_0__ int64 0 9.97k ⌀ |
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{"url":"https:\/\/web2.0calc.com\/questions\/in-right-triangle-abc-with-angle-b-90-degrees-we","text":"+0\n\n# In right triangle ABC with angle B = 90 degrees, we have 2 sin A = 3 cos A. What is tan A?\n\n0\n36\n1\n\nIn right triangle\u00a0$$ABC$$\u00a0with\u00a0$$\\angle B = 90^\\circ$$,\u00a0we have\u00a0$$2\\sin A = 3\\cos A$$.\u00a0What is\u00a0$$\\tan A$$?\n\nMar 14, 2021","date":"2021-04-21 08:21:01","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9928705096244812, \"perplexity\": 669.7340381206883}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-17\/segments\/1618039526421.82\/warc\/CC-MAIN-20210421065303-20210421095303-00174.warc.gz\"}"} | null | null |
{"url":"http:\/\/pdfcast.org\/pdf\/probability-examples","text":"This is not the document you are looking for? Use the search form below to find more!\n\nReport\n\n# Probability Examples\n\nDocument Description\nProbability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we are not certain. The proposition of interest is usually of the form \"Will a specific event occur?\" The attitude of mind is of the form \"How certain are we that the event will occur?\" The certainty we adopt can be described in terms of a numerical measure and this number, between 0 and 1, we call probability.[2] The higher the probability of an event, the more certain we are that the event will occur. Thus, probability in an applied sense is a measure of the confidence a person has that a (random) event will occur. The concept has been given an axiomatic mathematical derivation in probability theory, which is used widely in such areas of study as mathematics, statistics, finance, gambling, science, artificial intelligence\/machine learning and philosophy to, for example, draw inferences about the expected frequency of events. Probability theory is also used to describe the underlying mechanics and regularities of complex systems.\nFile Details\n\u2022 File size: 264.34kb\n\u2022 Pages: 4\n\u2022 Tags: number sense definition, square root of 7, how to solve identities, explain differentiation, number sets algebra 1 worksheets\n\u2022 content preview\nSubmitter\nEmbed Code:\n\nRelated Documents\n\n## Conditional Probability\n\nby: mahesh4528, 3 pages\n\nWhen we are solving Conditional Probability Examples ,we deal with two events, say A and B , sometimes these events are so related to each other , that the probability will depend on whether the ...\n\n## Conditional Probability\n\nby: mahesh4528, 3 pages\n\nWhen we are solving Conditional Probability Examples ,we deal with two events, say A and B , sometimes these events are so related to each other , that the probability will depend on whether the ...\n\n## Conditional Probability\n\nby: mahesh4528, 3 pages\n\nWhen we are solving Conditional Probability Examples ,we deal with two events, say A and B , sometimes these events are so related to each other , that the probability will depend on whether the ...\n\n## Conditional Probability\n\nby: mahesh4528, 4 pages\n\nWhen we are solving Conditional Probability Examples ,we deal with two events, say A and B , sometimes these events are so related to each other , that the probability will depend on whether the ...\n\n## Conditional Probability\n\nby: storysubmission11, 3 pages\n\nWhen we are solving Conditional Probability Examples ,we deal with two events, say A and B , sometimes these events are so related to each other , that the probability will depend on whether the ...\n\n## Evolution and Probability\n\nby: carolyn, 4 pages\n\nSome of the most impressive-sounding criticisms of the conventional theory of biological evolution involve probability. Such arguments have been raised, not just by religious fundamentalists, but by ...\n\n## Probability Calculator\n\nby: mahesh4528, 3 pages\n\nSteps for Probability We have n(s) = total possible outcome n(E) = number of event occurs p(A) = probability of A =$\\frac{n(E)}{n(s)}$ . Multiple Event Probability (X & Y are independent) p(x or ...\n\n## A shocking experiment: New evidence on probability weighting and common ratio violations\n\nby: shinta, 9 pages\n\nWe study whether probability weighting is observed when individuals are presented with a series of choices between lotteries consisting of real non-monetary adverse outcomes, electric shocks. ...\n\n## Frequency formats, probability formats, or problem structure? A test of the nested-sets hypothesis in an extensional reasoning task\n\nby: shinta, 13 pages\n\nFive experiments addressed a controversy in the probability judgment literature that centers on the efficacy of framing probabilities as frequencies. The natural frequency view predicts that ...\n\n## Time preference and its relationship with age, health, and survival probability\n\nby: shinta, 19 pages\n\nAlthough theories from economics and evolutionary biology predict that one\u2019s age, health, and survival probability should be associated with one\u2019s subjective discount rate (SDR), ...\n\nContent Preview\nProbability Examples\nProbability Examples\nProbability is ordinarily used to describe an attitude of mind towards some proposition\nof whose truth we are not certain. The proposition of interest is usually of the form\n\"Will a specific event occur?\" The attitude of mind is of the form \"How certain are we\nthat the event will occur?\"\nThe certainty we adopt can be described in terms of a numerical measure and this\nnumber, between 0 and 1, we call probability.[2] The higher the probability of an\nevent, the more certain we are that the event will occur.\nThus, probability in an applied sense is a measure of the confidence a person has\nthat a (random) event will occur.\nThe concept has been given an axiomatic mathematical derivation in probability theory,\nwhich is used widely in such areas of study as mathematics, statistics, finance, gambling,\nscience, artificial intel igence\/machine learning and philosophy to, for example, draw\ninferences about the expected frequency of events. Probability theory is also used to\ndescribe the underlying mechanics and regularities of complex systems.\n\nTutorcircle.com\nPageNo.:1\/4\n\nThe scientific study of probability is a modern development. Gambling shows that there\nhas been an interest in quantifying the ideas of probability for mil ennia, but exact\nmathematical descriptions arose much later.\nThere are reasons of course, for the slow development of the mathematics of probability.\nWhereas games of chance provided the impetus for the mathematical study of probability,\nfundamental issues are stil obscured by the superstitions of gamblers.\nAccording to Richard Jeffrey, \"Before the middle of the seventeenth century, the term\n'probable' (Latin probabilis) meant approvable, and was applied in that sense, univocal y,\nto opinion and to action. A probable action or opinion was one such as sensible people\nwould undertake or hold, in the circumstances.\"[8] However, in legal contexts especial y,\n'probable' could also apply to propositions for which there was good evidence.\nAside from elementary work by Girolamo Cardano in the 16th century, the doctrine of\nprobabilities dates to the correspondence of Pierre de Fermat and Blaise Pascal (1654).\nChristiaan Huygens (1657) gave the earliest known scientific treatment of the subject\nJakob Bernoul i's Ars Conjectandi (posthumous, 1713) and Abraham de Moivre's Doctrine\nof Chances (1718) treated the subject as a branch of mathematics.\nSee Ian Hacking's The Emergence of Probability and James Franklin's The Science of\nConjecture for histories of the early development of the very concept of mathematical\nprobability.\nIn a deterministic universe, based on Newtonian concepts, there would be no probability if\nal conditions are known, (Laplace's demon). In the case of a roulette wheel, if the force of\nthe hand and the period of that force are known, the number on which the bal wil stop\nwould be a certainty.\n\nTutorcircle.com\nPageNo.:2\/4\n\nOf course, this also assumes knowledge of inertia and friction of the wheel, weight,\nsmoothness and roundness of the bal , variations in hand speed during the turning and so\nforth. A probabilistic description can thus be more useful than Newtonian mechanics for\nanalyzing the pattern of outcomes of repeated rolls of roulette wheel.\nPhysicists face the same situation in kinetic theory of gases, where the system, while\ndeterministic in principle, is so complex (with the number of molecules typically the order\nof magnitude of Avogadro constant 6.02*1023) that only statistical description of its\nproperties is feasible.\nProbability theory is required to describe nature.[20] A revolutionary discovery of early\n20th century physics was the random character of al physical processes that occur at\nsub-atomic scales and are governed by the laws of quantum mechanics.\nThe objective wave function evolves deterministically but, according to the Copenhagen\ninterpretation, it deals with probabilities of observing, the outcome being explained by a\nwave function collapse when an observation is made.\nHowever, the loss of determinism for the sake of instrumentalism did not meet with\nuniversal approval. Albert Einstein famously remarked in a letter to Max Born: \"I am\nconvinced that God does not play dice\".\nLike Einstein, Erwin Schrodinger, who discovered the wave function, believed quantum\nmechanics is a statistical approximation of an underlying deterministic reality. In modern\ninterpretations, quantum decoherence accounts for subjectively probabilistic behavior.\n\nTut\nTu o\nt rc\nr i\nc rc\nr l\nc e\nl .\ne c\n. o\nc m\nPa\nP ge\ng\ne No\nN ..::2\/\n3 3\n\/4\n\nThankYou\nTutorCircle.com\n\n# Document Outline\n\n\u2022 \uffbf\n\nProbability Examples\n\nShare Probability Examples to:\n\nexample:\n\nhttp:\/\/myblog.wordpress.com\/\nor\nhttp:\/\/myblog.com\/\n\nShare Probability Examples as:\n\nFrom:\n\nTo:\n\nShare Probability Examples.\n\nEnter two words as shown below. If you cannot read the words, click the refresh icon.\n\nShare Probability Examples as:\n\nCopy html code above and paste to your web page.","date":"2014-10-31 14:14:50","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7413213849067688, \"perplexity\": 2368.791244550191}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2014-42\/segments\/1414637900019.55\/warc\/CC-MAIN-20141030025820-00239-ip-10-16-133-185.ec2.internal.warc.gz\"}"} | null | null |
Q: Slelling and karking Long ago I read a book that coined the terms 'slelling' (illegally sampling someone else's DNA) and 'karking' (altering DNA).
I reckon we're gonna need these verbs in general conversation real soon now :-\
For the life of me I can't remember the book or author; I do remember that the protagonists were at college in a city with streets of ice and skated everywhere.
Anyone know the book?
A: This is David Zindell's Neverness, which was set in a city of the same name.
I can't find my copy at the moment, so details are hazy in my memory, but the story involves a student at the university named Mallory Ringess. The one thing I do remember is that there were Neanderthals living near the city, and Mallory, his parents and his roommate (who may have been called Bardo) transformed themselves into Neanderthal form and went to live with them for a time.
If I remember right, the people who stole others' DNA were called "slell-neckers".
There was also a trilogy of sequels, The Broken God, The Wild and War in Heaven, collectively know as A Requiem for Home Sapiens.
| {
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using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using Newtonsoft.Json;
using Newtonsoft.Json.Linq;
using UnityEngine;
namespace JsonDotNet.Extras.CustomConverters
{
public class Vector3Converter : JsonConverter
{
public override void WriteJson(JsonWriter writer, object value, JsonSerializer serializer)
{
JToken t = JToken.FromObject(value);
if (t.Type != JTokenType.Object)
{
t.WriteTo(writer);
}
else
{
var o = (JObject)t;
IList<string> propertyNames = o.Properties().Where(p => p.Name == "x" || p.Name == "y" || p.Name == "z").Select(p => p.Name).ToList();
o.AddFirst(new JProperty("Keys", new JArray(propertyNames)));
o.WriteTo(writer);
}
}
public override object ReadJson(JsonReader reader, Type objectType, object existingValue, JsonSerializer serializer)
{
throw new NotImplementedException("Unnecessary because CanRead is false. The type will skip the converter.");
}
public override bool CanRead
{
get { return false; }
}
public override bool CanConvert(Type objectType)
{
return objectType == typeof(Vector3);
}
}
}
| {
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Hippie by the Sea: I'm in the pursuit of happiness.
I'm in the pursuit of happiness.
it's absolutely beautiful that way.
There are so many things that are out of our control.
but then there are those little things called decisions.
decide to not let it bother you, decide to accept where you are.
To literally "love the skies you are under".
Then there are those amazing moments life affords you, blesses you with.
amazing moments of clarity that you have if you slow down enough.
Don't compromise, embrace it, all of it.
don't just be happy, pursue it.
This post dedicated to my love, my mother who tells me on a regular basis that life is always greener on the other side. Things are not always greener over here. Life is still life. But I'll love it just the same and look for green patches when I can. | {
"redpajama_set_name": "RedPajamaC4"
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Q: SQL Query- Table as invalid object I have a SQL query
Select
temp1.domainname,
temp1.employeeid,
temp1.alletec_plantcode,
temp1.name,
temp1.alletec_customerengineer1name,
temp1.alletec_cityname,
temp1.alletec_regionname,
temp1.alletec_ce1name,
temp1.alletec_casecalltypename,
count(temp1.alletec_ce1name) as TOTALMIFASSIGN ,
(select count( MIF.alletec_ce1name) from temp1 where temp1.alletec_casecalltypename='aa') as CMMIFASSIGN
from (
Select
User1.domainname,
User1.employeeid,
User1.alletec_plantcode,
BU.name,
MIF.alletec_customerengineer1name,
MIF.alletec_cityname,
MIF.alletec_regionname,
MIF.alletec_ce1name,
Incident.alletec_casecalltypename
From
FilteredSystemUser As User1 Inner Join FilteredBusinessUnit As BU ON User1.businessunitid=BU.businessunitid
Inner join Filteredalletec_mif As MIF ON MIF.alletec_ce1=User1.systemuserid
Inner join FilteredIncident As Incident On Incident.alletec_serialnomif=MIF.alletec_mifid
where MIF.alletec_ce1name='Amit Chauhan' AND MIF.alletec_cityname='Gurgaon' and MIF.alletec_regionname='North' and Incident.statecodename='Resolved'
group by User1.domainname,
User1.employeeid,
User1.alletec_plantcode,
BU.name,
MIF.alletec_cityname,
MIF.alletec_regionname,
MIF.alletec_ce1name,
MIF.alletec_customerengineer1name,
Incident.alletec_casecalltypename
) As temp1
group by
temp1.domainname,
temp1.employeeid,
temp1.alletec_plantcode,
temp1.name,
temp1.alletec_customerengineer1name,
temp1.alletec_cityname,
temp1.alletec_regionname,
temp1.alletec_ce1name,
temp1.alletec_casecalltypename
The particular query above showing the temp1 as Invalid Object in the count query as i am require to place further filtration on it. Cant we use the above query in Aggregate function. Kindly suggest the an alternative to it.
Thanks.
A: The problem seem to be that the table alias temp1 isn't available when the query processor tries to resolve it. One solution that should work would be to wrap the query in a common table expression (cte). I believe this will work.
Try this:
;with temp1 (
domainname, employeeid,
alletec_plantcode, name,
alletec_customerengineer1name,
alletec_cityname, alletec_regionname,
alletec_ce1name, alletec_casecalltypename
)
as (
Select
User1.domainname,
User1.employeeid,
User1.alletec_plantcode,
BU.name,
MIF.alletec_customerengineer1name,
MIF.alletec_cityname,
MIF.alletec_regionname,
MIF.alletec_ce1name,
Incident.alletec_casecalltypename
From
FilteredSystemUser As User1 Inner Join FilteredBusinessUnit As BU ON User1.businessunitid=BU.businessunitid
Inner join Filteredalletec_mif As MIF ON MIF.alletec_ce1=User1.systemuserid
Inner join FilteredIncident As Incident On Incident.alletec_serialnomif=MIF.alletec_mifid
where MIF.alletec_ce1name='Amit Chauhan' AND MIF.alletec_cityname='Gurgaon' and MIF.alletec_regionname='North' and Incident.statecodename='Resolved'
group by User1.domainname,
User1.employeeid,
User1.alletec_plantcode,
BU.name,
MIF.alletec_cityname,
MIF.alletec_regionname,
MIF.alletec_ce1name,
MIF.alletec_customerengineer1name,
Incident.alletec_casecalltypename
)
Select
temp1.domainname,
temp1.employeeid,
temp1.alletec_plantcode,
temp1.name,
temp1.alletec_customerengineer1name,
temp1.alletec_cityname,
temp1.alletec_regionname,
temp1.alletec_ce1name,
temp1.alletec_casecalltypename,
(select count(temp1.alletec_ce1name) from temp1) as TOTALMIFASSIGN
from temp1
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,833 |
Lithocarpus kostermansii is a species of plant in the family Fagaceae. It is a tree endemic to Java in Indonesia. It is an endangered species threatened by habitat loss.
References
kostermani
Endemic flora of Java
Trees of Java
Endangered flora of Asia
Taxonomy articles created by Polbot | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,459 |
{"url":"https:\/\/www.askiitians.com\/forums\/Modern-Physics\/a-special-kind-of-light-bulb-emits-monochromatic-l_201345.htm","text":"#### Thank you for registering.\n\nOne of our academic counsellors will contact you within 1 working day.\n\nClick to Chat\n\n1800-1023-196\n\n+91-120-4616500\n\nCART 0\n\n\u2022 0\nMY CART (5)\n\nUse Coupon: CART20 and get 20% off on all online Study Material\n\nITEM\nDETAILS\nMRP\nDISCOUNT\nFINAL PRICE\nTotal Price: Rs.\n\nThere are no items in this cart.\nContinue Shopping\n\n# A special kind of light bulb emits monochromatic light of wavelength 700nm. Electrical energy supply to it at the rate of 60W and the bulb is 50% efficient at converting that energy to light energy. How many photons are emitted by the bulb during its life time of 1 day? (Give me solution using formula )\n\nArun\n25763 Points\n3 years ago\nDear Preeti\n\nAs we know\nn *h*\u00a0$\\nu$\u00a0= P\nn = P\/h$\\nu$\u00a0= P$\\lambda$\/ hc\nOn putting the values\nn = 2.1 * 10^20 photons\/sec\nSince efficiency = 50%\nHence\n50% of n = 1.05 * 10^20 photons\/sec","date":"2021-05-12 08:29:40","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 3, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.4844834804534912, \"perplexity\": 5678.357797624236}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-21\/segments\/1620243991685.16\/warc\/CC-MAIN-20210512070028-20210512100028-00070.warc.gz\"}"} | null | null |
Remarks delivered at the INET Conference at King's College
Cambridge, United Kingdom, April 9, 2010
"Anatomy of a Crisis"
Economic theory has modeled itself on theoretical physics. It has sought to establish timelessly valid laws that govern economic behavior and can be used reversibly both to explain and to predict events. But instead of finding laws capable of being falsified through testing, economics has increasingly turned itself into an axiomatic discipline consisting of assumptions and mathematical deductions – similar to Euclidean geometry.
Rational expectations theory and the efficient market hypothesis are products of this approach. Unfortunately they proved to be unsound. To be useful, the axioms must resemble reality. Euclid's axioms met that condition; rational expectations theory doesn't. It postulates that there is a correct view of the future to which the views of the participants tend to converge. But the correct view is correct only if it is universally adopted by all the participants — an unlikely prospect. Indeed, if it is unrealistic to expect all participants to subscribe to the theory of rational expectations, it's irrational for any participant to adopt it. Anyhow, rational expectations theory was pretty conclusively falsified by the crash of 2008 which caught most participants and most regulators unawares. The crash of 2008 also falsified the Efficient Market Hypothesis because it was generated by internal developments within the financial markets, not by external shocks, as the hypothesis postulates.
The failure of these theories brings the entire edifice of economic theory into question. Can economic phenomena be predicted by universally valid laws? I contend that they can't be, because the phenomena studied have a fundamentally different structure from natural phenomena. The difference lies in the role of thinking. Economic phenomena have thinking participants, natural phenomena don't. The thinking of the participants introduces an element of uncertainty that is absent in natural phenomena. The uncertainty arises because the participants' thinking does not accurately represent reality.
In human affairs thinking serves two functions: a cognitive one and a causal one. The two functions interfere with each other: the independent variable of one function is the dependent variable of the other. And when the two functions operate simultaneously, neither function has a truly independent variable. I call this interference reflexivity.
Reflexivity introduces an element of uncertainty both into the participants' understanding and into the situation in which they participate. It renders the situation unpredictable by timelessly valid laws. Such laws exist, of course, but they don't determine the course of events.
Economic theory jumped through many hoops trying to eliminate this element of uncertainty. It started out with the assumption of perfect knowledge. But as Frank Knight showed in his book, "Risk, Uncertainty, and Profit" published in 1921, in conditions of perfect knowledge there would be no room for profits.
The assumption of perfect knowledge was replaced by the assumption of perfect information. When that proved insufficient to explain how financial markets anticipate the future, economists developed the theory of rational expectations. That is when economic theory parted company with reality. Some great thinkers, including Friedrich Hayek in his Nobel Prize speech, kept reminding economists of the importance of uncertainty but advances in quantitative modeling led to the neglect of this Knightian uncertainty. That is because quantitative methods cannot take into account uncertainty that cannot be quantified. Collateralized Debt Obligations and Credit Default Swaps and risk management methods produced by these quantitative approaches played a nefarious role in the crash of 2008.
The meltdown of the financial system in 2008 forces us to go back to the drawing board and look for a more realistic approach. I believe that we have to start with recognizing a fundamental difference between human and natural phenomena.
This means that financial markets should not be treated as a physics laboratory but as a form of history. The course of events is time-bound and one-directional. Predictions and explanations are not reversible. Some timelessly valid generalizations can serve to explain events but not to predict them.
I have started to develop a set of generalizations along these lines by introducing the concept of reflexivity. Reflexivity can be interpreted as a two-way feedback mechanism between the participants' expectations and the actual course of events. The feedback may be positive or negative. Negative feedback serves to correct the participants' misjudgments and misconceptions and brings their views closer to the actual state of affairs until, in an extreme case, they actually correspond to each other. In a positive feedback a distortion in the participants' view causes mispricing in financial markets, which in turn affects the so-called fundamentals in a self-reinforcing fashion, driving the participants' views and the actual state of affairs ever further apart. What renders the outcome uncertain is that a positive feedback cannot go on forever, yet the exact point at which it turns negative is inherently unpredictable. Such initially self-reinforcing but eventually self-defeating, boom-bust processes are just as characteristic of financial markets as the tendency towards equilibrium.
Instead of a universal and timeless tendency towards equilibrium, equilibrium turns out to be an extreme case of negative feedback. At the other extreme, positive feedback produces bubbles. Bubbles have two components: a trend that prevails in reality and a misconception relating to that trend. The trend that most commonly causes a bubble is the easy availability of credit and the most common misconception is that the availability of credit doesn't affect the value of the collateral. Of course it does, as we have seen in the recent housing bubble. But that's not sufficient to fully explain the course of events.
I have formulated a specific hypothesis for the crash of 2008 which holds that it was the result of a "super-bubble" that started forming in 1980 when Ronald Reagan became President of the United States and Margaret Thatcher was Prime Minister of the United Kingdom. The prevailing trend in the super-bubble was also the ever-increasing use of credit and leverage; but the misconception was different. It was the belief that markets correct their own excesses. Reagan called it the "magic of the marketplace"; I call it market fundamentalism. Since it was a misconception, it gave rise to bubbles. So the super-bubble was composed of a number of smaller bubbles — and punctuated by a series of financial crises. Each time the authorities intervened and saved the system by taking care of the failing institutions and injecting more credit when necessary. So the smaller bubbles served as successful tests of a false belief, helping the super-bubble to grow bigger by reinforcing both credit creation and market fundamentalism.
It should be emphasized that this hypothesis was not sufficient to predict the outcome of individual crises. For instance, I predicted that the emerging market crisis of 1997 would lead to a collapse of global capitalism and I was wrong. Nor is it sufficient to fully explain actual outcomes. For that, one needs to take into account the specific historical circumstances. The hypothesis only helps to select the relevant circumstances.
Let me illustrate this by examining the origins of the super-bubble. For this, I need to go back beyond 1980 at least to the early 1970s.
At the end of World War II when I entered the financial markets, banks and financial markets were strictly regulated and international movements of financial capital were practically at a standstill. The restrictions were relaxed gradually, but at a glacial pace. As late as the beginning of the 1970s, the American banking system was still frozen into immobility. The industry was highly fragmented and regimented. A dull business attracted dull people who were more concerned with job security than with profits. Bank shares were traded by appointment. But I detected some signs of life. Walter Wriston at Citibank trained a new breed of profit oriented bankers who fanned out from Citibank to other banks.
Then in 1972, Citibank held a dinner meeting for security analysts – an unheard of event. I was not invited but it prompted me to publish a report entitled "The Case for Growth Banks" in which I argued that some banks were poised to embark on balanced growth by equity leveraging, i.e.: selling shares at a premium. The bouquet of bank shares I recommended did, in fact, rise by some 50% within a year.
Then came the first oil shock of 1973. The stock market tanked, ruling out equity leveraging. At the same time the recycling of petrodollars was left to the money center banks. They formed holding companies and established subsidiaries in London to escape the restrictions of the Glass-Steagall Act. That was the beginning of the eurodollar markets and of large-scale lending to emerging economies. It soon turned into a boom. Countries like Brazil experienced rapid growth, fuelled by foreign credit. The misconception in the lending boom was that the debt ratios which measured the credit-worthiness of the borrowing countries were independent of the flow of credit. The relationship was, of course, reflexive.
Then came the second oil shock in 1979 and the determined effort of the Federal Reserve under Paul Volcker to bring inflation under control. The Fed fund rate shot up into the high teens and the boom turned into a bust. In 1982 Mexico threatened to default. This was the onset of the first major financial crisis the response to which fuelled the growth of a super-bubble.
The international banking system would have collapsed if the authorities had not banded together to save it. They established what I called the "collective system of lending". The central banks ordered the banks under their control to roll over their loans and the international financial authorities extended enough additional credit to the heavily indebted countries to enable them to remain current on interest payments and redemptions. The IMF imposed harsh conditions on the debtor countries while the regulatory restrictions on the banks were actually relaxed in order to allow them to earn their way out of a hole. After several years, when the banks built up sufficient reserves, the debtor countries were encouraged to reorganize their debts by issuing so called Brady bonds and the banks had to take some losses. The net result was a lost decade for Latin America but a big boost to the international banking system. Financial markets were deregulated and globalized. This stood in stark contrast with earlier financial crises of the nineteenth and twentieth centuries when each time a crisis occurred, regulations were tightened in order to prevent a recurrence. That is how central banking and market regulations had developed and became an integral part of the financial system.
What set this occasion apart from previous ones? Undoubtedly it was the market fundamentalist belief that markets are self-correcting, and best left to their own devices. But the need of the banks to earn their way out of a hole also played a part. This was the specific historical context in which the super-bubble developed.
The system that emerged was called the Washington Consensus. It was characterized by what was called "moral hazard," but was actually an asymmetry between center and periphery. The countries at the periphery of the system were subject to harsh market discipline; but when the system itself was endangered, all bets were off. This gave the banks at the center a competitive advantage and they gradually came to dominate the global financial system.
The globalization spread like a virus. Since financial capital is an essential ingredient of production, once the U.S and the United Kingdom embraced market fundamentalist principles, other countries could not resist them. The financial sector of the U.S. and U.K. grew like Topsy, accounting for more than a third of corporate profits towards the end of the super-bubble in 2006.
In the absence of systemic reforms, the international banking crisis of 1982 repeated itself fifteen years later with only minor variations. Because banks had learned a lesson from 1982. The collective system of lending taught them that it is better to securitize loans and sell them to others than to keep them on their books because that way the central bank could not compel them to roll over loans that have gone sour. By the time the next emerging markets crisis struck in 1997, most of the loans had been securitized, greatly complicating the task of the international authorities. As a result, there was no collective system of lending except in the case of South Korea and there were no Brady bonds. The periphery countries had to bear an even larger share of the losses than in 1982.
Deregulation allowed financial innovators to introduce new forms of synthetic securities at will. Securitization was further encouraged by the misguided rule in Basel II which allowed banks to hold securities on their balance sheets without any reserve requirements because the securities were readily saleable. This may be true for individual banks but not for the banking system as a whole, as the LTCM crisis in 1998 demonstrated. Since the synthetic securities were designed on the basis of false principles, they played a major role in the crash of 2008. But I shall leave the examination of what happened in 2008 to the other speakers.
The point I am trying to make is that developments in the financial markets cannot be understood without considering them in a historical context. Financial markets have changed out of all recognition during my lifetime. Things that would have been inconceivable 50 years ago have become commonplace. Conversely, it seems inconceivable today that the economy could function without derivatives and other complicated instruments. Yet some of these instruments have had a destabilizing effect this is not properly understood. We really have to rethink our view of the financial markets quite profoundly; recognizing that instead of perfect knowledge and perfect information our understanding is inherently imperfect and that applies to market participants and regulators and social scientists alike.
But what is imperfect can be improved, and right now there is plenty of room for improvement – both in rethinking economics and rethinking regulations. I am afraid the current discussions miss the main point: namely that the recent financial crisis was not only a market failure but also a regulatory failure. And what matters now is not so much who regulates, but how. Regulators ought to undertake a course of critical self-examination – Chinese style. But that will be the subject of another panel.
Cambridge, United Kingdom, September 6, 2021 | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,592 |
Q: Pandas: Can I use the round() method on the mode of a column? I am trying to format the output of a 'year' which is stored as a float, to be rounded to 0 decimal places, and to remove the dataframe info.
# Display earliest, most recent, and most common year of birth
print('Earliest year of birth:')
min_yob = df.birth_year.min()
print(round(min_yob))
print('Max year of birth: ')
max_yob = df.birth_year.max()
print(round(max_yob))
print('Most common year of birth: ')
mod_yob = df.birth_year.mode()
print(round(mod_yob))
The output that I get for this is as follows:
Earliest year of birth:
1899
Max year of birth:
2016
Most common year of birth:
0 1989.0
dtype: float64
If I convert mod_yob to an int, it will display correctly, but I'm not sure why round() isn't working here.
Most common year of birth:
1989
Perhaps I'm going about this the wrong way.
A: If a single value in the Series is of type float - which it appears is the case in your scenario - then the mode function will output float values. Best to ensure that the Series of values are all of type integers before the fact.
df["birth_year"] = df["birth_year"].astype(int)
If you have NaN's present, try this:
df["birth_year"] = df["birth_year"].astype(pd.Int64Dtype())
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,889 |
Q: Sandbox Game Center Turn Event Notifications Not Consistent I'm making a turn-based game, using the Game Center Turn-Based Gaming functionality. I'm using the simulator and my iPhone to test notifications of turn events. The results are very inconsistent. About 75% of the time when I make a move on the simulator and pass the turn I don't get any notifications on my iPhone. It seems that this function doesn't get called:
handleTurnEventForMatch:didBecomeActive:
I set the GKEventHandlerDelegate in the code that authenticates the local user and it seems to be set correctly. The fact that I get notifications once in a while suggests that this isn't where the problem lies.
Does anyone have any idea what could be the problem here? Could this be a problem with the Sandbox Game Center Server? Do they limit the amount of notifications you can send in a short amount of time or something like that?
Edit: I just found some posts of people experiencing the same problem, but no solution. No very recent posts though. Any updates?
Edit2: I also find that after a move, when I open the GKTurnBasedMatchmakerViewController on the other device it still says that it's the turn of the other player. When I close and reopen it the information is correct. Also, when I open the game GKTurnBasedMatchmakerViewController when it's still showing incorrect information, it does gives me the correct match object with the updated information. Could this be related in any way?
A: I thought I would share my solution with you, in the hopes that it is the same problem you are experiencing. As you can see from my comment to your question, I was having the same issue as you. It turned out that my settings in iTunes Connect were the culprit, unbeknownst to me. What you should check for is that you still have Game Center enabled in BOTH places that iTunes Connect requires. First, after going into "Manage my Apps" in iTunes Connect, select your app and on the right menu, and click "Manage Game Center". Make sure this is currently enabled. I would even recommend disabling it, and re-enabling it again for good measure.
In addition (and this is what I was missing), you must enable it in one more place. Go back to the previous screen when you first select the app. The top right is where you can select Manage Game Center, but we are looking on the bottom in the "Versions" section. Click the View Details button for your app, and make sure the Game Center button is enabled near the bottom. Again, I would disable and re-enable it here as well. Give it about 10 minutes, clear all your open turns and matches from the Game Center app (this might be an optional step), and build & run again. Hopefully, you will receive turn notifications properly again.
A: I got the same problem. However when i tested it on multiple devices it was working fine... May be some problem with the simulator
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 92 |
{"url":"http:\/\/lambda-the-ultimate.org\/node\/4941","text":"## The Avail programming language\n\nThe Avail programming language was released yesterday, but it didn't receive the attention it deserves.\n\nAvail is a multi-paradigmatic general purpose programming language whose feature set emphasizes support for articulate programming\nI believe the authors are being a bit modest!\n\nThe most interesting thing about Avail is it's programmable (turing-complete) type system that enables strongly typed multiple method dispatch. To me, Avail appears to be a happy marriage between smalltalk, maude, and (dare i say) coq?\n\nAlthough, I'm not totally sold on the persuasive use of unicode - yet.\n\n## Comment viewing options\n\n### What's new?\n\nI saw this last night and all I concluded after looking at it for a bit is that I hate this documentation format. Does someone have enough of an understanding to compare and contrast this to the usual suspects? Does it have parametric polymorphism (see the example in the red box at the bottom of this link)?\n\n### Only two mentions of\n\nOnly two mentions of polymorphism on the whole site, so they're clearly using non-standard terminology. I believe it does support parametric polymorphism however, as described on the type parameterization page:\n\nA type that depends upon one or more other values is incomplete. Given an incomplete type A that depends upon a value B, it is said that B is a type parameter of A or, alternatively, that A is parametric on B. A type is considered complete if it either 1) does not have any type parameters or 2) has an actual value bound to each of its type parameters.\n\nConsider the type set. In Avail, this type is incomplete. It has a type parameter, the element type, that restricts the membership of each instance to values of a particular type. The type set of {\u00e2\u20ac\u00a6}\u00e1\u00b5\u20ac is the most general completion, imposing no membership restrictions on its instances. The type set of {\u00e2\u20ac\u00a6, \u00e2\u02c6\u20191, 0, 1, \u00e2\u20ac\u00a6}\u00e1\u00b5\u20ac, on the other hand, has as instances only those sets whose members are exclusively integers.\n\n### I don't think so\n\nI don't think it directly supports it as you're not allowed to have a value that is of incomplete type and there are no type variables that I see. So I believe you can have a type family List[_], and have values of type List[Int], but not functions of type a -> List[a]. You can get something similar with Semantic Restrictions, but verbose and with nothing checked until run-time when all of the types are monomorphic.\n\n### \"incomplete type\"\n\nThe \"incomplete type\" concept is for discussing Avail; it's technically not a property of the type system itself, as only complete types occur. Perhaps \"type family\" would be more appropriate, considering the confusion.\n\n### As an example, the type\n\nAs an example, the type produced by the expression \"tuple\" (without quotes), is the same type produced by the expression \"<any\u00e2\u20ac\u00a6|>\" or \"<any\u00e2\u20ac\u00a6|0..\u00e2\u02c6\u017e>\", or even \"<any\u00e2\u20ac\u00a6|0..42> \u00e2\u02c6\u00aa <any\u00e2\u20ac\u00a6|43..\u00e2\u02c6\u017e>\". We talk about tuple being an incomplete type (or type family), even though it's an actual type that lives at the top of a lattice of tuple types.\n\nThe same doesn't quite hold for variable types, however. A variable is parameterized by its content type, the kinds of things it can hold. Because we can read from variables, the variable type covaries with its content type. Because we can write to them, the variable type also has to contravary with its content type. So the type graph is actually just an infinitude of incomparable variable types. But we still want to talk about variable types in general, so we say \"variable\" is an incomplete type.\n\nAs it turns out, that's a bit of a fib for variables, since we decided to split the read and write capabilities into two orthogonal parameterizations, forming a lovely diamond lattice... where pretty much only the middle horizontal line of mutually disjoint types occurs in practice! But as a consequence we get a most-general variable type, from which you can read \u00e2\u0160\u00a4 and write \u00e2\u0160\u00a5. So technically that variable type is more general than everything in the variable type family, and therefore is a suitable representative.\n\n### Semantic restriction, take 2\n\nWith 1 day of studying the language, this is what I think is the idea.\n\nSemantic restrictions are user-defined methods that are called during compile time to strengthen (or specialise) types of the generic method at the call-site.\n\nHere is a generic method that enlists (or rather entuples) a generic element, which is further restricted to be the \u00e2\u20ac\u02dcinstantiation\u00e2\u20ac\u2122 type, as per following snippet:\n\nMethod \"enlist_\" is [ x : any | <x> ];\nSemantic restriction \"enlist_\" is [ x : any's type | <x\u00e2\u20ac\u00a6|1>] : type;\n\na : <integer\u00e2\u20ac\u00a6|1..>; \/* type: a tuple of integers with at least 1 element *\/\na := enlist 1; \/* returns <1>\na := enlist 1 ++ enlist 2; \/* returns <1,2>\na := enlist \"a\"; \/* fails to compile *\/\n\n\n### I'm not sure whether you're\n\nI'm not sure whether you're referring to the use of \/* *\/ comments, the descriptive narrative, or even the color of the code bubble. Please clarify.\n\nAnd yes, it has very deep parametric polymorphism. As just a quick example, the type <[0..10], {string\u00e2\u20ac\u00a6|2..2}, float\u00e2\u20ac\u00a6|2..5> is the type of tuples that have between two and five elements, where the first element is an integer between zero and ten inclusive, the second element is a set containing exactly two strings, and the third through fifth elements, should the tuple have that many elements, are all floats. The tuple <7, {\"foo\", \"bar\"}, 1.0, 2.0> is a member of that type. That type can be used as a parameter type in a method, which vouches at compile time for the operations performed on the elements of the tuple at runtime. If the parameter name is t, the expression |t[2]| + |t| is statically permitted in the method, and has a static type of [4..7].\n\nThe type system is described quite thoroughly on the site, and well worth a look if this piques your interest.\n\n### Thanks\n\nThat's an impressive example, but I don't consider it parametric polymorphism -- where's the parameter? Consider a 'sort' function. Is there an easy way (preferably inference!) to give this the type \"forall a. List(a) -> List(a)\" with 'a' being the type parameter? It doesn't have to take that exact form, but it should let you call sort with List(Int) and have the system infer that the result is List(Int). Or can you only type it \"List(Any) -> List(Any)\" and then impose a semantic restriction to recover something equivalent to parametric polymorphism?\n\n### The type parameters are a\n\nThe type parameters are a little bit disguised in the syntax. Or more correctly, they're presented with more pleasant, specific mechanisms than what you may be used to.\n\n<[0..10], {string\u00e2\u20ac\u00a6|2..2}, float\u00e2\u20ac\u00a6|2..5>\n\nversus\n\nTuple[IntRange[2,5],IntRange[0,10] * Set[IntRange[2,2], String] * Float]\n\nI really can't think of a more concise way of expressing the almost-equivalent of the Avail type using the functional notations that I have any familiarity with.\n\nBut you're right about the functions not being parametrically typed. I actually rejected that approach almost right away, in deference to semantic restrictions. Most functions don't really need it, but when one does need it in the presence of deeply parameterized types like Avail's, one generally can't just pass the parameter type straight through. Yes for some things like the function application operation \"_(\u00c2\u00ab_\u00e2\u20ac\u00a1,\u00c2\u00bb)\", parentheses \"(_)\", tuple construction \"<\u00c2\u00ab_\u00e2\u20ac\u00a1,\u00c2\u00bb>\", etc., but not for things like tuple subscripting \"_[_]\", where knowledge about the index's range influences the resulting type. That just can't be done with traditional parametric polymorphism. And those cases not only dominate the type narrowing that happens in Avail, but I couldn't figure out any other way to solve it without having to use really weak types. Suggestions?\n\nThe one other aspect of semantic restrictions is to ensure the types of the inputs are mutually appropriate. Parametric polymorphism had the same kinds and degree of usage holes here as for deriving the output types. For example, tuple subscripting \"_[_]\" takes a tuple and an index, but the semantic restriction checks statically that the possible values for the index are in bounds for the tuple type (and then produces the type union of the possible tuple element types). Technically, having the programmer ensure the tuple indices are always statically in bounds at every call site was far too onerous, so we weakened the requirement so that if any of the possible indices would be in range, we allow the call site -- but then we check the bounds at runtime. That's a general policy of the Avail library: any time we discover that satisfying the static precondition of some operation is systematically too difficult, we weaken it to only fail the cases that would always fail.\n\n### So you have parameterized\n\nSo you have parameterized types but not quantified types. Just FYI, I think most people and the academic literature understand 'parametric polymorphism' to mean the latter (see the Wikipedia article for example).\n\nThe kinds of usage holes you describe for parametric polymorphism are commonly addressed under the name dependent type. For example, you might write give a type such as \"vector-concat : Vec(n), Vec(m) -> Vec (m+n)\".\n\nI, like many people around here, have my own work-in-progress language design and it explores a similar region of design space (customizable types), so I have some strong opinions. (I haven't yet published anything about it in a form other than forum comment, though). I think parametric polymorphism is pretty important because it allows you to work in an abstract setting (let T be a type, let f : T -> T, etc.) and then package up the work you've done into a function or module in a minimal hassle way.\n\nSimilarly, for the kinds of precise types you're talking about, I think dependent types are important. How often are you going to want to hardcode a type like the one you gave above? It seems to me that usually you'll be working in a setting where you have some variables around (number of objects in the world, etc) and you'll want to reference those variables in your types. So rather than <[0..10], ...> you'll want <[0..n], ...> where n is a variable. It sounds like you've written a good bit of a code in this language. Have you not noticed that desire?\n\nBut you're right that the problems of type membership that you might like to solve quickly become very difficult (or formally undecidable) when you have full dependent types or even the kind of parameterized types that you have. The way I look at it, the type system serves more than one role. One of them is to infer a coarse shape for variables (Integer vs. String) so that we can do things like overload resolution or type-driven meta-programing. This can be done efficiently and mostly automatically -- check out the Hindley\u00e2\u20ac\u201cMilner type system if you aren't already familiar with it or one of the languages that uses something like it (e.g. ML, Haskell, ...).\n\nThen we have types that embody more precise assertions about our code. These are useful for lightweight theorem proving, and it's nice to be able to make them very precise, but important that we not be required to actually prove that the precise types are used correctly. So I take a similar position to the one you do with semantic restrictions in this regard - theorem proving is hard and programmers are busy enough. I would like the language to support such proof, though, when desired.\n\n### Thank you for the\n\nThank you for the terminology clarification. I've noticed the desire for a mechanism for parameterizing methods, but it's directly inconsistent with one of the simplest requirements of my language: That types, even function types, always be comparable. I don't believe it's possible with parameterized types, since I think it requires second-order logic (or solving the Halting Problem, depending on the power of the type system) just to decide if one function type is a specialization of another. Note that this is much harder than determining if a particular function satisfies a particular statically expressed parametric type (which is usually very quick, but takes exponential time in the exceedingly unlikely worst case, if I recall).\n\nFunctions have to be first-class objects in Avail, and as such they have to have types that can be compared with other types. So the manipulation and checking of what would in other languages be parametric types must instead be dealt with via Avail's semantic restrictions.\n\nAn alternative, imperative view of semantic restrictions is that of a simple cast mechanism. Avail's type system can't prove that a method \"foo_\" with body [x : integer | x - x] always produces zero, but a semantic restriction can at least state it, and the rest of the program can then rely on it. Then the subexpression \"10 \u00c3\u00b7 foo 5\" could fail to parse because of a mandatory division-by-zero error. Possible divisions by zero are permitted, but there's no way this division can succeed. Again, the semantic restriction on \"_\u00c3\u00b7_\" determines this policy. If you want different rules, just define a different \"_\u00c3\u00b7_\" method and don't import the library one. Embedding this policy directly in a programmer-inaccessible type system would be doomed to failure because of the sheer number of policy decisions.\n\nI have read in the past about the Hindley-Milner type system, and the Milner-Mycroft type inferencing algorithm, but I think semantic restrictions actually provide the missing piece of the puzzle. Arbitrary code is provided to perform the required semi-unification, so there are more cases that can fall to its might than to a type system which has a pre-coded algorithm for resolution.\n\nPerhaps there's a way to do both? What do you think?\n\n### What do you mean and why?\n\n[...] one of the simplest requirements of my language: That types, even function types, always be comparable.\n\nWhat do you mean that types must be comparable? Do you mean that given two types you must be able to decide if one is a super-type of the other? If so, why is that a requirement? Is it to obtain that 'first order type safety' property you mentioned? If so, I don't follow the logic here. If you limit yourself to unquantified types, typing becomes easier to solve, but if you want to express quantification over types or values (which in my experience you very frequently do!) you have to do so as semantic restrictions which aren't checked. If you added quantified types, the general case of typing gets harder, but only in the cases you couldn't express before. Furthermore, there are plenty of useful quantified cases (like sort) that you will be able to express without resorting to writing semantic conditions manually, and in fact they can often be inferred without writing any code and they will actually be verified by the type system, unlike semantic conditions.\n\nWith regard to your example, is \"this code always fails\" really a useful thing to report? This is catching the kind of bug that testing will catch 90% of the time. Isn't the more useful thing to report either \"here's an example case where this type\/assertion fails\" or \"couldn't find an example of this code failing, but couldn't prove that it succeeds, either\"?\n\n### Unlike functional languages,\n\nUnlike functional languages, Avail has objects and methods (multi-methods, actually). To dispatch a method, one has to decide which matching method implementation is most specific, i.e., has the most specific type signature. Quantified types should be able to decidably select which methods are applicable, but not which method is most specific. This requirement is the core reason why functional languages cannot have an extensible object system. I'm sure there has been some very clever work done in the last few decades on this subject (I haven't been paying much attention), but I would be very surprised if someone found a way which is promising.\n\nSemantic restrictions are themselves written as Avail code, so they have *some* internal safety. Avail has reflective types as well (they're objects), and those metatypes have a covariance relation to the types that they're metatypes of (a relation I call metacovariance), so the broad strokes are hard to get wrong in a semantic restriction. The details can often be tricky, however. But that's because one is usually dealing with far more precision than quantified types. That's because writing semantic restrictions in the equivalent of creating a type *system*, not a collection of types. If they're correct (equivalent to a type system being correct), then the resulting program is free of runtime type errors (ditto). Perhaps there's a way of specifying the equivalent of quantified types in a more direct way for methods that need them?\n\nConsider function application: \"_(\u00c2\u00ab_\u00e2\u20ac\u00a1,\u00c2\u00bb)\". The semantic restriction ensures the first argument (the function) is itself a function that statically accepts the tuple of (comma-separated) arguments captured in the second argument. Similarly the generalized currying partial-application method \"_(\u00c2\u00ab\u00c2\u00ab_\u00c2\u00bb\u00c2\u00ab_\u00c2\u00bb?\u00e2\u20ac\u00a1,\u00c2\u00bb)\" does the same thing for the supplied arguments and place-holder underscores, but produces a new function with the exact correct type. E.g.,\n\nf ::= [a : integer, b : byte, c : double | a + b + c];\ng ::= f(_,255,_);\nh : [integer, double]\u00e2\u2020\u2019double := g;\nPrint: g(3000,3.6);\n\/* produces 3258.6 *\/\n\n\nIn this example, g is statically known to be a function of type [integer, double]\u00e2\u2020\u2019double. Avail has no built-in support for currying, and yet it is able to define it safely within the language. BUT - because we're defining something so powerful, the type safety is for its use, not its implementation. Similar to a functional language having a type system at least partially implemented in C.\n\nAs for sorting, there are plenty of ways to address it. It's really no harder than the function application. The source is available for download, or if you don't want to install it you can look up the documentation for almost every method in the library through the stacks interface. I just looked for the word \"sort\" and it shows various methods and restrictions, but the semantic restriction doesn't yet mention the requirement that the function must (statically) accept any two elements of the tuple. Oops. It is a dev release, after all :-). And perhaps you can find a way to express the restriction more generally. Remember, this is Avail, so you don't have to directly invoke \"Semantic restriction_is_\" to have that effect.\n\nSo... the specific example of detecting division by zero could catch certain kinds of unlikely bugs, but really it was intended as an accessible example. Division by zero is often used in this kind of example because it's related to a tidy piece of math (taught early) that nobody sensibly refutes. A better fit for what you're looking for might be tuple subscripting, detecting when someone is definitely accessing an element that's beyond the range of a tuple. Again, you can use the stacks documentation link to find \"_[_]\", then in the Semantic restrictions area look at the entry for tuple meta, natural number's type.\n\nThe Avail team's practical experience is that the type system is sufficiently strong that after battling (negotiating with? having an enlightening discussion with?) the compiler and type system, the resulting code not only is almost always correct, but has flushed out and corrected many initial design limitations ahead of time. Give it a try and see what you think.\n\n### Dispatch\n\nTo dispatch a method, one has to decide which matching method implementation is most specific, i.e., has the most specific type signature.\n\nI hinted at this in an earlier post, but can you give an example where it's important to dispatch on a precise type? For example, why would I ever want to select different behavior when an argument has type Nat{>2} vs. when it has type Nat{>5 and even}? I don't think I would. So my position is that we can infer the coarse shape of values in a decidable way and use that for dispatch.\n\nCould you explain or link to documentation on how to parse _(\u00c2\u00ab\u00c2\u00ab_\u00c2\u00bb\u00c2\u00ab_\u00c2\u00bb?\u00e2\u20ac\u00a1,\u00c2\u00bb)?\n\n### Unfortunately, we're still\n\nUnfortunately, we're still missing an in-depth description of this syntax on the website. If I recall, MessageSplitter.java has some hefty comments describing the clauses that can occur, but let me spec this particular name here...\n\n_(\u00c2\u00ab\u00c2\u00ab_\u00c2\u00bb\u00c2\u00ab_\u00c2\u00bb?\u00e2\u20ac\u00a1,\u00c2\u00bb)\n\nThe first _ corresponds with a place that an argument goes, in this case some subexpression that produces a function. I'm sure you got that one.\n\nThe open-parenthesis is a literal character that should occur after that argument. As with all individual operator characters or runs of alphanumerics, whitespace is permitted around it (and is required between two alphanumeric tokens).\n\nNow it gets interesting. The outer guillemet group \"\u00c2\u00ab\u00c2\u00ab_\u00c2\u00bb\u00c2\u00ab_\u00c2\u00bb?\u00e2\u20ac\u00a1,\u00c2\u00bb\" has two components, the part before the double-dagger (\u00e2\u20ac\u00a1) and the literal comma (,) that comes after it. This means we can have a repetition of the left side zero or more times, separated by commas. The thing on the left is \"\u00c2\u00ab_\u00c2\u00bb\u00c2\u00ab_\u00c2\u00bb?\". This is itself composed of two parts, a repetition of an argument (which will be constrained by the method signature to have zero or one occurrences), and \u00c2\u00ab_\u00c2\u00bb?\". The question mark after the guillemet group means the group is optional, and its presence or absence should be indicated by pushing the constant true or the constant false, respectively. Inside that optional guillemet group is a back-tick () and the underscore character. The back-tick escapes a metacharacter, treating it like an ordinary operator. In this case it means we accept an actual underscore token (_) at the call site.\n\nThe semantic restriction ensures we have the right number of comma-separated arguments-or-underscores for the type of function being curried, and that the argument expressions have suitably strong types for the function's expected argument types. The semantic restriction also makes sure there is at least one underscore (otherwise the call site is talking about a simple function invocation, \"_(\u00c2\u00ab_\u00e2\u20ac\u00a1,\u00c2\u00bb)\".\n\nThe semantic restriction also strengthens the resulting function type, stating that the returned function must produce the same type as the original function, but takes arguments that correspond to the input types where the arguments occurred.\n\nThe implementation should (does) agree with the semantic restriction, and constructs a simple function with the expected input and output types -- at runtime -- with the assistance of a convenient primitive that does that tiny bit of work.\n\nLet me know if I missed anything or was unclear. There are a few other interesting guillemet group modifiers described in MessageSplitter.java. We also have a few more powerful ones planned, for example circled numbers to be used by renames that re-order arguments.\n\nFinally, your question about dispatching on values... Introspection sort, fibonacci, and a host of other operations lend themselves to dispatching by count. Even the currying operation above relies on a zero-or-one sized tuple type to treat a guillemet group as an optional thing rather than a repeated thing.\n\n### Introspection sort,\n\nIntrospection sort, fibonacci, and a host of other operations lend themselves to dispatching by count.\n\nIt's not clear that this is \"important\" though, which was one of Matt's criteria. If a subset of numbers are really important, you probably want to keep them distinct from the set of numbers used to represent them anyway, which means they'll have their own type descriptor on which to dispatch. It's not clear that moving program terms into dispatching logic is really that beneficial (which is probably one reason predicate dispatching hasn't caught on).\n\nThe only places I'd use subset constraints are in partial functions for the range that's undefined (like 0 in division), in which case I want a compile-time error for values that are out of range. I don't see the advantage in dispatching on counts on the range for which a function is defined.\n\n### Predicate dispatching (like\n\nPredicate dispatching (like the groundbreaking work in Cecil) was an interesting idea, but I don't like the idea of an object changing its type. I think that's my big issue with it. There's also the (solvable) problem of how to specify a partial order of \"named\" predicates (like Eiffel's named invariants) so that the dispatch logic can choose the most specific case.\n\nAvail allows a version of this directly, but only by checking all of the invariants at instantiation time and attaching \"natures\" to the object, which are later used to influence dispatch. Note that the type of an object can't change, but the (immutable) state of the object still influences its type in an arbitrary way.\n\nAvail's object model also allows embedded mutable variables to be shared between multiple objects, providing an interesting type-safe non-delegating form of instance inheritance like S\u00e1\u00b4\u2021\u00ca\u0178\u00ea\u0153\u00b0's (that's Self if my font trick doesn't work) parent slots. It wasn't designed with that in mind, but I noticed the correspondence between the two ideas in the late '90s.\n\n### Thanks for the explanation\n\nThanks for the explanation of your syntax. That makes sense.\n\nFinally, your question about dispatching on values... Introspection sort, fibonacci, and a host of other operations lend themselves to dispatching by count.\n\nMy question was about dispatching on types. I infer from your answer, though, that your rule is to dispatch to the most specific function (this is where you need the subtype comparisons) for which the arguments are each of the corresponding parameters' type. I guess that makes sense. It's quite different from how my setup works, but it sounds like an interesting language. Good luck!\n\nI leave you with the suggestion to try to bolt on a syntax for quantified types in a way that allows you to write e.g. 'a -> a' and get back 'Any -> Any' with an implicit semantic restriction. :)\n\n### Thanks for that! We\n\nThanks for that! We certainly generate semantic restrictions like that in other circumstances, but this particular one would be a handy shortcut for those places where quantified typing is all that's needed.\n\n### Open source?\n\nSeveral parts are interesting to me. I'll edit this post later to add more comments, unless I find out it's planned to be proprietary, in which case I'll say less. Is part of the project open source? (Is it possible you mean s\/persuasive\/pervasive\/`?)\n\nEdit: yes, the first paragraph at the top of www.availlang.org says it's open source with a 3-clause BSD license, which sounds good to me. The virtual machine, and presumably the whole dev environment, runs on top of Java. So it seems a Java extension of sorts. (This obviates my personal interest, since I'm only interested in lightweight processes and coroutines in code that can run in C or C++.)\n\n### fibers\n\nAvail doesn't extend java but does run on top of the jvm - like scala and clojure.\n\nAvail has light-weight fibers (via reified immutable continuations) that are multiplexed on top of jvm threads.\n\n### Java core\n\nJava is the implementation language, that's all. Well, and we also have POJO (plain old Java object) interface which hasn't gotten a great deal of use. POJOs have a sufficiently precise type in Avail that we're even able to unerase generics, potentially making it safer to use Java objects in Avail than in Java!\n\nThe Avail VM used to be written in Smalltalk and C++ (most of the C++ part being mechanically translated at will from Smalltalk), and it was only in the last few years that we ported it onto Java. We're quite likely to move it entirely off Java and onto either LLVM or Clang (leaning toward Clang for its licence freedom) in the next year or two.\n\n### use of fibers is cool\n\nSemantics carry over from implementation to runtime, so it's not a clean disconnect. Care and feeding of the jvm runtime is a runtime production deployment dependency, until you do it another way. I'm just observing, not complaining. When you move, I might be able to use it in something that cannot embed a jvm runtime.\n\nI like your use of fibers and coroutines, which is my main interest in the language. (I'm fairly indifferent to what a language looks like. I care mainly about runtime behavior. If a language's runtime behavior cannot be described except in terms of something very complex about syntax and type models, the resulting second class status of runtime behavior makes a language less desirable, as a matter of personal taste.) I'll keep track and look for an opportunity to use it later.\n\n### We appreciate your concern\n\nWe appreciate your concern about the runtime dependency, which is why we moved it to Java in the first place -- from Smalltalk and C++, neither of which was as ubiquitous as Java (on desktops, our initial broad target). Java's pretty clearly a temporary step for us.\n\nYou may want to check out VariableDescriptor.java and its subclasses to see how and when we elide locks and memory consistency without impacting the actual visible consistency. There are some pretty novel ideas in there, that are basically only applicable to Avail's other novel design choices. The Synchronization.avail module and its neighbors show how we cribbed park\/unpark from Java (in a way that makes sense for fibers multiplexed through a work pool), and built up various constructs from that and atomic variable operations. If you think you see a bug or significant inefficiency in there, please let us know.\n\nYour comment about semantics over syntax (to simplify) are exactly where Avail is coming from. The programmer essentially can't define anything new in the language, because it's already there. Even the way the object hierarchy works is semantically fixed by the VM, and all one can do is choose how to use it.\n\n### semantic restrictions\n\nI believe parametric polymorphism can be achieved by 'strengthening' any type via semantic restrictions.\n\n### Yeah\n\nIt does look like that would work, but it would be verbose (a big semantic restriction clause following each polymorphic function). And if you read at the bottom of that page they explain that the type system is customizable but makes no attempt at soundness, so I'm guessing the influence of Coq was minimal :).\n\nThe emphasis of the language appears to be on customization: syntax and type system. Seems like someone posted to LtU a couple of years ago about their similar customizable type system, and I've seen several similar extensible syntax proposals. I don't think either of these is going to be enough to sell the language, and I'm not sure what else novel there is. I'm with Sean that I'll wait for someone to explain more clearly what else is novel and why it's an improvement.\n\nEdit: But I notice there is a team of 6 people working on this, so maybe I'm being too dismissive.\n\n### Soundness\n\nCoq style correctness was out of scope for Avail's feature set. My goal at the time wasn't to provide a system so straight-jacketed that only the relatively narrow subset of correct programs would compile (although a lot has happened on that front since 1993!).\n\nInstead we strove for first-order type safety. That is, any expression is sound as long as all relevant semantic restrictions happen to be correct. So it's second-order type unsafe. It seems to be a pretty good compromise, although we run into wrong semantic restrictions from time to time. But some of those semantic restrictions (see the one for integer exponentiation, \"_^_\") feel like they're far too complex and powerful to be the consequence of a formal proof. If you can figure out how, we're certainly looking to grow our team.\n\n### Four days in and the team is\n\nFour days in and the team is already growing...\n\n### Anyone can choose their own\n\nAnyone can choose their own clothes to wear and call themselves a designer (Buxton?). Anyone can choose their own language features for a language and call themselves a language designer.\n\nI'm not seeing a coherent cohesive story in their documentation, just a bunch of features that are poorly presented, but maybe my presentations standards are too high. If it's worthwhile, I'm sure someone else will pick it up, understand it better than me, and blog about it eventually.\n\n### They certainly have an\n\nThey certainly have an impressive feature list. Type system-wise, looks like they have an undecidable sets-as-types interpretation supporting type union, intersection and subtyping\/subset relations. Not a fan of the syntax though.\n\n### Anyone figured out what this is about?\n\nA mechanism for observing expressions rather than values, thereby enabling a novel method of efficient, transactional change propagation that is more natural and powerful than the observer pattern or functional reactive programming.\n\n### \"Observing expressions\n\n\"Observing expressions rather than values\"? Sounds like fexprs...\n\n### Observerless\n\nVariables are first-class objects in Avail, whether they're module variables, local (and then perhaps captured) variables, or variables that are explicitly instantiated for some reason. Everything else is immutable. That factoring allows us to focus time on provided features that are tied to tracking mutable state, without having to deal with the variety of special cases that other languages might leave separate (instance variables, local variables, globals\/statics).\n\nNow stay with me. We also (currently) implement every Avail value as an instance of the final Java class AvailObject, which has a field called descriptor, of type Descriptor. Descriptor has many subclasses, instances of which may make the object behave like a set (SetDescriptor), a tuple (TupleDescriptor), a tuple of bytes with packed representation (ByteTupleDescriptor), Latin-1 strings (ByteStringDescriptor), etc. We also have IndirectionDescriptor to forward all messages to effect a change of representation (e.g., the expression \"a\" = \"a\" not only produces true, but will probably replace one of the AvailObjects' descriptors with an IndirectionDescriptor, plugging a forwarding pointer into a suitable place in the AvailObject. Subsequent compares won't need to examine the characters -- after traversal of indirections they'll simply be the same Java AvailObject and be trivially equal.\n\nThere are several Descriptor subclasses for representing variables. Some of these implement variables that can be reached from multiple fibers (and therefore multiple Java Threads running on separate cores). Some are for variables that are currently only reachable from one fiber (and don't require locks *or* memory consistency). And some specify a kind of variable that's supposed to be tracked in some way. A fiber (and cached state in the Interpreter that's running it at that moment) keeps flags that indicate whether the fiber is running in a special state.\n\nAvail provides primitives to enable\/disable a flag that causes every read access of any variable to cause that variable to be added to a (fiber-specific, weak) set of variables. Another primitive allows a \"write reactor\" to be added to a variable, which is just a nullary function that runs whenever the variable changes. Combining these two mechanisms, we can run an expression like [a!+b*7] while tracking variable reads, thereby collecting the variables a and b. We can then add a write reactor to a and b, which, say, causes a text field somewhere (theoretically - Avail has no graphical UI yet) to be updated with the new result of running the function [a!+b*7]. Thus, a slider control that's wired up to change b will cause the write reactor to trip, which will cause a!+b*7 to be computed and updated in the text field. Note that this works even if a and b didn't lexically appear in the body of the function, but were accessed by a helper method ten layers down.\n\nWe call this pattern \"Observerless\" because we don't have to specify which variables need to have change tracking added to them -- we specify a function that produces a value to display somewhere, and let the execution machinery tell us when the function would produce a different value.\n\nTechnically, we also keep track of variables written before being read, and exclude them from the set of dependency variables when running the [a!+b*7] function.\n\nWe will eventually have to do something special for primitives that do things like getting the current time from the OS, or read from a file. Case-by-case, I believe.\n\n### This is weird\n\nThis is similar to what I've been looking into, but if you're going to make a language be verbose like this for readability, why make it so unreadable?\n\n### Depends on the DSL\n\nJ is terse and readable (for experts). Java is both verbose and readable (for novices). What determines readability anyway? Both J and Java can be embedded in Avail as DSLs. Does that make Avail unreadable?\n\n### I'm just looking at the\n\nI'm just looking at the examples on the website. It seems as though different parts of the language waffle between literate constructs and absurdly terse functional programming constructs. The literate seems at times to be awkward:\n\nUse newGame as the implied game and do\n\nAnd the terseness shows up elsewhere:\n\n\"_st|nd|rd|th Fibonacci number\"\n\nAs a programmer that's used a fairly wide variety of languages, I expect to be able to read code and at least have a vague idea of what's going on - especially in the basic intro programs.\n\nMaybe it's an issue with the tutorials or with my ability to grok code rather than the language itself, but if I can't look at code at get it, there's no way someone who's only ever seen Java will be able to.\n\n### The grammar\n\nThe grammar of Avail is actually pretty easy to discuss, both verbally and textually. Things like \"_+_\" we pronounce \"blank plus blank\". So if you don't like some specific syntax, we'd love to hear discussion of specific syntactic improvements. You can also play around with it until you have a form you like, then tell us about your experiences. The easiest way to play is to define your own module that imports the Avail module, then uses negated imports (like -\"_+_\") to block some things and renamed imports (\"_+_\" \u00e2\u2020\u2019 \"_aggregate with_\") to change other things. There's no need to mess around with parsers or parser generators or even awkward BNF expressions.\n\n### APL versus COBOL\n\nFrom what I've seen from the website, the standard library code (in some parts) is very close to natural language. But such language is too verbose for my taste (although I can appreciate the possible (user) benefits).\n\nIn particular, I think the syntactic declaration of Methods and their semantic restriction (the most used constructs) should be shortened. I don't like to type much, if I can help it.\nAll this is personal preference of course: I'm recovering from my own programming language Enchilada which can be considered as an APL-ish postfix language.\n\nThat said, I truly believe that Avail is worthy to be seriously considered and studied for its unique merits.","date":"2017-12-15 02:34:57","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.45362967252731323, \"perplexity\": 1258.6756854382536}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-51\/segments\/1512948563083.64\/warc\/CC-MAIN-20171215021156-20171215041156-00184.warc.gz\"}"} | null | null |
Q: Checks for the true condition in for loop to store in datatable A for loop for the true condition sends a parameter to stored procedure and then required result is stored in a DataTable. Now my problem is at every loop the value in the DataTable gets refreshed with new value and previous value gets lost. How to retain all the value of true condition in the DataTable?
for (int i = 0; i < gridview1.Rows.Count; i++)
{
string yojnaNo = "";
CheckBox chl = (CheckBox)gridview1.Rows[i].Cells[0].FindControl("CheckBox1");
if (chl != null)
{
if (chl.Checked == true)
{
int rowsNo = (Convert.ToInt16(chl.ToolTip) - 1); //Convert.ToInt16(SrNo);
yojnaNo = ((Label)gridview1.Rows[rowsNo].FindControl("lblYojnaNo")).Text;
sc.Add(yojnaNo);
}
}
objyojnadetail4.YojnaNo = sc;
DataTable city = objyojnadetail4.Selectcity();
}
A: You can put the instantiation of city on the outside and add each rows into it in every loop. You won't lose the value, instead, the city will getting bigger each loop.
DataTable city = new DataTable(); // instantiate here
for (int i = 0; i < gridview1.Rows.Count; i++)
{
string yojnaNo = "";
CheckBox chl = (CheckBox)gridview1.Rows[i].Cells[0].FindControl("CheckBox1");
if (chl != null)
{
if (chl.Checked == true)
{
int rowsNo = (Convert.ToInt16(chl.ToolTip) - 1); //Convert.ToInt16(SrNo);
yojnaNo = ((Label)gridview1.Rows[rowsNo].FindControl("lblYojnaNo")).Text;
sc.Add(yojnaNo);
}
}
objyojnadetail4.YojnaNo = sc;
// this foreach loop may loop on anything the objyojnadetail4.Selectcity() provides
// what was important is that, in this loop you insert each data into rows in city.
foreach(var singleItem in objyojnadetail4.Selectcity().Rows)
{
city.Rows.Add(singleItem);
}
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,501 |
\section{Introduction}
In this article, we examine the \emph{existence} and \emph{uniqueness} of viscosity solutions, of a system of possibly fully nonlinear obstacle problems. Specifically, the system consists of variational type inequalities accompanied with interconnected obstacles, which furthermore satisfies initial and boundary conditions.\\
Moreover, let $ \Omega\subset\mathbb{R}^n $ be a bounded domain, i.e open, connected and bounded set in $ \mathbb{R}^n $ and let $ i\in\left\lbrace 1,2,\dots,m \right\rbrace $ for some positive integer $ m. $ The problem (IBVP) is stated as follows:\\
\begin{gather}
\min\biggl\{ F\bigl( y,x,u_{i}(y,x),D_{x}u_{i}(y,x),D_{xx}^2 u_{i}(y,x)\bigl), u_{i}(y,x)-\max_{j\neq i}\bigl( u_{j}(y,x)-c_{ij}(y,x)\bigl)\biggl\}=0,\ \forall\left(y,x \right)\in\Omega_{L}\nonumber\\
u_{i}(0,x)=g_{i}(x), x\in\bar{\Omega},\ u_i(y,x)=f_i(y,x), (y,x)\in(0,L)\times\partial{\Omega}\nonumber
\end{gather}
where $ \Omega_{L}:=(0,L)\times\Omega. $\\
The solution is considered as a vector mapping, $$ u:=\left( u_1,u_2,\dots,u_m\right):[0,L]\times\bar{\Omega}\rightarrow\mathbb{R}^m $$ where the component functions $ u_i $ are interconnected through the obstacle $ \mathcal{M}_i $, in such a way that the component function $ u_i $ will be above the obstacle
$$ \mathcal{M}_i u(y,x):=\max_{j\neq i}\left( u_{j}(y,x)-c_{ij}(y,x)\right) $$
This problem is included in the general area of the so called optimal switching problems and
occurs, for instance, when one models a facility which splits its production in m modes. In optimal switching problems there are basically two main approaches i.e. the probabilistic approach and the deterministic approach. For the first approach we refer the reader to \cite{AF12}, \cite{DH09}, \cite{DHP10}, \cite{HT07} and \cite{HZ10} for a well developed theory of stochastic differential equations governed by Brownian motion and the connection to multi-modes optimal switching problems. In the local setting, the second (deterministic) approach deals with the well developed theory of variational inequalities within the general area of partial differential equations (see \cite{A16}, \cite{ADPS09}, \cite{AH09}, \cite{EF79}, \cite{HM}, \cite{LNO14}, and references therein).
The main purpose of the present article is to study the elliptic fully nonlinear analogue of variational type inequalities accompanied with interconnected obstacles (which is originated in the fundamental work of Evans and Friedman \cite{EF79}). For this purpose we use the notion of viscosity solutions to fully nonlinear equations which provides a powerful way to prove existence and uniqueness in a very general setting. Our techniques could be considered within the classical approach which has been very successful the last two decades when seeking existence in problems with thick or thin obstacles (see for instance \cite{kiamlee1}, \cite{kiamlee2}, \cite{ms1}, \cite{ms2} and references therein) and are greatly inspired by the parabolic analogue of a similar problem which has been recently studied by N. Lundstr\"{o}m and M. Olofsson in \cite{LNOO4}. Although we follow the approach in \cite{LNOO4}, especially when dealing with the corresponding Perron method and comparison principles, we need to pay extra attention in the technical parts of the proofs. This has to do with the fact that our problem seems to be more close to obstacle type problems for extension operators of degenerate form (see the pioneer work of Caffarelli-Silvestre in \cite{CS}).
Throughout the article, the symbolisms $ D_x u_{i}\equiv D u_{i} \in R^n $, $ D_{xx}^2 u_{i}\equiv D^2 u_{i}\in\mathcal{S}^n $ refer respectively to the gradient and the hessian matrix of $ u_i $ with respect to the variable $ x $, where $ \mathcal{S}^n $ is the set of real symmetric matrices $ n\times n $. Additionally, the operator $ F:\left[0,L \right] \times\mathbb{R}^n\times\mathbb{R}\times\mathbb{R}^n\times\mathcal{S}^n\to\mathbb{R} $ is considered as a posibbly fully nonlinear for which, further assumptions will be set for the study of the problem. The modifier ``\textbf{possibly fully nonlinear}'' stands for ``\textbf{$ F $ may not be linear in any of its arguments}", but of course linear forms of $ F $ are not excluded. I.e $ F $ is either fully nonlinear or a quasilinear operator.
\section{Assumptions, Definitions and Main Results}
\subsection{Assumptions}
Let $ \Omega\subset\mathbb{R}^n $ be an open, connected and bounded set, with closure $ \bar{\Omega} $ and boundary $ \partial{\Omega}=\bar{\Omega}/\Omega $ and let $ y\in[0,L]. $
\subsubsection{ Assumptions for the operator $ F $}
We suppose that the operator $ F:[0,L]\times\mathbb{R}^n\times\mathbb{R}\times\mathbb{R}^n\times\mathcal{S}^n\to\mathbb{R}, $ is a \emph{continuous} mapping which satisfies the following:
\begin{itemize}
\item [( F1)] For each $ \left( y,x,p,X\right)\in[0,L]\times\mathbb{R}^n\times\mathbb{R}^n\times\mathcal{S}^n, $ the mapping $ r\mapsto F\left( y,x,r,p,X\right)-\gamma r $ is strictly increasing for some constant $ \gamma>0 $. In specific, for each $ r, s\in \mathbb{R} $ with $ r<s $, holds:
$$\gamma (s-r)\leq F\left( y,x,s,p,X\right)-F\left( y,x,r,p,X\right) $$
\end{itemize}
There exists a \emph{continuous} function $ \omega :[0,\infty) \to [0,\infty)$ with $ \omega(0+)=0, $ such that:
\begin{itemize}
\item [( F2)] $ \forall\left(y,x,r\right)\in [0,L]\times\mathbb{R}^n\times\mathbb{R}, \forall p,q\in\mathbb{R}^n,\forall X,Y\in\mathcal{S}^n $
$$ \left| F\left(y,x,r,p,X \right)-F\left(y,x,r,q,Y \right) \right|\leq\omega\big(\left| p-q\right|+\norm{X-Y})$$
\item [( F3)] $ \forall\left( y,r,p\right)\in[0,L)\times\mathbb{R}\times\mathbb{R}^n, \forall x,\tilde{x}\in\mathbb{R}^n,\forall X,Y\in\mathcal{S}^n, \forall\epsilon>0 $, holds
$$ F\left(y,\tilde{x},r,p,Y \right)-F\left(y,x,r,p,X \right)\leq\omega\Big(\frac{1}{\epsilon}\left|x-\tilde{x} \right|^2+\left|x-\tilde{x} \right|\left( \left|p \right|+1 \right)\Big) $$
whenever
$$\begin{pmatrix} X & 0 \\0 & -Y \end{pmatrix} \leq\frac{3}{\epsilon}\begin{pmatrix} I & -I \\-I & I \end{pmatrix}$$
\end{itemize}
From the above conditions for the operator $ F $, we extract that $ F $ is \emph{ (degenerate) elliptic}. In specific,
$$ \forall \left(y,x,r,p \right)\in[0,L]\times\mathbb{R}^n\times\mathbb{R}\times\mathbb{R}^n,\forall X,Y\in S(n), X\leq Y\Rightarrow F\left(y,x,r,p,X \right)\geq F\left(y,x,r,p,Y \right) $$
Consequently, $ F $ is a \emph{decreasing} function with respect of the fifth variable.
\subsubsection{ Assumptions for the obstacle functions$ \left(c_{i,j} \right) $}
For the obstacle functions, $ \left(c_{i,j} \right) $ we make the following assumptions:
\begin{itemize}
\item [(O1)] $ c_{i,j}\in C^{1,2}\left( [0,L]\times\bar{\Omega}\right), \forall i,j\in\left\lbrace 1,2,\dots,m \right\rbrace $
\item [(O2)] $ c_{i,i}(y,x)=0,\forall \left( y,x\right)\in[0,L] \times\bar{\Omega}, \forall i\in\left\lbrace 1,2,\dots,m\right\rbrace$
\item[(O3)] For each finite sequence of indices $ i_j\in\left\lbrace 1,2,\dots,m\right\rbrace ,\forall j\in\left\lbrace 1,2,\dots,k\right\rbrace , \forall\left(y,x \right)\in[0,L]\times\bar{\Omega}$ holds
$$c_{i_1,i_2}\left( y,x\right) +c_{i_2,i_3}\left( y,x\right) +\cdots+c_{i_{k-1},i_k}\left( y,x\right) +c_{i_k,i_1}\left( y,x\right) >0 $$
\end{itemize}
For the existence of viscosity solutions of the problem $ (IBVP) $ we demand furthermore the condition that follows:
\begin{itemize}
\item[(O4)] $ \forall\left( y,x\right)\in[0,L]\times\bar{\Omega}, \forall i,j,k\in\left\lbrace 1,2,\dots,m\right\rbrace $ holds:
$$c_{i,k}\left(y,x\right) \leq c_{i,j}\left(y,x\right) +c_{j,k}\left(y,x\right) $$
\item[(O5)] Finally, we demand for the boundary data $ f_i,\ i\in\{1,2,\dots,m\} $ to be continuous on $ (0,L)\times\partial\Omega $. Moreover, we demand for the data $ g_i,i\in\left\lbrace 1,2,\dots,m \right\rbrace $ to be continuous, i.e $g_i\in C\left( \bar{\Omega}\right) $ and to be compatible with the obstacle functions, in the following way:\\ $ \forall x\in\bar{\Omega}, \forall i,j\in\left\lbrace 1,2,\dots,m \right\rbrace $
$$g_i(x)\geq g_j(x)-c_{i,j}\left(0,x \right) $$
\end{itemize}
\subsection{Definitions}
First, the notions of \emph{subjets} and \emph{superjets} are defined, from which we define later the notion of \emph{Viscosity Solutions} for the problem $(IBVP)$.
\begin{definition}[\emph{Subjets} and \emph{Superjets}]
Let $ h:[0,L]\times\bar{\Omega}\to\mathbb{R} $ be a mapping. For $\left( y_0, x_{0}\right)\in(0,L)\times\Omega $ we define the sets $ J^{2,+}h(y_0,x_{0}) $ and $ J^{2,-}h(y_0,x_{0}) $, which are called respectively SuperJet of $ h $ on $\left( y_0,x_{0} \right) $ and SupJet of $ h $ on $\left( y_0,x_{0} \right) $.
\begin{itemize}
\item $\forall\left(\alpha,p,X \right)\in\mathbb{R}\times\mathbb{R}^{n}\times S(n), \left(\alpha,p,X \right)\in J^{2,+}h(y_0,x_{0}) $ if and only if
$$ h(y,x)\leq h(y_0,x_{0})+\alpha\left(y-y_0 \right)+\left\langle p,x-x_{0} \right\rangle +\frac{1}{2} \left\langle X(x-x_{0}),x-x_{0} \right\rangle \\ + o\left(\left|y-y_0 \right|+ \left| x-x_{0} \right|^{2} \right),$$
while $ \left(0,L \right) \times\Omega\ni\left( y,x\right) \to\left(y_0,x_{0}\right) $
\item $\forall\left(p,X \right)\in\mathbb{R}^{n}\times S(n), \left(p,X \right)\in J^{2,-}h(y_0,x_{0}) $ if and only if
$$ h(y,x)\geq h(y_0,x_{0})+\alpha\left(y-y_0 \right)+\left\langle p,x-x_{0} \right\rangle +\frac{1}{2} \left\langle X(x-x_{0}),x-x_{0} \right\rangle \\ + o\left(\left|y-y_0 \right| + \left| x-x_{0} \right|^{2} \right),$$
while $ \left(0,L \right) \times\Omega\ni\left( y,x\right) \to\left(y_0,x_{0}\right) $
\end{itemize}
Correspondingly, we define the closures of the $ J^{2,+}h(y_0,x_{0}) $ and $ J^{2,-}h(y_0,x_{0}) $. In specific:
\begin{itemize}
\item $ \left(\alpha, p,X \right) \in \bar{J}^{2,+}h(y_0,x_{0}) $ if and only if, $ \exists \left( y_k,x_{k}\right)\in\left(0,L \right)\times \Omega,\exists\left(\alpha_k, p_{k},X_{k}\right)\in J^{2,+}h(y_k,x_{k}), $
$$ \left(y_k, x_{k},h(y_k, x_{k}),\alpha_k,p_{k},X_{k}\right)\rightarrow\left( y_0,x_{0},h(y_0,x_{0}),\alpha,p,X\right)$$
\item $ \left(\alpha, p,X \right) \in \bar{J}^{2,-}h(y_0,x_{0}) $ if and only if, $ \exists \left( y_k,x_{k}\right)\in\left(0,L \right)\times \Omega,\exists\left(\alpha_k, p_{k},X_{k}\right)\in J^{2,-}h(y_k,x_{k}), $
$$ \left(y_k, x_{k},h(y_k, x_{k}),\alpha_k,p_{k},X_{k}\right)\rightarrow\left( y_0,x_{0},h(y_0,x_{0}),\alpha,p,X\right)$$
\end{itemize}
\end{definition}
Next we provide the definition of \emph{ Viscosity Solution} of the problem $ (IBVP) $.
\begin{definition}[\emph{ Viscosity Solutions}]
Let $ \textbf{u}=\left(u_{1},u_{2},\dots,u_{m} \right):\left[ 0,L\right] \times\bar{\Omega}\rightarrow\mathbb{R}^{m} $ be a mapping. Then, $ \textbf{u}:[0,L]\times\bar{\Omega}\rightarrow\mathbb{R}^{m} $ is called:
\begin{itemize}
\item[($\alpha$)] \emph{ viscosity subsolution} of the problem $ (IBVP) $ if and only if, it is upper semicontinuous on \\ $ \bar{\Omega}_L:=[0,L]\times\bar{\Omega}, $ i.e $ u_i\in USC(\bar{\Omega}_{L}) $ and \\
$ \forall i\in\left\lbrace 1,2,\dots, m \right\rbrace,\forall \left(y,x \right)\in(0,L)\times\Omega,\forall\left(\alpha, p,X \right)\in \bar{J}^{2,+}u_i(y,x),$
$$ \min\left\lbrace F\left(y,x,u_i(t,x),p,X\right),u_i(y,x)-\mathcal{M}_i\textbf{ u}(y,x)\right\rbrace \leq 0 $$ \textrm{ and } $ \forall i\in\left\lbrace 1,2,\dots, m \right\rbrace, u_{i}(0,x)\leq g_{i}(x),\ x\in\bar{\Omega},\quad \land \quad u_i(y,x)\leq f_i(y,x),\ (y,x)\in (0,L)\times\partial{\Omega} $
\item[($ \beta $)] \emph{ viscosity supersolution} of the problem $ (IBVP) $ if and only if, it is lower semicontinuous\\ on $ \bar{\Omega}_L:=[0,L]\times\bar{\Omega}, $ i.e $ u_i\in LSC(\bar{\Omega}_{L}) $ and\\
$ \forall i\in\left\lbrace 1,2,\dots, m \right\rbrace,\left(y,x \right)\in(0,L)\times\Omega,\forall\left(\alpha, p,X \right)\in \bar{J}^{2,-}u_i(y,x),$
$$\min\left\lbrace F\left(y,x,u_i(y,x),p,X\right),u_i(y,x)-\mathcal{M}_i\textbf{ u}(y,x)\right\rbrace \geq 0 $$ \textrm{ and } $ \forall i\in\left\lbrace 1,2,\dots, m \right\rbrace,u_{i}(0,x)\geq g_{i}(x),\ x\in\bar{\Omega},\quad \wedge\quad u_i(y,x)\geq f_i(y,x),\ (y,x)\in (0,L)\times\partial{\Omega} $
\item[($\gamma$)] \emph{ viscosity solution} of the problem $ (IBVP) $ if and only if,
$ \textbf{u} $ is bounded on $ [0,L]\times\bar{\Omega} $, continuous on $ [0,L)\times\bar{\Omega} $ and satisfies the following: \\
$ \forall i\in\left\lbrace 1,2,\dots, m \right\rbrace,\left(y,x \right)\in(0,L)\times\Omega,\forall\left(\alpha, p,X \right)\in \bar{J}^{2,-}u_i(y,x),$
$$\min\left\lbrace F\left(y,x,u_i(y,x),p,X\right),u_i(y,x)-\mathcal{M}_i\textbf{ u}(y,x)\right\rbrace = 0 $$ \textrm{ and } $ \forall i\in\left\lbrace 1,2,\dots, m \right\rbrace,u_{i}(0,x)= g_{i}(x),\ x\in\bar{\Omega},\quad \wedge\quad u_i(y,x)= f_i(y,x),\ (y,x)\in (0,T)\times\partial{\Omega} $
\end{itemize}
\end{definition}
\subsection{ Main Results}
\subsection*{Uniqueness of Solution}
In this article, when we refer to the notion of uniqueness of viscosity solution of $ (IBVP), $ we mean that, for every two functions $ u, w:[0,L]\times\bar{\Omega}\to\mathbb{R}^m $ that satisfy the definition of viscosity solution, have to be identical on the set $ [0,L)\times\bar{\Omega}. $ I.e
\begin{gather}
\forall i\in\{1,2,\dots,m\},\ \forall(y,x)\in[0,L)\times\bar{\Omega},\ u_i(y,x)=w_i(y,x)
\end{gather}
For this task, it is sufficent to prove an appropriate Comparison Principle.
\begin{theorem}[ Comparison Principle]
\label{Comparison Principle}
Let the axioms $ (F1)-(F3) $ and $ (O_1)-(O_3)$ hold. If $ u:[0,T]\times\bar{\Omega}\to\mathbb{R}^m $ and $ v:[0,L]\times\bar{\Omega}\to\mathbb{R}^m $ are subsolution and supersolution respectively of the problem $ (IBVP) $, then $$ u(y,x)\leq v(y,x), \forall (y,x)\in [0,L)\times\bar{\Omega}. $$
\end{theorem}
\begin{proof}[ Proof]
Throughout of the proof, we consider that functions $ u$ and $ v $ are subsolution and supersolution respectively of the problem $ (IBVP). $ The proof is implemented through the following steps. \\
\textbf{Step 1}\ From the assumption, the $ v_i\in LSC\left( [0,L]\times\bar{\Omega}\right). $ So equivalently the $ -v_i\in USC\left( [0,L]\times\bar{\Omega}\right). $ Then the $ u_i-v_i\in USC\left( [0,L]\times\bar{\Omega}\right), \forall i\in\left\lbrace 1,2,\dots,m \right\rbrace $. Because the $ u_i-v_i $ is upper semicontinuous on the compact set $ [0,L]\times\bar{\Omega}, $ from Maximum Principle for upper semicontinuous functions, will exists $ \left(y_i,x_i \right)\in [0,L]\times\bar{\Omega} $ such that $ \sup_{[0,L]\times\bar{\Omega}}\left(u_i-v_i \right)\left( y,x\right)=u_i(y_i,x_i) -v_i(y_i,x_i) .$ Let $ i_0\in\left\lbrace 1,2,\dots,m\right\rbrace $, such that:
$$\max_{k\in\left\lbrace 1,2,\dots,m\right\rbrace}\sup_{[0,L]\times\bar{\Omega}}\left( u_k-v_k\right)(t,x)=\sup_{[0,L]\times\bar{\Omega}}\left( u_{i_0}-v_{i_0}\right)(y,x)=u_{i_0}(y_{i_0},x_{i_0})-v_{i_0}(y_{i_0},x_{i_0})\equiv\delta $$
We define
$$I:=\big\{ i\in\{ 1,2,\dots,m\} \ : \ u_i\left( y_{i_0},x_{i_0}\right) -v_i\left( y_{i_0},x_{i_0}\right) =\delta \big\}.$$
Clearly we see that $ I\neq\emptyset $ because $ i_0\in I $ and in addition, from the definition of $ I $, we get that \\ $ \forall i\in I, \sup_{[0,L]\times\bar{\Omega}}\left(u_i-v_i \right)\left( y,x\right) =\delta. $
Let us assume the opposite of what we seek to prove. As a result of this, we receive that
\begin{gather}
\exists \left( \tilde{y},\tilde{x}\right) \in [0,L)\times\bar{\Omega}: \rceil \left(u\left( \tilde{y},\tilde{x}\right) \leq v\left(\tilde{y},\tilde{x} \right) \right)\nonumber\\
\Longleftrightarrow \exists \left( \tilde{y},\tilde{x}\right) \in [0,L)\times\bar{\Omega},\ \exists \tilde{i}\in\left\lbrace 1,2,\dots,m\right\rbrace: u_{\tilde{i}}\left(\tilde{y},\tilde{x} \right)-v_{\tilde{i}}\left( \tilde{y},\tilde{x}\right) >0\nonumber
\end{gather}
So we get
\begin{gather}
\delta\geq\sup_{[0,L]\times\bar{\Omega}}\left( u_{\tilde{i}}-v_{\tilde{i}}\right) \left( y,x\right) \geq u_{\tilde{i}}\left(\tilde{y},\tilde{x} \right)-v_{\tilde{i}}\left( \tilde{y},\tilde{x}\right) >0\nonumber
\end{gather}
In consequence, $ \delta>0. $ \\
Due to the fact that $ u,v $ are subsolution and supersolution respectively, we receive from their definition, that
\begin{gather}
u_i(0,x)\leq g_i(x)\leq v_i(0,x),\ \forall x\in\bar{\Omega}\\
u_i(y,x)\leq f_i(y,x)\leq v_i(y,x),\ \forall (y,x)\in(0,L]\times\partial{\Omega}
\end{gather}
From the above inequalities, we conclude that $ y_{i_0}>0 $ and $ x_{i_0}\in\Omega. $ In the continuity of our analysis, it is sufficent to further consider that ( the explanation is given in the end of the proof) that, $ y_{i_0}<L. $
We choose a random $ i\in I $ in the analysis that follows. For each $ \epsilon>0, $ we define the following mappings:
$$\phi_{\epsilon},\Phi_{\epsilon}:[0,L]\times\bar{\Omega}\times\bar{\Omega}\to\mathbb{R}$$
defined as
\begin{gather}
\phi_{\epsilon}(y,x,\tilde{x}):=\frac{1}{2\epsilon}\abs{x-\tilde{x}}^2+\abs{x-x_{i_0}}^4+\abs{\tilde{x}-x_{i_0}}^4+\abs{y-y_{i_0}}^2\nonumber\\
\Phi_{\epsilon}(y,x,\tilde{x}):=u_i(y,x)-v_i(y,\tilde{x})-\phi_{\epsilon}(y,x,\tilde{x})\nonumber
\end{gather}
The $ \Phi_{\epsilon} $ as an upper semicontinuous on the compact set $ [0,L]\times\bar{\Omega}\times\bar{\Omega}, $ attains a maximum value. Let $ \left(y_{\epsilon},x_{\epsilon},\tilde{x}_{\epsilon} \right)\in [0,L]\times\bar{\Omega}\times\bar{\Omega} $ such that
\begin{gather}
\Phi_{\epsilon}\left(y_{\epsilon},x_{\epsilon},\tilde{x}_{\epsilon} \right) =\max_{[0,L]\times\bar{\Omega}\times\bar{\Omega}} \Phi_{\epsilon}\left(y,x,\tilde{x}\right)\label{eq_3}
\end{gather}
Next, an important estimation of the distance between the terms of $ x_{\epsilon} $ and $ \tilde{x}_{\epsilon} $ is proved, where we will subsequently see that it is used to prove the limiting behaviour of an appropriate sequence of points which is extracted from the following family of points $ \left(y_{\epsilon},x_{\epsilon},\tilde{x}_{\epsilon} \right)_{\epsilon>0}. $
\begin{claim}\label{claim_1}
$$\forall \epsilon>0,\quad \frac{1}{2\epsilon}\abs{x_{\epsilon}-\tilde{x}_{\epsilon}}^2\leq u_i(y_{\epsilon},x_{\epsilon})-v_i(y_{\epsilon},\tilde{x}_{\epsilon})-\delta $$
\end{claim}
\begin{proof}[ Proof]
We observe that
\begin{gather}
\Phi_{\epsilon}\left(y_{i_0},x_{i_0},x_{i_0} \right)=u_i\left(y_{i_0},x_{i_0} \right) - v_i\left(y_{i_0},x_{i_0} \right)=\delta\quad \left( \text{because}\ i\in I \right) \nonumber
\end{gather}
but at the same time, from $ \left( \ref{eq_3}\right) $ we conclude that $ \Phi_{\epsilon}\left(y_{\epsilon},x_{\epsilon},\tilde{x}_{\epsilon} \right)\ge\Phi_{\epsilon}\left(y_{i_0},x_{i_0},x_{i_0} \right)=\delta $. Therefore, we receive that
\begin{gather}
u_i(y_{\epsilon},x_{\epsilon})-v_i(y_{\epsilon},\tilde{x}_{\epsilon})-\phi_{\epsilon}(y_{\epsilon},x_{\epsilon},\tilde{x}_{\epsilon})\ge\delta\nonumber\\
\Longleftrightarrow u_i(y_{\epsilon},x_{\epsilon})-v_i(y_{\epsilon},\tilde{x}_{\epsilon})-\Big{(}\frac{1}{2\epsilon}\abs{x_{\epsilon}-\tilde{x}_{\epsilon}}^2+\abs{x_{\epsilon}-x_{i_0}}^4+\abs{\tilde{x}_{\epsilon}-x_{i_0}}^4+\abs{y_{\epsilon}-y_{i_0}}^2\Big{)} \ge\delta\nonumber\\
\Longrightarrow u_i(y_{\epsilon},x_{\epsilon})-v_i(y_{\epsilon},\tilde{x}_{\epsilon})-\frac{1}{2\epsilon}\abs{x_{\epsilon}-\tilde{x}_{\epsilon}}^2\ge\delta\nonumber\\
\Longleftrightarrow \frac{1}{2\epsilon}\abs{x_{\epsilon}-\tilde{x}_{\epsilon}}^2\leq u_i(y_{\epsilon},x_{\epsilon})-v_i(y_{\epsilon},\tilde{x}_{\epsilon})-\delta\label{eq_4}
\end{gather}
\end{proof}
We consider a random sequence $ \left( \epsilon_{\tilde{k}}\right)_{\tilde{k}\in\mathbb{N}} $ in $ (0,\infty) $ such that $ \lim_{\tilde{k}\to\infty}\epsilon_{\tilde{k}}=0. $ Hence, a corresponding sequence of points $ \left\lbrace \left( y_{\epsilon_{\tilde{k}}},x_{\epsilon_{\tilde{k}}},\tilde{x}_{\epsilon_{\tilde{k}}}\right)_{\tilde{k}\in\mathbb{N}}\right\rbrace \subset[0,L]\times\bar{\Omega}\times\bar{\Omega}$ will exists. Since the $ [0,L]\times\bar{\Omega}\times\bar{\Omega} $ is compact, it will simultaneously also be sequentially compact. Therefore, the sequence of points $ \left( y_{\epsilon_{\tilde{k}}},x_{\epsilon_{\tilde{k}}},\tilde{x}_{\epsilon_{\tilde{k}}}\right)_{\tilde{k}\in\mathbb{N}} $ has a subsequence, let's say $ \left( y_{\epsilon_k},x_{\epsilon_k},\tilde{x}_{\epsilon_k}\right)_{k\in\mathbb{N}}, $ such that,\\ $ \left( y_{\epsilon_k},x_{\epsilon_k},\tilde{x}_{\epsilon_k}\right)\xrightarrow{k\to\infty}\left(\hat{y},\hat{x},\hat{\tilde{x}} \right) \in[0,L]\times\bar{\Omega}\times\bar{\Omega}, $ and in addition the following will hold $ \lim_{k\to\infty}\epsilon_k=0.\\ \quad \left(\text{because}\left( \epsilon_k\right)_{k\in\mathbb{N}}\ \text{is a subsequence of} \left(\epsilon_{\tilde{k}} \right)_{\tilde{k}\in\mathbb{N}} \right).$\\
\begin{claim}\label{claim_2}
$$\lim_{k\to\infty}\abs{x_{\epsilon_k}-\tilde{x}_{\epsilon_k}}=0 \text{ and } \hat{x}=\hat{\tilde{x}}$$
\end{claim}
\begin{proof}[ Proof]
We set $ M:=\max_{[0,L]\times\bar{\Omega}}u(y,x)-\min_{[0,L]\times\bar{\Omega}}v(y,\tilde{x}) $. Then $ \forall k\in\mathbb{N} $ we have \\ $ u\left( y_{\epsilon_k},x_{\epsilon_k}\right)- v\left( y_{\epsilon_k},\tilde{x}_{\epsilon_k}\right) \leq M. $ From $ (\ref{eq_4}) $, the following holds
\begin{align}
0\leq\abs{x_{\epsilon_k}-\tilde{x}_{\epsilon_k}}^2&\leq 2\ \epsilon_k \left( u_i\left(y_{\epsilon_k},x_{\epsilon_k} \right)-v_i\left(y_{\epsilon_k},\tilde{x}_{\epsilon_k} \right)-\delta \right) \nonumber\\
&\leq 2\ \epsilon_k \left( M-\delta\right) <^{\left(\delta>0 \right) } 2\ \epsilon_k M\xrightarrow{k\to\infty} 0\label{eq_5}
\end{align}
From $ (\ref{eq_5}) $ we get $ \lim_{k\to\infty}\abs{x_{\epsilon_k}-\tilde{x}_{\epsilon_k}}=0.$ But at the same time, $ x_{\epsilon_k}\xrightarrow{k\to\infty}\hat{x} $ and $ \tilde{x}_{\epsilon_k}\xrightarrow{k\to\infty}\hat{y} $. Therefore, $ \abs{x_{\epsilon_k}-\tilde{x}_{\epsilon_k}}\xrightarrow{k\to\infty} \abs{\hat{x}-\hat{\tilde{x}}}.$
In consequence, from the uniqueness of limit, we have $ \hat{x}=\hat{\tilde{x}}. $
\end{proof}
\begin{claim}\label{claim_3}
$$ \lim_{k\to\infty}\frac{\abs{x_{\epsilon_k}-\tilde{x}_{\epsilon_k}}^2}{ 2\ \epsilon_k} =0$$
\end{claim}
\begin{proof}[ Proof]
Because $ u\in USC\left([0,L]\times\bar{\Omega} \right), $ from the sequential criterion for upper semicontinuous functions, the following holds, $ \limsup_{k\to\infty} u\left( y_{\epsilon_k},x_{\epsilon_k}\right)\leq u\left( \hat{y},\hat{x}\right) $. Correspondigly and for\\ $ v\in LSC\left([0,L]\times\bar{\Omega} \right) $, from the sequential criterion for lower semicontinuous functions, the following holds $ \liminf_{k\to\infty} v\left( y_{\epsilon_k},\tilde{x}_{\epsilon_k}\right)\ge v\left( \hat{y},\hat{\tilde{x}}\right). $
\begin{align}
0\leq\liminf_{k\to\infty}\frac{\abs{x_{\epsilon_k}-\tilde{x}_{\epsilon_k}}^2}{ 2\ \epsilon_k}&\leq\limsup_{k\to\infty}\frac{\abs{x_{\epsilon_k}-\tilde{x}_{\epsilon_k}}^2}{ 2\ \epsilon_k}\nonumber\\
&\leq^{(\ref{eq_4})} \limsup_{k\to\infty}(u_i( y_{\epsilon_k},x_{\epsilon_k})-v_i( y_{\epsilon_k},\tilde{x}_{\epsilon_k})) -\delta\nonumber\\
&\leq\limsup_{k\to\infty} u_i( y_{\epsilon_k},x_{\epsilon_k})+\limsup_{k\to\infty}(-v_i( y_{\epsilon_k},\tilde{x}_{\epsilon_k})) -\delta\nonumber\\
&=\limsup_{k\to\infty} u_i(y_{\epsilon_k},x_{\epsilon_k})-\liminf_{k\to\infty}(v_i( y_{\epsilon_k},\tilde{x}_{\epsilon_k})-\delta\nonumber\\
&\leq u_i( \hat{y},\hat{x})-v_i( \hat{y},\hat{\tilde{x}})-\delta\nonumber\\ &=^{(\hat{x}=\hat{\tilde{x}} )}u_i(\hat{y},\hat{x})-v_i(\hat{y},\hat{x})-\delta\leq 0 \nonumber
\end{align}
where the last inequality holds from the fact that, $ i\in I \text{\ and thus,}\ \delta=\sup_{[0,L]\times\bar{\Omega}}(u_i-v_i ) ( y,x). $ Therefore, we conclude that
$$\liminf_{k\to\infty}\frac{\abs{x_{\epsilon_k}-\tilde{x}_{\epsilon_k}}^2}{ 2\ \epsilon_k}=0=\limsup_{k\to\infty}\frac{\abs{x_{\epsilon_k}-\tilde{x}_{\epsilon_k}}^2}{ 2\ \epsilon_k}$$
thus, $ \lim_{k\to\infty}\frac{\abs{x_{\epsilon_k}-\tilde{x}_{\epsilon_k}}^2}{ 2\ \epsilon_k}=0 $
\end{proof}
\begin{claim}\label{claim_4}
$$ \lim_{k\to\infty}y_{\epsilon_k}=y_{i_0}, \lim_{k\to\infty}x_{\epsilon_k}=x_{i_0}, \lim_{k\to\infty}\tilde{x}_{\epsilon_k}=x_{i_0} $$
\end{claim}
\begin{proof}[ Proof]
We had seen on Claim $ (\ref{claim_1} ) $ that $ \Phi_{\epsilon}( y_{\epsilon},x_{\epsilon},\tilde{x}_{\epsilon})\ge\delta $. Therefore, it applies that: \\
$ \Phi_{\epsilon_k}( y_{\epsilon_k},x_{\epsilon_k},\tilde{x}_{\epsilon_k})\ge\delta $. Thus, we obtain that:
$$u_i(y_{\epsilon_k},x_{\epsilon_k})-v_i(y_{\epsilon_k},\tilde{x}_{\epsilon_k})-\Big{( }\frac{1}{2\epsilon_k}\abs{x_{\epsilon_k}-\tilde{x}_{\epsilon_k}}^2+\abs{x_{\epsilon_k}-x_{i_0}}^4+\abs{\tilde{x}_{\epsilon_k}-x_{i_0}}^4+\abs{y_{\epsilon_k}-y_{i_0}}^2\Big{)} \ge\delta$$
From the previous inequality, we receive that:
\begin{gather}
u_i(y_{\epsilon_k},x_{\epsilon_k})-v_i(y_{\epsilon_k},\tilde{x}_{\epsilon_k})-\delta\ge \abs{y_{\epsilon_k}-y_{i_0}}^2 \nonumber\\
\wedge\ u_i(y_{\epsilon_k},x_{\epsilon_k})-v_i(y_{\epsilon_k},\tilde{x}_{\epsilon_k})-\delta\ge \abs{x_{\epsilon_k}-x_{i_0}}^4 \nonumber\\
\wedge\ u_i(y_{\epsilon_k},x_{\epsilon_k})-v_i(y_{\epsilon_k},\tilde{x}_{\epsilon_k})-\delta\ge \abs{\tilde{x}_{\epsilon_k}-x_{i_0}}^4 \label{eq_6}
\end{gather}
We will prove the first of the three limits $ \lim_{k\to\infty}y_{\epsilon_k}=y_{i_0} $, because the rest of them are proved analogously. We observe that, similarly with the proof of the Claim $ \ref{claim_3} $, the following holds:
\begin{gather}
\limsup_{k\to\infty}( u_i(y_{\epsilon_k},x_{\epsilon_k})-v_i(y_{\epsilon_k},\tilde{x}_{\epsilon_k})) -\delta\leq 0 \label{eq_7}
\end{gather}
Furthermore, from the first inequality of $ (\ref{eq_6}) $ we have that:
\begin{gather}
\limsup_{k\to\infty}\left( u_i(y_{\epsilon_k},x_{\epsilon_k})-v_i(y_{\epsilon_k},\tilde{x}_{\epsilon_k})\right) -\delta\ge\limsup_{k\to\infty}\abs{y_{\epsilon_k}-y_{i_0}}^4\ge 0 \label{eq_8}
\end{gather}
From $(\ref{eq_7}) $ and $ (\ref{eq_8}) $ we receive that $ \limsup_{k\to\infty}\abs{y_{\epsilon_k}-y_{i_0}}^4=0 $. Additionally, because\\ $\limsup_{k\to\infty}\abs{y_{\epsilon_k}-y_{i_0}}^4\ge \liminf_{k\to\infty}\abs{y_{\epsilon_k}-y_{i_0}}^4 \ge 0 $, it follows that:
$$\limsup_{k\to\infty}\abs{y_{\epsilon_k}-y_{i_0}}^4=0=\liminf_{k\to\infty}\abs{y_{\epsilon_k}-y_{i_0}}^4$$
Thus, $ \lim_{k\to\infty}\abs{t_{\epsilon_k}-y_{i_0}}^4=0 $ equivalently $ \lim_{k\to\infty}y_{\epsilon_k}=y_{i_0} $
\end{proof}
From the above Claim, combined with the uniqueness of limit, we have that $ y_{i_0}=\hat{y} $ and $ x_{i_0}=\hat{x} $. Furthermore, due to the fact that $ x_{i_0}\in\Omega $ and $ y_{i_0}\in(0,L) $, it follows that $ x_{i_0},y_{i_0} $ are in addition and internal points of $ \Omega $ and $(0,L) $ respectively. Then, from the Claim $ (\ref{claim_4}) $, an index $ k^*\in\mathbb{N} $ will exist, such that:
\begin{gather} \forall k\ge k^*,\ x_{\epsilon_k}\in\Omega\ \wedge\ \tilde{x}_{\epsilon_k}\in\Omega\ \wedge\ y_{\epsilon_k}\in (0,L). \label{eq_8(1)}
\end{gather}
\begin{claim}\label{claim_5}
$$ \limsup_{k\to\infty}u_i\left(y_{\epsilon_k},x_{\epsilon_k} \right)=u_i\left(y_{i_0},x_{i_0} \right) \text{ and } \liminf_{k\to\infty}v_i\left(y_{\epsilon_k},\tilde{x}_{\epsilon_k} \right)=v_i\left(y_{i_0},x_{i_0} \right) $$
\end{claim}
\begin{proof}[ Proof]
From the Claim $ (\ref{claim_4}) $ it holds that$ \left(y_{\epsilon_k},x_{\epsilon_k} \right)\xrightarrow{k\to\infty}\left( y_{i_0},x_{i_0}\right) $ and $ \left(y_{\epsilon_k},\tilde{x}_{\epsilon_k} \right)\xrightarrow{k\to\infty}\left( y_{i_0},x_{i_0}\right). $ Because $ u_i $ and $ v_i $ are upper and lower semicontinuous respectively, from the sequential criterion, we obtain that:\\
$ \limsup_{k\to\infty}u_i\left( y_{\epsilon_k},x_{\epsilon_k}\right)\leq u_i\left(y_{i_0},x_{i_0} \right) $ and $ \liminf_{k\to\infty} v_i\left( y_{\epsilon_k},\tilde{x}_{\epsilon_k}\right)\geq v_i\left(y_{i_0},x_{i_0} \right). $ At first, we will prove that $\limsup_{k\to\infty}u_i\left(y_{\epsilon_k},x_{\epsilon_k} \right)=u_i\left(y_{i_0},x_{i_0} \right)$. We suppose by contradiction, that the output is not true. Then, it is necessary from the above that $\limsup_{k\to\infty}u_i\left(y_{\epsilon_k},x_{\epsilon_k} \right)<u_i\left(y_{i_0},x_{i_0} \right)$. As previously, we have $ \forall k\in\mathbb{N} $
\begin{gather}
u_i\left( y_{i_0},x_{i_0}\right) -v_i\left(y_{i_0},x_{i_0}\right)=\Phi_{\epsilon_k}\left(y_{i_0},x_{i_0},x_{i_0} \right)\leq \Phi_{\epsilon_k}\left(y_{\epsilon_k},x_{\epsilon_k},\tilde{x}_{\epsilon_k} \right) \leq u_i\left( y_{\epsilon_k},x_{\epsilon_k}\right) -v_i\left(y_{\epsilon_k},\tilde{x}_{\epsilon_k} \right)\label{eq_9}
\end{gather}
Then:
\begin{align}
u_i\left( y_{i_0},x_{i_0}\right) -v_i\left(y_{i_0},x_{i_0}\right)&=\limsup_{k\to\infty}\left( u_i\left( y_{i_0},x_{i_0}\right)-v_i\left(y_{i_0},x_{i_0}\right) \right)\nonumber\\
&\leq^{(\ref{eq_9})}\limsup_{k\to\infty}\left( u_i\left( y_{\epsilon_k},x_{\epsilon_k}\right) -v_i\left(y_{\epsilon_k},\tilde{x}_{\epsilon_k} \right)\right) \nonumber\\
&\leq \limsup_{k\to\infty}\left( u_i\left( y_{\epsilon_k},x_{\epsilon_k}\right)\right) -\liminf_{k\to\infty}\left( v_i\left(y_{\epsilon_k},\tilde{x}_{\epsilon_k} \right)\right)\nonumber\\
&<u_i\left(y_{i_0},x_{i_0} \right)-\liminf_{k\to\infty}\left( v_i\left(y_{\epsilon_k},\tilde{x}_{\epsilon_k} \right)\right)\nonumber
\end{align}
Thus, $ v_i\left(y_{i_0},x_{i_0}\right)>\liminf_{k\to\infty}\left( v_i\left(y_{\epsilon_k},\tilde{x}_{\epsilon_k} \right)\right) $, which is a contradiction, because $ v_i $ is lower semicontinuous. Therefore, we have $\limsup_{k\to\infty}u_i\left(y_{\epsilon_k},\tilde{x}_{\epsilon_k} \right)=u_i\left(y_{i_0},x_{i_0} \right)$. \\
Correspondigly, if we make the assumption that the following is not true,\\ $ \liminf_{k\to\infty}v_i\left(y_{\epsilon_k},\tilde{x}_{\epsilon_k} \right)=v_i\left(y_{i_0},x_{i_0} \right), $ then from the above we receive that $\liminf_{k\to\infty}v_i\left(y_{\epsilon_k},\tilde{x}_{\epsilon_k} \right)>v_i\left(y_{i_0},x_{i_0} \right). $ Like before, we get that
\begin{align}
u_i\left( y_{i_0},x_{i_0}\right) -v_i\left(y_{i_0},x_{i_0}\right)&\leq\limsup_{k\to\infty}\left( u_i\left( y_{\epsilon_k},x_{\epsilon_k}\right)\right) -\liminf_{k\to\infty}\left( v_i\left(y_{\epsilon_k},\tilde{x}_{\epsilon_k} \right)\right)\nonumber\\
&<\limsup_{k\to\infty}\left(u_i\left( y_{\epsilon_k},x_{\epsilon_k}\right)\right)-v_i\left(y_{i_0},x_{i_0}\right)
\end{align}
Hence, $ u_i\left( y_{i_0},x_{i_0}\right)<\limsup_{k\to\infty}\left(u_i\left( y_{\epsilon_k},x_{\epsilon_k}\right)\right) $ which is a contradiction, because $ u_i $ is upper semicontinouous. In consequence, $ \liminf_{k\to\infty}v_i\left(y_{\epsilon_k},x_{\epsilon_k} \right)=v_i\left(y_{i_0},x_{i_0} \right). $
\end{proof}
\textbf{ Step 2}: $ \left(\text{ Obstacle Avoidance} \right) $ \\
First, we will prove that exists at least one component $ u_j $ of the subsolution $ u $ which lies strictly above its obstacle $ \mathcal{ M}_{j}u $ at point $ \left(y_{i_0},x_{i_0} \right). $
In specific, we have the following claim.
\begin{claim}\label{claim_6}
\begin{gather}
\exists\ j_0\in I,\ u_{j_0}\left(y_{i_0},x_{i_0} \right)>\mathcal{ M}_{j_0} u\left( y_{i_0},x_{i_0}\right)\label{eq_2-1}
\end{gather}
\end{claim}
\begin{proof}[ Proof]
Let's make the assumption that the above claim is false. Therefore:
\begin{gather}
\forall i\in I, u_i\left( y_{i_0},x_{i_0}\right)\leq\mathcal{M}_i u\left(y_{i_0},x_{i_0} \right) =\max_{j\neq i}\left( u_j\left( y_{i_0},x_{i_0}\right) -c_{i,j}\left( y_{i_0},x_{i_0}\right) \right)\nonumber
\end{gather}
We randomly select a fixed index $ l_1\in I $. Subsequently, an index $ l_2\in\left\lbrace 1,2,\dots,l_1-1,l_1+1,\dots,m\right\rbrace $ exists, such that:
\begin{gather}
u_{l_1}\left( y_{i_0},x_{i_0}\right)\leq u_{l_2}\left( y_{i_0},x_{i_0}\right) -c_{l_1,l_2}\left( y_{i_0},x_{i_0}\right) =\max_{j\neq l_1}\left( u_j\left( y_{i_0},x_{i_0}\right) -c_{l_1,j}\left( y_{i_0},x_{i_0}\right)\right) \nonumber\\
\iff u_{l_1}\left( y_{i_0},x_{i_0}\right)+c_{l_1,l_2}\left( y_{i_0},x_{i_0}\right)\leq u_{l_2}\left( y_{i_0},x_{i_0}\right)\label{eq_2-1_1}
\end{gather}
Due to the fact that $ v $ is a supersolution, it follows that:
\begin{gather}
v_{l_1}\left( y_{i_0},x_{i_0} \right)\geq\mathcal{M}_{l_1} v\left( y_{i_0},x_{i_0}\right)
=\max_{j\neq l_1}\left( v_j\left( y_{i_0},x_{i_0}\right) -c_{l_1,j}\left( y_{i_0},x_{i_0}\right)\right)
\geq v_{l_2}\left( y_{i_0},x_{i_0}\right) -c_{l_1,l_2}\left( y_{i_0},x_{i_0}\right) \nonumber
\end{gather}
Thus:
\begin{gather}
v_{l_2}\left(y_{i_0},x_{i_0}\right) -c_{l_1,l_2}\left( y_{i_0},x_{i_0}\right)\leq v_{l_1}\left( y_{i_0},x_{i_0} \right)\label{eq_2-1_2}
\end{gather}
By adding $ (\ref{eq_2-1_1}) $ and $ (\ref{eq_2-1_2}) $ we obtain:
\begin{gather}
u_{l_1}\left( y_{i_0},x_{i_0}\right)-v_{l_1}\left(y_{i_0},x_{i_0}\right)\leq u_{l_2}\left( y_{i_0},x_{i_0}\right)-v_{l_2}\left( y_{i_0},x_{i_0}\right)\label{eq_2-1_3}
\end{gather}
Because $ l_1\in I, $ we have that $ \delta=u_{l_1}\left( y_{i_0},x_{i_0}\right)-v_{l_1}\left(y_{i_0},x_{i_0}\right) $ and at the same time we have that:
$$u_{l_2}\left( y_{i_0},x_{i_0}\right)-v_{l_2}\left( y_{i_0},x_{i_0}\right)\leq\sup_{[0,L]\times\bar{\Omega}}\left(u_{l_2}-v_{l_2} \right)\left(y,x \right)\leq\max_{j\in\left\lbrace1,2,\dots,m \right\rbrace } \sup_{[0,L]\times\bar{\Omega}}\left(u_j-v_j \right)\left(y,x \right)=\delta $$
Considering the above, combined with $ (\ref{eq_2-1_3}), $ we get that $ u_{l_2}\left( y_{i_0},x_{i_0}\right)-v_{l_2}\left( y_{i_0},x_{i_0}\right)=\delta. $ Therefore,\\ $ l_2\in I\setminus\left\lbrace l_1\right\rbrace . $
Next, we repeat the above procedure, considering that $ l_1 $ is replaced with $ l_2 $, and we obtain the existence of $ l_3\in I\setminus\left\lbrace l_2 \right\rbrace $ such that:
\begin{gather}
u_{l_2}\left( y_{i_0},x_{i_0} \right) \leq u_{l_3}\left( y_{i_0},x_{i_0} \right) - c_{l_2,l_3}\left(y_{i_0},x_{i_0} \right) \label{eq_2-1_4}
\end{gather}
Combining $ (\ref{eq_2-1_1}) $ and $ (\ref{eq_2-1_4}) $ we obtain:
\begin{gather}
u_{l_1}\left( y_{i_0},x_{i_0}\right)+c_{l_1,l_2}\left( y_{i_0},x_{i_0}\right)\leq u_{l_2}\left( y_{i_0},x_{i_0}\right)\leq u_{l_3}\left( y_{i_0},x_{i_0} \right) - c_{l_2,l_3}\left(y_{i_0},x_{i_0} \right)\nonumber
\end{gather}
Thus, we have that:
\begin{gather}
u_{l_1}\left( y_{i_0},x_{i_0}\right)+c_{l_1,l_2}\left(y_{i_0},x_{i_0} \right)+c_{l_2,l_3}\left(y_{i_0},x_{i_0} \right)\leq u_{l_3}\left( y_{i_0},x_{i_0} \right)
\end{gather}
Repeating this procedure, as many times as needed, we construct in this way a finite sequence of indexes $ \left\lbrace l_1,l_2,\dots,l_p,l_1\right\rbrace $ with $ l_j\neq l_{j+1}, \forall j=1,2,\dots,p-1 $ such that:
$$u_{l_1}\left( y_{i_0},x_{i_0}\right)+c_{l_1,l_2}\left(y_{i_0},x_{i_0} \right)+c_{l_2,l_3}\left(y_{i_0},x_{i_0} \right)+\dots,c_{l_p,l_1}\left(y_{i_0},x_{i_0} \right)\leq u_{l_1}\left( y_{i_0},x_{i_0} \right)$$
from which it is extracted that:
$$c_{l_1,l_2}\left(y_{i_0},x_{i_0} \right)+c_{l_2,l_3}\left(y_{i_0},x_{i_0} \right)+\dots,c_{l_p,l_1}\left(y_{i_0},x_{i_0} \right)\leq 0$$
which leads to a contradiction with respect to the axiom $ (O_3) $. In consequence, at least one index $ j_0\in I $ exists, such that: $ u_{j_0}\left(y_{i_0},x_{i_0} \right)>\mathcal{ M}_{j_0} u\left( y_{i_0},x_{i_0}\right). $
\end{proof}
Next, throughtout the analysis that follows, for the $ j_0\in I $ of the Claim $ (\ref{claim_6}), $ all the results that were proved at step 1, can be applied, keeping the same symbols for the sequence $ \left( y_{\epsilon},x_{\epsilon},\tilde{x}_{\epsilon}\right) $.\\
Using Claim $ (\ref{claim_6}) $, it is proved within the next Claim $ (\ref{claim_7}), $ that a subsequence exists, $ \left(\mu_k \right)_{k\in\mathbb{N}} $, of the sequence $\left( \epsilon_k\right)_{k\in\mathbb{N}}, $ and an index $ \hat{k}\in\mathbb{N} $ such that $ \forall k\ge\hat{k},\\u_{j_0}\left(y_{\mu_k},x_{\mu_k} \right)>\mathcal{ M}_{j_0} u\left( y_{\mu_k},x_{\mu_k}\right). $ As will be demonstrated later, the previous Claim, has the defining impact to finilize negative sign of function $ F $, when it acts on the sequence:\\ $ \left( y_{\mu_k},x_{\mu_k},u_{j_0}\left(y_{\mu_k},x_{\mu_k} \right),D_x\phi_{\mu_k}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k} \right),X \right)_{k\in\mathbb{N}}, $ given that the following is true:\\$ \left( \alpha,D_x\phi_{\mu_k}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k} \right),X\right) \in\bar{J}^{2,+}u_{j_0}\left( y_{\mu_k},x_{\mu_k}\right). $ A corresponding information holds for the non negativity of function $ F. $
\begin{claim}\label{claim_7}
There exists a subsequence $(\mu_k)_{k\in\mathbb{N}}$ of $(\epsilon_k)_{k\in\mathbb{N}} $ and an index $\hat{k}\in\mathbb{N}$ such that $\forall k\ge\hat{k}$
\begin{itemize}
\item The sequence of points $ \left(y_{\mu_k},x_{\mu_k}, \tilde{x}_{\mu_k} \right) $ is inside of $ (0,L)\times\Omega\times\Omega. $
\item If $ \left( \alpha,D_x\phi_{\mu_k}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k}\right),X \right) \in\bar{J}^{2,+}u_{j_0}\left( y_{\mu_k},x_{\mu_k}\right) $, then
\begin{gather}
F\left( y_{\mu_k},x_{\mu_k},u_{j_0}\left(y_{\mu_k},x_{\mu_k} \right),D_x\phi_{\mu_k}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k} \right),X \right) \leq 0 \label{eq_2-2}
\end{gather}
\item If $ \left( \alpha,-D_{\tilde{x}}\phi_{\mu_k}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k} \right),Y\right) \in\bar{J}^{2,-}v_{j_0}\left( y_{\mu_k},\tilde{x}_{\mu_k}\right) $, then
\begin{gather}
F\left( y_{\mu_k},\tilde{x}_{\mu_k},v_{j_0}\left(y_{\mu_k},\tilde{x}_{\mu_k} \right),-D_{\tilde{x}}\phi_{\mu_k}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k} \right),Y \right) \ge 0 \label{eq_2-3}
\end{gather}
\end{itemize}
\end{claim}
\begin{proof}[ Proof]
From the Claim $ (\ref{claim_6}), $ we get that $u_{j_0}\left(y_{i_0},x_{i_0} \right)>\mathcal{ M}_{j_0} u\left( y_{i_0},x_{i_0}\right)$. In addition, we have that $ \forall j\in{1,2,\dots,m} $, the function $ u_j $ is upper semicontinuous. Therefore, from the sequential criterion for upper semicontinuous functions, we have that $ \limsup_{k\to\infty}u_j\left(y_{\epsilon_k},x_{\epsilon_k} \right)\leq u_j\left(y_{i_0},x_{i_0} \right). $ In specific, if $ j\in I $, then from the Claim $ (\ref{claim_5}) $ we get that $ \limsup_{k\to\infty}u_j\left(y_{\epsilon_k},x_{\epsilon_k} \right)= u_j\left(y_{i_0},x_{i_0} \right). $
We observe that
\begin{align}
\mathcal{ M}_{j_0}u\left( y_{i_0},x_{i_0}\right)&=\max_{j\ne j_0}\left(u_j\left(y_{i_0},x_{i_0} \right) -c_{j_0,j}\left(y_{i_0},x_{i_0} \right) \right)\nonumber\\
&=\max_{j\ne j_0}\left(\limsup_{k\to\infty} u_j\left(y_{\epsilon_k},x_{\epsilon_k} \right) -c_{j_0,j}\left(y_{i_0},x_{i_0} \right) \right)\nonumber\\
&=\max_{j\ne j_0}\limsup_{k\to\infty} \left(u_j\left(y_{\epsilon_k},x_{\epsilon_k} \right) -c_{j_0,j}\left(y_{\epsilon_k},x_{\epsilon_k} \right) \right)\nonumber\\
&\geq\limsup_{k\to\infty}\max_{j\ne j_0} \left(u_j\left(y_{\epsilon_k},x_{\epsilon_k} \right) -c_{j_0,j}\left(y_{\epsilon_k},x_{\epsilon_k} \right) \right)\nonumber\\
&=\limsup_{k\to\infty}\mathcal{ M}_{j_0}u\left( y_{\epsilon_k},x_{\epsilon_k}\right)
\end{align}
Thus, $ \mathcal{ M}_{j_0}u\left( y_{i_0},x_{i_0}\right)\geq \limsup_{k\to\infty}\mathcal{ M}_{j_0}u\left( y_{\epsilon_k},x_{\epsilon_k}\right) $ and because \\ $\limsup_{k\to\infty}u_{j_0}\left(y_{\epsilon_k},x_{\epsilon_k} \right)=^{\left(j_0\in I \right)} u_{j_0}\left(y_{i_0},x_{i_0} \right)>\mathcal{ M}_{j_0} u\left( y_{i_0},x_{i_0}\right) $, we obtain that:
\begin{gather}
\limsup_{k\to\infty}u_{j_0}\left(y_{\epsilon_k},x_{\epsilon_k} \right)>\limsup_{k\to\infty}\mathcal{ M}_{j_0}u\left( y_{\epsilon_k},x_{\epsilon_k}\right)
\end{gather}
From the theory of the limsup, there exists a strictly increasing sequence of natural numbers $ \left( \sigma_k\right)_{k\in\mathbb{N}} $ such that,
\begin{gather}
\lim_{k\to\infty} u_{j_0}\left(y_{\epsilon_{\sigma_k}},x_{\epsilon_{\sigma_k}} \right)>\lim_{k\to\infty}\mathcal{ M}_{j_0}u\left(y_{\epsilon_{\sigma_k}},x_{\epsilon_{\sigma_k}}\right)\label{eq_2-4}
\end{gather}
Then, from the above inequality of limits, we can extract an index $ \tilde{k}\in\mathbb{N} $ such that $ \forall k\geq\tilde{k} $
\begin{gather}
u_{j_0}\left(y_{\epsilon_{\sigma_k}},x_{\epsilon_{\sigma_k}} \right)>\mathcal{ M}_{j_0}u\left(y_{\epsilon_{\sigma_k}},x_{\epsilon_{\sigma_k}}\right)\label{eq_2-5}
\end{gather}
from now on, for practical reasons, the subsequence, $ \left( \epsilon_{\sigma_k}\right) _{k\in\mathbb{N}} $ of $ \left( \epsilon_k\right)_{k\in\mathbb{N}}, $ will be denoted as $\left( \mu_k\right)_{k\in\mathbb{N}} $.\\
From $ (\ref{eq_8(1)}) $ and $ (\ref{eq_2-5}) $, if we set $ \hat{k}=\max\left\lbrace k^*,\tilde{k}\right\rbrace $, we have that $ \forall k\geq\hat{k} $
\begin{gather}
u_{j_0}\left(y_{\mu_k},x_{\mu_k} \right)>\mathcal{ M}_{j_0}u\left(y_{\mu_k},x_{\mu_k}\right)\label{eq_2-6}\\
\text{and}\nonumber\\
x_{\mu_k}\in\Omega\ \wedge\ y_{\mu_k}\in\Omega\ \wedge y_{\mu_k}\in(0,L)\label{eq_2-7}
\end{gather}
Next, we consider a random $ k\geq\hat{k} $. We have the following cases:
\begin{itemize}
\item Let $ \left( \alpha,D_x\phi_{\mu_k}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k}\right),X \right) \in\bar{J}^{2,+}u_{j_0}\left( y_{\mu_k},x_{\mu_k}\right) $. Because $ u $ is subsolution of the problem $ (IBVP) $ and $ \left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k}\right)\in (0,L) \times\Omega\times\Omega $ from $ (\ref{eq_2-7}) $, it follows that:
$$\min\left\lbrace F\left(y_{\mu_k},x_{\mu_k},u_{j_0}\left(y_{\mu_k},x_{\mu_k} \right),D_x\phi_{\mu_k}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k}\right),X \right),u_{j_0}\left( y_{\mu_k},x_{\mu_k}\right) -\mathcal{ M}_{j_0}u\left(y_{\mu_k},x_{\mu_k} \right) \right\rbrace \leq 0$$
But from the $ (\ref{eq_2-6}) $ it is necessary that:
\begin{gather}
F\left(y_{\mu_k},x_{\mu_k},u_{j_0}\left(y_{\mu_k},x_{\mu_k} \right),D_x\phi_{\mu_k}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k}\right),X \right)\leq 0 \label{eq_2-8}
\end{gather}
\item Let $ \left( \alpha,-D_{\tilde{x}}\phi_{\mu_k}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k} \right),Y\right) \in\bar{J}^{2,-}v_{j_0}\left( y_{\mu_k},\tilde{x}_{\mu_k}\right) $. Because $ v $ is a supersolution of the problem $ (IBVP) $ and $ \left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k}\right)\in (0,L) \times\Omega\times\Omega $ from $ (\ref{eq_2-7}) $, it follows that:
\begin{gather}
\min\biggl\{ F\left(y_{\mu_k},\tilde{x}_{\mu_k},v_{j_0}\left(y_{\mu_k},\tilde{x}_{\mu_k} \right),-D_{\tilde{x}}\phi_{\mu_k}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k}\right),Y \right),\\ v_{j_0}\left( y_{\mu_k},\tilde{x}_{\mu_k}\right) -\mathcal{ M}_{j_0}v\left(y_{\mu_k},\tilde{x}_{\mu_k} \right) \biggl\} \geq 0
\end{gather}
Therefore, we conclude from the above that:
\begin{gather}
F\left(y_{\mu_k},\tilde{x}_{\mu_k},v_{j_0}\left(y_{\mu_k},\tilde{x}_{\mu_k} \right),-D_{\tilde{x}}\phi_{\mu_k}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k}\right),Y \right)\geq 0 \label{eq_2-9}
\end{gather}
\end{itemize}
\end{proof}
\textbf{Step 3}: (Approaching the Contradiction)
In this step, we demonstrate that we end up with a contadiction. In order to reach this conclution, a special form of lemma is required, the so-called maximum principle for semicontinuous functions. In specific, the Lemma $ (\ref{Lemma_3.1}) $, which is presented below, is a special result case, which corresponds to Theorem 8.3 of \cite{CIL}.\\
At first, we have to mention that from Claim $ (\ref{claim_7}), $ the subsequence of maximum points $ \left( y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k}\right)_{k\in\mathbb{N}} $ of functions $\Phi_{\mu_k}\left(y,x,\tilde{x} \right) := u_{j_0}\left( y,x\right)-v_{j_0}\left(y,\tilde{x} \right) -\phi_{\mu_k}\left(y,x,\tilde{x} \right), $ on the set, $ \left[0,L \right] \times\bar{\Omega}\times\bar{\Omega}, $ is located finally within the set $ (0,L)\times\Omega\times\Omega. $ In consequence, the points that maximize the function $\Phi_{\mu_k}\left(y,x,\tilde{x} \right) := u_{j_0}\left( y,x\right)-v_{j_0}\left(y,\tilde{x} \right) -\phi_{\mu_k}\left(y,x,\tilde{x} \right) $ defined on the set $ (0,L)\times\Omega\times\Omega, $ are the same with points that maximize $ \Phi_{\mu_k} $ defined on the set $ \left[0,L \right] \times\bar{\Omega}\times\bar{\Omega} $.
\begin{lemma}\label{Lemma_3.1}
Let $ \left( y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k}\right)_{k\geq\hat{k}} $ be a maximum point of
\begin{gather}
\Phi_{\mu_k}\left(y,x,\tilde{x} \right) :=u_{j_0}\left( y,x\right)-v_{j_0}\left(y,\tilde{x} \right) -\phi_{\mu_k}\left(y,x,\tilde{x} \right)\nonumber
\end{gather}
on the set $ (0,L)\times\Omega\times\Omega. $ Then, for each $ \theta>0, $ there exists $ X_{\theta},Y_{\theta}\in \mathcal{S}^n $ such that:
\begin{itemize}
\item \begin{gather}
\left(\alpha,D_x\phi_{\mu_k}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k} \right),X_{\theta} \right)\in \bar{J}^{2,+}u_{j_0}\left(y_{\mu_k},x_{\mu_k} \right)\label{3_1}\\
\left(\tilde{\alpha},-D_{\tilde{x}}\phi_{\mu_k}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k} \right),Y_{\theta} \right)\in \bar{J}^{2,-}v_{j_0}\left(y_{\mu_k},\tilde{x}_{\mu_k} \right)\label{3_2}
\end{gather}
\item \begin{gather}
\begin{pmatrix}
X_{\theta} & O \\
O & -Y_{\theta}
\end{pmatrix}
\leq A+\theta A^2
\end{gather}
\end{itemize}
where $ A:=D^2_x\phi_{\mu_k}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k} \right) $ is the hessian matrix of $ \phi_{\mu_k}\left(y,x,\tilde{x} \right) $ with respect of $ x $ and $ \tilde{x}. $
\end{lemma}
At first we observe that for the vectors $ x=\left(x^{(1)},x^{(2)},\dots,x^{(n)} \right) $ and $ \tilde{x}=\left(\tilde{x}^{(1)},\tilde{x}^{(2)},\dots,\tilde{x}^{(n)} \right) $ on $ \mathbb{R}^n $ holds
\begin{gather}
D_x\phi_{\mu_k}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k} \right) =\frac{1}{\mu_k}\left( x_{\mu_k}-\tilde{x}_{\mu_k}\right) +4\abs{x_{\mu_k}-x_{i_0}}^2\left(x_{\mu_k}-x_{i_0} \right) \nonumber\\
-D_{\tilde{x}}\phi_{\mu_k}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k} \right) =\frac{1}{\mu_k}\left( x_{\mu_k}-\tilde{x}_{\mu_k}\right) -4\abs{\tilde{x}_{\mu_k}-x_{i_0}}^2\left(\tilde{x}_{\mu_k}-x_{i_0} \right)\nonumber\\
\frac{\partial^2\phi_{\mu_k}}{\partial x^{(m)\ 2}}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k} \right) =\frac{1}{\mu_k}+4 \left( 2\left(x_{\mu_k}^{(m)} -x_{i_0}^{(m)}\right)^2+\abs{x_{\mu_k}-x_{i_0}}^2 \right) \nonumber\\
\frac{\partial^2\phi_{\mu_k}}{\partial \tilde{x}^{(m)\ 2}}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k} \right) =\frac{1}{\mu_k}+4 \left( 2\left(\tilde{x}_{\mu_k}^{(m)} -x_{i_0}^{(m)}\right)^2+\abs{\tilde{x}_{\mu_k}-x_{i_0}}^2 \right) \nonumber\\
\frac{\partial^2\phi_{\mu_k}}{\partial x^{(m)}\partial x^{(\lambda)}}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k} \right) =8\left(x_{\mu_k}^{(m)} -x_{i_0}^{(m)}\right)\left(x_{\mu_k}^{(\lambda)} -x_{i_0}^{(\lambda)}\right),\ \lambda\neq m\nonumber\\
\frac{\partial^2\phi_{\mu_k}}{\partial \tilde{x}^{(m)}\partial \tilde{x}^{(\lambda)}}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k} \right) =8\left(\tilde{x}_{\mu_k}^{(m)} -x_{i_0}^{(m)}\right)\left(\tilde{x}_{\mu_k}^{(\lambda)} -x_{i_0}^{(\lambda)}\right),\ \lambda\neq m\nonumber\\
\frac{\partial^2\phi_{\mu_k}}{\partial x^{(m)}\partial \tilde{x}^{(m)}}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k} \right) =-\frac{1}{\mu_k}=\frac{\partial^2\phi_{\mu_k}}{\partial \tilde{x}^{(m)}\partial x^{(m)}}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k} \right)\nonumber\\
\frac{\partial^2\phi_{\mu_k}}{\partial x^{(m)}\partial \tilde{x}^{(\lambda)}}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k} \right) =0=\frac{\partial^2\phi_{\mu_k}}{\partial \tilde{x}^{(\lambda)}\partial x^{(m)}}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k} \right),\ \lambda\neq m\nonumber
\end{gather}
From the above, the matrix $ A:=D^2_x\phi_{\mu_k}(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k} ) $ is written as:
\begin{gather}
A=\frac{1}{\mu_k}\begin{array}{cc}
I & -I \\
-I & I
\end{array}
+ \hat{\tilde{B}}( y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k}) \nonumber
\end{gather}
where $ \hat{\tilde{B}}\in\mathbb{R}^{2n\times 2n}, $ is an appropriate matrix, of which its elements $ \hat{\tilde{B}}_{i,j}(y,x,\tilde{x}), $ are continuous functions of non fractional type, such that $ \hat{\tilde{B}}_{i,j}(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k})= \hat{\tilde{h}}_{i,j}(\abs{x_{\mu_k}-x_{i_0}},\abs{\tilde{x}_{\mu_k}-x_{i_0}}) $ with $ \hat{\tilde{h}}_{i,j} $ to be an appropriate continuous function of two variables on $ \mathbb{R}^2. $ Based on, $ A $ the matrix $ A^2 $ is defined as:
\begin{gather}
A^2=\frac{2}{\mu_k^2}\begin{array}{cc}
I & -I \\
-I & I
\end{array}
+ \tilde{B}( y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k}) \nonumber
\end{gather}
where $ \tilde{B}\in\mathbb{R}^{2n\times 2n} $ is an appropriate matrix, of which its elements $ \tilde{B}_{i,j}(y,x,\tilde{x}), $ are continuous functions, of non fractional type, such that $ \tilde{B}_{i,j}(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k})= \tilde{h}_{i,j}(\abs{x_{\mu_k}-x_{i_0}},\abs{\tilde{x}_{\mu_k}-x_{i_0}}) $ with $ \tilde{h}_{i,j} $ to be an appropriate continuous function of two variables on $ \mathbb{R}^2. $\\
By selecting $ \theta=\mu_k, $ from Lemma $ (\ref{Lemma_3.1}) $ we obtain the existence of symmetric matrices $ X_{\mu_k} $ and $ Y_{\mu_k} $ such that:
\begin{gather}
\left(\alpha,D_x\phi_{\mu_k}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k} \right),X_{\mu_k} \right)\in \bar{J}^{2,+}u_{j_0}\left(y_{\mu_k},x_{\mu_k} \right) \text{ and }\\ \left(\tilde{\alpha},-D_{\tilde{x}}\phi_{\mu_k}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k} \right),Y_{\mu_k} \right)\in \bar{J}^{2,-}v_{j_0}\left(y_{\mu_k},\tilde{x}_{\mu_k} \right) \label{3_3}
\end{gather}
with \begin{gather}
\begin{array}{cc}
X_{\mu_k} & O \\
O & -Y_{\mu_k}
\end{array}
\leq A+\theta A^2=\frac{3}{\mu_k} \begin{array}{cc}
I & -I \\
-I & I
\end{array}
+ B( y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k}) \label{3_4}
\end{gather}
where $ B\in\mathbb{R}^{2n\times 2n} $ is an appropriate matrix, of which its elements $ B_{i,j}(y,x,\tilde{x}) $ are continuous functions of non fractional type, such that $ B_{i,j}(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k})= h_{i,j}\left(\abs{x_{\mu_k}-x_{i_0}},\abs{\tilde{x}_{\mu_k}-x_{i_0}}\right) $ with $ h_{i,j} $ be an appropriate continuous function of two variables on $ \mathbb{R}^2$, such that, $ h_{i,j}(0,0)=0 $. Then, we obtain \\ $ \forall i,j\in\left\lbrace 1,2,\dots,2n\right\rbrace \lim_{k\to\infty} B_{i,j}(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k})=0. $
\begin{claim}\label{claim_8}
For each $ \xi>0 $ exists natural number $ k_{\xi}\geq\hat{k} $, such that, $ \forall k\geq k_{\xi} $
$ (\ref{3_4}) $ holds and
\begin{gather}
\begin{pmatrix}
X_{\mu_{k}} & O \\
O & -Y_{\mu_{k}}
\end{pmatrix}
\leq\frac{3}{\mu_{k}} \begin{pmatrix}
I & -I \\
-I & I
\end{pmatrix}
+\xi \begin{pmatrix}
I & O \\
O & I
\end{pmatrix}
\iff \begin{pmatrix}
X_{\mu_{k}}-\xi I & O \\
O & -\left(Y_{\mu_{k}}+\xi I \right)
\end{pmatrix}
\leq\frac{3}{\mu_{k}}\begin{pmatrix}
I & -I \\
-I & I \label{3_5}
\end{pmatrix}
\end{gather}
\end{claim}
\begin{proof}[ Proof]
Let $ \xi>0 $ be a random fixed number. At first, we will prove that there exists a natural number $ k_{\xi}\geq \hat{k} $ such that $ \forall k\geq k_{\xi} $:
\begin{gather}
B\left( y_{\mu_{k}},x_{\mu_{k}},\tilde{x}_{\mu_{k}} \right)< \xi \begin{pmatrix}
I & O \\
O & I
\end{pmatrix}\nonumber\\
\iff \forall z\in\mathbb{R}^{2n\times 1}\setminus\left\lbrace 0 \right\rbrace ,\
z^{T}B\left( y_{\mu_{k}},x_{\mu_{k}},\tilde{x}_{\mu_{k}} \right) z-\xi\ z^{T} \begin{pmatrix}
I & O \\
O & I
\end{pmatrix}
z< 0\label{3_6}
\end{gather}
Because the matrix $ M:=\xi \begin{pmatrix}
I & O \\
O & I
\end{pmatrix} $ is positive definite, it follows that it exists $ \lambda>0 $ such that, $ \forall z\in\mathbb{R}^{2n\times 1}\setminus\left\lbrace 0 \right\rbrace,\ z^T M z\geq \lambda\norm{z}^2_2 $. Furthermore, $ \forall z\in\mathbb{R}^{2n\times 1},\ \forall k\geq \hat{k}$ holds that:
\begin{gather}
z^T B\left( y_{\mu_{k}},x_{\mu_{k}},\tilde{x}_{\mu_{k}} \right) z=\left\langle z,B\left( y_{\mu_{k}},x_{\mu_{k}},\tilde{x}_{\mu_{k}} \right) z\right\rangle\nonumber\\ \leq^{(C-S)}\norm{z}_2\norm{B\left( y_{\mu_{k}},x_{\mu_{k}},\tilde{x}_{\mu_{k}} \right) z}_2\nonumber\\
\leq\norm{B\left( y_{\mu_{k}},x_{\mu_{k}},\tilde{x}_{\mu_{k}} \right)}\norm{z}^2_2\nonumber
\end{gather}
Because $\forall i,j\in\left\lbrace1,2,\dots,2n \right\rbrace B_{i,j}\left( y_{\mu_{k}},x_{\mu_{k}},\tilde{x}_{\mu_{k}} \right)\xrightarrow{k\to\infty} 0 $ holds that $ \norm{B\left( y_{\mu_{k}},x_{\mu_{k}},\tilde{x}_{\mu_{k}} \right)}\xrightarrow{k\to\infty}0 $. In consequence, an index $ k_{\xi}\geq \hat{k} $ will exist, such that $ \forall k\geq k_{\xi},\ \norm{B\left( y_{\mu_{k}},x_{\mu_{k}},\tilde{x}_{\mu_{k}} \right)}<\lambda $. Hence, we obtain that $ \forall z\in\mathbb{R}^{ 2n\times 1 }\setminus\left\lbrace 0 \right\rbrace,\ \forall k\geq k_{\xi} $
\begin{gather}
z^T M z\geq \lambda\norm{z}^2_2>\norm{B\left( y_{\mu_{k_{\xi}}},x_{\mu_{k_{\xi}}},\tilde{x}_{\mu_{k_{\xi}}} \right)}\norm{z}^2_2\geq z^T B\left( y_{\mu_{k_{\xi}}},x_{\mu_{k_{\xi}}},\tilde{x}_{\mu_{k_{\xi}}} \right) z
\end{gather}
Therefore, $ (\ref{3_6}) $ is proved. Then, through $ (\ref{3_4}) $, using the transition identity for matrix ordering, we obtain the desirable result.
\end{proof}
Next, throughout our analysis, we consider that $ k\geq k_{\xi}. $ From Claim $ (\ref{claim_7}), $ we obtain that $ \forall k\geq k_{\xi} $
\begin{gather}
F\left( y_{\mu_k},x_{\mu_k},u_{j_0}\left(y_{\mu_k},x_{\mu_k} \right),D_x\phi_{\mu_k}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k} \right),X \right) \leq 0\nonumber\\
F\left( y_{\mu_k},\tilde{x}_{\mu_k},v_{j_0}\left(y_{\mu_k},\tilde{x}_{\mu_k} \right),-D_{\tilde{x}}\phi_{\mu_k}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k} \right),Y \right) \ge 0\nonumber
\end{gather}
from which we receive that:
\begin{gather}
0\leq F\left( y_{\mu_k},\tilde{x}_{\mu_k},v_{j_0}\left(y_{\mu_k},\tilde{x}_{\mu_k} \right),-D_{\tilde{x}}\phi_{\mu_k}\left(x_{\mu_k},\tilde{x}_{\mu_k},\tilde{x}_{\mu_k} \right),Y \right)-\nonumber\\
-F\left( y_{\mu_k},x_{\mu_k},u_{j_0}\left(y_{\mu_k},x_{\mu_k} \right),D_x\phi_{\mu_k}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k} \right),X \right)\label{3_8}
\end{gather}
From the axiom $ (F2), $ for $ X=Y $ and $ p:=-D_{\tilde{x}}\phi_{\mu_k}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k} \right) , q:=D_x\phi_{\mu_k}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k} \right), $ we obtain that
\begin{gather}
-F\left( y_{\mu_k},x_{\mu_k},u_{j_0}\left( y_{\mu_k},x_{\mu_k}\right),q,X \right) \leq\omega\left(\abs{p-q} \right)-F\left( y_{\mu_k},x_{\mu_k},u_{j_0}\left( y_{\mu_k},x_{\mu_k}\right),p,X \right) \nonumber
\end{gather}
I.e.,
\begin{gather}
-F\left( y_{\mu_k},x_{\mu_k},u_{j_0}\left( y_{\mu_k},x_{\mu_k}\right),D_x\phi_{\mu_k}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k} \right) ,X \right) \leq\omega\left(4\left(\abs{x_{i_0}-x_{\mu_k}}^3+\abs{x_{i_0}-\tilde{x}_{\mu_k}} ^3\right) \right) \nonumber\\-F\left( y_{\mu_k},x_{\mu_k},u_{j_0}\left( y_{\mu_k},x_{\mu_k}\right),-D_{\tilde{x}}\phi_{\mu_k}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k} \right),X \right) \label{3_7}
\end{gather}
From $ (\ref{3_8}) $ and $(\ref{3_7}) $ we obtain that:
\begin{align}
0&\leq F\left( y_{\mu_k},\tilde{x}_{\mu_k},v_{j_0}\left(y_{\mu_k},\tilde{x}_{\mu_k} \right),-D_{\tilde{x}}\phi_{\mu_k}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k} \right),Y \right)-\nonumber\\
&-F\left(y_{\mu_k},x_{\mu_k},u_{j_0}\left(y_{\mu_k},x_{\mu_k} \right),D_x\phi_{\mu_k}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k} \right),X\right)\nonumber\\
&\leq F\left( y_{\mu_k},\tilde{x}_{\mu_k},v_{j_0}\left(y_{\mu_k},\tilde{x}_{\mu_k} \right),-D_{\tilde{x}}\phi_{\mu_k}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k} \right),Y \right)+\omega\left(4\left(\abs{x_{i_0}-x_{\mu_k}}^3+\abs{x_{i_0}-\tilde{x}_{\mu_k}} ^3\right) \right)-\nonumber\\
&-F\left( y_{\mu_k},x_{\mu_k},u_{j_0}\left( y_{\mu_k},x_{\mu_k}\right),-D_{\tilde{x}}\phi_{\mu_k}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k}\right),X \right)\label{3_9}
\end{align}
From the claim $ (\ref{claim_1}) $, it follows that $ \forall k\geq k_{\xi},\
$
\begin{gather}
0<\delta\leq\delta+\frac{1}{2\mu_k}\abs{x_{\mu_k}-\tilde{x}_{\mu_k}}^2\leq u_{j_0}\left(y_{\mu_k},x_{\mu_k} \right) -v_{j_0}\left(y_{\mu_k},\tilde{x}_{\mu_k} \right)\nonumber
\end{gather}
Setting on the axiom $ (F1),$ for $ s=u_{j_0}\left( y_{\mu_k},x_{\mu_k}\right) , r=v_{j_0}\left(y_{\mu_k},\tilde{x}_{\mu_k} \right) $, from the previous, we get that $ r<s, $ thus we obtain
\begin{align}
&F\left( y_{\mu_k},\tilde{x}_{\mu_k},v_{j_0}\left(y_{\mu_k},\tilde{x}_{\mu_k} \right),-D_{\tilde{x}}\phi_{\mu_k}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k} \right),Y \right)\nonumber\\
&\leq F\left( y_{\mu_k},\tilde{x}_{\mu_k},u_{j_0}\left(y_{\mu_k},x_{\mu_k} \right),-D_{\tilde{x}}\phi_{\mu_k}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k} \right),Y \right)-\gamma\left(u_{j_0}\left(y_{\mu_k},x_{\mu_k} \right)-v_{j_0}\left(y_{\mu_k},\tilde{x}_{\mu_k} \right) \right) \label{3_10}
\end{align}
From $ (\ref{3_9}) $ and $ (\ref{3_10}) $ we obtain that:
\begin{align}
0&\leq F\left( y_{\mu_k},\tilde{x}_{\mu_k},v_{j_0}\left(y_{\mu_k},\tilde{x}_{\mu_k} \right),-D_{\tilde{x}}\phi_{\mu_k}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k} \right),Y \right)-\nonumber\\
&-F\left(y_{\mu_k},x_{\mu_k},u_{j_0}\left(y_{\mu_k},x_{\mu_k} \right),D_x\phi_{\mu_k}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k} \right),X\right)\nonumber\\
&\leq F\left( y_{\mu_k},\tilde{x}_{\mu_k},u_{j_0}\left(y_{\mu_k},x_{\mu_k} \right),-D_{\tilde{x}}\phi_{\mu_k}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k} \right),Y \right)-\gamma\left(u_{j_0}\left(y_{\mu_k},x_{\mu_k} \right)-v_{j_0}\left(y_{\mu_k},\tilde{x}_{\mu_k} \right) \right)\nonumber\\
&+\omega\left(4\left(\abs{x_{i_0}-x_{\mu_k}}^3+\abs{x_{i_0}-\tilde{x}_{\mu_k}} ^3\right) \right)
-F\left( y_{\mu_k},x_{\mu_k},u_{j_0}\left( y_{\mu_k},x_{\mu_k}\right),-D_{\tilde{x}}\phi_{\mu_k}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k}\right),X \right)
\end{align}
Hence,
\begin{align}
0&\leq F\left( y_{\mu_k},\tilde{x}_{\mu_k},u_{j_0}\left(y_{\mu_k},x_{\mu_k} \right),-D_{\tilde{x}}\phi_{\mu_k}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k} \right),Y \right)-\nonumber\\
&-F\left( y_{\mu_k},x_{\mu_k},u_{j_0}\left( y_{\mu_k},x_{\mu_k}\right),-D_{\tilde{x}}\phi_{\mu_k}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k}\right),X \right)-\nonumber\\
&-\gamma\left(u_{j_0}\left(y_{\mu_k},x_{\mu_k} \right)-v_{j_0}\left(y_{\mu_k},\tilde{x}_{\mu_k} \right) \right)+\omega\left(4\left(\abs{x_{i_0}-x_{\mu_k}}^3+\abs{x_{i_0}-\tilde{x}_{\mu_k}} ^3\right) \right)
\label{3_11}
\end{align}
Next, we set $ X:=X_{\mu_k}-\xi I,\ Y:=Y_{\mu_k}+\xi I,\ p:=-D_{\tilde{x}}\phi_{\mu_k}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k} \right). $ From $ (\ref{3_5}) $ of the Claim $ (\ref{claim_8}) $, using the axiom $ (F3) $, we have
\begin{gather}
F\left( y_{\mu_k},\tilde{x}_{\mu_k},u_{j_0}\left( y_{\mu_k},x_{\mu_k}\right),p,Y_{\mu_k}+\xi I \right)-F\left( y_{\mu_k},x_{\mu_k},u_{j_0}\left( y_{\mu_k},x_{\mu_k}\right),p,X_{\mu_k}-\xi I \right)\nonumber\\
\leq\omega\left(\frac{1}{\mu_k}\abs{x_{\mu_k}-\tilde{x}_{\mu_k}}^2+\abs{x_{\mu_k}-\tilde{x}_{\mu_k}}\left( \abs{p}+1\right) \right) \label{3_12}
\end{gather}
In addition, with the double use of axiom $ (F2) $, setting $ p=q $, we obtain:
\begin{align}
F\left( y_{\mu_k},x_{\mu_k},u_{j_0}\left( y_{\mu_k},x_{\mu_k}\right),p,X_{\mu_k}-\xi I \right)-F\left( y_{\mu_k},x_{\mu_k},u_{j_0}\left( y_{\mu_k},x_{\mu_k}\right),p,X_{\mu_k} \right) &\leq\omega\left(\norm{X_{\mu_k}-\left(X_{\mu_k}-\xi I \right) } \right)\nonumber\\
&=\omega\left( \xi\right) \label{3_13}
\end{align}
\begin{align}
F\left( y_{\mu_k},\tilde{x}_{\mu_k},u_{j_0}\left( y_{\mu_k},x_{\mu_k}\right),p,Y_{\mu_k} \right)-F\left( y_{\mu_k},y_{\mu_k},u_{j_0}\left( y_{\mu_k},x_{\mu_k}\right),p,Y_{\mu_k}+\xi I \right)
&\leq\omega\left(\norm{Y_{\mu_k}-\left(Y_{\mu_k}+\xi I \right) } \right)\nonumber\\
&=\omega\left( \xi\right) \label{3_14}
\end{align}
By summing $ (\ref{3_13}) $ and $ (\ref{3_14}) $, we obtain that:
\begin{gather}
F\left( y_{\mu_k},x_{\mu_k},u_{j_0}\left( y_{\mu_k},x_{\mu_k}\right),p,X_{\mu_k}-\xi I \right)-F\left( y_{\mu_k},x_{\mu_k},u_{j_0}\left( y_{\mu_k},x_{\mu_k}\right),p,X_{\mu_k} \right)\nonumber\\
+F\left( y_{\mu_k},\tilde{x}_{\mu_k},u_{j_0}\left( y_{\mu_k},x_{\mu_k}\right),p,Y_{\mu_k} \right)-F\left( y_{\mu_k},y_{\mu_k},u_{j_0}\left( y_{\mu_k},x_{\mu_k}\right),p,Y_{\mu_k}+\xi I \right)
\leq 2\omega\left( \xi\right) \nonumber\\
\iff F\left( y_{\mu_k},y_{\mu_k},u_{j_0}\left( y_{\mu_k},x_{\mu_k}\right),p,Y_{\mu_k} \right)-F\left( y_{\mu_k},x_{\mu_k},u_{j_0}\left( y_{\mu_k},x_{\mu_k}\right),p,X_{\mu_k} \right)\nonumber\\
\leq F\left( y_{\mu_k},\tilde{x}_{\mu_k},u_{j_0}\left( y_{\mu_k},x_{\mu_k}\right),p,Y_{\mu_k}+\xi I \right)-F\left( y_{\mu_k},x_{\mu_k},u_{j_0}\left( y_{\mu_k},x_{\mu_k}\right),p,X_{\mu_k}-\xi I \right)+2\omega\left( \xi\right)
\label{3_15}
\end{gather}
By combining $ (\ref{3_12}) $ and $ (\ref{3_15}) $ we receive that:
\begin{gather}
F\left( y_{\mu_k},\tilde{x}_{\mu_k},u_{j_0}\left( y_{\mu_k},x_{\mu_k}\right),p,Y_{\mu_k} \right)-F\left( y_{\mu_k},x_{\mu_k},u_{j_0}\left( y_{\mu_k},x_{\mu_k}\right),p,X_{\mu_k} \right)\leq 2\omega\left( \xi\right) +
\nonumber\\ +\omega\left(\frac{1}{\mu_k}\abs{x_{\mu_k}-\tilde{x}_{\mu_k}}^2+\abs{x_{\mu_k}-\tilde{x}_{\mu_k}}\left( \abs{p}+1\right) \right)\label{3_16}
\end{gather}
Finally, through a combination of $ (\ref{3_11}) $ and $ (\ref{3_16}) $, we obtain that:
\begin{gather}
0\leq \omega\left(4\left(\abs{x_{i_0}-x_{\mu_k}}^3+\abs{x_{i_0}-\tilde{x}_{\mu_k}} ^3\right) \right)
+2\omega\left(\xi \right)-\gamma\left(u_{j_0}\left(y_{\mu_k},x_{\mu_k} \right)-v_{j_0}\left(y_{\mu_k},\tilde{x}_{\mu_k} \right) \right)+\nonumber\\ +\omega\left(\frac{1}{\mu_k}\abs{x_{\mu_k}-\tilde{x}_{\mu_k}}^2+\abs{x_{\mu_k}-\tilde{x}_{\mu_k}}\left( \abs{-D_{\tilde{x}}\phi_{\mu_k}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k} \right)}+1\right) \right)
\end{gather}
Then, through a property of $ \limsup $, we obtain,
\begin{gather}
0\leq\limsup_{k\to\infty}\left(-\gamma\left(u_{j_0}\left(y_{\mu_k},x_{\mu_k} \right)-v_{j_0}\left(y_{\mu_k},\tilde{x}_{\mu_k} \right) \right)\right) + \limsup_{k\to\infty}\left( \omega\left(4\left(\abs{x_{i_0}-x_{\mu_k}}^3+\abs{x_{i_0}-\tilde{x}_{\mu_k}} ^3\right) \right)\right) +\nonumber\\
+2\omega\left(\xi \right)
+\limsup_{k\to\infty}\left( \omega\left(\frac{1}{\mu_k}\abs{x_{\mu_k}-\tilde{x}_{\mu_k}}^2+\abs{x_{\mu_k}-\tilde{x}_{\mu_k}}\left( \abs{-D_{\tilde{x}}\phi_{\mu_k}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k} \right)}+1\right) \right)\right) \label{3_17}
\end{gather}
In specific, we have the following estimations for each quantity that contains $ \limsup: $
\begin{align}
\limsup_{k\to\infty}\left(-\gamma\left(u_{j_0}\left(y_{\mu_k},x_{\mu_k} \right)-v_{j_0}\left(y_{\mu_k},\tilde{x}_{\mu_k} \right) \right)\right)
&\leq-\gamma\ \limsup_{k\to\infty}\left(u_{j_0}\left(y_{\mu_k},x_{\mu_k} \right)-v_{j_0}\left(y_{\mu_k},\tilde{x}_{\mu_k} \right) \right) \nonumber\\
&\leq -\gamma \left( \limsup_{k\to\infty}\left(u_{j_0}\left(y_{\mu_k},x_{\mu_k} \right)\right)+\limsup_{k\to\infty}\left(-v_{j_0}\left(y_{\mu_k},\tilde{x}_{\mu_k} \right)\right) \right) \nonumber\\
&=-\gamma \left( \limsup_{k\to\infty}\left(u_{j_0}\left(y_{\mu_k},x_{\mu_k} \right)\right)-\liminf_{k\to\infty}\left(v_{j_0}\left(y_{\mu_k},\tilde{x}_{\mu_k} \right)\right) \right) \nonumber\\
&=-\gamma \left( u_{j_0}\left(y_{i_0},x_{i_0} \right)-v_{j_0}\left(y_{i_0},x_{i_0}\right)\right) \nonumber\\
&=-\gamma\ \delta
\end{align}
It is true that for the sequence of real numbers $ b_k:=4\left(\abs{x_{i_0}-x_{\mu_k}}^3+\abs{x_{i_0}-\tilde{x}_{\mu_k}} ^3\right), $ that $ b_k\xrightarrow{k\to\infty}0 $. Then, from the sequential criterion of continuity of function $ \omega $, and in addition with the limit relation of a function and continuity, we have
$$\lim_{k\to\infty}\omega(b_k)=\omega(0)=\lim_{x\to 0^+}\omega(x)\equiv\omega(0^+)=0$$
Thus, we obtain $ \limsup_{k\to\infty}\omega(b_k)=0 $. In a similar way, it is proved that $ \limsup_{k\to\infty}\omega(m_k)=0 $ where $ m_k:=\frac{1}{\mu_k}\abs{x_{\mu_k}-\tilde{x}_{\mu_k}}^2+\abs{x_{\mu_k}-y\tilde{x}_{\mu_k}}\left( \abs{-D_{\tilde{x}}\phi_{\mu_k}\left(y_{\mu_k},x_{\mu_k},\tilde{x}_{\mu_k} \right)}+1\right). $
From the above, and $ (\ref{3_17}), $ we arrive at a conclusion where:
$$\gamma \delta\leq 2\omega(\xi)$$
The above inequality was proved for a random $ \xi>0 $. Then we have\\
$ \gamma \delta=\lim_{\xi\to 0^+} \gamma\ \delta\leq\lim_{\xi\to 0^+} 2\ \omega(\xi)=2\ \omega(0^+)=0.$
In consequence, $ \gamma\ \delta\leq 0, $ which is a contratiction, since $ \gamma, \delta>0. $\\
\\
The proof of the above result, was established in the case that $ y_{i_0}<L. $ In the case that we have $ y_{i_0}= L,$ (see step 1), then we can process this in a different way, by using an appropriate modification of $ u $.\\
\\
Specifically, while having in mind that the functions $ u $ and $ v $ are viscosity subsolutions and supersolutions respectively for the problem $ \left( IBVP\right) $, we define the following modified function for a random parameter $ \theta>0. $
\begin{gather}
u^{\theta}:[0,L)\times\bar{\Omega}\to\mathbb{R}^m\nonumber\\
u^{\theta}(t,x):=u(y,x)+h^{\theta}(y,x)\nonumber\\
\end{gather}
where $ h^{\theta}(y,x) $ is an appropriate function such that the following identities hold: $ \forall i\in\left\lbrace 1,2,\dots,m\right\rbrace $
\begin{itemize}
\item [($\alpha$)] $\forall(y,x)\in\left[ 0,L\right)\times\bar{\Omega},\ \lim_{\theta\to 0^+}u^{\theta}_i(y,x)=u_i(y,x)$
\item[($\beta$)]$ \lim_{t\to L^-}u_i^{\theta}(y,x)=-\infty$ uniformly on $ \bar{\Omega} $\\
with the following interpretation: We set $ (g^i_y)_{t\in\left[ 0,L\right) }$ to be a family of functions\\ $ g^i_y(x):=u_i^{\theta}(y,x),\ x\in\bar{\Omega} $ for which we demand to converge uniformly on $ \bar{\Omega}, $ i.e.,
$$\forall M>0,\exists\ \delta(M)>0,\ \forall y\in(L-\delta,L),\ \forall x\in\bar{\Omega}: u^{\theta}_i(y,x)<-M$$
\end{itemize}
Next, we will see a Proposition that provides us a sufficent condition, such that the limit of a sum of two functions is $ -\infty $, uniformly on $ \bar{\Omega} $ while $ y\to L^-. $ Furthermore, it is proved that, if an upper semicontinuous function satisfies the above limiting behaviour, then it is necessary that it will attain a maximum point in $ \left[0,L \right)\times\bar{\Omega}. $
\begin{prop}\label{prop_4_1}
Let $ \kappa,\rho:\left[ 0,L\right)\times\bar{\Omega}\to\mathbb{R} $ mappings, such that $ \kappa $ is upper bounded and \\ $ \lim_{y\to L^-}\rho(y,x)=-\infty $ uniformly on $\bar{\Omega}. $ Then, $ \lim_{y\to L^-}\left(\kappa(y,x)+\rho(y,x) \right) =-\infty $ uniformly on $ \bar{\Omega}. $
\end{prop}
\begin{proof}[ Proof]
Let $ M>0 $ be random. Due to the fact that the function $ \kappa $ is upper bounded, there will be a positive constant $ C>0,$ such that, $ \forall (y,x)\in \left[0,L \right)\times\bar{\Omega},\ \kappa(y,x)\leq C. $ Furthermore, because $ \lim_{y\to L^-}\left(\kappa(y,x)+\rho(y,x) \right) =-\infty $, in specific for $ \tilde{M}=M+C>0 $, it will exists $ \delta(\tilde{M}) >0,\\ \forall y\in (L-\delta,L),\ \forall x\in \bar{\Omega},\rho(y,x)<-\tilde{M}=-M-C$. Then $ \forall y\in(L-\delta,L),\ \forall x\in\bar{\Omega} $
$$\kappa(y,x)+\rho(y,x)\leq M+\rho(y,x)<C+(-M-C)=-M$$
In consequence, we have the desirable result.
\end{proof}
\begin{prop}\label{prop_4_2}
Let $ \kappa:\left[ 0,L\right)\times\bar{\Omega}\to\mathbb{R} $ be an upper semicontinuous with the identity $ \lim_{y\to L^-}\kappa(y,x)=-\infty $ uniformly on $\bar{\Omega}. $ Then exists $ (\tilde{y},\tilde{x}) \in \left[0,L \right)\times\bar{\Omega} $ such that $ \kappa(\tilde{y},\tilde{x})=\max_{\left[0,L \right)\times\bar{\Omega} }\kappa(y,x) $
\end{prop}
\begin{proof}[ Proof]
Let $ \lim_{y\to L^-}\kappa(y,x)=-\infty, $ uniformly on $ \bar{\Omega}. $ We consider $ y^*\in\left[0,L \right) $ and $ x^*\in\bar{\Omega} $ random. We define $ M:=\abs{\kappa(y^*,x^*)}+1>0. $ Because $ \lim_{y\to L^-}\kappa(y,x)=-\infty, $ uniformly on $ \bar{\Omega} $, it holds that
\begin{gather}\label{prop_4_2_eq_1}
\exists\tilde{\delta}(M_{x^*})>0,\ \forall y\in(L-\tilde{\delta},L),\ \forall x\in\bar{\Omega},\ u(y,x)<-M_{x^*}=-\abs{\kappa(y^*,x^*)}-1
\end{gather}
We select $ \delta_0\in(0,\tilde{\delta}), $ such that $ y^*<L-\delta_0. $ Then the $ \kappa $ as an upper semicontinuous on the compact $ \left[0,L-\delta_0 \right] \times\bar{\Omega} $ will attain maximum value. Thus,
\begin{gather}\label{prop_4_2_eq_2}
\exists(\tilde{y},\tilde{x})\in\left[ 0,L-\delta_0\right]\times\bar{\Omega},\ \forall(y,x)\in\left[ 0,L-\delta_0\right]\times\bar{\Omega},\ \kappa(\tilde{y},\tilde{x})\geq\kappa(y,x)
\end{gather}
Because $ y^*<L-\delta_0 $ and $ x^*\in\bar{\Omega} $, will apply that $ \forall y\in(L-\delta_0,L)\subset(L-\tilde{\delta},L),\ \forall x\in\bar{\Omega}, $
\begin{gather}
\kappa(\tilde{y},\tilde{x})\geq^{(\ref{prop_4_2_eq_2})}\kappa(y^*,x^*)\geq-\abs{\kappa(y^*,x^*)}>-\abs{\kappa(y^*,x^*)}-1>^{(\ref{prop_4_2_eq_1})}\kappa(y,x)\nonumber
\end{gather}
Therefore, it is shown that $ \forall y\in\left[0,L \right)\times\bar{\Omega},\ \kappa(\tilde{y},\tilde{x}) \geq\kappa(y,x). $
\end{proof}
An example of a function $ u^{\theta}:\left[0,L \right)\times\bar{\Omega}\to\mathbb{R}^m $ that satisfies the assumptions $ (\alpha) $ and $ (\beta) $, is
\begin{gather}
u^{\theta}_i(y,x):=u_i(y,x)-\frac{\theta}{L-y},\ \forall (y,x)\in \left[ 0,L\right) \times\bar{\Omega}
\end{gather}
Specifically, $ h^{\theta}(y,x):=-\frac{\theta}{L-y},\ \forall (y,x)\in \left[ 0,L\right) \times\bar{\Omega}. $ \\
We observe that $ h $ as continuous, will be simultaneously upper and lower semicontinuous and furthermore, as $ y $ tends to $ L^- $, its limit will be $ -\infty $, uniformly on $ \bar{\Omega}$. Furthermore, we have that $ u_i^{\theta}(y,x)-v_i(y,x)=u_i(y,x)-v_i(y,x)+h^{\theta}(y,x). $ The first statement is that $ u_i-v_i $ is upper bounded on $ \left[ 0,L\right)\times\bar{\Omega}. $ Indeed,
\begin{align}
\forall i\in\left\lbrace 1,2,,\dots,m \right\rbrace,\ \sup_{[0,L)\times\bar{\Omega}}(u_i-v_i)(y,x)&\leq \sup_{[0,L]\times\bar{\Omega}}(u_i-v_i)(y,x)\nonumber\\
&\leq \max_{j\in\{1,2,\dots,m \}}\sup_{[0,L]\times\bar{\Omega}}(u_i-v_i)(y,x)\nonumber\\
&=\sup_{[0,L]\times\bar{\Omega}}(u_{i_0}-v_{i_0})(y,x)=u_{i_0}(L,x_{i_0})-v_{i_0}(L,x_{i_0})=\delta
\end{align}
Then from Proposition $ (\ref{prop_4_1}) $, we obtain that the limit of $ u_i^{\theta}-v_i$ as $ y\to L^- $ is $ -\infty $ uniformly on $ \bar{\Omega}. $ Having in mind that $ u_i^{\theta}-v_i $ is upper semicontinuous as a sum of upper semicontinuous functions, we obtain from Proposition $ (\ref{prop_4_2}) $ that there will be a maximum point of $ u_i^{\theta}-v_i $ inside of $ \left[0,L \right)\times\bar{\Omega}. $\\
\\
Next, it is proved, through the claims below, that $ u^{\theta}:\left[0,L \right)\times\bar{\Omega}\to\mathbb{R}^m $ is a viscosity subsolution of an appropriate modified system.
\newpage
\begin{claim}\label{Isxyrismos_epektasi}
$\forall (y,x)\in(0,L)\times\Omega,\ \forall p\in\mathbb{R}^n,\ \forall X\in\mathcal{S}^n,\ \forall i\in\left\lbrace 1,2,\dots,m\right\rbrace $ holds that:
\begin{itemize}
\item[( i)] $ u_i(y,x)>\mathcal{ M}_i u(y,x)\iff u_i^{\theta}(y,x)>\mathcal{ M}u^{\theta}(y,x) $
\item[( ii)] $F\left(y,x,u_i^{\theta}(y,x),p,X \right)\leq F\left(y,x,u_i(y,x),p,X \right)$
\end{itemize}
\end{claim}
\begin{proof}[ Proof]
\begin{itemize}
\item[(i)] Let $ (t,x)\in(0,L)\times\Omega $ be a random point. We observe that $ \forall i\in \left\lbrace 1,2,\dots,m \right\rbrace $
\begin{align}
u_i(y,x)-\mathcal{ M}_i u(y,x)&=u_i(y,x)-\frac{\theta}{L-y}+\frac{\theta}{L-y}-\max_{j\ne i}\left(u_j(y,x)-c_{i,j}(y,x) \right) \nonumber\\
&=\left( u_i(y,x)-\frac{\theta}{L-y}\right) -\max_{j\ne i}\left[\left( u_j(y,x)-\frac{\theta}{L-y}\right)-c_{i,j}(y,x) \right] \nonumber\\
&=u_i^{\theta}(y,x)-\max_{j\ne i}\left( u_j^{\theta}(y,x)-c_{i,j}(y,x) \right)\nonumber\\
&=u_i^{\theta}(y,x)-\mathcal{M}_i u^{\theta}(y,x)
\end{align}
Therefore, the desired result is immediate.
\item[(ii)] At first, we see that $ \forall (y,x)\in(0,T)\times\Omega,\ u_i^{\theta}(y,x)<u_i(y,x). $ We set $ r:=u_i^{\theta}(y,x) $ and \\$ s:=u_i(y,x). $ From axiom $ (F_1), $ we obtain that:
\begin{gather}
\gamma(s-r)\leq F(y,x,s,p,X)-F(y,x,r,p,X)\nonumber\\
\text { i.e, }\ \gamma(u_i(y,x)-u_i^{\theta}(y,x))\leq F(y,x,(u_i(t,x),p,X)-F(y,x,u_i^{\theta}(y,x),p,X)\nonumber\\
\begin{align} \text{ equivalently, }\ F(y,x,u_i^{\theta}(y,x),p,X)&\leq F(y,x,(u_i(t,x),p,X)- \gamma(u_i(y,x)-u_i^{\theta}(y,x))\nonumber\\
&=F(y,x,(u_i(y,x),p,X)-\frac{\gamma\theta}{L-y}\nonumber\\
&<F(y,x,(u_i(y,x),p,X)
\end{align}
\end{gather}
where the last inequality holds, due to the $ \frac{\gamma \theta}{L-y}>0. $
\end{itemize}
\end{proof}
\begin{claim}
We consider the following modified system which depends on the positive parameter $ \theta>0 $
$$\begin{cases}
\min\biggl\{ F\bigl( y,x,q_{i}(y,x),D q_{i}(y,x),D^2 q_{i}(y,x)\bigl), q_{i}(y,x)-\max_{j\neq i}\bigl( q_{j}(y,x)-c_{ij}(y,x)\bigl)\biggl\}=0, \left(y,x \right)\in\Omega_{L}
q_{i}(0,x)=g_{i}(x)-\frac{\theta}{L},\ x\in\bar{\Omega}
q_{i}(y,x)=f_i(y,x)-\frac{\theta}{L-y}, (y,x)\in(0,L)\times\partial{\Omega}
\end{cases}$$
Then, $ u^{\theta}:\left[0,L \right)\times\bar{\Omega}\to\mathbb{R}^m $ and $ v:\left[0,L \right)\times\bar{\Omega}\to\mathbb{R}^m, $ are viscosity subsolutions and supersolutions respectively of the above problem.
\end{claim}
\begin{proof}[Proof]
At first, we will prove that $ u^{\theta}:\left[0,L \right)\times\bar{\Omega}\to\mathbb{R}^m $ is a viscosity subsolution of the above system. Indeed, let $ i\in \left\lbrace 1,2,\dots,m \right\rbrace,\ (y,x)\in(0,L) $ be random and $ (\alpha,p,X)\in \bar{J}^{2,+}u_i(y,x). $ Then equivalently, we have $ \left(\alpha+\frac{\theta}{\left( T-t\right)^2} ,p,X \right)\in \bar{J}^{2,+}u^{\theta}_i(y,x). $ Due to the fact that function $ u $ is a viscosity subsolution of $ (IBVP) $, then the following holds:
$$\min\left\lbrace F(y,x,u_i(y,x),p,X),\ u_i(y,x)-\mathcal{M}_i u(y,x)\right\rbrace\leq 0 $$
From the above, we extract the following cases:
\begin{itemize}
\item[(i)] Let $ \min\left\lbrace F(y,x,u_i(y,x),p,X),\ u_i(y,x)-\mathcal{M}_i u(y,x)\right\rbrace=u_i(y,x)-\mathcal{M}_i u(y,x). $\\ Then, $u_i(y,x)-\mathcal{M}_i u(y,x)\leq 0. $
From the claim $ (\ref{Isxyrismos_epektasi}) (i)$, it is necessary that\\ $ u_i^{\theta}(y,x)-\mathcal{ M}u^{\theta}(y,x)\leq 0. $ Then,
$$\min\left\lbrace F(y,x,u_i^{\theta}(t,x),p,X),\ u_i^{\theta}(y,x)-\mathcal{M}_i u^{\theta}(y,x)\right\rbrace\leq u_i^{\theta}(y,x)-\mathcal{M}_i u^{\theta}(y,x)\leq 0 $$
\item[(ii)]Let $ \min\left\lbrace F(y,x,u_i(y,x),p,X),\ u_i(y,x)-\mathcal{M}_i u(y,x)\right\rbrace=F(y,x,u_i(y,x),p,X). $\\ Then, $ F(y,x,u_i(y,x),p,X)\leq 0. $
From claim $ (\ref{Isxyrismos_epektasi}) (ii)$, we obtain that:
$$F(y,x,u_i^{\theta}(y,x),p,X) \leq F(y,x,u_i(y,x),p,X)$$
Then
$$\min\left\lbrace F(y,x,u_i^{\theta}(y,x),p,X),\ u_i^{\theta}(y,x)-\mathcal{M}_i u^{\theta}(y,x)\right\rbrace\leq F(y,x,u_i^{\theta}(y,x),p,X)\leq 0 $$
Regarding the initial and boundary conditions, we obtain that:
\begin{gather}
\forall x\in\bar{\Omega},\ u^{\theta}_i(0,x)=u_i(0,x)-\frac{\theta}{L}\leq^{(u subsolution )} g_i(x)-\frac{\theta}{L}\nonumber\\
\forall y\in(0,L),\ \forall x\in\partial{\Omega},\ u^{\theta}_i(y,x)=u_i(y,x)-\frac{\theta}{L-y}\leq^{(u subsolution )}f_i(y,x)-\frac{\theta}{L-y}\nonumber
\end{gather}
\end{itemize}
Therefore, function $ u^{\theta} $ is a viscosity subsolution of the above problem. As for $ v, $ it is clear that it is a viscosity supersolution of the above problem.
\end{proof}
From the above remarks, we follow the same steps as we did for functions $ u $ and $ v $, with a difference, that in place of function $ u $, we place function $ u^{\theta} $. In this way, we receive the result $ u_i^{\theta}(y,x)\leq v_i(y,x), \forall i\in \left\lbrace 1,2,\dots,m \right\rbrace, \forall \left(y,x \right)\in \left[0,L \right) \times\bar{\Omega}. $ Then, we fix a random point, $ (y,x)\in \left[0,L \right)\times\bar{\Omega}, $ and we leave parameter $ \theta $ to tend at 0. From property $ (\alpha) $, we obtain the desirable result.
\end{proof}
Using now the Comparison Principle, we succeed the desired uniqueness property of viscosity solutions, in the sense tha was described at the beginning.
\begin{theorem}[ Uniqueness of the Solution of the Problem $\left( IBVP \right) $]
The problem $\left( IBVP\right) $ has a unique viscosity solution.
\end{theorem}
\begin{proof}[Proof]
We consider $ \textbf{h}:=\left( h_{1},h_{2},\dots,h_{m} \right):[0,L]\times\bar{\Omega}\to\mathbb{R} $ and $ \textbf{w}:=\left( w_{1},w_{2}\dots,w_{m} \right):[0,L]\times\bar{\Omega}\to\mathbb{R}$ two random viscosity solutions of problem $ (IBVP). $ We will prove that $ \textbf{h}=\textbf{w} $ on $ [0,L)\times\bar{\Omega}. $ Indeed, from the fact that $ \textbf{h ,w} $ are viscosity solutions of problem $ (IBVP), $ we can construct\footnote{Let $ V:[0,L]\times\bar{\Omega}\to\mathbb{R}^m $ with $ V=( V_1,V_2,\dots, V_m) $ be a bounded function\\ (i.e $\forall i\in\{1,2,\dots,m\},\ \sup\{V_i(y,x)\ :\ (y,x)\in[0,L]\times\bar{\Omega}\}<+\infty $ and $ \inf\{V_i(y,x)\ :\ (y,x)\in[0,L]\times\bar{\Omega}\}>-\infty $) and continuous on $ [0,L)\times\bar{\Omega}. $ Then, for each $ i\in\{1,2,\dots,m\}, $ the following functions
$$\hat{V}^1_i(y,x):=\begin{cases}
V_i(y,x) & \ \ (y,x)\in[0,T)\times\bar{\Omega}\cr
M & y=L,\ x\in\bar{\Omega}
\end{cases}
\ \text{and} \ \hat{V}^2_i(y,x):=\begin{cases}
V_i(y,x) &\quad (y,x)\in[0,L)\times\bar{\Omega}\\
m & y=L,\ x\in\bar{\Omega} \\
\end{cases}$$
where $ M:=\max_{i\in\{1,2,\dots,m\}}\sup\{V_i(t,x)\ :\ (y,x)\in [0,L]\times\bar{\Omega}\}$ and $ m:=\min_{i\in\{1,2,\dots,m\}}\inf\{V_i(y,x)\ :\ (y,x)\in [0,L]\times\bar{\Omega}\} $, are upper semicontinuous and lower semicontinuous on $[0,L]\times\bar{\Omega}$ respectively.
}
appropriate modifications $ \hat{\textbf{h}}^j:[0,L]\times\bar{\Omega}\to\mathbb{R}^m, j=1,2 $ and $ \hat{\textbf{w}}^j:[0,L]\times\bar{\Omega}\to\mathbb{R}^m, j=1,2 $ of $ \textbf{h}$ and $ \textbf{w} $ respectively, such that
\begin{gather}\label{visco_modi}
\hat{\textbf{h}}^j=\textbf{h},\ \text{and}\
\hat{\textbf{w}}^j=\textbf{w},\ \text{on}\ [0,L)\times\bar{\Omega},\ j=1,2
\end{gather}
satisfying also the identity that $ \hat{\textbf{h}}^{1},\ \hat{\textbf{w}}^{1} $ are upper semicontinuous functions on $ [0,T]\times\bar{\Omega} $ and $ \hat{\textbf{h}}^{2},\ \hat{\textbf{w}}^{2} $ are lower semicontinuous on $ [0,L]\times\bar{\Omega} $. From the definitions of these modifications, it is extracted that $ \hat{\textbf{h}}^1,\ \hat{\textbf{w}}^1 $ are viscosity subsolutions of $ (IBVP) $ and $ \hat{\textbf{h}}^2,\ \hat{\textbf{w}}^2 $ are viscosity supersolutions of $ (IBVP) $. Using now, the Comparison Principle for the following couples of viscosity subsolutions and viscosity supersolutions of $ (IBVP), $ in specific, $ \left(\hat{\textbf{h}}^1, \hat{\textbf{w}}^2\right) $ and $ \left( \hat{\textbf{w}}^1,\hat{\textbf{h}}^2\right) $, we deduce that
\begin{gather}
\hat{\textbf{h}}^1\leq\hat{\textbf{w}}^2,\ \text{and}\
\hat{\textbf{w}}^1\leq\hat{\textbf{h}}^2,\ \text{on}\ [0,L)\times\bar{\Omega}, \nonumber
\end{gather}
Combining the $ (\ref{visco_modi}) $ and the above inequalities, we obtain that $ \textbf{h}=\textbf{w} $ on $ [0,L)\times\bar{\Omega}. $
\end{proof}
\subsection*{Existence of Viscosity Solution }
Throughout the section, we assume that the assumptions $ (F1)-(F3) $ and $ (O_1)-(O_5) $ hold. Furthermore, we assume the following axiom for the operator $ F $ and an axiom that controls further the relation between the obstacles functions $ c_{i,j} $, the initial data $ g_i $ and the boundary data $ f_{i} $ as well:
\begin{gather}
(F4)\ \forall (y,x,r,p,X)\in [0,L]\times\mathbb{R}^n\times\mathbb{R}\times\mathbb{R}^n\times\mathcal{S}^n,\ F(y,x,r,p,X)\geq r\nonumber\\
\end{gather}
\begin{align}
\left( O_6 \right)\ \min_{i,j\in\left\lbrace 1,2,\dots,m \right\rbrace }\{ c_{i,j}(y,x) \mid y\in[0,L],\ x\in\partial\Omega\}& \geq\max_{i\in\left\lbrace 1,2,\dots,m \right\rbrace } \{f_i(y,x) \mid y\in[0,L],\ x\in\partial\Omega\}-\nonumber\\
&-\min_{i\in\left\lbrace 1,2,\dots,m \right\rbrace} \{g_i(x) \mid x\in\partial\Omega\}\nonumber
\end{align}
\begin{gather}
\left( O_7 \right)\ \forall y\in(0,L),\ \forall x,\tilde{x}\in\Omega,\ g_i(x)+c_{i,j}(y,\tilde{x})\geq 0
\end{gather}
\begin{theorem}[ Existence of Viscosity Solution for the Problem $\left( IBVP \right)$]\label{Existence Theorem}
The problem $\left( IBVP\right) $ has at least one viscosity solution.
\end{theorem}
In order to prove the above theorem, it is sufficent to prove the following propositions:
\begin{prop}\label{Perronmethod}
Assume that for each $ i\in\left\lbrace 1,2,\dots,m\right\rbrace $ and $ \hat{x}\in\bar{\Omega} $ there exists a family of continuous viscosity sub- and supersolutions, $ \left\lbrace u^{i,\hat{x},\epsilon}\right\rbrace_{\epsilon>0} $ and $ \left\lbrace v^{i,\hat{x},\epsilon}\right\rbrace_{\epsilon>0} $ respectively to (IBVP) such that
$$\sup_{\epsilon>0}u_i^{\hat{x},\epsilon}(0,\hat{x})=g_i(\hat{x})=\inf_{\epsilon>0}v_i^{i,\hat{x},\epsilon}(0,\hat{x})$$
Then, there exists a a viscosity solution of (IBVP).
\end{prop}
With this result given, what remains is to construct appropriate barriers, i.e families of viscosity sub- and supersolutions. More specifically, we prove the following:
\begin{prop}\label{existence of barriers}
There exists $ \kappa>0 $ such that, for each $ i\in\left\lbrace 1,2,\dots,m \right\rbrace,\ \hat{x}\in\bar{\Omega} $ and for any given, nonnegative, continuous function $ \phi:\bar{\Omega}\rightarrow\mathbb{R}, $ on $ \bar{\Omega}, $ with the identity: $ \exists\delta_{\hat{x}}>0,\ \forall x\in\Omega_{\delta_{\hat{x}}}:= B_{\rho_2}\left( \hat{x},\delta_{\hat{x}}\right)\cap\Omega,\ \phi(x)\geq\phi(\hat{x}) $, the functions $ U^{\hat{x},\epsilon}:=\left( U_1^{\hat{x},\epsilon},U_2^{\hat{x},\epsilon},\dots,U_m^{\hat{x},\epsilon} \right) $ and $\ V^{i,\hat{x},\epsilon}:=\left( V_1^{i,\hat{x},\epsilon},V_2^{i,\hat{x},\epsilon},\dots,V_m^{i,\hat{x},\epsilon} \right) $,
\begin{gather}
U_j^{\hat{x},\epsilon}(y,x):=g_j(\hat{x})-A(\phi(x)-\phi(\hat{x}))-B\exp(\kappa \phi(x))\abs{x-\hat{x}}^2-\epsilon-Ct\nonumber\\
V_j^{i,\hat{x},\epsilon}(y,x):=g_i(\hat{x})+A(\phi(x)-\phi(\hat{x}))+B\exp(\kappa \phi(x))\abs{x-\hat{x}}^2+\epsilon+Ct+ c_{i,j}(y,x)\nonumber
\end{gather}
are viscosity sub- and supersolutions of the (IBVP) respectively. Moreover,
$$\sup_{\epsilon>0}U_i^{\hat{x},\epsilon}(0,\hat{x})=g_i(\hat{x})=\inf_{\epsilon>0}V_i^{i,\hat{x},\epsilon}(0,\hat{x})$$
\end{prop}
Combining Propositions $ (\ref{Perronmethod}) $ and $ (\ref{existence of barriers}) $ above, we extract the existence part of Theorem $ \ref{Existence Theorem}. $ We start with the proof of Proposition $ \ref{existence of barriers} $
\begin{proof}(\textbf{of Proposition $ \ref{existence of barriers} $})
At first, we consider a random $ i\in\left\lbrace 1,2,\cdots,m \right\rbrace $ and we prove that $ V^{i,\hat{x},\epsilon} $ is viscosity supersolution of (IBVP). Let $ j\in\left\lbrace 1,2,\dots,m\right\rbrace $ be a random index. We observe that function $ V_j^{i,\hat{x},\epsilon}:[0,L]\times\bar{\Omega}\to\mathbb{R} $ as a continuous, is lower semicontinuous on $ [0,L]\times\bar{\Omega}. $ The aim is to prove that function $ V_j^{i,\hat{x},\epsilon} $ satisfies the definition of viscosity supersolution.\\
The proof consists of the following claims:
\begin{claim}\label{c_1}
For a given $ A\in\mathbb{R} $ and any $ \kappa>0, $ there exists an appropriate number $ \tilde{B}\in\mathbb{R}^{+} $ such that\\ $ \forall j\in\left\lbrace 1,2,\dots,m\right\rbrace $,
\begin{gather}
V_j^{i,\hat{x},\epsilon}(0,x)> g_j(x),\ \forall x\in\bar{\Omega}\nonumber\\
g_i(\hat{x})=\inf_{\epsilon>0}V_i^{i,\hat{x},\epsilon}(0,\hat{x})\nonumber
\end{gather}
\end{claim}
\begin{proof}
First, we observe that $ V_j^{i,\hat{x},\epsilon}(0,\hat{x})\ge g_j(\hat{x}). $ Indeed, $ \forall\epsilon>0 $ we have that
\begin{gather}
V_j^{i,\hat{x},\epsilon}(0,\hat{x})=g_i(\hat{x})+c_{i,j}(0,\hat{x})+\epsilon\nonumber
\end{gather}
but by the use of axiom $ (O_5) $ we get that $ g_i(\hat{x})+c_{i,j}(0,\hat{x})\ge g_j(\hat{x}),\ \forall i,j\in\left\lbrace 1,2,\dots,m \right\rbrace. $ Consequently,
\begin{gather}
V_j^{i,\hat{x},\epsilon}(0,\hat{x})\geq g_j(\hat{x})+\epsilon>g_j(\hat{x}),\ \forall\epsilon>0\nonumber
\end{gather}
As a result from the above we receive that $ V_j^{i,\hat{x},\epsilon}(0,\hat{x})> g_j(\hat{x}). $ Next, we proceed to the proof of non-negativity of $ V_j^{i,\hat{x},\epsilon}(0,\cdot)-g_j:\bar{\Omega}\to\mathbb{R} $.
From the last inequality and from the fact that, $ V_j^{i,\hat{x},\epsilon}(0,\cdot)-g_j$ is continuous on $\bar{\Omega} $ and especially at the point $ \hat{x}, $ we can select an appropriate radious $ r_{\hat{x}} $ such that $ \forall x\in B(\hat{x}, r_{\hat{x}}) \cap \bar{\Omega},\ V_j^{i,\hat{x},\epsilon}(0,x)-g_j(x)>0. $ It remains to show that\\ $ \forall x\in\bar{\Omega}\setminus B(\hat{x}, r_{\hat{x}}),\ V_j^{i,\hat{x},\epsilon}(0,x)-g_j(x)>0. $ Clearly we see that the set $ \bar{\Omega}\setminus B(\hat{x}, r_{\hat{x}}) $ is closed in $ \mathbb{R}^n, $ since $ \left( \bar{\Omega}\setminus B(\hat{x}, r_{\hat{x}})\right)^{c}=\bar{\Omega}^{c}\cup B(\hat{x}, r_{\hat{x}}) $ is open in $ \mathbb{R}^n. $ Moreover, the set $ \bar{\Omega}\setminus B(\hat{x}, r_{\hat{x}}) $ is bounded as a subset of the bounded set $ \bar{\Omega}. $ Consequently, $ \bar{\Omega}\setminus B(\hat{x}, r_{\hat{x}}) $ is compact set, as closed and bounded set in $ \mathbb{R}^n. $ Let $ x\in\bar{\Omega}\setminus B(\hat{x}, r_{\hat{x}}) $, be a random point. We observe that
\begin{align}
V_j^{i,\hat{x},\epsilon}(0,x)-g_j(x)&=g_i(\hat{x})+A(\phi(x)-\phi(\hat{x}))+\tilde{B}\exp(\kappa \phi(x))\abs{x-\hat{x}}^2+\epsilon+c_{i,j}(0,x)-g_j(x)\nonumber\\
&=K_{i,j}(x) +\epsilon
\end{align}
where $K_{i,j} $ is the following function
\begin{gather}
K_{i,j}:\bar{\Omega}\setminus B(\hat{x}, r_{\hat{x}})\to\mathbb{R},\ K_{i,j}(x):=k_1^{i,j}(x)+\tilde{B}\ k_2(x),\ \forall x\in\bar{\Omega}\nonumber
\end{gather}
with
\begin{align}
k_1^{i,j}(x)&:=g_i(\hat{x})-g_j(x)+c_{i,j}(0,x)+A(\phi(x)-\phi(\hat{x})),\ \forall x\in\bar{\Omega}\setminus B(\hat{x}, r_{\hat{x}}) \nonumber\\
k_2(x)&:=\exp(\kappa \phi(x))\abs{x-\hat{x}}^2,\ \forall x\in\bar{\Omega}\setminus B(\hat{x}, r_{\hat{x}})\nonumber
\end{align}
For our task, i.e $ \forall x\in\bar{\Omega}\setminus B(\hat{x}, r_{\hat{x}}),\ V_j^{i,\hat{x},\epsilon}(0,x)-g_j(x)>0, $ it is sufficent to show that \\$ K_{i,j}\geq 0,\ \forall x\in \bar{\Omega}\setminus B(\hat{x}, r_{\hat{x}}). $ Clearly, we see that $ k_2 $ is non-negative function on $ \bar{\Omega}\setminus B(\hat{x}, r_{\hat{x}}). $
The functions $ k_1^{i,j},\ k_2 $ as continuous on the compact set, attain a minimum value. In spesific, let
\begin{gather}
k_2^{\min}:=\min\left\lbrace k_2(x)\ :\ x\in\bar{\Omega}\setminus B(\hat{x}, r_{\hat{x}})\right\rbrace\ \text{and}\ k_1^{\min}:=\min_{i,j\in\left\lbrace 1,2,\dots,m\right\rbrace }\left\lbrace \min\left\lbrace k_1^{i,j}(x)\ :\ x\in\bar{\Omega}\setminus B(\hat{x}, r_{\hat{x}})\right\rbrace\right\rbrace \nonumber
\end{gather}
It is obvious that $ k_2^{\min}>0, $ since $ \hat{x}\notin \bar{\Omega}\setminus B(\hat{x}, r_{\hat{x}}) $ and the non-negative expression $ \exp(\kappa\phi(x))\abs{x-\hat{x}}^2 $ is zero iff $ x=\hat{x} $. Then, $ \forall i,j\in\left\lbrace 1,2,\dots,m\right\rbrace,\ \forall x\in \bar{\Omega}\setminus B(\hat{x}, r_{\hat{x}}) $ and $ B\geq 0 $ we have
\begin{gather}
\Delta_{\tilde{B}}:=k_1^{\min}+\tilde{B}\ k_2^{\min}\leq k_1^{i,j}(x)+\tilde{B}\ k_2(x)\equiv K_{i,j}(x)\nonumber
\end{gather}
Consequently, in order to get the desired inequality, $ K_{i,j}(x)\geq 0,\ \forall x\in \bar{\Omega}\setminus B(\hat{x}, r_{\hat{x}}) $, it is sufficent to show that $ \Delta_{\tilde{B}}\geq 0 $ for some appropriate $ \tilde{B}\geq 0. $ Indeed, the quantity $ \Delta_B:=k_1^{\min}+B\ k_2^{\min}\geq 0 $ is true iff\\ $ \tilde{B}\geq \max\left\lbrace 0, -\frac{ k_1^{\min}}{k_2^{\min}}\right\rbrace. $\\
Finally, we proceed to the proof of $ g_i(x)=\inf_{\epsilon>0}V_i^{i,\hat{x},\epsilon}(0,\hat{x}) $. We have already seen, that \\$ V_i^{i,\hat{x},\epsilon}(0,\hat{x})=g_i(\hat{x})+\epsilon>g_i(\hat{x}),\ \forall\epsilon>0. $ As a result from that, we receive that $ g_i(\hat{x}) $ is a lower bound of the non empty set $ \Delta:=\left\lbrace V_i^{i,\hat{x},\epsilon}(0,\hat{x})\ :\ \ \epsilon>0 \right\rbrace $. Consequently, the $ \inf\Delta $ is well defined. We claim that $ g_i(\hat{x})=\inf\Delta. $ By the characterization of infimum, it is sufficent to show that,
\begin{gather}
\forall \tilde{\epsilon}>0,\exists z_{\tilde{\epsilon}}\in\Delta: z_{\tilde{\epsilon}}<g_i(\hat{x})+\tilde{\epsilon}\nonumber
\end{gather}
Indeed, considering a random $ \tilde{\epsilon}>0, $ we choose any $ \epsilon\in\left( 0,\tilde{\epsilon} \right),$ for example $ \epsilon=\tilde{\epsilon}/2 $ and set $ z_{\tilde{\epsilon}}:=g_i(\hat{x})+\tilde{\epsilon}/2\in\Delta $. Clearly, we receive see that $ z_{\tilde{\epsilon}}<g_i(\hat{x})+\tilde{\epsilon}$. As a result from the last, we get the desired equality.
\end{proof}
\begin{claim}\label{c_2}
For a given $ A\in\mathbb{R} $ and any given $\kappa>0 $ there exists appropriate number $ B\geq\tilde{B}, $ such that, for all $(t,x)\in (0,L)\times\Omega, $ and all $ (\alpha,p,X)\in \bar{J}^{2,-}V_j^{i,\hat{x},\epsilon}(t,x) $ the following holds:\\
\begin{gather}
\min\left\lbrace F(y,x,V_j^{i,\hat{x},\epsilon}(y,x),D_x V_j^{i,\hat{x},\epsilon}(y,x),D^2_{xx}V_j^{i,\hat{x},\epsilon}(y,x)),\ V_j^{i,\hat{x},\epsilon}(y,x)-\mathcal{M}_j V^{i,\hat{x},\epsilon}(y,x) \right\rbrace \geq 0\nonumber
\end{gather}
\end{claim}
\begin{proof}
Initially, we prove that $ \forall (t,x)\in(0,T)\times\Omega, $
\begin{gather}
F(y,x,V_j^{i,\hat{x},\epsilon}(y,x),D_x V_j^{i,\hat{x},\epsilon}(y,x),D^2_{xx}V_j^{i,\hat{x},\epsilon}(y,x))\geq 0
\end{gather}
From the axiom $ (F4) $, we have obtalin that, $ \forall (y,x)\in(0,L)\times\Omega, $
\begin{gather}
F(y,x,V_j^{i,\hat{x},\epsilon}(t,x),D_x V_j^{i,\hat{x},\epsilon}(y,x),D^2_{xx}V_j^{i,\hat{x},\epsilon}(y,x))\geq V_j^{i,\hat{x},\epsilon}(y,x)
\end{gather}
From the last, it is sufficent to prove that there exists an appropriate number $ B\geq \tilde{B} $ such that \\ $ V_j^{i,\hat{x},\epsilon}(y,x)\geq 0,\ \forall (y,x)\in(0,L)\times\Omega. $ First, we have that
\begin{gather}
V_j^{i,\hat{x},\epsilon}(y,x)=L_1(y,x)+L_2(x)\nonumber
\end{gather}
where
\begin{gather}
L_1(y,x):=g_i(\hat{x})+\epsilon+C t+c_{i,j}(y,x)\nonumber\\
L_2(x):= A \left( \phi(x)-\phi(\hat{x})\right) + B\exp\left( \kappa\phi(x)\right)\abs{x-\hat{x}}^2\nonumber
\end{gather}
From the axiom $ \left(O_7 \right) $, we obtain immediately that $ L_1(y,x)\geq 0,\ \forall (y,x)\in(0,L)\times\Omega. $ Moreover, from the characteristics of function $ \phi, $ there exists $ \delta_{\hat{x}}>0 $ such that $ \forall x\in\Omega_{\delta_{\hat{x}}}:=B_{\rho_2}\left(\hat{x},\delta \right)\cap\Omega, \\ \phi(x)-\phi(\hat{x}) \geq 0. $ Consequently, we receive that $ \forall (y,x)\in(0,L)\times\Omega_{\delta_{\hat{x}}},\\ V_j^{i,\hat{x},\epsilon}(y,x)=L_1(y,x)+L_2(x)\geq 0. $ It remains to prove that $ \forall (y,x)\in(0,L)\times\left( \Omega\setminus\Omega_{\delta_{\hat{x}}}\right) ,\\ V_j^{i,\hat{x},\epsilon}(y,x)\geq 0. $ Since, it holds $ L_1(y,x)\geq 0,\ \forall (y,x)\in(0,L)\times\Omega, $ it is sufficent to show that $ \forall x\in\Omega\setminus\Omega_{\delta_{\hat{x}}},\ L_2(x)\geq 0. $ We observe that the set $ \bar{\Omega}\setminus\Omega_{\delta_{\hat{x}}} $ is closed and bounded subset of $ \mathbb{R}^n. $ Equivalently, is compact subset of $ \mathbb{R}^n. $ Since the functions $N_1(x)=A\left( \phi(x)-\phi(\hat{x}) \right) $ and\\ $ N_2(x)=\exp\left(\kappa\phi(x) \right)\abs{x-\hat{x}}^2 $ are continuous on that compact set, will be lower bounded each of them. Also, since $ \bar{\Omega}\setminus\Omega_{\delta_{\hat{x}}}\supset\Omega\setminus\Omega_{\delta_{\hat{x}}}, $ both functions will be lower bounded and on $ \Omega\setminus\Omega_{\delta_{\hat{x}}}. $ We set
\begin{gather}
d_{N_1}:=\inf\left\lbrace N_1(x)\ :\ x\in\Omega\setminus\Omega_{\delta_{\hat{x}}}\right\rbrace \ \text{and}\
d_{N_2}:=\inf\left\lbrace N_2(x)\ :\ x\in\Omega\setminus\Omega_{\delta_{\hat{x}}}\right\rbrace\nonumber
\end{gather}
From the definition of function $ N_2, $ we see that its zero point is only the point $ \hat{x}, $ which is excluded from $ \bar{\Omega}\setminus\Omega_{\delta_{\hat{x}}}. $ Furthermore,
\begin{gather}\label{d_n ineq}
d_{N_2}\geq \tilde{d}_{N_2}:=\inf\left\lbrace N_2(x)\ :\ x\in\bar{\Omega}\setminus\Omega_{\delta_{\hat{x}}} \right\rbrace
\end{gather}
where $ \tilde{d}_{N_2}=N_2(\tilde{x}) $ for some $ \tilde{x}\in\bar{\Omega}\setminus\Omega_{\delta_{\hat{x}}}, $ since $ N_2 $ is continuous on the compact set $ \bar{\Omega}\setminus\Omega_{\delta_{\hat{x}}}. $ Definitely, by the definition of $ N_2 $ we have $\tilde{d}_{N_2} \geq 0.$ In specific, $ \tilde{d}_{N_2} > 0 $, since if we suppose that $ N_2(\tilde{x})=0, $ we will have a zero point $ \tilde{x}\neq\hat{x} $, which is a contradiction. Finally, from $ (\ref{d_n ineq}) $ we get $d_{N_2}>0. $
Then, we receive that $ \forall x\in\Omega\setminus\Omega_{\delta_{\hat{x}}}, $
\begin{gather} \label{d_n ineq_2}
L_2(x)=N_1(x)+B N_2(x)\geq\inf\left\lbrace N_1(x)+B N_2(x)\ :\ x\in\Omega\setminus\Omega_{\delta_{\hat{x}}}\right\rbrace \geq d_{N_1}+ B d_{N_2}
\end{gather}
If $ d_{N_1}\geq 0 $ then from $ (\ref{d_n ineq_2}) $, we get the desired inequality $ L_2(x)\geq 0,\ \forall x\in\Omega\setminus\Omega_{\delta_{\hat{x}}}. $ If $ d_{N_1}<0 $, then from $ (\ref{d_n ineq_2}) $, if se set $ B\geq\max\left\lbrace B,-\frac{d_{N_1}}{d_{N_2}} \right\rbrace $, we receive again the desired inequality.\\
Next, we proceed to the proof of
\begin{gather}
V_j^{i,\hat{x},\epsilon}(y,x)-\mathcal{M}_j V^{i,\hat{x},\epsilon}(y,x)\geq 0,\ \forall (y,x)\in(0,L)\times\Omega\nonumber
\end{gather}
We have that $ \forall (y,x)\in (0,L)\times\Omega, $
\begin{flalign}
V_j^{i,\hat{x},\epsilon}(y,x)-\mathcal{M}_j V^{i,\hat{x},\epsilon}(y,x)&= V_j^{i,\hat{x},\epsilon}(y,x)-\max_{\lambda\neq j}\left(V_{\lambda}^{i,\hat{x},\epsilon}(y,x) -c_{j,\lambda}(y,x)\right) \nonumber\\
&= c_{i,j}(y,x)-\max_{\lambda\neq j}\left(c_{i,\lambda}(y,x)-c_{j,\lambda}(y,x)\right) \nonumber\\
&= c_{i,j}(y,x)-\left( c_{i,\tilde{\lambda}}(y,x)-c_{j,\tilde{\lambda}}(y,x)\right)\ \left( \text{for some}\ \tilde{\lambda}\neq j\right) \nonumber\\
&= c_{i,j}(t,x)- c_{i,\tilde{\lambda}}(y,x)+c_{j,\tilde{\lambda}}(y,x)\label{claim_c2_eq1}
\end{flalign}
where the second equality is a result from the definition of function $ V_j^{i,\hat{x},\epsilon}. $ Using now the axiom $ (O_4) $, we receive that
\begin{gather}
c_{i,j}(y,x)+c_{j,\tilde{\lambda}}(y,x) \geq c_{i,\tilde{\lambda}}(y,x)\nonumber\\
\iff c_{i,j}(y,x)+c_{j,\tilde{\lambda}}(y,x) - c_{i,\tilde{\lambda}}(y,x)\geq 0\label{claim_c2_eq2}
\end{gather}
Finally, from $ (\ref{claim_c2_eq1}) $ and $ (\ref{claim_c2_eq2}) $ we get desired result.
\end{proof}
\begin{claim}\label{c_3}
For a given $ A\in\mathbb{R}, $ there exists $ \kappa>0 $ such that
\begin{gather}\label{c_3_2}
\forall j\in\left\lbrace 1,2,\dots,m\right\rbrace \forall (y,x)\in \left( 0,L\right]\times\partial\Omega,\ V_j^{i,\hat{x},\epsilon}(y,x)\geq f_j(y,x)
\end{gather}
\end{claim}
\begin{proof}
From the claim \ref{c_1}, we have seen that for a given number $ A\in\mathbb{R} $, we selected appropriate number $ B\in\mathbb{R}^{+} $ such that the inequality $ V_j^{i,\hat{x},\epsilon}(0,x)> g_j(x),\ \forall x\in\bar{\Omega} $ holds. In this claim, we proceed to the verification of an appropriate $ \kappa>0 $, such that $ \forall (y,x)\in \left( 0,L\right]\times\partial\Omega,\\ V_j^{i,\hat{x},\epsilon}(y,x)\geq f_j(y,x). $ We will notice below, that the proof of the last inequality is independed from the choice of the constant $ C. $ For this task, we are going to use the axiom $ (O_6). $ To prove the above inequality, we distinguish the following cases with respect to the position of $ \hat{x}. $
\begin{itemize}
\item[a)]$\hat{x}\in\partial\Omega$\\
We see that $ \forall y\in\left( 0,L\right] $
\begin{align}
V_j^{i,\hat{x},\epsilon}(y,\hat{x})&=g_i(\hat{x})+c_{i,j}(y,\hat{x})+Cy+\epsilon\nonumber\\
&>^{(C\geq0,\epsilon>0)}\min_{\lambda\in\left\lbrace 1,2,\dots,m \right\rbrace }\{g_{\lambda}(x)\mid\ x\in\partial\Omega\}+ \min_{d,k\in\left\lbrace 1,2,\dots,m \right\rbrace }\{c_{d,k}(y,x)\mid\ y\in[0,L] x\in\partial\Omega\}\nonumber\\
&\geq^{(O_6)} \max_{\lambda\in\left\lbrace 1,2,\dots,m \right\rbrace } \{f_{\lambda}(y,x) \mid y\in[0,L],\ x\in\partial\Omega\}\geq f_j(y,\hat{x})\nonumber
\end{align}
\item [b)] $ \hat{x}\in\Omega $\\
Due to the fact that, $ \hat{x}\in\Omega $ we receive that $ \hat{x}\notin\partial\Omega, i.e \left\lbrace \hat{x} \right\rbrace \cap\partial\Omega=\emptyset $ (since $ \Omega $ is open). Also, from the compactness of $ \partial\Omega $ and the closed set $ \left\lbrace \hat{x}\right\rbrace $, we receive that
\begin{gather}
d\equiv dist(\hat{x},\partial\Omega)=dist(\left\lbrace\hat{x} \right\rbrace ,\partial\Omega)>0
\end{gather}
Moreover, due to the continuity of $ \phi $ on the compact set $ \partial\Omega, $ the function $ \phi $ attains a minimum value. We set $ \phi_{\min}:=\min\{\phi(x)\mid x\in\partial\Omega\}=\phi(\tilde{x}) $, for some $ \tilde{x}\in\partial\Omega. $ Also, $ \phi_{\min}>0, $ since $ \phi(x)>0 $ on $ \bar{\Omega}. $ By the same argument, the function $ \Gamma_A:=A (\phi(x)-\phi(\hat{x})) $ attains a minimum value, which is denoted as $ \Gamma_A^{\min}. $
Then, $ \forall y\in(0,L],\ \forall x\in\partial\Omega $ we observe that
\begin{align}
V_j^{i,\hat{x},\epsilon}(y,x)&=g_i(\hat{x})+ A\left(\phi(x)-\phi(\hat{x}) \right)+ B \exp(\kappa\phi(x))\abs{x-\hat{x}}^2+C y+ c_{i,j}(y,x)+\epsilon\nonumber\\
&>^{(\exp\text{is increasing func.} )}_{(\epsilon>0)}\min_{\lambda\in\{1,2,\dots,m\}}\{g_{\lambda}(x)\mid x\in\partial\Omega\}+\nonumber\\
&+\min_{d,k\in\left\lbrace 1,2,\dots,m \right\rbrace }\{c_{d,k}(y,x)\mid\ t\in[0,L] x\in\partial\Omega\}+ \Gamma_A^{\min}+B\ d^2\exp(k\phi_{\min})\nonumber\\
&\geq^{(O_6)}\max_{\lambda\in\left\lbrace 1,2,\dots,m \right\rbrace } \{f_{\lambda}(y,x) \mid y\in[0,L],\ x\in\partial\Omega\}+\Gamma_A^{\min}+B\ d^2\exp(k\phi_{\min})\nonumber\\
&\geq f_j(y,x)+\Gamma_A^{\min}+B\ d^2\exp(k\phi_{\min})\nonumber
\end{align}
From the last inequality, in order to show that $ V_j^{i,\hat{x},\epsilon}(y,x)>f_j(y,x), $ it is sufficent to show that the quantity $ J:= \Gamma_A^{\min}+B\ d^2\exp(k\phi_{\min})$ is non-negative. Indeed,\\
$$ J\geq 0 \iff \kappa\geq\kappa_0:=\frac{-\Gamma_A^{\min}}{B\ d^2\ \phi_{\min}}$$
As a result from the last inequality, we can select for $ \kappa $, to be every value in the interval\\ $ \left[ \max\{1,\kappa_0\},\infty\right) $, since we want $ \kappa $ to be positive.
\end{itemize}
In both cases, we observe that, for any given number A, the determination of the constant $ C, $ does not play any role for the satisfaction of our defined target: $ V_j^{i,\hat{x},\epsilon}(t,x)>f_j(t,x),\ \forall t\in(0,T],\\ \forall x\in\partial\Omega. $
\end{proof}
Following an analogous procedure, as we did for the function $ V^{i,\hat{x},\epsilon}:[0,L]\times\bar{\Omega}\to\mathbb{R}^m $, for any given number $ A\in\mathbb{R}, $ we prove that there exists constants $ \tilde{B}\in\mathbb{R}^{+} $ and $ \tilde{\kappa}>0 $ such that the function $ U^{\hat{x},\epsilon}:[0,L]\times\bar{\Omega}\to\mathbb{R}^m $ is a viscosity subsolution of the problem $ (IBVP) $ and moreover satisfies $ \sup_{\epsilon>0}U_i^{\hat{x},\epsilon}(0,\hat{x})=g_i(\hat{x}),\ \forall i\in\{1,2,,\dots,m\}. $ Additionally, for any given number $ A\in\mathbb{R} $, we can choose appropriate constants $ B\in\mathbb{R}^{+} $ and $ \kappa>0 $ such that, both functions\\ $ V^{i,\hat{x},\epsilon}:[0,L]\times\bar{\Omega}\to\mathbb{R}^m $ and $ U^{\hat{x},\epsilon}:[0,L]\times\bar{\Omega}\to\mathbb{R}^m $ satisfy respectively the definition of viscosity supersolution and subsolution for the problem $ (IBVP). $
\end{proof}
\begin{proof}(\textbf{of Proposition $ \ref{Perronmethod} $})
Throughout the proof, we fix a random index $ i\in\{1,2,\dots,m\}. $ Let $ \epsilon_o>0 $ be a random point. Then from Proposition $ \ref{existence of barriers} $ we receive that the function\\ $ V_j^{i,\hat{x},\epsilon_o}:[0,L]\times\bar{\Omega}\to\mathbb{R},\ j\in\{1,2,\dots,m\}, $ attains a maximum value as a continuous on a compact set. Let $ M_{\epsilon_o}^j:=\max_{[0,L]\times\bar{\Omega}}V_j^{i,\hat{x},\epsilon_o} $ and $ M_{\epsilon_o}:=\max\{M_{\epsilon_o}^j\ :\ \ j=1,2,\dots,m \} $. Next, we define the following function, $ w:[0,L]\times\bar{\Omega}\to\mathbb{R}^m $, where $ w=(w_1,w_2,\dots,w_m) $ with $ w_i:[0,L]\times\bar{\Omega}\to\mathbb{R},\\ i\in\{1,2,\dots,m\}, $ defined as
\begin{gather}
w_i(y,x):=\sup\{u_i(y,x)\ :\ \ u\in\mathcal{F}\}, \forall(y,x)\in[0,L]\times\bar{\Omega}
\end{gather}
where
\begin{gather}
\mathcal{F}:=\{u=(u_1,u_2,\dots,u_m):[0,L]\times\bar{\Omega}\to\mathbb{R} \ : \ u\ \text{subsolution of (IBVP)}, u_i\leq M_{\epsilon_o}\ \text{on}\ [0,L]\times\bar{\Omega}\}\nonumber
\end{gather}
Initially, we prove that the above function is well defined and bounded as well. Later, it is proved that the function $ w_i $ satisfies the definition of viscosity solution.
\begin{claim}\label{well defined}
The function $ w_i:[0,L]\times\bar{\Omega}\to\mathbb{R} $ is well defined and bounded as well.
\end{claim}
\begin{proof}
By saying well defined, we mean that, the function $ w_i:[0,L]\times\bar{\Omega}\to\mathbb{R}$ attains a real value for each $ (t,x)\in[0,L]\times\bar{\Omega}.$ We define for each $ (t,x)\in[0,L]\times\bar{\Omega},\ i\in\{1,2,\dots,m\} $ the following set:
$$A^i_{y,x}:=\{u_i(y,x)\ :\ \ u\in\mathcal{F}\}$$
We observe that the above set is non empty. Indeed, we know from the proposition $ \ref{existence of barriers} $ that there exists a family of continuous subsolutions $ (U_i^{\hat{x},\epsilon})_{\epsilon>0} $ of the problem (IBVP). Since the function\\ $ V_i^{i,\hat{x},\epsilon_o} $ is a supersolution of (IBVP), then from the Comparison Principle, we receive that \\ $ U_i^{\hat{x},\epsilon}(t,x)\leq V_i^{i,\hat{x},\epsilon_o}(y,x), $ on $ [0,L)\times\bar{\Omega} $. Furthermore, from the last inequality and the fact that $ U_i^{\hat{x},\epsilon}, V_i^{i,\hat{x},\epsilon_o} $ are continuous on the point $ (T,x)\in[0,L]\times\bar{\Omega} $, we receive that $ U_i^{\hat{x},\epsilon} (L,x)\leq V_i^{i,\hat{x},\epsilon_o} (L,x) $. In conclution, we have that $ U_i^{\hat{x},\epsilon} (y,x)\leq V_i^{i,\hat{x},\epsilon_o} (y,x)\leq M_{\epsilon_o}. $ As a result the family $ (U_i^{\hat{x},\epsilon})_{\epsilon>0} $, is contained in $ A^i_{y,x}:=\{u_i(y,x)\ :\ u\in\mathcal{F} \} $. It remains to prove that the set $ A^i_{y,x} $ is upper bounded for each $(y,x)\in[0,T]\times\bar{\Omega}.$ Then, by the axiom of Completeness we receive that $ w_i(y,x):=\sup A^i_{y,x} $ is located in $ \mathbb{R}. $ Let $ (y_o,x_o) $ be a random point in $ [0,L]\times\bar{\Omega}. $ Then, $ u_i(y_o,x_o)\leq M_{\epsilon_o}, \forall u\in\mathcal{F}. $ From the last we see that $ A^i_{y_o,x_o} $ is upper bounded from $ M_{\epsilon_o} $. Moreover, we have that\\ $ w_i(y_o,x_o)=\sup A^i_{y_o,x_o}\leq M $. I.e $ w_i $ is upper bounded by $ M_{\epsilon_o}. $
We proceed to the proof that function $ w_i $ is lower bounded. Indeed, we choose a random $ \epsilon^{*}>0. $ Then, from the fact that $ U_i^{\hat{x},\epsilon^*}\in\mathcal{F}, $ is continuous on the compact set $ [0,L]\times\bar{\Omega}, $ will attain a minimum value $ m:=\min_{(y,x)\in[0,L]\times\bar{\Omega}}U_i^{\hat{x},\epsilon^*}(y,x) $. Then, we have that $ \forall (y,x)\in[0,L]\times\bar{\Omega}, $
\begin{gather}
m\leq U_i^{\hat{x},\epsilon^*}(y,x)\leq w_i(y,x):=\sup\{u_i(y,x)\ :\ u\in\mathcal{F} \}
\end{gather}
Consequently $ w_i $ is lower bounded.
\end{proof}
We proceed to the proof that function $ w_i:[0,L]\times\bar{\Omega}\to\mathbb{R} $ is a viscosity solution of (IBVP).
For this task, we will need the notions of lower and upper semicontinuous envelopes\footnote{Let $ (X,\rho) $ be a metric space, $ A\subset X $ and $ f:A\to[-\infty,\infty]. $ We define as lower semicontinuous envelope of the function $ f $, the function
\begin{gather}
f_{*}:A\to[-\infty,\infty],\ f_{*}(x):=\lim_{\delta\to 0}\inf\{f(y)\ :\ y\in A\cap B_{\rho}(x,\delta)\},\ \forall x\in A\nonumber
\end{gather}
Analogously, we define as upper semicontinuous envelope of the function $ f $, the function
\begin{gather}
f^{*}:A\to[-\infty,\infty],\ f^{*}(x):=\lim_{\delta\to 0}\sup\{f(y)\ :\ y\in A\cap B_{\rho}(x,\delta) \},\ \forall x\in A\nonumber
\end{gather}
} of $ w. $ In specific, we denote $ \forall i\in\{1,2,\dots,m\} $
\begin{gather}
w_{*,i}:[0,T]\times\bar{\Omega}\to\bar{\mathbb{R}}\nonumber\ \text{and}\ w_{i}^{*}:[0,T]\times\bar{\Omega}\to\bar{\mathbb{R}}\nonumber
\end{gather}
to be the lower and the upper semicontinuous envelope of function $ w $ respectively. By their definition, it is extrtacted that the lower semicontinuous envelope $ w_{*,i} $ of $ w_i $ is the largest lower semicontinuous function that it is dominated by function $ w_i $. Moreover, the upper semicontinuous function $ w_{i}^{*} $ is the smallest upper semicontinuous function that dominates the function $ w_i $. So we have
\begin{gather}
w_{*,i}\leq w_i,\ \text{on}\ [0,L]\times\bar{\Omega}\ \text{and}\ \forall h\in LSC([0,L]\times\bar{\Omega}),\ h\leq w_i\Rightarrow h\leq w_{*,i}\label{perron_1} \\
w_{i}^{*}\geq w_i,\ \text{on}\ [0,L]\times\bar{\Omega}\ \text{and}\ \forall h\in USC([0,T]\times\bar{\Omega}),\ h\geq w_i\Rightarrow h\geq w_{i}^{*}\label{perron_2}
\end{gather}
Also, we observe that $ w_{*,i} $ and $ w_i^{*} $ take real values and in particular, both envelopes are bounded functions by the numbers $ m $ and $ M_{\epsilon_o}. $ I.e $ Im( w_{i,*})\subset [m,M_{\epsilon_o}] $ and $ Im(w_i^{*})\subset[m,M_{\epsilon_o}]. $ The last is a consequence from the definition of $ w_{*,i} $ and $ w_i^{*} $ and the fact that $ w_i $ is bounded function.
\begin{claim}\label{bound_wi}
For the function $ w_i:[0,L]\times\bar{\Omega}\to\mathbb{R}, $ the folowing holds:
$$\forall\epsilon>0,\ \forall(t,x)\in[0,T)\times\bar{\Omega},\ w_i(y,x)\leq V_i^{i,\hat{x},\epsilon}(y,x)$$
\end{claim}
\begin{proof}
Let $ \epsilon>0 $ be a random point, that we fix and let $ u\in\ F $ be random function. Then, from the comparison principle, since $ V_i^{i,\hat{x},\epsilon} $ is a supersolution of $ (IBVP) $, it is extracted that $ u_i(y,x)\leq V_i^{i,\hat{x},\epsilon}(y,x),\ \forall (y,x)\in[0,L)\times\bar{\Omega}. $ By setting a fixed point $ (y_o,x_o)\in[0,L)\times\bar{\Omega},$ we receive $ u_i(y_o,x_o)\leq V_i^{i,\hat{x},\epsilon}(y_o,x_o),\ \forall u\in\mathcal{F}. $ From the last, we see that the real number $ V_i^{i,\hat{x},\epsilon}(y_o,x_o) $ is an upper bound of the set $ A^i_{(y_o,x_o)} $. By the definition of $ w_i $, we conclude that\\ $ w_i(y_o,x_o)=\sup A^i_{(y_o,x_o)}\leq V_i^{i,\hat{x},\epsilon}(y_o,x_o).$ Because $ (y_o,x_o)\in[0,L)\times\bar{\Omega} $ is a random point, we have the desired result $ w_i(y,x)\leq V_i^{i,\hat{x},\epsilon}(y,x),\forall (y,x)\in [0,L)\times\bar{\Omega}.
\end{proof}
\begin{claim}\label{viscosity_property}
The function $ w:[0,L]\times\bar{\Omega}\to\mathbb{R}^m $ satisfies the axioms of viscosity solution for (IBVP).
\end{claim}
\begin{proof}
Let $ i\in\{1,2,\dots,m\} $ be a random index. From $ (\ref{perron_1}) $ and $ (\ref{perron_2}) $ we receive that
\begin{gather}\label{visco_1}
w_{*,i}\leq w_i \leq w^*_{i},\ \text{on}\ [0,L]\times\bar{\Omega}
\end{gather}
The main idea, is to prove that $ w_{*,i} $ is a \textit{supersolution} of (IBVP) and $ w^*_{i} $ is a \textit{subsolution} of (IBVP). Then, from the application of Comparison Principle, we receive that
\begin{gather}\label{visco_2}
w^*_{i}\leq w_{*,i}\ \text{on}\ [0,L)\times\bar{\Omega}
\end{gather}
Combining $ (\ref{visco_1}) $ and $ (\ref{visco_2}) $, we obtain that $ w_i=w^*_{i}=w_{*,i},\ \text{on}\ [0,L)\times\bar{\Omega}. $ From the last equality, we conclude that the function $ w_i:[0,L]\times\bar{\Omega}\to\mathbb{R} $ satisfies the axioms of viscosity solution. \\
First, we establish the subsolution property of $ w^*_{i}. $
\begin{itemize}
\item We proceed to the proof that $ w^{*}_i(0,x) \leq g_i(x),\ \forall x\in\bar{\Omega}.$\\
From the claim $ (\ref{bound_wi}) $, we receive that $ \forall\epsilon\in(0,\epsilon_o],\ \forall(y,x)\in[0,L)\times\bar{\Omega},\ w_i(y,x)\leq V_i^{i,\hat{x},\epsilon}(y,x). $ The function $ V_i^{i,\hat{x},\epsilon},\ \epsilon\in(0,\epsilon_o], $ as continuous on $ [0,L]\times\bar{\Omega}, $ is upper semicontinuous on that set. In the case that, there exists at least one $ \epsilon^*\in(0,\epsilon_o] $ and a point $ \tilde{x}\in\bar{\Omega}$ such that $ w_i(T,\tilde{x})>V_i^{i,\hat{x},\epsilon^*}(L,\tilde{x}) $, then we can do an appropriate modification\footnote{We set as a modification of function $ V_i^{i,\hat{x},\epsilon^*} $ the function $ \tilde{V}_i^{i,\hat{x},\epsilon^*}:[0,L]\times\bar{\Omega}\to\mathbb{R} $ defined as\\ $ \tilde{V}_i^{i,\hat{x},\epsilon^*}(y,x) =
\begin{cases}
V_i^{i,\hat{x},\epsilon^*}(t,x) &\quad (y,x)\in[0,T)\times\bar{\Omega}\\
M_{\epsilon_o} & y=L,\ x\in\bar{\Omega} \\
\end{cases} $, where $ M_{\epsilon_o}=\max_{(y,x)\in [0,L]\times\bar{\Omega}} V_i^{i,\hat{x},\epsilon_o}(y,x), $ which also satisfies \\ $ M_{\epsilon_o}>=\max_{(y,x)\in [0,L]\times\bar{\Omega}} V_i^{i,\hat{x},\epsilon}(y,x),\ \forall\epsilon\in(0,\epsilon_o]. $ Then, the function $ \tilde{V}_i^{i,\hat{x},\epsilon^*} $ remains upper semicontinuous on its domain and satisfies at the same time $ \tilde{V}_i^{i,\hat{x},\epsilon^*}\geq V_i^{i,\hat{x},\epsilon*} $ on $ [0,L]\times\bar{\Omega},\ \forall\epsilon>0 $.} of function $ V_i^{i,\hat{x},\epsilon^*} $, let this modification denoted as $\tilde{V}_i^{i,\hat{x},\epsilon^*} $ which remains upper semicontinuous on $ [0,L]\times\bar{\Omega} $ and satisfies at the same time $ w_i(y,x)\leq \tilde{V}_i^{i,\hat{x},\epsilon^*}(y,x),\ \forall (y,x) \in [0,L]\times\bar{\Omega}. $ Then, from the relation $ (\ref{perron_2}) $, the following holds:
\begin{gather}
w_i^{*}(y,x)\leq \left( \tilde{V}_i^{i,\hat{x},\epsilon^*}\right) ^*(y,x),\ \forall(y,x)\in[0,L]\times\bar{\Omega},
\end{gather}
Specifically, for $ \hat{x}\in\bar{\Omega} $ randomly selected, we receive that
$$w_i^{*}(0,\hat{x})\leq \left(\tilde{V}_i^{i,\hat{x},\epsilon^*}\right)^* (0,\hat{x})=\left( V_i^{i,\hat{x},\epsilon^*}\right)^* (0,\hat{x})\stackrel{V_i^{i,\hat{x},\epsilon^*}\text{continuous} }{=}V_i^{i,\hat{x},\epsilon^*}(0,\hat{x}),$$
where the second inequality is a known result\footnote{Let $ (X,\rho) $ be a metric space, $ A\subset X,\ x_o\in A $ and $ f,g:A\to\mathbb{R} $ mappings. If there exists $ \delta_0>0, $ such that\\ $ \forall x\in B_{\rho}\left(x_o,\delta_0 \right)\cap A,\ f(x)=g(x) $ then, $ f^*(x_o)=g^*(x_o) $ and $ f_*(x_o)=g_*(x_o) $}. In conclusion the above pathological case, is solved. For the rest $ \epsilon\in(0,\epsilon_o], $ which satisfy the inequality $ w_i(y,x)\leq V_i^{i,\hat{x},\epsilon}(y,x), \\ \forall (y,x)\in[0,L]\times\bar{\Omega} $, we follow a similar procedure to prove that $ w^*_i(0,\hat{x})\leq V_i^{i,\hat{x},\epsilon}(0,\hat{x}) $. In specific, from the last inequality, from the relation $ (\ref{perron_2}) $, the following holds:
\begin{gather}
w_i^{*}(y,x)\leq \left(V_i^{i,\hat{x},\epsilon}\right) ^*(y,x),\ \forall(y,x)\in[0,L]\times\bar{\Omega}
\end{gather}
Then for the previous selected $ \hat{x}\in\bar{\Omega} $, we receive that
$$w_i^{*}(0,\hat{x})\leq \left(V_i^{i,\hat{x},\epsilon}\right)^* (0,\hat{x})=\stackrel{V_i^{i,\hat{x},\epsilon}\text{continuous} }{=}V_i^{i,\hat{x},\epsilon}(0,\hat{x})$$
Consequently, from the above analysis, we receive that
$$w_i^{*}(0,\hat{x})\leq V_i^{i,\hat{x},\epsilon}(0,\hat{x}),\ \forall\epsilon\in(0,\epsilon_o]$$
From the above inequality, we conclude that the real number $ w_i^{*}(0,\hat{x}) $ is a lower bound of the set $ \{V_i^{i,\hat{x},\epsilon}(0,\hat{x})\ : \\epsilon\in(0,\epsilon_o] \} $. Consequently, from the definition of infimum, we obtain that
\begin{gather}
w_i^{*}(0,\hat{x})\leq\inf_{\epsilon\in(0,\epsilon_o]}V_i^{i,\hat{x},\epsilon}(0,\hat{x})=\inf_{\epsilon>0}V_i^{i,\hat{x},\epsilon}(0,\hat{x})\stackrel{\text{Proposition}\ \ref{existence of barriers}}{=}g_i(\hat{x})
\end{gather}
where the $ \inf_{\epsilon\in(0,\epsilon_o]}V_i^{i,\hat{x},\epsilon}(0,\hat{x})=\inf_{\epsilon>0}V_i^{i,\hat{x},\epsilon}(0,\hat{x}) $ holds since the function $ V_i^{i,\hat{x},\epsilon}(0,\hat{x}) $ is of the form $ V_i^{i,\hat{x},\epsilon}(0,\hat{x}) =h+\epsilon $, where $ h $ is a constant number.
Because $ \hat{x}\in\bar{\Omega} $ was a random point, the desired inequality is established.
\item We proceed to the proof that $\forall \left(y,x \right)\in(0,L)\times\Omega,\forall\left(\alpha, p,X \right)\in \bar{J}^{2,+}w^{*}_i(y,x),$
$$\min\{F\left(y,x,w_i^{*}(y,x),p,X\right),w^{*}_i(y,x)-\mathcal{M}_i\textbf{w}^{*}(y,x)\} \leq 0$$
Let $ (\hat{y},\hat{x})\in(0,L)\times\Omega $ be a random point, and $ (p,X)\in \bar{J}^{2,+}w^*_i(\hat{y},\hat{x}). $ By the definition of $ w_i^*, $ there exists a sequence
\begin{gather}\label{convergence of envelope}
(y_n,x_n,u_i^n\left(y_n,x_n \right) ) \xrightarrow{n\to\infty} \left(\hat{y},\hat{x},w^*_i(\hat{y},\hat{x})\right)
\end{gather}
where $ (y_n,x_n)\in[0,L]\times\bar{\Omega} $ and $ \left( u_i^n \right)_{n\in\mathbb{N}} $ an appropriate sequence of subsolutions of $ (IBVP) $. Furthermore, if $ z_n=(\tilde{y}_n,\tilde{x}_n) $ is a sequence of points in $ [0,L]\times\bar{\Omega} $ such that\\ $ z_n\xrightarrow{n\to\infty} (\tilde{y}_o,\tilde{x}_o)\in[0,L]\times\bar{\Omega}, $ then $ \limsup_{n\to\infty}u_i^n(z_n)\leq w^*_i(\tilde{y}_o,\tilde{x}_o). $ Indeed, due to the fact that $ w^*_i $ is upper semicontinuous function, then by the sequential criterion, we receive that
\begin{gather}
\limsup_{n\to\infty} w^*_i(z_n)\leq w^*_i(\tilde{y}_o,\tilde{x}_o)\label{sequential_criterion_w^*_i}
\end{gather}
Moreover, by the definition of $ w^*_i, $ and $ w_i $ it follows that
\begin{gather}
\forall n\in\mathbb{N},\ w^*_i(z_n)\geq w_i(z_n),\ w_i(z_n)\geq u_i^n(z_n)
\end{gather}
From the last inequalities, we receive
\begin{gather}
\limsup_{n\to\infty} w^*_i(z_n)\geq\limsup_{n\to\infty} w_i(z_n)\label{perron_condition_1}\\
\limsup_{n\to\infty} w_i(z_n)\geq\limsup_{n\to\infty} u_i^n(z_n)\label{perron_condition_2}
\end{gather}
By the combination of $ (\ref{sequential_criterion_w^*_i}),\ (\ref{perron_condition_1}) $ and $ (\ref{perron_condition_2}) $ it is extracted, that $ \limsup_{n\to\infty}u^n_i(z_n)\leq w^*_i(\tilde{t}_o,\tilde{x}_o). $ Then, we get from Proposition 4.3 of $ \cite{CIL} $ that there exists a sequence $ (\hat{y}_n,\hat{x}_n)\in [0,L]\times\bar{\Omega} $ and \\$ (p_n,X_n)\in \bar{J}^{2,+}u^n_i(\hat{y}_n,\hat{x}_n) $ such that,
\begin{gather}
\left(\hat{y}_n,\hat{x}_n,u^n_i(\hat{y}_n,\hat{x}_n),p_n,X_n \right) \xrightarrow{n\to\infty}\left( \hat{y},\hat{x},w^*_i(\hat{y},\hat{x}),p,X\right)
\end{gather}
Because the point $ (\hat{y},\hat{x}) $ is an internal point of $ (0,L)\times\Omega, $ thus there exists a $ m_o\in\mathbb{N} $ such that,\\ $ \forall n\geq m_o, (\hat{y}_n,\hat{x}_n)\in (0,L)\times\Omega $. Also, from the continuity of $ F $, we receive that
\begin{gather}\label{continuity_of_F}
F\left(\hat{y}_n,\hat{x}_n,u^n_i(\hat{y}_n,\hat{x}_n),p_n,X_n \right) \xrightarrow{n\to\infty} F\left( \hat{y},\hat{x},w^*_i(\hat{y},\hat{x}),p,X\right)
\end{gather}
From the fact that $ w_j^* $ is an upper semicontinuous function, we receive that
\begin{gather}\label{w_j_estimation}
w_j^*(\hat{y},\hat{x})\geq\limsup_{n\to\infty} w_j^*(\hat{y}_n,\hat{x}_n) =\limsup_{n\to\infty} w_j^*(\hat{y}_{n+m_o},\hat{x}_{n+m_o})
\end{gather}
Then, we have that
\begin{align}
\max_{j\neq i}\left( w_j^*(\hat{y},\hat{x})-c_{i,j}(\hat{y},\hat{x})\right) &\geq^{(\ref{w_j_estimation})}\max_{j\neq i}\left(\limsup_{n\to\infty} w_j^*(\hat{y}_{n+m_o},\hat{x}_{n+m_o})-c_{i,j}(\hat{y},\hat{x})\right)\nonumber\\
&\stackrel{(c_{i,j}\ contin.)}{=}\max_{j\neq i}\left(\limsup_{n\to\infty}\left( w_j^*(\hat{y}_{n+m_o},\hat{x}_{n+m_o})-c_{i,j}(\hat{y}_{n+m_o},\hat{x}_{n+m_o})\right)\right) \nonumber\\
&\geq\limsup_{n\to\infty}\max_{j\neq i}\left( w_j^*(\hat{y}_{n+m_o},\hat{x}_{n+m_o})-c_{i,j}(\hat{y}_{n+m_o},\hat{x}_{n+m_o})\right) \nonumber\\
&\geq\limsup_{n\to\infty}\left( \max_{j\neq i}\left( u_j^{n+m_o}(\hat{y}_{n+m_o},\hat{x}_{n+m_o})-c_{i,j}(\hat{y}_{n+m_o},\hat{x}_{k_n+m_o})\right) \right)
\end{align}
From the last inequality, it follows that $ \forall n\in\mathbb{N}, $
\begin{align}
a_n:&=u^{n+m_o}_i(\hat{y}_{n+m_o},\hat{x}_{n+m_o})-\max_{j\neq i}\left( w_j^*(\hat{y},\hat{x})-c_{i,j}(\hat{y},\hat{x})\right)\nonumber\\
& \leq u^{n+m_o}_i(\hat{y}_{n+m_o},\hat{x}_{n+m_o})-\nonumber\\
&-\limsup_{n\to\infty}\left( \max_{j\neq i}\left( u_j^{n+m_o}(\hat{y}_{n+m_o},\hat{x}_{n+m_o})-c_{i,j}(\hat{y}_{n+m_o},\hat{x}_{n+m_o})\right)\right) :=b_n\nonumber
\end{align}
Thus, $ \forall n\in\mathbb{N},\ a_n\leq b_n $ and as a result from the last, we conclude that $ \lim_{n\to\infty}a_n\leq\lim_{n\to\infty}b_n $. The last inequality, it is true, since from the expressions of the sequences $ (a_n)_{n\in\mathbb{N}} $ and $ (b_n)_{n\in\mathbb{N}}, $ it is clear that their limits exists. If we translate the last limit inequality, we observe that
\begin{align}
& w_i^*(\hat{y},\hat{x})-\max_{j\neq i}\left( w^*_j(\hat{y},\hat{x})-c_{i,j}(\hat{y},\hat{x})\right)\nonumber\\
& \leq w_i^*(\hat{y},\hat{x})-\limsup_{n\to\infty}\left(\max_{j\neq i}\left( u^{n+m_o}_j(\hat{y}_{n+m_o},\hat{x}_{n+m_o}) -c_{i,j}(\hat{y}_{n+m_o},\hat{x}_{n+m_o})\right) \right) \nonumber
\end{align}
From the definition of $ \limsup_{n\to\infty}, $ there exists a strictly increasing sequence of natural numbers $ (k_n)_{n\in\mathbb{N}} $, such that
\begin{align}
&\limsup_{n\to\infty}\left(\max_{j\neq i}\left( u^{n+m_o}_j(\hat{t}_{n+m_o},\hat{x}_{n+m_o}) -c_{i,j}(\hat{y}_{n+m_o},\hat{x}_{n+m_o})\right) \right)\nonumber\\ &=\lim_{n\to\infty}\max_{j\neq i}\left(u_j^{k_n+m_o} \left( \hat{y}_{k_n+m_o},\hat{x}_{k_n+m_o}\right) -c_{i,j}\left( \hat{y}_{k_n+m_o},\hat{x}_{k_n+m_o}\right) \right) \nonumber
\end{align}
At the same time, we observe that
\begin{gather}
w^*_i(\hat{y},\hat{x})=\lim_{n\to\infty}u^n_i(\hat{y}_n,\hat{x}_n)=\lim_{n\to\infty} u^{k_n+m_o}_i(\hat{y}_{k_n+m_o},\hat{x}_{k_n+m_o})
\end{gather}
Combining the last two limit equalities, we receive that
\begin{align}
& w_i^*(\hat{y},\hat{x})-\max_{j\neq i}\left( w^*_j(\hat{y},\hat{x})-c_{i,j}(\hat{y},\hat{x})\right)\nonumber\\
&\leq \lim_{n\to\infty}\left(u^{k_n+m_o}_i(\hat{y}_{k_n+m_o},\hat{x}_{k_n+m_o})- \max_{j\neq i}\left(u_j^{k_n+m_o} \left( \hat{y}_{k_n+m_o},\hat{x}_{k_n+m_o}\right) -c_{i,j}\left( \hat{y}_{k_n+m_o},\hat{x}_{k_n+m_o}\right) \right) \right) \nonumber
\end{align}
Thus,
\begin{align}
&\min\left\lbrace F\left( \hat{y},\hat{x},w^*_i(\hat{y},\hat{x}),p,X\right) ,w_i^*(\hat{y},\hat{x})-\max_{j\neq i}\left( w^*_j(\hat{y},\hat{x})-c_{i,j}(\hat{y},\hat{x})\right)\right\rbrace \nonumber\\
&\leq\min\biggl\{ F\left( \hat{y},\hat{x},w^*_i(\hat{y},\hat{x}),p,X\right) ,\lim_{n\to\infty}\biggl(u^{k_n+m_o}_i(\hat{y}_{k_n+m_o},\hat{x}_{k_n+m_o})-\nonumber\\
&\qquad\qquad-\max_{j\neq i}\left(u_j^{k_n+m_o} \left( \hat{y}_{k_n+m_o},\hat{x}_{k_n+m_o}\right) - c_{i,j}\left( \hat{y}_{k_n+m_o},\hat{x}_{k_n+m_o}\right)\right)\biggl)\biggl\} \nonumber\\
&=^{(\ref{continuity_of_F})}\min\biggl\{ \lim_{n\to\infty}F(\hat{y}_{k_n+m_o},\hat{x}_{k_n+m_o},u^{k_n+m_o}_i(\hat{t}_{k_n+m_o},\hat{x}_{k_n+m_o}),p_{k_n+m_o},X_{k_n+m_o}),\nonumber\\ &\quad\lim_{n\to\infty}\biggl(u^{k_n+m_o}_i(\hat{y}_{k_n+m_o},\hat{x}_{k_n+m_o})-\max_{j\neq i}\left(u_j^{k_n+m_o} \left( \hat{y}_{k_n+m_o},\hat{x}_{k_n+m_o}\right) - c_{i,j}\left( \hat{y}_{k_n+m_o},\hat{x}_{k_n+m_o}\right) \right) \biggl) \biggl\}\nonumber\\
&=\lim_{n\to\infty} \min\biggl\{ F\left( \hat{y}_{k_n+m_o},\hat{x}_{k_n+m_o},u^{k_n+m_o}_i(\hat{y}_{k_n+m_o},\hat{x}_{k_n+m_o}),p_{k_n+m_o},X_{k_n+m_o}\right) ,\nonumber\\
&\qquad u^{k_n+m_o}_i(\hat{t}_{k_n+m_o},\hat{x}_{k_n+m_o})
-\max_{j\neq i}\left(u_j^{k_n+m_o} \left( \hat{y}_{k_n+m_o},\hat{x}_{k_n+m_o}\right) - c_{i,j}\left( \hat{y}_{k_n+m_o},\hat{x}_{k_n+m_o}\right) \right)\biggl\}\leq 0
\end{align}
where the last inequality is true, since the sequence of functions $ \left( u^n_i\right)_{n\in\mathbb{N}} $ are subsolutions of $ (IBVP). $ Consequently, the desired inequality is established.
\item We proceed to the proof that $ \forall (y,x)\in(0,L)\times\partial{\Omega},\ w_i^*(y,x)\leq f_i(y,x) $\\
Let $ (\hat{y},\hat{x})\in(0,L)\times\partial{\Omega} $ a random point. Similarly, as before, by the definition of $ w^*_i $ there exists a sequence
\begin{gather}
\left( y_n,x_n,u^n_i(y_n,x_n)\right)\xrightarrow{n\to\infty}\left( \hat{y},\hat{x},w^*_i(\hat{y},\hat{x}) \right)
\end{gather}
where $ \left( y_n,x_n\right)\in(0,L)\times\partial{\Omega} $ and $ u^n_i $ appropriate sequence of subsolution of $ (IBVP) $. Since $ u^n_i $ are subsolutions, we receive that $ \forall n\in\mathbb{N} $
\begin{gather}
u^n_i(y_n,x_n) \leq f_i(y_n,x_n)
\end{gather}
Consequently, by taking $ n\to\infty $, we receive the desired inequality.
\end{itemize}
In particular, since $ w^{*} $ is a subsolution of $ (IBVP), $ then by the definition of $ w $, it is extracted that $ w^{*}\leq w, $ on $ [0,L]\times\bar{\Omega}. $ Also from $ (\ref{visco_1}), $ we have $ w\leq w^{*} $ on $ [0,L]\times\bar{\Omega}. $ From the last, we conclude that $ w=w^{*} $ on $ [0,L]\times\bar{\Omega}. $\\
Next we prove that $ w_{*} $ is a supersolution of $ (IBVP), $ following a classical argument by contradiction. In specific, we show that, if $ w_{*} $ is not a supersolution, then there exists a subsolution strictly greater than $ w^{*}, $ where the last function, is equal with $ w. $ This is a contradiction, by the definition of $ w. $\\
First, we notice that, $ w_{*} $ satisfies $ w_{*,i}(0,x)\geq g_i(x),\ \forall x\in\bar{\Omega}. $ Indeed, by the definition of $ w_i $, we have that $ w_i(y,x)\geq U_i^{\hat{x},\epsilon}(y,x),\ \forall (y,x)\in[0,L]\times\bar{\Omega},\ \forall\epsilon>0. $ Due to the fact that $ U_i^{\hat{x},\epsilon} $ is continuous on $ [0,L]\times\bar{\Omega}, $ it is extracted that the specific function lower semicontinuous on $ [0,L]\times\bar{\Omega}. $ From $ (\ref{perron_1}) $, we receive that $ w_{*,i}(y,x)\geq U_i^{\hat{x},\epsilon}(y,x) $ on $ [0,L]\times\bar{\Omega},\ \forall\epsilon>0. $ In particular, we get that $ w_{*,i}(0,\hat{x})\geq U_i^{\hat{x},\epsilon}(0,\hat{x}),\ \forall\epsilon>0. $ From the last we conclude that the real number $ w_{*,i}(0,\hat{x})(0,\hat{x}) $ is an upper bound of the set $ \{U_i^{\hat{x},\epsilon}(0,\hat{x})\ :\ \epsilon>0\}. $ Consequently,
\begin{gather}
w_{*,i}(0,\hat{x})(0,\hat{x})\geq\sup_{\epsilon>0}U_i^{\hat{x},\epsilon}(0,\hat{x})\stackrel{ \text{Proposition}\ \ref{existence of barriers}}{=}g_i(\hat{x})
\end{gather}
Assume now that $ w_{*,i} $ is not a supersolution $ (IBVP). $ Then, there exists at least one index\\ $ i_o\in\{1,2,\dots,m\} $ such that the function $ w_{*,i_o} $ does not satisfy the supersolution property. Then there are two main possible cases.
\begin{itemize}
\item $ \exists (\hat{y},\hat{x})\in(0,L)\times\Omega $ and $ (\alpha_o,p_o,X_o) \in \bar{J}^{2,-}w_{*,i_o}(\hat{y},\hat{x})$, such that
\begin{gather}\label{main_case_w_*}
\min\{F\left(\hat{y},\hat{x},w_{*,i_o}(\hat{y},\hat{x}),p_o,X_o \right),w_{*,i_o}(\hat{y},\hat{x}) -\mathcal{M}_{i_o}w_{*}(\hat{y},\hat{x})\}<0
\end{gather}
For $ \delta\in(0,\delta_o), $ let $ \beta:(0,\delta_o)\to\mathbb{R} $ be a random real positive function with the identity $ \lim_{\delta\to 0}\beta(\delta)=0. $ Furthermore, let $ h:[0,L]\times\bar{\Omega}\times(0,\delta_o)\to\mathbb{R} $ be a random function, such that the following identities hold:
\begin{itemize}
\item $ \lim_{(\delta,y,x)\to(0^+,\hat{y},\hat{x})}D_x h(y,x,\delta)=0 $ and $ \lim_{(\delta,y,x)\to(0^+,\hat{y},\hat{x})}D^2_{xx} h(y,x,\delta)=0 $ \nonumber
\item $h$ is continuous and bounded on its domain
\item $ \forall\delta\in(0,\delta_o),\ \forall V\subset [0,L]\times\bar{\Omega}\ \text{bounded}, \sup_{(y,x)\in V}h(y,x,\delta)>0$\nonumber
\end{itemize}
Gradually, during the following proof, we will demand extra conditions for functions $ \beta $ and $ h. $
Now, we construct the following family of functions \\ $ \tilde{w}^{\delta}:[0,L]\times\bar{\Omega}\to\mathbb{R}^m $, where $ \tilde{w}^{\delta}=\left( \tilde{w}^{\delta}_1,\tilde{w}^{\delta}_2,\dots, \tilde{w}^{\delta}_m\right) $ with
\begin{align}
\tilde{w}^{\delta}_{i}(y,x):=&w_{*,i}(\hat{y},\hat{x})+ h(y,x,\delta)\beta(\delta)+\alpha_o (y-\hat{t})+\left\langle p_o,(x-\hat{x}) \right\rangle+\nonumber\\
&+\frac{1}{2}\left\langle X_o\ (x-\hat{x}), (x-\hat{x})\right\rangle,\ i\in\left\lbrace1,2,\dots,m \right\rbrace\nonumber
\end{align}
For each $ i\in\left\lbrace 1,2,\dots,m\right\rbrace $ fixed, we define the function $ S_i(\delta,y,x):= \tilde{w}^{\delta}_{i}(y,x). $ We observe that
\begin{gather}
\lim_{(\delta,y,x)\to(0^+,\hat{y},\hat{x})} S_i(\delta,y,x)=w_{*,i}(\hat{y},\hat{x})\nonumber\\
\text{and}\ \lim_{(\delta,y,x)\to(0^+,\hat{y},\hat{x})} D_x S_i(\delta,y,x)=\lim_{(\delta,y,x)\to(0^+,\hat{y},\hat{x})} D_x\tilde{w}^{\delta}_i(y,x)=p_o\nonumber\\
\lim_{(\delta,t,x)\to(0^+,\hat{y},\hat{x})} D_{xx}^2 S_i(\delta,y,x)=\lim_{(\delta,y,x)\to(0^+,\hat{y},\hat{x})} D_{xx}^2\tilde{w}^{\delta}_i(y,x)=X_o
\end{gather}
\begin{claim}\label{claim_subsolution property}
There exists sufficiently small positive parameters, $ \delta^*\in(0,\delta_o),$ and $ R\in(0,R_o], $ such that the function $ \tilde{w}_{i_o}\equiv\tilde{w}^{\delta^*}_{i_o}:[0,L]\times\bar{\Omega}\to\mathbb{R}, $ satisfies the subsolution property of
\begin{gather}
\min\{F(y,x,\tilde{w}_{i_o}(y,x),D_x \tilde{w}_{i_o}(y,x),D^2_{xx} \tilde{w}_{i_o}(y,x)), \tilde{w}_{i_o}(y,x)-\mathcal{M}_{i_o} w^*(y,x)\}=0
\end{gather}
in $ Q_R:=\left\lbrace (t,x)\in[0,L]\times\bar{\Omega}\ :\ \abs{y-\hat{y}}+\abs{x-\hat{x}}^2\leq R\right\rbrace $ and also $ \tilde{w}_{i_o}\leq M $ on $ Q_R. $
\end{claim}
\begin{proof}
Initially, we observe that
\begin{align}
\mathcal{M}_{i_o}w_*(\hat{t},\hat{x})&=\max_{\lambda\neq i_o}\left( w_{*,\lambda}(\hat{y},\hat{x}) -c_{i_o,\lambda}(\hat{y},\hat{x})\right)\nonumber\\
&= \max_{\lambda\neq i_o}\left(\lim_{(\delta,y,x)\to(0^+,\hat{t},\hat{x})}\left( \tilde{w}^{\delta}_{\lambda}(y,x)-c_{i_o,\lambda}(y,x) \right)\right) \nonumber\\
&=\lim_{(\delta,t,x)\to(0^+,\hat{y},\hat{x})}\max_{\lambda\neq i_o}\left(\tilde{w}^{\delta}_{\lambda}(y,x)-c_{i_o,\lambda}(y,x) \right)\nonumber\\
&= \lim_{(\delta,y,x)\to(0^+,\hat{y},\hat{x})}\mathcal{M}_{i_o}\tilde{w}^{\delta}(y,x)
\end{align}
Thus, from the above, we receive
\begin{gather}
w_{*,i_o}\left(\hat{y},\hat{x}\right)-\mathcal{M}_{i_o}w_*(\hat{y},\hat{x})=\lim_{(\delta,y,x)\to(0^+,\hat{y},\hat{x})}\left(\tilde{w}^{\delta}_{i_o}(y,x)-\mathcal{M}_{i_o}\tilde{w}^{\delta}(y,x)\right)
\end{gather}
Moreover, from continuity of $ F $, we have that
\begin{gather}
F\left(\hat{y},\hat{x},w_{*,i_o}(\hat{y},\hat{x}),p_o,X_o \right)=\lim_{(\delta,y,x)\to(0^+,\hat{y},\hat{x})}F\left(y,x,\tilde{w}^{\delta}_{i_o}(y,x),D_x\tilde{w}^{\delta}_{i_o}(t,x),D^2_{xx}\tilde{w}^{\delta}_{i_o}(y,x) \right)
\end{gather}
Considering the above limits, we have that
\begin{align}
0&>^{(\ref{main_case_w_*})}\min\left\lbrace F\left(\hat{y},\hat{x},w_{*,i_o}(\hat{y},\hat{x}),p_o,X_o \right), w_{*,i_o}(\hat{y},\hat{x})-\mathcal{M}_{i_o}w_*(\hat{y},\hat{x})\right\rbrace \nonumber\\
&=\min\Biggl\{\lim_{(\delta,y,x)\to(0^+,\hat{y},\hat{x})}F\left(y,x,\tilde{w}^{\delta}_{i_o}(y,x),D_x\tilde{w}^{\delta}_{i_o}(y,x),D^2_{xx}\tilde{w}^{\delta}_{i_o}(y,x) \right),\nonumber\\ &\qquad\qquad\lim_{(\delta,y,x)\to(0^+,\hat{y},\hat{x})}\left( \tilde{w}^{\delta}_{i_o}(y,x)-\mathcal{M}_{i_o}\tilde{w}^{\delta}(y,x)\right) \Biggl\} \nonumber\\
&=\lim_{(\delta,y,x)\to(0^+,\hat{y},\hat{x})}\min\Biggl\{ F\left(y,x,\tilde{w}^{\delta}_{i_o}(y,x),D_x\tilde{w}^{\delta}_{i_o}(y,x),D^2_{xx}\tilde{w}^{\delta}_{i_o}(y,x) \right),\nonumber\\ &\qquad\qquad\qquad\qquad\qquad\qquad\tilde{w}^{\delta}_{i_o}(y,x)-\mathcal{M}_{i_o}\tilde{w}^{\delta}(y,x)\Biggl\}
\end{align}
Consequently, there exists $ \tilde{\delta}\in(0,\delta_o)>0 $ and $ \tilde{R}\in(0,R_o], $ such that $ \forall\delta\in(0,\tilde{\delta}) $ and \\$ \forall (y,x)\in Q_{\tilde{R}}, $
\begin{gather}
\min\left\lbrace F\left(y,x,\tilde{w}^{\delta}_{i_o}(y,x),D_x\tilde{w}^{\delta}_{i_o}(y,x),D^2_{xx}\tilde{w}^{\delta}_{i_o}(y,x) \right), \tilde{w}^{\delta}_{i_o}(y,x)-\mathcal{M}_{i_o}\tilde{w}^{\delta}(y,x)\right\rbrace <0
\end{gather}
We now fix $ \delta^*\in(0,\tilde{\delta}) $, and from the above inquality it holds that, $ \forall (y,x)\in Q_{\tilde{R}} $
\begin{gather}\label{important_relation_b}
\min\left\lbrace F\left(y,x,\tilde{w}^{\delta^*}_{i_o}(y,x),D_x\tilde{w}^{\delta^*}_{i_o}(y,x),D^2_{xx}\tilde{w}^{\delta^*}_{i_o}(y,x) \right), \tilde{w}^{\delta^*}_{i_o}(y,x)-\mathcal{M}_{i_o}\tilde{w}^{\delta^*}(y,x)\right\rbrace <0
\end{gather}
We claim now, that there exists a sufficently small $ R\in(0,\tilde{R}) $ such that, $ \forall (y,x)\in Q_R,$
\begin{gather}
\mathcal{M}_{i_o}\tilde{w}^{\delta^*}(y,x) \leq\mathcal{M}_{i_o}w^*(y,x)
\end{gather}
Indeed, we first notice that
\begin{align}
\mathcal{M}_{i_o}\tilde{w}^{\delta^*}(y,x)&=\max_{\lambda\neq i_o}\left\lbrace\tilde{w}^{\delta^*}_{\lambda}(y,x)-c_{i_o,\lambda}(y,x) \right\rbrace\nonumber\\
&=\max_{\lambda\neq i_o}\{ w_{*,\lambda}(\hat{y},\hat{x})+ h(y,x,\delta^*)\beta(\delta^*)+\alpha_o (y-\hat{y})+\left\langle p_0, x-\hat{x}\right\rangle+\nonumber\\
&+\frac{1}{2}\left\langle X_o (x-\hat{x}),(x-\hat{x}) \right\rangle-c_{i_o,\lambda}(y,x) \}\nonumber\\
&\equiv\max_{\lambda\neq i_o}\left\lbrace K(y,x)-c_{i_o,\lambda}(y,x) \right\rbrace\label{important_relation_o}
\end{align}
where
\begin{gather}
K(y,x):=w_{*,\lambda}(\hat{y},\hat{x})+h(y,x,\delta^*)\beta(\delta^*)+\alpha_o (y-\hat{y})+\left\langle p_0, x-\hat{x}\right\rangle+\frac{1}{2}\left\langle X_o (x-\hat{x}),(x-\hat{x}) \right\rangle\nonumber
\end{gather}
Due to the fact that, $ w_{*,\lambda} $ is lower semicontinuous at point $ (\hat{y},\hat{x}) $, for fixed $ \epsilon^*>0 $, there exists $ r_{\lambda}>0 $ such that, $ \forall(y,x)\in B_{\rho_2}\left( (\hat{y},\hat{x}),r_{\lambda} \right)\cap \left( [0,L]\times\bar{\Omega}\right), $
\begin{gather}
w_{*,\lambda}(\hat{y},\hat{x})<w_{*,\lambda}(y,x)+\epsilon^*\leq w^*_{\lambda}(y,x)+\epsilon^*
\end{gather}
We define $ r_o:=\min\left\lbrace r_{\lambda}\ :\ \lambda\in\left\lbrace 1,2,\dots,m\right\rbrace\setminus\left\lbrace i_o \right\rbrace \right\rbrace $. Then $ \forall (t,x)\in B_{\rho_2}\left( (\hat{y},\hat{x}),r_o\right) \cap \left( [0,L]\times\bar{\Omega}\right), $
\begin{gather}
w_{*,\lambda}(\hat{y},\hat{x})\leq w^*_{\lambda}(y,x)+\epsilon^*
\end{gather}
We select $ R\in (0,\tilde{R}), $ such that $ Q_R\subset B_{\rho_2}\left( (\hat{y},\hat{x}),r_o\right) \cap \left( [0,L]\times\bar{\Omega}\right). $
From the above estimations, \\ $\forall \lambda\in\left\lbrace 1,2,\dots,m \right\rbrace\setminus \left\lbrace i_o \right\rbrace\ \text{and}\ \forall (y,x)\in Q_R $
\begin{align}
K(y,x)&\leq w^*_{\lambda}(y,x)+\epsilon^*+ h(y,x,\delta^*)\beta(\delta^*)+\alpha_o (y-\hat{y})+\left\langle p_0, x-\hat{x}\right\rangle+\nonumber\\
&+\frac{1}{2}\left\langle X_o (x-\hat{x}),(x-\hat{x}) \right\rangle\nonumber\\
&\leq w^*_{\lambda}(y,x)+\epsilon^*+\sup\left\lbrace h(y,x,\delta^*)\ : \ \ (y,x)\in Q_R\right\rbrace \beta(\delta^*)+\sup\big{\lbrace}\alpha_o (y-\hat{t})+\left\langle p_0, x-\hat{x}\right\rangle+\nonumber\\
&+\frac{1}{2}\left\langle X_o (x-\hat{x}),(x-\hat{x}) \right\rangle\ : \ \ (y,x)\in Q_R \big{\rbrace}
\end{align}
We set
\begin{gather}
\Lambda:=\sup\big{\lbrace}\alpha_o (y-\hat{y})+\left\langle p_0, x-\hat{x}\right\rangle
+\frac{1}{2}\left\langle X_o (x-\hat{x}),(x-\hat{x}) \right\rangle\ \ :\ \ (y,x)\in Q_R \big{\rbrace}\nonumber
\end{gather}
The main target at this point, is to guarantee the following
\begin{gather}\label{negativity_rel}
\epsilon^*+\sup\{h(y,x,\delta^*)\ : \ (y,x)\in Q_R\} +\Lambda\leq 0\
\text{and}\ \beta(\delta^*) h(\hat{y},\hat{x},\delta^*)>0
\end{gather}
setting appropriate extra conditions for functions $ h $ and $ \beta. $ The second condition will be needed in the proof of the following claim.\\
For this task, we have the following cases
\begin{itemize}
\item Let $ \Lambda\leq 0 $. In this case, we can choose $ \epsilon^*>0 $ such that $ \epsilon^*+\Lambda<0 $ holds. Equivalently, $ 0<-\epsilon^*-\Lambda. $ In order to hold $ (\ref{negativity_rel}) $, it is sufficent to define the value of function $ \beta $ at point $ \delta
^* $ in a way that the following holds
\begin{gather}
0<\beta(\delta^*)\leq\frac{-\epsilon^*-\Lambda}{\sup\{h(y,x,\delta^*)\ | \ (y,x)\in Q_R\}}\equiv k_o\ \text{and}\ h(\hat{y},\hat{x},\delta^*)>0
\end{gather}
\item Let $ \Lambda> 0 $. In this case, we receive automatically that $ -\epsilon^*-\Lambda<0. $ Then again, we observe that, if we restrict the value of $ \beta $ at point $ \delta^* $ to be inside the interval $ (-\infty,k_o) $ and moreover demand $ h(\hat{t},\hat{x},\delta^*)<0 $, we conclude that $ (\ref{negativity_rel}) $ holds.
\end{itemize}
From, both cases, we obtain that, $ \forall (y,x)\in Q_R,\ K(y,x)\leq w^*_{\lambda}(y,x) $. As a result, we receive that
\begin{align}
\mathcal{M}_{i_o}\tilde{w}^{\delta^*}(y,x)&\leq^{(\ref{important_relation_o})}\max_{\lambda\neq i_o}\left\lbrace K(y,x)-c_{i_o,\lambda}(y,x) \right\rbrace\nonumber\\
&\leq\max_{\lambda\neq i_o}\left\lbrace w^*_{\lambda}(y,x)-c_{i_o,\lambda}(y,x)\right\rbrace=\mathcal{M}_{i_o}w^*(y,x)
\end{align}
Equivalently, we obtain $ \forall (t,x)\in Q_R\subset Q_{\tilde{R}}, $
\begin{gather}
\tilde{w}^{\delta^*}_{i_o}(y,x)-\mathcal{M}_{i_o}w^*(y,x) \leq \tilde{w}^{\delta^*}_{i_o}(y,x)-\mathcal{M}_{i_o}\tilde{w}^{\delta^*}(y,x)
\end{gather}
From the last inequality combined with $ (\ref{important_relation_b}), $ we receive that, $ \forall (y,x)\in Q_R $ that
\begin{gather}
\min\left\lbrace F\left(y,x,\tilde{w}^{\delta^*}_{i_o}(y,x),D_x\tilde{w}^{\delta^*}_{i_o}(y,x),D^2_{xx}\tilde{w}^{\delta^*}_{i_o}(y,x) \right), \tilde{w}^{\delta^*}_{i_o}(y,x)-\mathcal{M}_{i_o}w^{*}(y,x)\right\rbrace <0
\end{gather}
In conclusion, the function $ \tilde{w}_{i_o}:=\tilde{w}^{\delta^*}_{i_o} $ satisfies in the classical sense the subsolution property.
\end{proof}
Next we define the following function $ \hat{u}:=(\hat{u}_1,\hat{u}_2,\dots,\hat{u}_m):[0,L]\times\bar{\Omega}\to\mathbb{R}^m $, where
\begin{gather}
\hat{u}_{i_o}(y,x):=\begin{cases}
\max\{w^*_{i_o}(y,x),\tilde{w}_{i_o}(y,x)\},\ & \text{ if }\ (y,x)\in Q_R\nonumber\\
w^*_{i_o}(y,x),\ & \text{otherwise}
\end{cases}
\ \text{and}\ \hat{u}_{j}(y,x):=w^*_{j}(y,x),\ \text{if}\ j\neq i_o
\end{gather}
\begin{claim}
The function $ \hat{u}_{i_o}:[0,L]\times\bar{\Omega}\to\mathbb{R} $ is upper semicontinuous.
\end{claim}
\begin{proof}
First, we notice that $ Q_R $ is a closed subset of $ \mathbb{R}\times\mathbb{R}^n $ and as a result, of the closedness of $ [0,L]\times\bar{\Omega} $, we receive that $ Q_R $ is closed in the metric subspace $ [0,L]\times\bar{\Omega}. $ Let $ (\tilde{y},\tilde{x})\in[0,L]\times\bar{\Omega} $ be a random point. There are three cases: $ (\tilde{y},\tilde{x})\in int\left( Q_R \right) $ or $ (\tilde{y},\tilde{x})\in \partial Q_R $ or \\ $ (\tilde{y},\tilde{x})\in K:=\left( [0,L]\times\bar{\Omega}\right) \setminus Q_R. $
\begin{itemize}
\item Let $ (\tilde{y},\tilde{x})\in int\left( Q_R\right). $ Since $ int\left( Q_R\right) $ is an open set, then there exists $ \tilde{\delta}>0 $ such that, $ B_{\rho_2}\left((\tilde{y},\tilde{x}),\tilde{\delta} \right)\subset int(Q_R). $ For a random $ \epsilon>0, $ due to the fact that $ w^*_{i_o} $ is upper semicontinuous, there exists $\delta_1>0, $ such that $ \forall (y,x)\in B_{\rho_2}\left( (\tilde{y},\tilde{x}),\delta_1\right)\cap\left([0,L]\times\bar{\Omega} \right), $
\begin{gather}
w_{i_o}^*(y,x)<w^*_{i_o}(\tilde{y},\tilde{x})+\epsilon \leq\hat{u}_{i_o} (\tilde{y},\tilde{x})+\epsilon
\end{gather}
Moreover, since $ \tilde{w}_{i_o} $ is continuous at $ (\tilde{y},\tilde{x}) $, it will be also an upper semicontinuous on that point. As a result, there exists $ \delta_2>0 $ such that, $ \forall (y,x)\in B_{\rho_2}\left( (\tilde{y},\tilde{x}),\delta_2\right)\cap\left([0,L]\times\bar{\Omega} \right), $
\begin{gather}
\tilde{w}_{i_o}(y,x)<\tilde{w}_{i_o}(\tilde{y},\tilde{x})+\epsilon\leq\hat{u}_{i_o} (\tilde{y},\tilde{x})+\epsilon
\end{gather}
Then, setting $ \delta_o:=\min\{\tilde{\delta},\delta_1,\delta_2\}>0 $ we see that, \\$ \forall(y,x)\in B_{\rho_2}\left( (\tilde{y},\tilde{x}),\delta_o\right)=B_{\rho_2}\left( (\tilde{y},\tilde{x}),\delta_o\right)\cap \left( [0,L]\times\bar{\Omega}\right), $
\begin{gather}
w_{i_o}^*(y,x)<\hat{u}_{i_o}(\tilde{y},\tilde{x})+\epsilon\ \text{and}\ \tilde{w}_{i_o}(y,x)<\hat{u}_{i_o}(\tilde{y},\tilde{x})+\epsilon
\end{gather}
And from the last, we obtain $ \hat{u}_{i_o}(y,x):=\max\{w^*_{i_o}(y,x),\tilde{w}_{i_o}(y,x)\}<\hat{u}_{i_o}(\tilde{t},\tilde{x}) +\epsilon$
\item Let $ (\tilde{y},\tilde{x})\in\partial Q_R $ As before, due to the fact that $ w^*_{i_o} $ and $ \tilde{w}_{i_o} $ are upper semicontinuous at point $ (\tilde{y},\tilde{x}) $, for $ \epsilon>0 $ randrom the following holds:
\begin{gather}
\exists \delta_1>0:\ \forall (y,x)\in B_{\rho_2}\left((\tilde{y},\tilde{x}),\delta_1 \right)\cap \left([0,L]\times\bar{\Omega} \right)\nonumber\\
w^*_{i_o}(y,x)<w^*_{i_o}(\tilde{t},\tilde{x})+\epsilon\leq\max\{w^*_{i_o}(\tilde{y},\tilde{x}),\tilde{w}_{i_o}(\tilde{y},\tilde{x})\}+\epsilon=\hat{u}_{i_o}(\tilde{y},\tilde{x})+\epsilon\label{relation_aghjk_1}\\
\exists \delta_2>0:\ \forall (y,x)\in B_{\rho_2}\left((\tilde{y},\tilde{x}),\delta_2 \right)\cap \left([0,L]\times\bar{\Omega} \right)\nonumber\\
\tilde{w}_{i_o}(y,x)<\tilde{w}_{i_o}(\tilde{y},\tilde{x})+\epsilon\leq \hat{u}_{i_o}(\tilde{y},\tilde{x})+\epsilon\label{relation_aghjk_2}
\end{gather}
We set $ \delta_o:=\{\delta_1,\delta_2\}>0. $ Then $ \forall (y,x)\in B_{\rho_2}\left( (\tilde{y},\tilde{x}),\delta_o\right)\cap\left([0,L]\times\bar{\Omega} \right) $ from $ (\ref{relation_aghjk_1}) $ and $ (\ref{relation_aghjk_2}) $ we obtain that
\begin{gather}
\hat{u}_{i_o}(y,x)=\max\{w^*_{i_o}(y,x),\tilde{w}_{i_o}(y,x)\}<\hat{u}_{i_o}(\tilde{y},\tilde{x})+\epsilon
\end{gather}
\item Let $ (\tilde{y},\tilde{x})\in K:=\left( [0,L]\times\bar{\Omega}\right) \setminus Q_R, $ by the definition of $ \hat{u}_{i_o} $ we obtain that $ \hat{u}_{i_o}(\tilde{y},\tilde{x})=w^*_{i_o}(\tilde{y},\tilde{x}). $ Because $ w^*_{i_o} $ is upper semicontinuous at point $ (\tilde{y},\tilde{x}), $ for a random $ \epsilon>0, $ there exists $\delta>0$ such that $ \forall (y,x)\in B_{\rho_2}\left( (\tilde{y},\tilde{x}),\delta \right)\cap\left( [0,L]\times\bar{\Omega}\right) \subset K, $
\begin{gather}
\hat{u}_{i_o}(y,x)=w^*_{i_o}(t,x)<w^*_{i_o}(\tilde{y},\tilde{x})+\epsilon=\hat{u}_{i_o}(\tilde{y},\tilde{x})+\epsilon\nonumber
\end{gather}
\end{itemize}
From the three cases, we conclude that $ \hat{u}_{i_o} $ is upper semicontinuous on its domain.
\end{proof}
Next, we proceed to the proof that $ \hat{u}:[0,L]\times\bar{\Omega}\to\mathbb{R} $ is a subsolution of $ (IBVP). $\\
\begin{claim}
The function $ \hat{u}:[0,L]\times\bar{\Omega}\to\mathbb{R}^m $ with $ \hat{u}=(\hat{u}_1,\hat{u}_2,\dots,\hat{u}_{i_o},\dots\hat{u}_m) $ is a subsolution of $ (IBVP) $
\end{claim}
\begin{proof}
Initially, we prove that $ \forall j\neq i_o, \hat{u}_j $ is a subsolution of $ (IBVP). $ We notice that
\begin{gather}\label{main relation_1*}
\forall (y,x)\in(0,L)\times\Omega,\ \hat{u}_j(y,x)-\mathcal{M}_j\hat{u}(y,x)\leq w^*_j(y,x)-\mathcal{M}_j w^*(y,x)
\end{gather}
Indeed, for $ (y,x)\in (0,L)\times\Omega $
\begin{equation*}
\hat{u}_j(y,x)-\mathcal{M}_j\hat{u}(y,x)\stackrel{\text{Def of}\ \hat{u}_j}{=}w^*_j(t,x)-\max\Bigl\{ \hat{u}_{\lambda}(y,x)-c_{j,\lambda}(y,x)\ :\ \lambda\in\{1,2,\dots,i_o,\dots,m\}\setminus\{j\} \Bigl\}\nonumber
\end{equation*}
But for each $ \lambda\in\{1,2,\dots,m\}\setminus\{i_o\},\ \hat{u}_{\lambda}=w^*_{\lambda}$ and $ \hat{u}_{i_o}\geq w^*_{i_o} $ on $ [0,L]\times\bar{\Omega}. $
Thus, we obtain
\begin{gather}
\max\Bigl\{ \hat{u}_{\lambda}(y,x)-c_{j,\lambda}(y,x)\ : \ \lambda\in\{1,2,\dots,i_o,\dots,m\}\setminus\{j\} \Bigl\}\nonumber\\
\geq\max\Bigl\{ w^*_{\lambda}(y,x)-c_{j,\lambda}(y,x)\ :\ \lambda\in\{1,2,\dots,i_o,\dots,m\}\setminus\{j\} \Bigl\}\nonumber
\end{gather}
Equivalently,
\begin{align}
\hat{u}_j(y,x)-\mathcal{M}_j\hat{u}(t,x)&=w^*_j (y,x)-\max\Bigl\{ \hat{u}_{\lambda}(y,x)-c_{j,\lambda}(y,x)\ :\ \lambda\in\{1,2,\dots,i_o,\dots,m\}\setminus\{j\} \Bigl\}\nonumber\\
&\leq w^*_j (y,x)-\max\Bigl\{ w^*_{\lambda}(y,x)-c_{j,\lambda}(y,x)\ :\ \lambda\in\{1,2,\dots,i_o,\dots,m\}\setminus\{j\} \Bigl\}\nonumber\\
&=w^*_j (y,x)-\mathcal{M}_j w^*(y,x)\nonumber
\end{align}
Then, for $ (y,x)\in(0,L)\times\Omega, $ and $ (a,p,X) \in \bar{J}^{2,+}\hat{u}_j(y,x)\stackrel{j\neq i_o}{=}\bar{J}^{2,+}w^*_j(y,x)$, from the definition of $ \hat{u}_j $ it follows, that
\begin{gather}\label{main relation_2*}
F\left( y,x,\hat{u}_j(y,x),p,X\right) =F\left( y,x,w^*_j(y,x),p,X\right)
\end{gather}
Thus, we obtain
\begin{align}
&\min\Big\{F\left( y,x,\hat{u}_j(y,x),p,X\right) ,\hat{u}_j(y,x)-\mathcal{M}_j\hat{u}(y,x)\Bigl\}\nonumber\\
&=^{(\ref{main relation_2*})}\min\Big\{F\left( y,x,w^*_j(y,x),p,X\right) ,\hat{u}_j(y,x)-\mathcal{M}_j\hat{u}(y,x)\Bigl\}\nonumber\\
&\leq^{(\ref{main relation_1*})} \min\Big\{F\left( y,x,w^*_j(y,x),p,X\right) ,w^*_j(y,x)-\mathcal{M}_j w^*(y,x)\Bigl\}\nonumber\\
&\leq 0\ \text{(since}\ w^*_j\ \text{is a subsolution of (IBVP)})
\end{align}
Moreover, since $ w^*_j $ is a subsolution of (IBVP), we have for $ t=0, \forall x\in\bar{\Omega},\\ \hat{u}_j(0,x)=w^*_j(0,x)\leq g_j(x)$
and $ \forall (y,x)\in (0,L) \times\partial\Omega,\ \hat{u}_j(y,x)=w^*_j(y,x)\leq f_j(y,x)$. \\
Next, we prove that $ \hat{u}_{i_o} $ is a subsolution of $ (IBVP) $.
First, we remind that due to the fact that $ (\hat{y},\hat{x}) \in (0,L)\times\Omega $, and $ (\hat{y},\hat{x}) $ is an internal point of $ Q_R, $ we are able to choose an appropriate radious $ R $ such that $ Q_R\subset(0,L)\times\Omega. $ Then by the definition of $ \hat{u}_{i_o}, $ and the fact that $ w^*_{i_o} $ is a subsolution of $ (IBVP ) $ we obtain that
\begin{itemize}
\item $\forall x\in\bar{\Omega},\ \hat{u}_{i_o}(0,x)=w^*_{i_o}(0,x)\leq g_{i_o}(x)$
\item $\forall (y,x)\in(0,L)\times\partial\Omega,\ \hat{u}_{i_o}(y,x)=w^*_{i_o}(y,x)\leq f_{i_o}(y,x)$
\end{itemize}
Now, let $ (y,x)\in(0,L)\times\Omega, $ and $ (a,p,X)\in \bar{J}^{2,+}\hat{u}_{i_o}(y,x) $. There are two main cases: \\$ (y,x)\in(0,L)\times\Omega\setminus Q_R $ or $ (y,x)\in Q_R $. In the first case, by the definition of $ \hat{u}_{i_o} $, we have that $ \hat{u}_{i_o}(y,x)=w^*_{i_o}(y,x) $ and, since $ w^*_{i_o} $ is a subsolution we have the desired inequality
\begin{gather}
\min\Bigl\{ F\left( y,x,\hat{u}_{i_o}(y,x),p,X\right),\hat{u}_{i_o}(y,x)-\mathcal{M}_{i_o}\hat{u}(y,x) \Bigl\}\leq 0\nonumber
\end{gather}
For the second case, we have
\begin{gather}
\hat{u}_{i_o}(y,x)=\max\{w^*_{i_o}(y,x),\tilde{w}_{i_o}(y,x)\}\nonumber
\end{gather}
In this situation, there are two possible subcases: $ \hat{u}_{i_o}(y,x)=w^*_{i_o}(y,x) $ or $ \hat{u}_{i_o}(y,x)=\tilde{w}_{i_o}(y,x) $. We notice that
\begin{align}
\mathcal{M}_{i_o}\hat{u}(y,x)=\max_{\lambda\neq i_o}\{\hat{u}_{\lambda}(y,x)-c_{i_o,\lambda}(y,x)\}&=\max_{\lambda\neq i_o}\{w^*_{\lambda}(y,x)-c_{i_o,\lambda}(y,x)\}\nonumber\\
&=\mathcal{M}_{i_o}w^*(y,x) \nonumber
\end{align}
Also, from $ (\ref{claim_subsolution property}) $, we receive that
\begin{gather}
\min\{F(y,x,\tilde{w}_{i_o}(y,x),D_x \tilde{w}_{i_o}(y,x),D^2_{xx} \tilde{w}_{i_o}(y,x)), \tilde{w}_{i_o}(y,x)-\mathcal{M}_{i_o} w^*(y,x)\}\leq 0\nonumber
\end{gather}
Using the above we obtain that
\begin{gather}
\min\{F(y,x,\hat{u}_{i_o}(y,x),p,X),\hat{u}_{i_o}(y,x)-\mathcal{M}_{i_o} \hat{u}(y,x)\}\leq 0\nonumber
\end{gather}
\end{proof}
In the next part, we reach to a contradiction. In specific, we prove that function $ \hat{u}_{i_o} $ is strictly greater than $ w_{i_o}, $ which is a contradiction by the definition of $ w_{i_o}. $ Previously, we saw that $ w^*_{i_o}=w_{i_o}. $ So it follows that $ (w*_{i_o})_*=w_{*,i_o} $. From the definition of $ (w*_{i_o})_* $ there exists a sequence
\begin{gather}\label{approaching sequence}
\left( y_n,x_n,w^*_{i_o}(y_n,x_n)\right) \xrightarrow{n\to\infty}\left( \hat{y},\hat{x},(w^*_{i_o})_{*}(\hat{y},\hat{x})\right)=\left( \hat{y},\hat{x},(w_{*,i_o})(\hat{y},\hat{x})\right)
\end{gather}
Then, it holds that, for each $ n\in\mathbb{N}, $
\begin{align}
\tilde{w}_{i_o}(y_n,x_n)&=w_{*,i_o}(\hat{y},\hat{x})+h(y,x,\delta^*)\beta(\delta^*)+\alpha_o (y_n-\hat{y})+\left\langle p_o,x_n-\hat{x}\right\rangle+\nonumber\\
&+\frac{1}{2}\left\langle X_o(x_n-\hat{x}),(x_n-\hat{x}) \right\rangle
\end{align}
Using now (\ref{negativity_rel}), and the fact that $ h $ is a continuous function, we obtain
\begin{gather}
\tilde{w}_{i_o}(y_n,x_n)\xrightarrow{n\to\infty}w_{*,i_o}(\hat{y},\hat{x})+h(\hat{y},\hat{x},\delta^*)\beta(\delta^*)>^{(\ref{negativity_rel})} w_{*,i_o}(\hat{y},\hat{x})=^{(\ref{approaching sequence})}\lim_{n\to\infty}w^*_{i_o}(y_n,x_n)
\end{gather}
Finally, we receive that,
\begin{gather}
\lim_{n\to\infty}\tilde{w}_{i_o}(y_n,x_n)>\lim_{n\to\infty}w^*_{i_o}(y_n,x_n)
\end{gather}
From the last, we receive that
\begin{gather}
\exists n_o\in\mathbb{N},\forall n\geq n_o, \tilde{w}_{i_o}(y_n,x_n)>w^*_{i_o}(y_n,x_n)
\end{gather}
Due to the fact that, $ (\hat{t},\hat{x}) $ is an internal point of $ Q_R $, there exists $ \tilde{n}\in\mathbb{N} $ such that\\ $ (y_n,x_n)\in Q_R $. We set $ n_o^*:=\max\{n_o,\tilde{n}\}, $ we observe by the definition of $ \hat{u}_{i_o} $, that $ \forall n\geq n_o^*, $
\begin{gather}
\hat{u}_{i_o}(y_n,x_n) :=\max\{w^*_{i_o}(y_n,x_n),\tilde{w}_{i_o}(y_n,x_n)\}=\tilde{w}_{i_o}(y_n,x_n)>w^*_{i_o}(y_n,x_n)=w_{i_o}(y_n,x_n)
\end{gather}
At this point, we have a contradiction from the definition of $ w_{i_o} $ and from the fact that $ \hat{u}_{i_o} $ is a subsolution of $ (IBVP). $\\
Next, we consider the case where $ w_{*,i_o} $ fails to be supersolution of the $ (IBVP) $ violating the boundary condition. I.e
\item $ \exists(\hat{y},\hat{x}) \in(0,L)\times\partial{\Omega}$ such that
\begin{gather}
w_{*,i_o}(\hat{y},\hat{x})<f_{i_o}(\hat{y},\hat{x})\nonumber
\end{gather}
From the definition of $ \tilde{w}_{i_o} $, we obtain that
$ \tilde{w}_{i_o}(\hat{y},\hat{x}) =w_{*,i_o}(\hat{y},\hat{x})+\beta(\delta)h(\hat{y},\hat{x},\delta)$. For the given functions $ \beta $ and $ h $, we select a sufficently small parameter $ \delta>0 $ such that
\begin{gather}\label{boundary condition_perron}
\tilde{w}_{i_o}(\hat{y},\hat{x})=w_{*,i_o}(\hat{y},\hat{x})+\beta(\delta)h(\hat{y},\hat{x},\delta)<f_{i_o}(\hat{y},\hat{x})
\end{gather}
Then, from continuity of $ \tilde{w}_{i_o} $ and $ f_{i_o} $ at point $ (\hat{y},\hat{x}) $ we receive from $ (\ref{boundary condition_perron}) $, that there exists an appropriate $ r>0 $ such that $ \forall (y,x)\in B_{\rho_2}\left((\hat{y},\hat{x}) ,r\right)\cap\left( (0,L)\times\partial\Omega\right) ,\ \tilde{w}_{i_o}(y,x)<f_{i_o}(y,x). $ As in the previous analysis, we select appropriate functions $ \beta\ \text{and} h $ and radious $ R, $ such that $ \tilde{w}_{i_o} $ satisfies the subsolution property of claim $ (\ref{claim_subsolution property}) $ for $ (y,x)\in\Omega_{L} $ sufficently close to $ (\hat{y},\hat{x}). $ Then, we deduce that $ \hat{u}_{i_o} $ is a subsolution of the problem $ (IBVP) $ which dominates $ w^*_{i_o}=w_{i_o}, $ which again is a contradiction relative to the definition of $ w_{i_o}. $
\end{itemize}
\end{proof}
\begin{claim}\label{wi_continuous}
The function $ w_i:[0,L]\times\bar{\Omega}\to\mathbb{R}$ is continuous on each point of $ [0,L)\times\bar{\Omega} $.
\end{claim}
\begin{proof}
From the claim $ \ref{viscosity_property} $, we have the result that $ w_{*,i}(y,x)=w_i(y,x)=w^*_i(y,x),\ \\ \forall(y,x)\in[0,L)\times\bar{\Omega} $. So the restriction of $ w_i $ on the set $ [0,L)\times\bar{\Omega} $, i.e $ w_i |_{[0,L)\times\bar{\Omega}} $ is continuous. Then, from the fact that for each point $ (y,x)\in[0,L)\times\bar{\Omega} $ there exists $ \delta>0 $ such that
$ B_{\rho_2}\left( (y,x),\delta\right) \cap \left([0,L)\times\bar{\Omega} \right) = B_{\rho_2}\left( (y,x),\delta\right) \cap \left([0,L]\times\bar{\Omega} \right) $, combined with the continuity of the restriction $ w_i|_{[0,L)\times\bar{\Omega}} $ on the point $ (y,x), $ we also receive
\footnote{Let $ (X,\rho) $ be a metric space, $ A\subset B\subset X,\ x_o\in A $ and $ f:B\to\mathbb{R}. $ If the following holds\\ \begin{gather}
\exists\delta>0,\ A\cap B_{\rho}(x_o,\delta)=B\cap B_{\rho}(x_o,\delta) \nonumber
\end{gather} and the restriction $ f|_A $ is continuous on $ x_o $, then the extension $ f $ also is continuous on $ x_o. $}
the continuity of $ w_i $ at $ (y,x). $
From the above, it is extracted that $ w_i:[0,L]\times\bar{\Omega}\to\mathbb{R} $ is a continuous function on each point of $ [0,L) \times\bar{\Omega}.$
\end{proof}
From the last two claims, it is extracted that the function $ w_i:[0,L]\times\bar{\Omega}\to\mathbb{R} $ satisfies the definition of viscosity solution of $ (IBVP). $
\end{proof}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,338 |
The East-West Link is the Victorian governments headline project: a toll road tunnel that connects Melbourne's Eastern suburbs to the Port of Melbourne. At a projected cost of $14 billion it is one of the largest road infrastructure projects in the world.
The project has been controversial since it was first proposed under a Labor government in 2007 and public support for the project continues to decline. The latest Age/Neilson poll puts support for the East West road tunnel at just 24%! More popular transport priorities are the Metro Rail project and the removal of level crossings. This demonstrates that Victorians want investment in public transport prioritised over the building of more roads.
The current Napthine Liberal/National coalition government is unpopular in the polls. This is no surprise considering the Liberals promised a rail line to the Melbourne Airport, as well as a rail line to the eastern suburbs of Doncaster, would be prioritised during this term. Both projects will fail to progress as the East West Link swallows the state's transport funding.
With an election looming, the struggling government needs to be seen to be addressing the jobs crisis created by the collapse of the manufacturing sector. However, Napthine's headline infrastructure project – The East West Link – will do almost nothing to address unemployment. The jobs that would be created by the project would be short-term construction jobs, not the kind of permanent skilled jobs being lost across the state.
The solution is simple. Scrapping the monumental waste that is the East West Link and investing instead in public transport would not only help soothe Melbourne's traffic congestion woes but could also revitalise the state's manufacturing industry.
A clear transport plan to provide all of Melbourne's suburbs – as well as regional Victoria – with integrated, efficient 24-hour public transport would see tens of thousands of jobs created in Victoria's flailing manufacturing sector. As the car manufacturers abandon Australia, their discarded factories and workforces should be retooled and retrained to manufacture trains, trams and buses.
A publicly owned and controlled revival of the manufacturing sector could not only ensure Victoria has the public transport infrastructure it needs for decades to come, but offer the only a real solution to the jobs crisis.
However, this obvious solution is directly counter to the interests of the road and oil lobby, and the neo-liberal ideology of those who do their bidding in government. Both major parties have wasted billions investing in roads, at the expense of public transport, because of their slavish commitment to big business and market capitalism. The fact that both the Liberal and the ALP are willing to waste public money on a road project that will cost taxpayers more than it will return in benefits highlights the lunacy of private interests dictating public policy.
In this context, the significance of the community campaign against the East West road tunnel (stage one of the East West Link) is huge. We have pointed to the profit motives behind major road projects and brought the question of public investment into public transport to the forefront of political discussion. Pressure is mounting on the ALP to commit to ripping up the contracts if they are signed before the election.
We have a real chance of derailing the destructive East West road tunnel.
The long term challenge is that none of the major parties want to take on a program of serious public investment into the manufacture and construction of public transport. Only the mass mobilisation of people will force change. We need to build a new force in politics, a new workers party that unashamedly puts the needs of ordinary people before those of big business. Ultimately only a socialist program of public investment and democratic planning can wash away the wasteful fouls of capitalism and create a sustainable world where decisions are made of the basis of human need, not private profit.
Earlier this week the Roads Minister Terry Mulder said that test drilling for the proposed East-West Link has ended. This is despite the Linking Melbourne Authority (LMA) previously planning to drill at least nine more bore holes across inner city Melbourne.
This climb down is a clear win for our campaign!
While Mulder pretends that the job has been completed on time, the truth is that the government had directed Yarra City Council to issue permits for at least nine more holes in local streets. At the same time the Melbourne City Council had issued a permit to drill three holes in a park adjacent to Princes Street in Carlton. None of these holes have been completed.
At a community meeting a last month Victoria Police Assistant Commissioner Andrew Crisp confirmed this by saying that the LMA had informed him that they would be "drilling into April" in order to complete the remaining holes.
This is not the first time that Mulder and the LMA have lied and said that the test drilling was finished. The LMA originally claimed that the test drilling was completed in November 2013. They then returned in the dead of night to drill four holes previously hampered by our community pickets in December 2013.
Subsequently they announced that about 16 more holes were required and that they would be completed by late January. Now, in early March, only a fraction of those holes have been drilled with the government now signalling that they have given up.
Clearly our pickets, occupations of drill sites, and lock ons have hampered the works. It seems the decision to halt the drilling came on the back of two disastrous polls for the government. One estimated that a mere 24% of people saw the East-West Link as a priority while another (in the pro-tunnel Herald Sun!) estimated that only 15% of people support the East-West Link.
It seems the government has decided to halt the works as they are beginning to realise that they have very little support for this project. The vast bulk of people support the demands of the picketers for more public transport.
The risk they ran if they were to continue the drilling is that our pickets and pro public transport alternatives would be in the spotlight leading to them bleeding even more support for the project. Clearly they have decided to cut their losses.
The most important lesson from this back down is that collective direct action works. If we stick together and build this campaign even bigger we can stop not just the preliminary works but the entire disastrous project!
Stay tuned for information about the next steps of our campaign. Like the No East West Tunnel – Take the pledge Facebook page, text 'tunnel' to 0432447036 for regular sms updates and follow the #TunnelPicket hash tag on social media.
Community radio station 3CR also has an archive of stories from the pickets which you can listen to here. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,752 |
\section*{The protocol}
\paragraph*{Protocol.} The protocol \cite{Mista_09} depicted in Fig.~\ref{fig1} consists of three steps.
Initially, a distributor David prepares modes $A$ and $C$ in momentum squeezed and position squeezed vacuum states,
respectively, with quadratures
$\hat x_{A,C}=e^{\pm r}\hat x_{A,C}^{(0)}$ and $\hat p_{A,C}=e^{\mp
r}\hat p_{A,C}^{(0)}$, whereas mode $B$ is in a vacuum state with
quadratures $\hat x_{B}=\hat x_{B}^{(0)}$ and $\hat p_{B}=\hat p_{B}^{(0)}$. Here, $r$ is the
squeezing parameter and the superscript $(0)$ denotes the
vacuum quadratures. David then exposes all the modes to suitably
tailored local correlated displacements \cite{Mista_12}:
\begin{eqnarray}\label{displacements}
\hat p_{A}&\rightarrow& \hat p_{A}- p,\quad \hat x_{C}\rightarrow
\hat x_{C}+ x,\nonumber\\
\hat x_{B}&\rightarrow& \hat x_{B}+\sqrt{2}x,\quad
\hat p_{B}\rightarrow \hat p_{B}+\sqrt{2} p.
\end{eqnarray}
The uncorrelated classical displacements $x$ and $p$ obey a
zero mean Gaussian distribution with the same variance
$(e^{2r}-1)/2$. The state has been prepared by LOCC across $A|B|C$
splitting and hence is fully separable.
In the second step, David passes modes $A$ and $C$ of the resource state to Alice and mode $B$ to Bob. Alice superimposes modes $A$ and $C$ on a balanced beam splitter BS$_{AC}$, whose output modes are denoted by $A'$ and $C'$. The beam splitter BS$_{AC}$ cannot create entanglement with mode $B$. Hence the state is separable with respect to $B|A'C'$ splitting. Moreover, the state also fulfils the positive partial transpose (PPT) criterion~\cite{Peres96, Horodecki96} with respect to mode $C'$ and
hence is also separable across $C'|A'B$ splitting
\cite{Werner_01}, as required (see Appendix).
In the final step, Alice sends mode $C'$ to Bob who superimposes it with his mode $B$ on another balanced beam splitter BS$_{BC}$. The presence of the entanglement between modes $A'$ and $B'$ is confirmed by the sufficient condition for entanglement \cite{Giovannetti_03,Dong_07}
\begin{eqnarray}\label{Duan}
\Delta_{\rm norm}^{2}(g\hat x_{A'}+\hat x_{B'})\Delta_{\rm norm}^{2}(g\hat p_{A'}-\hat p_{B'})<1,
\end{eqnarray}
where $g$ is a variable gain factor. Minimizing the left-hand side of~Ineq.~(\ref{Duan}) with respect to $g$, we get fulfilment of the criterion for any $r>0$, which confirms successful entanglement distribution.
\paragraph*{Experiment.} The experimental realization is divided into three steps: state preparation, measurement,
and data processing. The corresponding setup is depicted in Fig.~\ref{Fig:SetupPaper}. From now on,
we will work with polarization variables described by Stokes observables (see, e.g., Refs.~\cite{Korolkova_02,Heersink_05}) instead of quadratures. We choose the state of polarization such that mean values of $\hat{S}_1$ and $\hat{S}_2$ equal zero while $\langle\hat{S}_3\rangle\gg0$. This configuration allows us to identify the ``dark'' $\hat S_1$-$\hat S_2$ plane with the quadrature phase space. $\hat{S}_{\theta}, \hat S_{\theta + \pi/2}$ in this plane correspond to $\hat S_1, \hat S_2$ renormalized with respect to $\hat {S}_3 \approx S_3$ and can be associated with the effective quadratures $\hat x,\hat p$.
We use the modified version of the protocol indicated in Fig.~\ref{fig1} by the dashed arrow showing the alternative position of displacement in mode $B$: The random displacement applied by David can be performed after the beam splitter interaction of $B$ and $C'$, even {\it a posteriori} after the measurement of mode $B'$. This is technically more convenient and emphasizes that the classical information is sufficient for the entanglement recovery after the interaction of mode $A$ with mode $C$ and mode $B$ with mode $C'$.
David prepares two identically polarization squeezed modes~\cite{Heersink_05, Leuchs99, Silberhorn01, Dong_07} and adds noise in the form of random displacements to the squeezed observables. The technical details on the generation of these modes can be found in the Appendix. The modulation patterns applied to modes $A$ and $C$ to implement the random displacements are realized using electro-optical modulators (EOMs) and are chosen such that the two-mode state $\hat \rho_{A'C'}$ is separable. By applying a sinusoidal voltage $V_{\text{mod}}$, the birefringence of the EOMs changes at a frequency of 18.2\,MHz. In this way, the state is modulated along the direction of its squeezed observable.
Two such identically prepared modes $A$ and $C$ are interfered on a balanced beam splitter (BS$_{AC}$) with a fixed relative phase of $\pi/2$ by controlling the optical path length of one mode with a piezoelectric transducer and a locking loop. This results in equal intensities of both output modes. In the final step, Bob mixes the ancilla mode $C'$ with a vacuum mode $B$ on another balanced beam splitter and performs a measurement on the transmitted mode $B'$.
\begin{figure}[ht]
\centering
\includegraphics[width=\linewidth]{setup_elec}
\caption[Sketch of the experimental setup.]
{Sketch of the experimental setup.} Used abbreviations: HWP, half-wave plate; QWP, quarter-wave plate; EOM, electro-optical modulator; BS, beam splitter; WS, Wollaston prism; and AD, analog-to-digital. State preparation: The polarization of two polarization squeezed states ($A$ and $C$) is modulated using EOMs and sinusoidal voltages from a function generator (dotted lines). The HWPs before the EOMs are used to adjust the direction of modulation to the squeezed Stokes variable, whereas the QWPs compensate for the stationary birefringence of the EOMs. Such prepared modes interfere with a relative phase of $\pi/2$ on a balanced beam splitter BS$_{AC}$. In the last step of the protocol, the mode $C'$ interferes with the vacuum mode $B$ on a second balanced beam splitter BS$_{BC}$. Measurement process: A rotatable HWP, followed by a WS and a pair of detectors, from which the difference signal is taken, allows us to measure all possible Stokes observables in the $\hat S_1$-$\hat S_2$ plane. To determine the two-mode covariance matrix $\gamma_{A'B'}$, all necessary combinations of Stokes observables are measured. Removing the second beam splitter of the state preparation allows us to measure the covariance matrix of the two-mode state $\hat\rho_{A'C'}$. Data acquisition To achieve displacements of the modes in the $\hat S_1$-$\hat S_2$ plane we electronically mix the Stokes signals with a phase matched electrical local oscillator and sample them by an analog-to-digital converter.}
\label{Fig:SetupPaper}
\end{figure}
The states involved are Gaussian quantum states and, hence,
are completely characterized by their first moments and the covariance matrix $\gamma$ comprising all second moments (see Appendix).
To study the correlations between modes $A'$ and $C'$ after BS$_{AC}$, multiple pairs of Stokes observables ($\hat S_{A',\theta},\hat S_{C',\theta}$) are measured.
The covariance matrix $\gamma_{A'C'}$ is obtained by measuring five pairs of observables: \((\hat S_{A',0^{\circ}},\hat S_{C',0^{\circ}})\), \((\hat S_{A',90^{\circ}},\hat S_{C',0^{\circ}})\), \((\hat S_{A',0^{\circ}},\hat S_{C',90^{\circ}})\), \((\hat S_{A',90^{\circ}},\hat S_{C',90^{\circ}})\), and $(\hat S_{A',45^{\circ}},\hat S_{C',45^{\circ}})$, which determine all of its 10 independent elements.
Here, $\theta$ is the angle in the $\hat S_1$-$\hat S_2$ plane between $\hat S_{0^{\circ}}$ and $\hat S_{\theta}$.
For the measurements of the different Stokes observables, we use two Stokes measurement setups, each comprising a rotatable half-wave plate, a Wollaston prism, and two balanced detectors. The difference signal of one pair of detectors gives one Stokes observable $\hat S_\theta$ in the $\hat S_1$-$\hat S_2$ plane, depending on the orientation of the half-wave plate. The signals are electrically down-mixed using an electric local oscillator at 18.2\,MHz, which is in phase with the modulation used in the state preparation step. With this detection scheme, the modulation translates to a displacement of the states in the $\hat S_1$-$\hat S_2$ plane. The difference signal is low pass filtered (1.9\,MHz), amplified, and then digitized using an analog-to-digital converter card (GaGe Compuscope 1610) at a sampling rate of $10^6$ samples/s. After the measurement process, we digitally low pass filter the data by an average filter with a window of 10 samples.
Because of the ergodicity of the problem, we are able to create a Gaussian mixed state computationally
from the data acquired as described above.
By applying $80$ different modulation depths $V_\text{mod}$ to each of the EOMs we acquire a set of $6400$ different modes. From this set of modes, we take various amounts of samples, weighted by a two-dimensional Gaussian distribution.
The covariance matrix $\gamma_{A'C'}$ for the two-mode state after BS$_{AC}$ has been measured to be
\begin{align}
\mbox{$
\gamma_{A'C'}= \begin{pmatrix}
20.90 & 1.102 & -7.796 & -1.679\\
1.102 & 25.30 & 1.000 & 14.63\\
-7.796 & 1.000 & 20.68 & 0.8010\\
-1.679 & 14.63 & 0.8010 & 24.65
\end{pmatrix}.$}
\end{align}
The estimation of the statistical errors of this covariance matrix $\gamma_{A'C'}$ can be found in the Appendix. A necessary and sufficient condition for the separability of a Gaussian state $\hat \rho_{XY}$ of two modes $X$ and $Y$ with the covariance matrix $\gamma_{XY}$ is given by the PPT criterion
\begin{eqnarray}\label{PPTcondition}
\gamma_{XY}^{(T_{Y})}+\imath\Omega_{2} &\geq 0, \quad \Omega_{2}&=\bigoplus_{i=1}^2\left(\begin{array}{cc}
0 & 1 \\
-1 & 0 \\
\end{array}\right)
\end{eqnarray}
where $\gamma_{XY}^{(T_Y)}$ is the matrix corresponding to the partial transpose of the state $\hat \rho_{XY}$ with respect to the mode $Y$ (see Appendix). Effects that could possibly lead to some non-Gaussianity of the utilized states are also discussed in detail in the Appendix.
The state described by $\gamma_{A'C'}$ fulfils the condition~(\ref{PPTcondition}) as the eigenvalues (39.84, 28.47, 13.85, and 9.371) of $(\gamma_{A'C'}^{(T_{C'})}+\imath\Omega_{2})$ are positive; hence, mode $C'$ remains separable after BS$_{AC}$.
The measured two-mode covariance matrix of the output state $\gamma_{A'B'}$ is given by
\begin{align}
\mbox{$
\gamma_{A'B'}= \begin{pmatrix}
19.95& 1.025& -4.758& -1.063\\
1.025& 22.92& 0.9699& 9.153\\
-4.758& 0.9699& 9.925& 0.2881\\
-1.063& 9.153& 0.2881& 11.65
\end{pmatrix}.$}
\end{align}
The statistical error of this measured covariance matrix is given in the Appendix.
The separability is proven by the PPT criterion (eigenvalues 28.24, 21.79, 8.646, and 5.756).
The postprocessing for the recovery of the entanglement is performed on the measured raw data of mode $B'$. Therefore, the displacement of the individual modes caused by the two modulators is calibrated. By means of this calibration, suitable displacements are applied digitally. The classical noise inherent in mode $B'$ is completely removed. A part of the classical noise associated with $\hat S_{A',0^{\circ}}$ is subtracted from $\hat S_{B',0^{\circ}}$, while the same fraction of the noise in $\hat S_{A',90^{\circ}}$ is added to $\hat S_{B',90^{\circ}}$. In this way, the noise partially cancels out in the calculation of the separability criterion~(\ref{Duan}) and allows us to reveal the entanglement.
We chose the fraction as in~Eq.~(\ref{displacements}), which is compatible with the separability of the transmitted mode $C'$ from the subsystem $(A'B)$ in the scenario with modulation on mode $B$ before the beam splitter BS$_{BC}$.
Only as Bob receives the classical information about the modulation on the initial modes $A$ and $C$ from David is he able to recover the entanglement between $A'$ and $B'$. Bob verifies that the product entanglement criterion~(\ref{Duan}) is fulfilled, as illustrated in Fig.~\ref{Fig:Duan}.
That proves the emergence of entanglement.
The used gain factor $g$ considers the slightly different detector
response and the intentional loss of 50\,\% at Bob's beam
splitter. The clearest confirmation of entanglement $0.6922\pm0.0002<1$ is shown for $g_\text{opt} = 0.4235\pm0.0005$ (Fig.~\ref{Fig:Duan}). This is the only step of the protocol, where entanglement emerges, thus demonstrating the remarkable possibility to entangle remote parties Alice and Bob solely by sending a separable auxiliary mode $C'$.
\begin{figure}
\includegraphics[width=0.9\linewidth]{Duan_product_combined.pdf}
\caption{Entanglement distributed between modes $A'$ and $B'$. The experimental values for the
criterion (\ref{Duan}) are depicted in dependence of the gain factor $g$. Because of the attenuation of mode
$B$ by 50\,\%, a gain factor of about 0.5 yields a value smaller
than 1, i.e., below the limit for entanglement (solid red line). The inset zooms into the interesting
section around the minimum. The depicted estimated errors are so small because of the large amount of data taken.}
\label{Fig:Duan}
\end{figure}
\paragraph*{Discussion}
The performance of the protocol can be explained using the structure of
the displacements (\ref{displacements}). Entanglement distribution without sending entanglement highlights vividly the important role played by classical information in quantum information protocols. Classical information lies in our knowledge about all the correlated displacement involved. This allows the communicating parties (or David on their behalf) to adjust the displacements locally to recover through clever noise addition quantum resources initially present in the input quantum squeezed states. Mode $C'$
transmitted from Alice to Bob carries on top of the sub-shot-noise quadrature of the input squeezed state the displacement noise which
is anticorrelated with the displacement noise of Bob's mode. Therefore,
when the modes are interfered on Bob's beam splitter,
this noise partially cancels out in the output mode $B'$ when
the light quadratures of both modes add. Moreover, the residual noise in Bob's
position (momentum) quadrature is correlated (anticorrelated) with
the displacement noise in Alice's position (momentum) quadrature in mode $A'$, again initially squeezed.
Because of this, the product of variances in criterion (\ref{Duan}) drops below the value for separable states, and thus entanglement between Alice and Bob's modes emerges. The difference between the theoretically proposed protocol~\cite{Mista_09} and the experimental demonstration reported in this Letter lies merely in the way classical information is used. In the original protocol, the classical information is retained by David and he is responsible for clever tailoring of correlated noise. Bob evokes the required noise cancellation by carrying out the final part of the global operation via superimposing his mode with the ancilla on BS$_{BC}$. In the experimentally implemented protocol, David shares part of his information with Bob, giving Bob a possibility to get entanglement {\it a posteriori}, by using his part of the classical information after the quantum operation is carried out. Thus entanglement distribution in our case is truly performed via a dual classical and quantum channel, via classical information exchange in combination with the transmission of separable quantum states.
There are other interesting aspects to this protocol, which may open new, promising avenues for research. Noise introduced into the initial states by displacements contains specific classical correlations. On a more fundamental level, these displacements can be seen as correlated dissipation
(including mode $C$ into the ``environment''). It is already known that dissipation to a common reservoir can even lead to the creation of entanglement~\cite{dissipation, dissipation2}.
Our scheme can be viewed as another manifestation of a positive role dissipation may play in quantum protocols.
The presence of correlated noise results in nonzero Gaussian discord at all stages of the protocol, a more general form of quantum correlations, which are beyond entanglement~\cite{Adesso_10}. The role of discord in entanglement distribution has recently been discussed theoretically~\cite{Streltsov_12, Chuan_12}. The requirements devised there are reflected in the particular separability properties of our global state after the interaction of modes $A$ and $C$ on Alice's beam splitter. The state $\hat \rho_{A' B C'}$ contains discord and entanglement across $A' \vert BC'$ splitting and is separable and discordant across $C' \vert A' B$ splitting as required by the protocol. Our work thus illustrates an interplay of entanglement and other quantum correlations, such as correlations described by discord, across different partitions of a multipartite quantum system.
L. M. acknowledges Project No. P205/12/0694 of GA\v{C}R. N. K. is grateful for the support provided by the A. von Humboldt Foundation. The project was supported by the BMBF Grant ``QuORep'' and by the FP7 Project QESSENCE. We thank Christoffer Wittmann and Christian Gabriel for fruitful discussions. C. P. and V. C. contributed equally to this work.\\
\paragraph*{Note added.} Recently, an experiment has been presented in Ref.~\cite{Vollmer13}, which is based on a similar protocol. The main difference consists in the fact that it starts with entanglement which is hidden and recovered with thermal states. For this implementation no knowledge about classical information has to be communicated to Bob, besides the used thermal state. By contrast the setup presented in this work exhibits entanglement only at the last step of the protocol. Thus both works give good insights on different aspects of the theoretically proposed protocol~\cite{Mista_09}. Another independent demonstration of a similar protocol based on discrete variables was recently presented in Ref~\cite{Fedrizzi13}.\\
\begin{appendix}
\section{Preparation of polarization squeezed states}
To prepare two identically, polarization squeezed modes we use a well known technique like in~\cite{Heersink_05, Leuchs99, Silberhorn01, Dong_07}. Each of these modes is generated by launching two orthogonally polarized femtosecond pulses ($\sim$200\,fs) with balanced powers onto the two birefringent axes of a polarization maintaining fiber (FS-PM-7811, Thorlabs, 13\,m). The pump source is a soliton-laser emitting light at a center wavelength of 1559\,nm and a repetition rate of 80\,MHz. By exploiting the optical Kerr effect of the fibers, the orthogonally polarized pulses are individually quadrature squeezed and subsequently temporally overlapped with a relative phase of $\pi/2$, resulting in a circular polarized light beam. The relative phase is actively controlled using an interferometric birefringence compensator including a piezoelectric transducer and a locking loop based on a 0.1\,\% tap-off signal after the fiber. In terms of Stokes observables (see~\cite{Heersink_05, Korolkova_02}) this results in states with zero mean values of $\hat S_1$ and $\hat S_2$, but a bright $\langle \hat S_3\rangle \gg 0$ component. These states exhibit polarization squeezing at a particular angle in the $\hat S_1$-$\hat S_2$-plane.
\section{Gaussian states}
We implement the entanglement distribution protocol using optical
modes which are systems in infinitely-dimensional Hilbert state
space. An $N$-mode system can be conveniently characterized by the
quadrature operators $\hat x_{j},\hat p_{k}$, $j,k=1,2,\ldots,N$ satisfying
the canonical commutation rules $[\hat x_j,\hat p_k]=i\delta_{jk}$ which can
be expressed in the compact form as
\begin{equation}\label{commutators}
[\hat \xi_j,\hat \xi_k]=\imath{\Omega_{N}}_{jk}.
\end{equation}
Here we have introduced the vector of quadratures
$\hat \xi=(\hat x_{1},\hat p_{1},\ldots,\hat x_{N},\hat p_{N})$ and
\begin{eqnarray}\label{Omega}
\Omega_{N}=\bigoplus_{i=1}^{N} J,\quad J=\left(\begin{array}{cc}
0 & 1 \\
-1 & 0\\
\end{array}\right),
\end{eqnarray}
is the symplectic matrix.
The present protocol relies on Gaussian quantum states. As any standard Gaussian distribution, a Gaussian
state $\hat \rho$ is fully characterized by the vector of its first moments
\begin{equation}\label{d}
d=\mbox{Tr}\left(\hat \rho\hat \xi\right),
\end{equation}
and by the covariance matrix $\gamma$ with elements
\begin{eqnarray}
\gamma_{jk}=\mbox{Tr}[\hat \rho\{\hat \xi_{j}-d_{j}\openone,\hat \xi_{k}-d_{k}\openone\}],
\end{eqnarray}
where $\{\hat{A},\hat{B}\}=\hat{A}\hat{B}+\hat{B}\hat{A}$ is the
anticommutator. A real symmetric positive-definite $2N\times2N$ matrix
$\gamma$ describes a covariance matrix of a physical quantum state if and
only if it satisfies the condition \cite{Simon_94}:
\begin{eqnarray}\label{Heisenberg}
\gamma+\imath\Omega_N\geq0.
\end{eqnarray}
The separability of Gaussian states can be tested using the positive
partial transpose (PPT) criterion. A single mode $j$ is separable
from the remaining $N-1$ modes if and only if the Gaussian state
$\hat \rho$ has a positive partial transposition $\hat \rho^{T_{j}}$ with
respect to the mode $j$ \cite{Simon_00,Werner_01}. On the level of the covariance matrices, the partial
transposition is represented
by a matrix $\Lambda_{j}=\left(\bigoplus_{i\ne
j=1}^{N-1}\openone^{(i)}\right)\oplus\sigma_{z}^{(j)}$, where
$\sigma_{z}^{(j)}=\mbox{diag}(1,-1)$ is the diagonal Pauli
$z$-matrix of mode $j$ and $\openone^{(i)}$ is the $2\times2$ identity matrix. The matrix $\gamma^{(T_{j})}$
corresponding to a partially transposed state $\hat \rho^{T_{j}}$
reads $\gamma^{(T_{j})}=\Lambda_{j}\gamma\Lambda_{j}^{T}$. In
terms of the covariance matrix, one can then express the PPT criterion
in the following form. A mode $j$ is separable from the remaining $N-1$ modes if and
only if \cite{Simon_00,Werner_01}
\begin{eqnarray}\label{HeisenbergTj}
\gamma^{(T_j)}+\imath\Omega_N\geq0.
\end{eqnarray}
The PPT criterion (\ref{HeisenbergTj}) is a sufficient condition
for separability only under the assumption of Gaussianity. In our experiment, however, non-Gaussian states
can be generated for which this criterion represents only a necessary condition for separability. Therefore it
can fail in detecting entanglement.
\section{Analysis of non-Gaussianity}
There are two sources of imperfections in our experimental set up that are
potential sources of non-Gaussianity. These are phase fluctuations and the modulation of the initial squeezed states before the first beam splitter. They are discussed in the following sections.
\subsection{Phase fluctuations}
The experiment includes an interference of the modes $A$ and $C$ on a beam splitter, which is the
first beam splitter $BS_{AC}$ in the protocol. Imperfect phase locking at this beam
splitter might cause a phase drift resulting in a non-Gaussian
character of the state $\hat \rho_{A'C'}$ after the beam splitter. The
phase fluctuations can be modelled by a random phase shift of mode
$A$ before the beam splitter described by a Gaussian
distribution $P(\phi)$ with zero mean and variance $\sigma^{2}$.
Denoting the operator corresponding to a beam splitter transformation as $\hat{\mathcal{U}}$ and the
phase shift $\phi$ on mode $A$ as $\hat V_{A}(\phi)$, the state $\hat \rho_{A'C'}$ can be linked to
the state $\hat \rho_{AC}$ before the onset of phase fluctuations as
\begin{equation}\label{rhoACprimed}
\hat \rho_{A'C'}=\int_{-\infty}^{\infty}P(\phi)\hat{\mathcal{U}}\hat V_{A}(\phi)\hat \rho_{AC}\hat V_{A}^{\dag}(\phi)\hat{\mathcal{U}}^{\dag}d\phi.
\end{equation}
Hence we can express the measured covariance matrix
$\gamma_{A'C'}$ given in Eq.~(3) of the main letter, and the vector of the first moments $d'$ of the state
$\hat \rho_{A'C'}$ in terms of the covariance matrix $\gamma_{AC}$ and
the vector of the first moments $d$ of the input state $\hat \rho_{AC}$.
For this it is convenient to define matrices $D$ and $D'$ of the first
moments with elements $D_{ij}=d_id_j$ and $D'_{ij}=d_i'd_j'$,
$i,j=1,\ldots,4$. Using Eq.~(\ref{rhoACprimed}) and after some algebra, one
gets the transformation rule for the
matrix of the first moments in the form $D'=U\Sigma D\Sigma U^{T}$,
where $U$ describes the beam splitter on the level of covariance
matrices. $\Sigma=\mbox{diag}(e^{-\frac{\sigma^2}{2}},e^{-\frac{\sigma^2}{2}},1,1)$
is a diagonal matrix. Similarly we get the covariance matrix
\begin{eqnarray}\label{gammaACprimed}
\gamma_{A'C'}=U\left(\Sigma\gamma_{AC}\Sigma+\pi\oplus0\right)U^{T},
\end{eqnarray}
where $0$ is the $2\times2$ zero matrix and
\begin{eqnarray}\label{pi}
\pi=\frac{(1-e^{-\sigma^2})^2}{2}(A+\alpha)+\frac{1-e^{-2\sigma^2}}{2}J(A+\alpha)J^T.
\end{eqnarray}
Here the matrix $A$ is the $2\times2$ matrix with elements
$A_{ij}=(\gamma_{AC})_{ij}$, $i,j=1,2$, $\alpha$ is
the $2\times2$ matrix with elements $\alpha_{ij}=2D_{ij}$,
$i,j=1,2$, and $J$ is defined in Eq.~(\ref{Omega}).
Similar to Ref.~\cite{Vollmer13} we can now invert the relation
(\ref{gammaACprimed}) and express the input covariance
matrix $\gamma_{AC}$ via the output covariance matrix
$\gamma_{A'C'}$ and the first moments after the beam splitter
$BS_{AC}$ as
\begin{eqnarray}\label{gammaAC}
\gamma_{AC}=\Sigma^{-1}U^{T}\gamma_{A'C'}U\Sigma^{-1}+\tilde{\pi}\oplus0,
\end{eqnarray}
where
\begin{eqnarray}\label{piprimed}
\tilde{\pi}=\frac{(1-e^{\sigma^2})^2}{2}(\tilde{A}+\tilde{\alpha})+\frac{1-e^{2\sigma^2}}{2}J(\tilde{A}+\tilde{\alpha})J^T.
\end{eqnarray}
The $2\times2$ matrices $\tilde{A}$ and $\tilde{\alpha}$
possess the elements $\tilde{A}_{ij}=(U^T\gamma_{A'C'}U)_{ij}$ and
$\tilde{\alpha}_{ij}=2(U^TD'U)_{ij}$, $i,j=1,2$.
Our estimate for the variance of the phase fluctuations is
$\sigma^2=0.02^\circ$ and the vector $d'$ of the measured mean values of the
state $\hat \rho_{A'C'}$ reads
\begin{equation}\label{dprimed}
d'=(-0.208,9.876,13.32,1.78).
\end{equation}
By substituting these experimental values for $\sigma^2$ and $d'$
in Eq.~(\ref{gammaAC}) and using the beam splitter with the
measured transmissivity $T=0.49$ we get a legitimate covariance
matrix $\gamma_{AC}$ before the phase fluctuations as can be
easily verified by checking the condition (\ref{Heisenberg}).
Provided that the state with the covariance matrix $\gamma_{AC}$
is classical it can be expressed as a convex mixture of products
of coherent states. Gaussian distributed phase fluctuations and a
beam splitter preserve the structure of the state, hence the state
after the first beam splitter cannot be entangled. The covariance
matrix $\gamma_{AC}$ determines a physical Gaussian quantum state.
Moreover, the covariance matrix possesses all eigenvalues greater
than one and therefore the state is not squeezed \cite{Simon_94}
which is in a full agreement with the fact that modulations of
modes $A$ and $C$ completely destroy the squeezing. It then
follows that this Gaussian state is classical and it therefore
transforms to a separable state after the first beam splitter.
The inversion (\ref{gammaAC}) thus allows us to associate a Gaussian
state before the phase fluctuations with the covariance matrix
$\gamma_{A'C'}$ measured after the first beam splitter. The
separability properties of the state after the beam splitter can
then be determined from the non-classicality properties of this
Gaussian state.
\subsection{Gaussianity of the utilized states}
We have paid great attention on the modulations on modes $A$ and $C$ to preserve Gaussian character of the state $\hat \rho_{A'C'}$. Our success can be visually inspected at the examples in Fig.~\ref{Fig:Hists}, which illustrates that both the modulation and the subsequent Gaussian mixing faithfully samples the required Gaussian shape.
\begin{figure}
\includegraphics[width=\linewidth]{CombinedHistos}
\caption{\textbf{Histogram plots for $\hat S_{A',0^\circ}$ of the Gaussian mixed state (blue) and three exemplary individual modes.} This figure illustrates the preparation of the Gaussian mixed state via post processing. Exemplarily, three of the 6400 displaced individual modes are visualized by their histograms (in green, black and red colour). The normalization is chosen such that they can be depicted in the same plot as the histogram of the mixed state (blue), which is normalized to its maximum value. By merging the data for all individual modes using a weighting with a two dimensional Gaussian distribution, the mixed state is achieved. Its Gaussianity is visualized by the Gaussian fit (red curve).}
\label{Fig:Hists}
\end{figure}
Besides this raw visual check we have also tested quantitatively Gaussianity of the involved states by measuring higher-order moments of the Stokes measurements on modes $A'$ and $C'$. Specifically, we have focused on the determination of the shape measures called skewness $S$ and kurtosis $K$ defined for a random variable $x$ as the following third and fourth standardized moments
\begin{equation}\label{SK}
S=\frac{\mu_{3}}{s^3},\quad K=\frac{\mu_{4}}{s^4},
\end{equation}
where $\mu_{k}=\langle(x-\langle x\rangle)^k\rangle$ is the $k$th central moment, $\langle x\rangle$ is the mean value and $s=\sqrt{\mu_{2}}$ is the standard deviation.
Skewness characterizes the orientation and the amount of skew of a given distribution and therefore informs us about its asymmetry in the horizontal direction. Gaussian distributions possess skewness of zero. The exemplary values of skewness for various measurement settings are summarized in the Table~\ref{Stable}.
\begin{table}[ht]
\caption{Skewness $S$ for Stokes measurements on modes $A'$ and $C'$ in different measurement directions.}
\centering
\begin{tabular}{| c | c | c |}
\hline Measurement & $S_{A',0^\circ}$ & $S_{A',90^\circ}$ \\
\hline Skewness$\times10^3$
& $6.240\pm0.781$ & $-1.478\pm0.563$ \\
\hline
\hline Measurement & $S_{C',0^\circ}$ & $S_{C',90^\circ}$ \\
\hline Skewness$\times10^3$ & $10.123\pm0.727$ & $1.106\pm0.830$ \\
\hline
\end{tabular}\label{Stable}
\caption{Kurtosis $K$ for Stokes measurements on modes $A'$
and $C'$ in different measurement directions.}
\centering
\begin{tabular}{| c | c | c |}
\hline Measurement & $S_{A',0^\circ}$ & $S_{A',90^\circ}$\\
\hline Kurtosis
& $2.971\pm2.211\times10^{-3}$ & $2.986\pm1.852\times10^{-3}$ \\
\hline
\hline Measurement & $S_{C',0^\circ}$ & $S_{C',90^\circ}$ \\
\hline Kurtosis
& $2.972\pm1.978\times10^{-3}$ & $2.992\pm1.568\times10^{-3}$ \\
\hline
\end{tabular}\label{Ktable}
\end{table}
The skewness can vanish also for the other symmetrical
distributions, which may, however, differ from a Gaussian
distribution in the peak profile and the weight of tails. These
differences can be captured by the kurtosis which is equal to 3
for Gaussian distributions. The exemplary values of kurtosis for
various measurement settings are summarized in the
Table~\ref{Ktable}.
The tables reveal that the measured probability distributions
satisfy within the experimental error the necessary Gaussianity
conditions $S=0$ and $K=3$. More sophisticated normality tests can
be performed, which is beyond the scope of the present manuscript.
\section{Statistical errors of the measured covariance matrices}
By dividing our dataset in 10 equal in size parts we can estimate the statistical errors of our measured covariance matrices $\gamma_{A'C'}$ and $\gamma_{A'B'}$ given in Eqs.~(3) and (5) of the main letter. We calculate the covariance matrix for each part and use the standard deviation as error estimation.
The covariance matrix $\gamma_{A'C'}$ including the statistical error turns out to be
\begin{widetext}
\begin{align}
\gamma_{A'C'}= \begin{pmatrix}
20.90\pm 0.0087 & 1.102\pm 0.0091 & -7.796\pm 0.0069 & -1.679\pm 0.0076\\
1.102\pm 0.0091 & 25.30\pm 0.013 & 1.000\pm 0.0071 & 14.63\pm 0.0091\\
-7.796\pm 0.0069 & 1.000\pm 0.0071 & 20.68\pm 0.0093 & 0.8010\pm 0.011\\
-1.679\pm 0.0076 & 14.63\pm 0.0091 & 0.8010\pm 0.011 & 24.65\pm 0.0073
\end{pmatrix}.
\end{align}
\end{widetext}
Similarly, the covariance matrix $\gamma_{A'B'}$ including the statistical error reads as
\begin{widetext}
\begin{align}
\gamma_{A'B'}= \begin{pmatrix}
19.95\pm 0.011& 1.025\pm 0.016& -4.758\pm 0.0050& -1.063\pm0.0051\\
1.025\pm 0.016& 22.92\pm 0.012& 0.9699\pm 0.0047& 9.153\pm0.0058\\
-4.758\pm 0.0050& 0.9699\pm 0.0047& 9.925\pm 0.0048& 0.2881\pm0.0047\\
-1.063\pm 0.0051& 9.153\pm 0.0058& 0.2881\pm 0.0047& 11.65\pm0.0038
\end{pmatrix}.
\end{align}
\end{widetext}
We could achieve such small statistical errors by recording sufficient large datasets.
\end{appendix}
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Een toenaam, soms ook wel bijnaam genoemd, is een bijvoeglijk naamwoord ter unieke identificatie van iets of iemand. Toenamen geven onderscheid en versterken identiteiten door de kennis en visie die naamkiezers in hun naamkeuzes doorgeven.
Toenamen in dagelijks gebruik
Toenamen zijn vaak associatief. Ze informeren en beïnvloeden de samenleving en naamdragers zelf. Grieken noemden de toenaam epitheton. Toenamen zijn vast, zoals bij de Noorse boskat, maar kunnen naar inzicht wijzigen.
Historische toenamen
Een historisch veel gebruikte toenaam is de Grote, waarmee veel figuren zijn gesierd. Bekende voorbeelden hiervan zijn Karel de Grote, Alexander de Grote, Constantijn de Grote en Alfred de Grote. De Stoute was een toevoeging die ten deel viel aan onder meer Karel de Stoute en Filips de Stoute. Deze toenaam duidt overigens niet de ondeugd aan van de dragers, maar wordt gebruikt in de betekenis van 'de stoutmoedige' ofwel 'de dappere' (de betekenissen zijn overigens wel aan elkaar verwant, iemand die ondeugend is, is ook dapper). In dezelfde categorie vallen Alexander de Goede, Karel de Goede en Filips de Goede. Onder de vele Karels met een toenaam bevinden zich ook Karel de Kale en Karel de Eenvoudige. In vrijheid kan men zich presenteren als Voor in Visie, Langgewenst Godsgeschenk, Pax, Goud, Associatief etc.
Uitdrukking
De term 'toenaam' komt voor in de Nederlandse uitdrukking met naam en toenaam, waarmee wordt aangegeven dat men iets of iemand tot in detail benoemt.
Zie ook
Epitheton
Referenties
Toenaam | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,493 |
Synod's small group reports released by Vatican
Press conference with Cardinal Christoph Schoenborn at the Holy See press office, Oct. 16, 2014. | Bohumil Petrik/CNA.
Vatican City, Oct 16, 2014 / 13:28 pm
After the synod's small groups released their reports Thursday asking for a substantial rewriting of the meeting's mid-term report, the Vatican has said the synod's final report will be prepared, and voted upon on Saturday morning.
Fr. Federico Lombardi, director of the Holy See press office, added that it is unlikely the synod's final relatio will be released Saturday evening.
"We are in a work in progress, and I would approach step by step toward Saturday. Given the several requests for amendments, I find it difficult (to suppose) that Saturday evening there will be a polished text ready for publication," Fr. Lombardi said Oct. 16.
Unlike previous synods, the synod fathers will not vote on a series of proposals that the Pope will take in consideration for the following post-synodal apostolic exhortation, but will vote rather on a comprehensive text that will serve as the basis for discussion and preparation for 2015 synod, also on the family.
Beyond Cardinal Peter Erdo and Archbishop Bruno Forte, General Rapporteur and Special Secretary of the Synod, and Cardinal Lorenzo Baldisseri, Secretary General of the Synod, the Pope had appointed a commission of six prelates to help in drafting the final document.
Yesterday, Pope Francis enlarged that commission, adding Cardinal Wilfrid Napier of Durban and Archbishop Denis Hart of Melbourne.
"Since someone noted that there were no representatives from Africa in the commission, the commission has been enlarged in order to include representatives of all the five continents," Fr. Lombardi commented.
He also announced the release of the texts of the ten reports of the small groups, and recounted that "there had been an explicit discussion within the synod about whether to make these relations public or not."
According to one of the synod fathers, Cardinal Baldisseri said that the relations were not going to be published, thus arousing the lively reaction of the assembly.
"Many bishops stood, and someone also banged their fist on the table. Cardinal Baldisseri said he wanted to listen to everyone, and he found that the majority of the synod fathers wanted the document to be published," the source maintained.
"This way, there will not be so much ground to change or bias our discussion in the final report," he maintained.
Glancing over the ten reports issued by the small groups, the need to substantially rewrite the mid-term report emerges.
There are three common and primary concerns of the synod fathers: the absence in the text of any reference to the Gospel of the Family, and more widely to Gospel references; the need to underscore and highlight positive examples of Christian families; and the request to take out, or at least clarify, the principle of graduality, which may lead to confusion.
"Gradualness should not make insipid the challenge of the Gospel to conversion, to 'go and sin no more', as Jesus said to the woman caught in adultery," reads the report of the third English-speaking small group, chaired by Archbishop Joseph Kurtz of Louisville. "The aim of recognizing gradualness should be to draw people closer to Christ."
"Truth and mercy are not mutually exclusive terms, and in proclaiming truth we also proclaim the most profound mercy – that of reconciliation and unity with God; on the other hand, it is in mercy that we find truth," the group added.
Cardinal Christoph Schoenborn of Vienna, coordinator of one of the French-speaking small groups, tried to balance the different positions while speaking with journalists during a media briefing.
"If some synod fathers say: 'Be careful, because we mustn't forget doctrine', on the other hand there is also a need for the accompaniment of many situations, those situations the Pope refers to as a field hospital."
As many of the synod fathers have lamented the absence of the word "sin" from the mid-term report, Cardinal Schoenborn clarified that "the discussions have also dealt with Confession."
For the Archbishop of Vienna, it is "evident there are tensions in the synod, there are different aspects to take in consideration: on one side, the doctrine, the clear Word of the Gospel; on the other, Jesus acting with mercy. How to join these two sides is the permanent challenge of the Church and pastors."
These tensions are proven by the some 600 requests for modification presented in the morning before the reading of the relations of the small groups.
All remaining focus is on the synod's report, which is to be voted on Saturday morning.
"If the new report does mirror the synod fathers' remarks, it is possible that the text will be dismissed by the assembly," one of the synod fathers commented to CNA. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,746 |
Q: Integer solutions to $x^2=2y^4+1$. Find all integer solutions to $x^2=2y^4+1$.
What I tried
The only solutions I got are $(\pm 1 ,0)$, I rewrote the question as : is $a_{n}$ a perfect square for $n>0$ were
$$a_0=0,\quad a_1=2, \quad a_{n+2}=6a_{n+1}-a_n.$$
I tried taking $\pmod{4}$ and $\pmod{12}$ but that lead me nowhere.
A: There are no solutions other than those trivial ones. First we note that $x$ must be odd. Let $x = 2k+1$. This gives $2k^2+2k = y^4$. Thus $y = 2j$ and we have $k(k+1) = 8j^4$. Note that $\gcd(k,k+1) = 1$ so we have two cases.
Case 1: $k+1 = 8a^4, k = b^4$.
This case is impossible since this implies $b^4 \equiv -1 \pmod 4$ but $-1$ is a not a quadratic residue $\pmod 4$.
Case 2: $k+1 = b^4, k = 8a^4$.
This case is equivalent to solving the diophantine equation $8a^4+1 = b^4$. An elementary solution is given here. In the referenced paper, it is prove that the only non trivial solution to this equation is $(a,b) = (1,3)$ which gives $k = 8, k+1 = 9$ but $9 = b^4$ is impossible.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,828 |
{"url":"https:\/\/socratic.org\/questions\/how-do-you-find-the-sum-of-the-series-4n-3-from-n-1-to-n-3","text":"# How do you find the sum of the series 4n^3 from n=1 to n=3?\n\nOct 2, 2016\n\n$144.$\n\n#### Explanation:\n\nWe will use the following Formula :\n\n$: {\\sum}_{j = 1}^{j = n} {j}^{3} = \\sum {n}^{3} = {1}^{3} + {2}^{3} + {3}^{3} + \\ldots \\ldots + {n}^{3} = \\frac{{n}^{2} {\\left(n + 1\\right)}^{2}}{4.}$\n\nReqd. Sum$= {\\sum}_{n = 1}^{n = 3} 4 {n}^{3} = 4 {\\sum}_{n = 1}^{n = 3} {n}^{3}$\n\n$= 4 \\left\\{\\frac{\\left({3}^{2}\\right) {\\left(3 + 1\\right)}^{2}}{4}\\right\\} = \\left(9\\right) \\left(16\\right) = 144$.","date":"2019-10-15 09:15:39","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 4, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9788942337036133, \"perplexity\": 2217.846049117562}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-43\/segments\/1570986657949.34\/warc\/CC-MAIN-20191015082202-20191015105702-00185.warc.gz\"}"} | null | null |
Supreme%20Court%20Rules%20EPA%20Cannot%20Require%20Existing%20Fossil%20Fuel%20Power%20Facilities%20to%20Shift%20to%20Lower%20CO2%20Emitting%20Sources%20of%20Electricity%20
https://www.whitecase.com/insight-alert/supreme-court-rules-epa-cannot-require-existing-fossil-fuel-power-facilities-shift
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Supreme Court Rules EPA Cannot Require Existing Fossil Fuel Power Facilities to Shift to Lower CO2 Emitting Sources of Electricity
Seth Kerschner |
Taylor Pullins |
Laura Mulry
On June 30, 2022, the Supreme Court ruled that the US Environmental Protection Agency (EPA) cannot use the Clean Air Act to require fossil fuel power facilities to implement a measure known as "generation shifting" without express authorization from Congress. Generation shifting was a measure EPA proposed that would have required a shift in electricity production from certain fossil fuel power generation sources, primarily fired by coal and natural gas, to other sources that emit less carbon dioxide.
The Supreme Court's ruling concerned a 2015 proposal where EPA relied on Section 111 of the Clean Air Act, which regulates emissions of certain pollutants from existing sources. Under Section 111(d), states set rules governing existing sources (such as power plants), but EPA determines the emissions limit with which they will have to comply. EPA establishes that limit by determining the "best system of emission reduction" for the type of source at issue.
In its 2015 proposal, EPA had determined that the "best system of emission reduction" for existing coal and natural gas plants entailed three types of measures: (1) heat rate improvements, (2) a shift in generation from existing coal plants to natural gas plants, and (3) a shift from coal and natural gas plants to renewable power sources (primarily wind and solar plants). To implement the shift to renewables, an operator could reduce a plant's own production of electricity, build or invest in a new or existing natural gas plant, wind farm, or solar installation, or purchase emission allowances or credits as part of a cap-and-trade regime. The first measure, heat rate improvements, was not at issue in this case. The Court explained that this type of measure was consistent with how EPA had historically set limits under Section 111. The second two measures in EPA's 2015 proposal, which involved generation shifting, were the focus of the Court's ruling because they focused on the overall power system rather than emissions performance of individual sources.
In its opinion, the Court explained that Congress did not provide clear authorization for EPA to use Section 111(d) of the Clean Air Act to devise emissions caps based on the generation shifting approach EPA proposed in 2015. The Court considered whether restructuring the country's overall mix of electricity generation can be a "best system of emission reduction" within the meaning of Section 111. The Court expressed skepticism regarding whether what it described as an "ancillary provision" or "the previously little-used backwater of Section 111(d)" could be used to force a shift in the power grid from one type of energy source to another. In considering this issue, the Court found that this is a "major questions" case, which refers to "extraordinary cases" in which the "history and the breadth of the authority" that a federal agency has asserted, and the "economic and political significance" of that assertion, require the federal agency to point to "clear congressional authorization" for the authority it claims. In assessing this issue, the Court found that EPA could not point to such authority. The Court found that EPA had admitted that issues of electricity transmission, distribution, and storage are not within its traditional expertise, and was therefore skeptical that Congress intended to delegate such a decision of such economic and political significance (i.e., "how much coal-based generation there should be over the coming decades") to any administrative agency. The Court also noted that Congress had already considered and rejected numerous times a cap-and-trade program for carbon dioxide emissions, therefore casting doubt on EPA's argument that Section 111(d) enabled it to effectively enact such a program. Notably, the Court did not hold that a "best system of emission reduction" refers exclusively to measures focused on emissions performance of individual sources such that all other actions do not qualify as "best system of emission reduction." Therefore, while the Court's ruling prohibits EPA from enacting a proposal that requires the type of generation shifting that EPA proposed in 2015, it does not mandate that EPA only focus on emissions performance of individual sources.
The Court also addressed an argument of mootness from the government. While a lower court had concluded that EPA's generation shifting proposal could be implemented pursuant to Section 111 of the Clean Air Act, the 2015 proposal was never implemented. The Biden Administration, therefore, urged the Court not to intervene arguing in part that there is no regulation to review, and the case is therefore moot, because EPA informed the courts that it had no plan to reinstate the 2015 proposed rule. The Court held that, despite the federal government's argument that EPA does not intend to enforce the 2015 proposed rule prior to creating a new rule under Section 111(d), the case remains justiciable because voluntary cessation of the implementation of a rule does not moot a case unless it is "absolutely clear that the allegedly wrongful behavior could not reasonably be expected to recur." The Court distinguished between standing and mootness and found that, because the federal government did not suggest that if the litigation was resolved in its favor it will not re-impose emissions limits predicated on generation shifting, EPA's position did not moot the dispute at hand.
It is unlikely that he Court's decision will deter EPA from moving forward with a new proposal to regulate carbon dioxide emissions from existing power plants. EPA was planning this anyway, and market forces have caused the United States power industry to engage in generation shifting to meet the emissions targets that EPA proposed in 2015 even in the absence of that proposal being implemented. For the regulated community, which has been faced with a level of uncertainty since EPA's 2015 proposal, the Court's decision may not provide clarity on what EPA could propose that would fit within the bounds of "best system of emission reduction." While the ruling limits EPA's authority to require generation shifting to regulate carbon dioxide emissions from power sources, it does not broadly inhibit EPA and the federal government's authority to regulate climate change generally. EPA retains authority to regulate greenhouse gas emissions at the source under Section 111 and through other Clean Air Act provisions.
In response to the Court's ruling, EPA Administrator Michael Regan commented, "[W]e are committed to using the full scope of EPA's authorities to protect communities and reduce the pollution that is driving climate change. We will move forward to provide certainty and transparency for the energy sector, which will support the industry's ongoing efforts to grow our clean energy economy."1 In addition, President Biden stated on Thursday that he will "continue using lawful executive authority, including the EPA's legally-upheld authorities," work with cities and states to pass laws, and "keep pushing for additional Congressional action, so that Americans can fully seize the economic opportunities, cost-saving benefits, and security of a clean energy future."2
1 https://www.epa.gov/newsreleases/epa-administrator-regan-issues-statement-west-virginia-v-environmental-protection.
2 https://www.whitehouse.gov/briefing-room/statements-releases/2022/06/30/statement-by-president-joe-biden-on-supreme-court-ruling-on-west-virginia-v-epa.
Seth Kerschner
Environment & Climate Change, Pharmaceuticals & Healthcare, Mergers & Acquisitions, ESG and Sustainability, Financial Institutions
Taylor Pullins
Partner|Houston
Mergers & Acquisitions, Environment & Climate Change, ESG and Sustainability, Energy Transition
Counsel|Los Angeles
Mergers & Acquisitions, Environment & Climate Change | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,914 |
Do good books make good movies?
Do you ever stop to really question whether you should see the screen adaptation of a book you've read and loved?
I think it would be fair to say that you would rarely ever read the book after you've seen the movie. But we often head off to the cinema after we've read a book only to come away disappointed.
I know I'm a very visual kind of person and always create images in my mind while reading a book. I recall through the early 90s seeing a minimalist stage production of Pride & Prejudice. Boxes stacked in strategic spots across the stage were all the props in the set. Even with the then emerging William McInnes playing Mr Darcy it was just wrong, ALL wrong.
A book is a shared conversation between a writer and a reader, and every conversation is different. It is up to the reader to re-create the writer's story, furnishing the visual characteristics of people and places, giving the emotional engagement between the characters importance or not. The reader's mind is the theatre in which a novelist's dialogue is mounted, creating a very individual performance according to the information that a reader brings to the experience.
The nuances of a book can rarely be recreated in a movie adaptation.
A movie is literal. The creative work that your mind employs with a book is all done for you in a movie. The scene is depicted, the characters are defined in the way they look, they way they react and the way they speak. There are limited options for you to interpret the outcome of scenarios in movies.
So, what do we think? Do our favourite books make good movies?
Vote in the poll below to indicate your opinion and tomorrow we'll look at several books that I've enjoyed and weigh them against the movie adaptation.
This entry was posted in Book Groups, Books I've Read and tagged books vs movies by wentowrite. Bookmark the permalink. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,882 |
package com.facebook.buck.jvm.java;
import com.facebook.buck.event.BuckEventBus;
import com.facebook.buck.event.BuckTracingEventBusBridge;
import com.facebook.buck.event.MissingSymbolEvent;
import com.facebook.buck.event.api.BuckTracing;
import com.facebook.buck.io.ProjectFilesystem;
import com.facebook.buck.jvm.java.tracing.TranslatingJavacPhaseTracer;
import com.facebook.buck.log.Logger;
import com.facebook.buck.model.BuildTarget;
import com.facebook.buck.rules.SourcePathResolver;
import com.facebook.buck.step.ExecutionContext;
import com.facebook.buck.util.HumanReadableException;
import com.google.common.base.Function;
import com.google.common.base.Functions;
import com.google.common.base.Joiner;
import com.google.common.base.Optional;
import com.google.common.base.Preconditions;
import com.google.common.base.Splitter;
import com.google.common.collect.FluentIterable;
import com.google.common.collect.ImmutableList;
import com.google.common.collect.ImmutableMap;
import com.google.common.collect.ImmutableSortedSet;
import com.google.common.collect.Iterables;
import com.google.common.collect.Lists;
import java.io.Closeable;
import java.io.File;
import java.io.IOException;
import java.io.PrintWriter;
import java.io.Writer;
import java.net.MalformedURLException;
import java.net.URL;
import java.net.URLClassLoader;
import java.nio.file.Path;
import java.nio.file.Paths;
import java.util.Enumeration;
import java.util.Iterator;
import java.util.List;
import java.util.Set;
import java.util.zip.ZipEntry;
import java.util.zip.ZipFile;
import javax.annotation.Nullable;
import javax.annotation.processing.Processor;
import javax.tools.Diagnostic;
import javax.tools.DiagnosticCollector;
import javax.tools.JavaCompiler;
import javax.tools.JavaFileObject;
import javax.tools.StandardJavaFileManager;
/**
* Command used to compile java libraries with a variety of ways to handle dependencies.
*/
public abstract class Jsr199Javac implements Javac {
private static final Logger LOG = Logger.get(Jsr199Javac.class);
private static final JavacVersion VERSION = JavacVersion.of("in memory");
private static final StandardJavaFileManagerFactory DEFAULT_FILE_MANAGER_FACTORY =
new StandardJavaFileManagerFactory() {
@Override
public StandardJavaFileManager create(
JavaCompiler compiler) {
return compiler.getStandardFileManager(null, null, null);
}
};
@Override
public JavacVersion getVersion() {
return VERSION;
}
@Override
public String getDescription(
ImmutableList<String> options,
ImmutableSortedSet<Path> javaSourceFilePaths,
Path pathToSrcsList) {
StringBuilder builder = new StringBuilder("javac ");
Joiner.on(" ").appendTo(builder, options);
builder.append(" ");
builder.append("@").append(pathToSrcsList);
return builder.toString();
}
@Override
public String getShortName() {
return "javac";
}
@Override
public ImmutableList<String> getCommandPrefix(SourcePathResolver resolver) {
throw new UnsupportedOperationException("In memory javac may not be used externally");
}
@Override
public ImmutableMap<String, String> getEnvironment(SourcePathResolver resolver) {
throw new UnsupportedOperationException("In memory javac may not be used externally");
}
protected abstract JavaCompiler createCompiler(
ExecutionContext context,
SourcePathResolver resolver);
@Override
public int buildWithClasspath(
ExecutionContext context,
ProjectFilesystem filesystem,
SourcePathResolver resolver,
BuildTarget invokingRule,
ImmutableList<String> options,
ImmutableSortedSet<Path> javaSourceFilePaths,
Path pathToSrcsList,
Optional<Path> workingDirectory,
Optional<Path> usedClassesFile,
Optional<StandardJavaFileManagerFactory> fileManagerFactory) {
JavaCompiler compiler = createCompiler(context, resolver);
StandardJavaFileManager fileManager =
fileManagerFactory.or(DEFAULT_FILE_MANAGER_FACTORY).create(compiler);
try {
Iterable<? extends JavaFileObject> compilationUnits;
try {
compilationUnits = createCompilationUnits(
fileManager,
filesystem.getAbsolutifier(),
javaSourceFilePaths);
} catch (IOException e) {
LOG.warn(e, "Error building compilation units");
return 1;
}
try {
return buildWithClasspath(
context,
filesystem,
invokingRule,
options,
javaSourceFilePaths,
pathToSrcsList,
compiler,
usedClassesFile,
fileManager,
compilationUnits);
} finally {
close(compilationUnits);
}
} finally {
try {
fileManager.close();
} catch (IOException e) {
LOG.warn(e, "Unable to close java filemanager. We may be leaking memory.");
}
}
}
private int buildWithClasspath(
ExecutionContext context,
ProjectFilesystem filesystem,
BuildTarget invokingRule,
ImmutableList<String> options,
ImmutableSortedSet<Path> javaSourceFilePaths,
Path pathToSrcsList,
JavaCompiler compiler,
Optional<Path> usedClassesFile,
StandardJavaFileManager fileManager,
Iterable<? extends JavaFileObject> compilationUnits) {
// write javaSourceFilePaths to classes file
// for buck user to have a list of all .java files to be compiled
// since we do not print them out to console in case of error
try {
filesystem.writeLinesToPath(
FluentIterable.from(javaSourceFilePaths)
.transform(Functions.toStringFunction())
.transform(ARGFILES_ESCAPER),
pathToSrcsList);
} catch (IOException e) {
context.logError(
e,
"Cannot write list of .java files to compile to %s file! Terminating compilation.",
pathToSrcsList);
return 1;
}
DiagnosticCollector<JavaFileObject> diagnostics = new DiagnosticCollector<>();
List<String> classNamesForAnnotationProcessing = ImmutableList.of();
Writer compilerOutputWriter = new PrintWriter(context.getStdErr());
ClassUsageTracker classTracker = new ClassUsageTracker();
final StandardJavaFileManager maybeWrappedFileManager =
usedClassesFile.isPresent() ?
classTracker.wrapFileManager(fileManager) :
fileManager;
JavaCompiler.CompilationTask compilationTask = compiler.getTask(
compilerOutputWriter,
maybeWrappedFileManager,
diagnostics,
options,
classNamesForAnnotationProcessing,
compilationUnits);
boolean isSuccess = false;
BuckTracing.setCurrentThreadTracingInterfaceFromJsr199Javac(
new BuckTracingEventBusBridge(context.getBuckEventBus(), invokingRule));
try {
try (
// TranslatingJavacPhaseTracer is AutoCloseable so that it can detect the end of tracing
// in some unusual situations
TranslatingJavacPhaseTracer tracer = TranslatingJavacPhaseTracer.setupTracing(
invokingRule,
context.getClassLoaderCache(),
context.getBuckEventBus(),
compilationTask);
// Ensure annotation processors are loaded from their own classloader. If we don't do
// this, then the evidence suggests that they get one polluted with Buck's own classpath,
// which means that libraries that have dependencies on different versions of Buck's deps
// may choke with novel errors that don't occur on the command line.
ProcessorBundle bundle = prepareProcessors(
context.getBuckEventBus(),
compiler.getClass().getClassLoader(),
invokingRule,
options)) {
compilationTask.setProcessors(bundle.processors);
// Invoke the compilation and inspect the result.
isSuccess = compilationTask.call();
} catch (IOException e) {
LOG.warn(e, "Unable to close annotation processor class loader. We may be leaking memory.");
}
} finally {
// Clear the tracing interface so we have no chance of leaking it to code that shouldn't
// be using it.
BuckTracing.clearCurrentThreadTracingInterfaceFromJsr199Javac();
}
for (Diagnostic<? extends JavaFileObject> diagnostic : diagnostics.getDiagnostics()) {
LOG.debug("javac: %s", DiagnosticPrettyPrinter.format(diagnostic));
}
if (isSuccess) {
if (usedClassesFile.isPresent()) {
ClassUsageFile.writeFromTrackerData(
filesystem,
filesystem.resolve(usedClassesFile.get()),
classTracker,
context.getObjectMapper());
}
return 0;
} else {
if (context.getVerbosity().shouldPrintStandardInformation()) {
int numErrors = 0;
int numWarnings = 0;
for (Diagnostic<? extends JavaFileObject> diagnostic : diagnostics.getDiagnostics()) {
Diagnostic.Kind kind = diagnostic.getKind();
if (kind == Diagnostic.Kind.ERROR) {
++numErrors;
handleMissingSymbolError(invokingRule, diagnostic, context, filesystem);
} else if (kind == Diagnostic.Kind.WARNING ||
kind == Diagnostic.Kind.MANDATORY_WARNING) {
++numWarnings;
}
context.getStdErr().println(DiagnosticPrettyPrinter.format(diagnostic));
}
if (numErrors > 0 || numWarnings > 0) {
context.getStdErr().printf("Errors: %d. Warnings: %d.\n", numErrors, numWarnings);
}
}
return 1;
}
}
private void close(Iterable<? extends JavaFileObject> compilationUnits) {
for (JavaFileObject unit : compilationUnits) {
if (unit instanceof Closeable) {
try {
((Closeable) unit).close();
} catch (IOException e) {
LOG.warn(e, "Unable to close zipfile. We may be leaking memory.");
}
}
}
}
private ProcessorBundle prepareProcessors(
BuckEventBus buckEventBus,
ClassLoader compilerClassLoader,
BuildTarget target,
List<String> options) {
String processorClassPath = null;
String processorNames = null;
Iterator<String> iterator = options.iterator();
while (iterator.hasNext()) {
String curr = iterator.next();
if ("-processorpath".equals(curr) && iterator.hasNext()) {
processorClassPath = iterator.next();
} else if ("-processor".equals(curr) && iterator.hasNext()) {
processorNames = iterator.next();
}
}
ProcessorBundle processorBundle = new ProcessorBundle();
if (processorClassPath == null || processorNames == null) {
return processorBundle;
}
// N.B. You might think that we could avoid some overhead by using the same classloader every
// time we create an instance of annotation processor. In an ideal world, that would work well,
// but many annotation processors aren't thread-safe, and they store state in class-static
// variables. In the interest of maximum safety, we'll create a new ClassLoader every time we
// need an annotation processor.
Iterable<String> rawPaths = Splitter.on(File.pathSeparator)
.omitEmptyStrings()
.split(processorClassPath);
URL[] urls = FluentIterable.from(rawPaths)
.transform(
new Function<String, URL>() {
@Override
public URL apply(String pathRelativeToProjectRoot) {
try {
return Paths.get(pathRelativeToProjectRoot).toUri().toURL();
} catch (MalformedURLException e) {
// The paths we're being given should have all been resolved from the file
// system already. We'd need to be unfortunate to get here. Bubble up a runtime
// exception.
throw new RuntimeException(e);
}
}
})
.toArray(URL.class);
processorBundle.classLoader = new URLClassLoader(
urls,
compilerClassLoader);
Iterable<String> names = Splitter.on(",")
.trimResults()
.omitEmptyStrings()
.split(processorNames);
for (String name : names) {
try {
LOG.debug("Loading %s from own classloader", name);
Class<? extends Processor> aClass =
Preconditions.checkNotNull(processorBundle.classLoader)
.loadClass(name)
.asSubclass(Processor.class);
processorBundle.processors.add(
new TracingProcessorWrapper(
buckEventBus,
target,
aClass.newInstance()));
} catch (ReflectiveOperationException e) {
// If this happens, then the build is really in trouble. Better warn the user.
throw new HumanReadableException(
"%s: javac unable to load annotation processor: %s",
target.getFullyQualifiedName(),
name);
}
}
return processorBundle;
}
private Iterable<? extends JavaFileObject> createCompilationUnits(
StandardJavaFileManager fileManager,
Function<Path, Path> absolutifier,
Set<Path> javaSourceFilePaths) throws IOException {
List<JavaFileObject> compilationUnits = Lists.newArrayList();
for (Path path : javaSourceFilePaths) {
String pathString = path.toString();
if (pathString.endsWith(".java")) {
// For an ordinary .java file, create a corresponding JavaFileObject.
Iterable<? extends JavaFileObject> javaFileObjects = fileManager.getJavaFileObjects(
absolutifier.apply(path).toFile());
compilationUnits.add(Iterables.getOnlyElement(javaFileObjects));
} else if (pathString.endsWith(SRC_ZIP) || pathString.endsWith(SRC_JAR)) {
// For a Zip of .java files, create a JavaFileObject for each .java entry.
ZipFile zipFile = new ZipFile(absolutifier.apply(path).toFile());
boolean hasZipFileBeenUsed = false;
for (Enumeration<? extends ZipEntry> entries = zipFile.entries();
entries.hasMoreElements();
) {
ZipEntry entry = entries.nextElement();
if (!entry.getName().endsWith(".java")) {
continue;
}
hasZipFileBeenUsed = true;
compilationUnits.add(new ZipEntryJavaFileObject(zipFile, entry));
}
if (!hasZipFileBeenUsed) {
zipFile.close();
}
}
}
return compilationUnits;
}
private void handleMissingSymbolError(
BuildTarget invokingRule,
Diagnostic<? extends JavaFileObject> diagnostic,
ExecutionContext context,
ProjectFilesystem filesystem) {
JavacErrorParser javacErrorParser = new JavacErrorParser(
filesystem,
context.getJavaPackageFinder());
Optional<String> symbol = javacErrorParser.getMissingSymbolFromCompilerError(
DiagnosticPrettyPrinter.format(diagnostic));
if (!symbol.isPresent()) {
// This error wasn't related to a missing symbol, as far as we can tell.
return;
}
MissingSymbolEvent event = MissingSymbolEvent.create(
invokingRule,
symbol.get(),
MissingSymbolEvent.SymbolType.Java);
context.getBuckEventBus().post(event);
}
private static class ProcessorBundle implements Closeable {
@Nullable
public URLClassLoader classLoader;
public List<Processor> processors = Lists.newArrayList();
@Override
public void close() throws IOException {
if (classLoader != null) {
classLoader.close();
}
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,813 |
\section{Introduction}
S.~S.~Magliveras has described in~\cite{A} a symmetric key
cryptosystem, called PGM (for Permutation Group Mappings), which is
based on \emph{logarithmic signatures} (also known as \emph{group
bases}) for finite permutation groups.
In~\cite{NewApproaches},
S.~S.~Magliveras, D.R.~Stinson and Tran van Trung have proposed two
public key cryptosystems $\mathrm{MST}_{1}$ and $\mathrm{MST}_{2}$,
which are based on logarithmic signatures. An implementation of a
symmetric block cipher TST based on these ideas is described
in~\cite{Hor, Impl}.
These cryptosystems are based on certain round (or encryption)
functions, the PGM transformations, which are the permutations on the
set $\Set{1, 2, \dots, \Order{G}}$, where $G$ is a finite group,
induced by
\emph{exact-transversal logarithmic signatures} on $G$
(these are also known as
\emph{transversal group bases}; see
Section~\ref{sec:prelim} for the relevant definitions).
In~\cite{AlgebraicProperties}, S.~S.~Magliveras and N.~D.~Memon
studied the algebraic properties of the group generated by the set
$\Eh$ of PGM transformations, in particular investigating its
size. This is because a small group here would make the cryptosystem
weak, and indeed questions about the size of the corresponding groups
have been asked (and answered) for DES \cite{KRS, CW, WDES}, AES
\cite{WAES}, and other cryptosystems.
S.~S.~Magliveras and N.~D.~Memon have proved
in~\cite{AlgebraicProperties} that the group is as big
as possible, subject to some restrictions.
\begin{theorem}[Magliveras-Memon]\label{thm:mm}
Let $G$ be a finite non-Hamiltonian group.
Suppose the order of $G$ is different from
\begin{equation*}
q, 1 + q^{2}, 1 + q^{3}, \dfrac{q^{k} - 1}{q - 1},
2^{k-1} (2^{k} \pm 1),
11, 12, 15, 22, 23, 24, 176, 276,
\end{equation*}
where $q$ is a prime power and $k$ is a positive integer.
Then the group $\Span{\Eh}$ generated by $\Eh$ is the full symmetric
group $\Sym(\Order{G})$.
\end{theorem}
(Here a group is said to be \emph{Hamiltonian} when all of its
subgroups are normal.)
S.~S.~Magliveras, D.R.~Stinson and
Tran van Trung suggest in~\cite{NewApproaches} that the above Theorem
may in fact hold in more general circumstances. The goal of this short
note is to show that this is indeed the case. We prove
\begin{theorem}\label{thm:c-dv}
Let $G$ be a nontrivial finite group. Suppose $G$ is not cyclic of
order a prime, or the square of a prime.
Then the group $\Span{\Eh}$ generated by $\Eh$
is the full symmetric group $\Sym(\Order{G})$.
\end{theorem}
In a cyclic group of prime order we
have $\Eh = \emptyset$, so the result does not hold. We
deal with the case of cyclic groups of order the
square of a prime in Section~\ref{sec:psquare}.
The key to our approach is an analysis of some PGM
transformations from the point of view of imprimitive group actions
(Section~\ref{sec:blocks}). We are then able to avoid a call to the
classification of $2$-transitive groups (which is where the list of
exceptions in Theorem~\ref{thm:mm} comes in), obtaining an
elementary proof of Theorem~\ref{thm:c-dv} (Section~\ref{sec:proof}).
We are grateful to Andrea Lucchini for a useful reference.
\section{Preliminaries}
\label{sec:prelim}
In this section we recall briefly the definitions we need
from~\cite{AlgebraicProperties, NewApproaches}, and set up some
notation for the rest of the paper. Two convenient references for the
theory of (permutation) groups we use are~\cite{Rob, Cam}. For a
positive integer $n$, we write
\begin{equation*}
\I_{n} = \Set{0, 1, \dots, n-1}
\end{equation*}
Let $G$ be a finite group. Let
\begin{equation}\label{eq:gamma}
\Set{1} = G_{0} < G_{1} < \dots < G_{s-1} < G_{s} = G
\end{equation}
be a chain of subgroups of $G$, with $s \ge 2$. (So there is no such
chain if $G$ is trivial, or if it has prime order.) An
\emph{exact-transversal logarithmic signature} (\etls) for $G$ with
respect to~\eqref{eq:gamma} is an $s$-tuple $\alpha = (\alpha_{1},
\alpha_{2}, \dots, \alpha_{s})$, where each $\alpha_{i}$ is a
bijection between $\I_{\Order{G_{i}:G_{i-1}}}$ and a complete set of
right coset representatives of $G_{i-1}$ in $G_{i}$, for $i = 1, \dots
s$. (These are called \emph{transversal group bases} in \cite{Hor,
Impl}.) In this paper we will only need the case when $s = 2$, so
that~\eqref{eq:gamma} becomes a chain
\begin{equation*
\Set{1} < H < G.
\end{equation*}
Writing $\mu = \Order{H}$ and $\lambda =
\Order{G:H}$ (so that $\lambda \mu = \Order{G}$), we have that
$\alpha_{1} : \I_{\mu} \to H$ is a bijection, and $\alpha_{2}$
is a bijection between $\I_{\lambda}$ and a complete set of right coset
representatives of $H$ in $G$.
Writing $n = \Order{G}$, an \etls\ $\alpha$ with respect to \HG\
establishes a bijection between
$\I_{n}$ and $G$, given by
\begin{equation}\label{eq:breve}
\begin{aligned}
\balpha :\ \I_{n} &\to G\\
x &\mapsto
\alpha_{1}(x_{1}) \cdot \alpha_{2}(x_{2}),
\end{aligned}
\end{equation}
where $x$ is written uniquely as $x = x_{2} + \lambda x_{1}$, with
$x_{2} \in \I_{\lambda}$, and $x_{1} \in \I_{\mu}$.
The map $\I_{n} \to \I_{\mu} \times \I_{\lambda}$, that
maps $x$ to the pair $(x_{1}, x_{2})$, is known
as a \emph{knapsack transformation}~\cite[Def.~2.19]{Hor}. A proof in
Section~\ref{sec:proof} would be slightly smoother using the
(equivalent) knapsack transformation in which the roles of $x_{1}$ and
$x_{2}$ are reversed. We prefer to stick to the conventions
of~\cite{NewApproaches, AlgebraicProperties}, though.
Once an \etls\ $\alpha$ is fixed (see the comments
in~Subsection~\ref{subsec:alpha}), one may consider the set of
permutations of $\I_{n}$ given by
\begin{equation}\label{eq:Eh}
\Eh
=
\Eh_{\alpha}
=
\Set{
\balpha \circ \bbeta^{-1} : \I_{n} \to \I_{n}
\mid
\text{$\beta$ an \etls\ for $G$}}.
\end{equation}
(Here and in the following, we compose maps left-to-right.) This is
the set of PGM transformations mentioned in the statements of
Theorems~\ref{thm:mm}~and
\ref{thm:c-dv}.
Note that if $\gamma$ is a further \etls, we have
\begin{equation*}
(\balpha \circ \bgamma^{-1})^{-1}
\circ
(\balpha \circ \bbeta^{-1})
=
\bgamma \circ \bbeta^{-1}.
\end{equation*}
It follows, as in~\cite{NewApproaches}, that the group generated by
$\Eh$ also contains all permutations
\begin{equation*}
\bgamma \circ \bbeta^{-1} : \I_{n} \to \I_{n},
\end{equation*}
where $\beta, \gamma$ are \etls\ for $G$.
We write $\Sym(X)$ (resp.\ $\Alt(X)$) for the symmetric (resp.\
alternating) group on a set $X$; we write $\Sym(n) = \Sym(\I_{n})$,
and similarly for $\Alt$. In particular, $\Eh \subseteq \Sym(n)$.
\section{Imprimitivity}
\label{sec:blocks}
In this section we analyze the permutations
$\balpha \circ \bbeta^{-1} \in \Sym(n)$, where $\alpha$ and $\beta$
are \etls\ of a certain form, from the point of view of
\emph{imprimitive group actions}.
Our arguments apply to the case when $\alpha$ is a fixed \etls\ with
respect to \HG, and $\beta$ is another \etls\ with respect to \HG,
obtained from $\alpha$ via certain transformations, which we now
describe.
We consider the partition of $G$ in
the right cosets of
$H$, and certain transformation on $G$ that move cosets to
cosets. The first such transformation is obtained by reordering the
coset representatives in $\alpha$. That is, given a permutation $\tau
\in \Sym(\lambda)$, we obtain a new \etls\ $\beta$ by setting
$\beta_{1} = \alpha_{1}$, and then $\beta_{2}(x_{2}) = \alpha_{2}(x_{2}
\tau)$, for $x_{2} \in \I_{\lambda}$. From~\eqref{eq:breve} we have $x
\bbeta = \beta_{1}(x_{1}) \cdot \beta_{2}(x_{2}) = \alpha_{1}(x_{1})
\cdot \alpha_{2}(x_{2} \tau)$. In other words $x
\bbeta = (x \breve{\tau}) \balpha$, where $x
\breve{\tau} = (x_{2} + \lambda x_{1}) \breve{\tau} = x_{2} \tau + \lambda
x_{1}$; that is, $\bbeta = \breve{\tau} \circ
\balpha$.
All these transformations $\breve{\tau}$ act \emph{imprimitively} on
$\I_{n}$. We recall that if a group $S$ acts transitively on a set
$X$, then we say that $S$ acts \emph{imprimitively} on $X$ (or simply
that $S$ is \emph{imprimitive}) if there is a partition $\mathcal{P}$ of $X$,
called a \emph{block system}, whose elements, called \emph{blocks},
satisfy the following properties:
\begin{enumerate}
\item $S$ maps an element of $\mathcal{P}$ onto another element of
$\mathcal{P}$ (it follows in particular that all elements of
$\mathcal{P}$ have the same order);
\item the elements of $\mathcal{P}$ are proper subsets of $X$, containing at
least two elements.
\end{enumerate}
One says that $S$ \emph{respects} the block system $\mathcal{P}$ on
$X$. (See for instance~\cite[Sect.~1.9]{Cam}~or \cite[Sect.~7.2]{Rob},
for further details. We regret that we are using the term
\emph{block} in a sense that is different by that of
\cite{AlgebraicProperties, NewApproaches}, but the terminology we use
is well established in the context of permutation groups.)
The
transitive group $S$ is said
to be \emph{primitive} if it is not imprimitive. It is an easy fact
that a $2$-transitive
group is primitive (see~\cite[7.2.4]{Rob}~or
\cite[Theorem~1.7]{Cam}).
In our context,
the $\breve{\tau}$ act on $\I_{n}$, respecting the block system on
$\I_{n}$ given by the blocks
\begin{equation}\label{eq:blocks}
B_{x_{2}} = \Set{x_{2} + \lambda x_{1} : x_{1} \in \I_{\mu}},
\end{equation}
for $x_{2} \in \I_{\lambda}$.
Now note that $\balpha \circ \bbeta^{-1} =
\balpha \circ \balpha^{-1} \circ \breve{\tau}^{-1} =
\breve{\tau}^{-1}$. We can then forget about $\alpha$ and $\beta$, and
consider only the $\breve{\tau}^{-1}$. We call these transformations
$\breve{\tau}^{-1}$ (or the $\breve{\tau}$, which is the same) the
\emph{blockwise permutations} of the block system $B_{i}$.
The second type of transformations arise from permutations within a
single coset. Choose a fixed coset
representative $\alpha_{2}(z_{0})$, for some $z_{0} \in \I_{\lambda}$,
and a fixed element $h \in H$, and
consider the $\beta$ that coincides with $\alpha$, but for
$\beta_{2}(z_{0}) = h \cdot \alpha_{2}(z_{0})$. We have $x \bbeta = x
\balpha$, except when $x = z_{0} + \lambda x_{1}$, when we have
\begin{align*}
x \bbeta = \beta_{1}(x_{1}) \cdot \beta_{2}(z_{0})
= \alpha_{1}(x_{1}) \cdot (h \cdot \alpha_{2}(z_{0}))
= (\alpha_{1}(x_{1}) \cdot h) \cdot \alpha_{2}(z_{0}).
\end{align*}
Now, given a group $H$, the group homomorphism $H \to \Sym(H)$ given
by $h \mapsto (k \mapsto k \cdot h)$ is called the \emph{regular
representation} of $H$. We write $\tau_{h}$ for the permutation of
$\I_{\mu}$ induced by the image of $h$ under the regular
representation, via the bijection $\alpha_{1} : \I_{\mu} \to H$. In
other words, for $x_{1} \in \I_{\mu}$ we write $\alpha_{1}(x_{1})
\cdot h = \alpha_{1}(x_{1} \tau_{h})$. In
this setting, we have $\bbeta =
\breve{\tau}_{z_{0}, h} \circ \balpha$, where $\breve{\tau}_{z_{0},
h} = \balpha \circ \bbeta^{-1}$ is the identity on all blocks, except
that on the block
$B_{z_{0}}$ it will act as $(z_{0} + \lambda x_{1})
\breve{\tau}_{z_{0}, h} = z_{0} + \lambda (x_{1} \tau_{h})$.
We call these transformations the \emph{regular
permutations} of the block $B_{z_{0}}$. Clearly, they also respect the
block system~\eqref{eq:blocks}.
Note that the combination of the two classes of transformations we
have described go under the name of \emph{monomial transformations}
in~\cite{AlgebraicProperties}. Monomial transformations alone would yield
$1$-transitivity; however, we will be proving a stronger statement in
Section~\ref{sec:proof}.
The third class of transformations occurs when we obtain $\beta$ from
$\alpha$ by permuting the elements of $H$, that is, by
taking $\beta_{2} = \alpha_{2}$, and then $\beta_{1}(x_{1}) = \alpha_{1}(x_{1}
\tau)$, where $\tau \in \Sym(\mu)$. This yields, proceeding
as above, transformations of the form $x
\breve{\tau} = (x_{2} + \lambda x_{1}) \breve{\tau}= x_{2} + \lambda
(x_{1} \tau)$. In other words, these \emph{diagonal permutations} act
with the same permutation at the same time on all the blocks. (Here
we regard elements in different blocks to be the same if they have the
same $x_{1}$ coordinate.) Again,
these transformations respect the block system~\eqref{eq:blocks}.
\section{Proof of Theorem~\ref{thm:c-dv}}
\label{sec:proof}
\subsection{$2$-transitivity}
\label{subsec:2-trans}
We begin with showing that $\Span{\Eh}$ acts $2$-transitively on $G$.
Fix a nontrivial, proper subgroup $H$ of $G$, and consider
the setting of Section~\ref{sec:blocks}. If $x, x' \in \I_{n}$ are in
different blocks, and $y, y' \in \I_{n}$ are also in different blocks,
there is a composition of blockwise and regular permutations that
carries $x$ onto $y$ and $x'$ onto $y'$. In fact, first use a
blockwise permutation to carry $x$ within the block to which $y$
belongs, and $x'$ within the block to which $y'$ belongs. (To avoid
complicating notation unnecessarily, we keep the names $x$ and $x'$
for the images of $x$ and $x'$ under this permutation.) Then use
regular permutations within the two blocks to carry $x$ onto $y$ and
$x'$ onto $y'$.
If $x$ and $x'$ are in the same block $B$, and $y$ and $y'$ are in the
same block $C$, first apply a blockwise permutation to carry $B$ onto
$C$, and then use a diagonal permutation (which induces the full
symmetric group on each block) to carry $x$ onto $y$ and $x'$ onto $y'$.
We are left with the case when $x$ and $x'$ are in the same block $B$,
while $y$ and $y'$ are in different blocks. Clearly the
transformations of Section~\ref{sec:blocks} are not enough here,
as a $2$-transitive group is primitive.
Given the above, however, it will be enough to find an element of
$\Eh$ that fixes $x'$, and moves $x$ out of $B$. By applying a
blockwise permutation, and a diagonal one, we may assume that $B =
B_{0}$, the zeroth block of $\alpha$, and $x' = 0$. Suppose thus
$\alpha$ is an \etls\ with respect to \HG, with $\alpha_{1}(0) =
\alpha_{2}(0) = 1$, so that the coset $H \alpha_{2}(0)$
is $H$, and $x' \balpha = 0 \balpha = 1$. Write $x \balpha = h \in
H$. We also consider another nontrivial, proper subgroup $K$, and an
\etls\ $\beta$ with respect to \KG, with $\beta_{1}(0) = \beta_{2}(0)
= 1$. Let $B_{i}'$ be the blocks relative to $\beta$. We will make
more precise choices of $H$, $K$ and $\beta$ later, according to the
properties of $G$.
We note first that since $G$ is nontrivial, and it is not cyclic of
order a prime or the square of a prime, it has at least two distinct
nontrivial, proper subgroups. Moreover, if all the nontrivial, proper
subgroups have the same order $p$, then $p$ is a prime number, and so
$G$ is a (non-cyclic) elementary abelian $p$-group of order $p^{2}$.
Accordingly, we distinguish two cases.
Suppose first that $G$ has two nontrivial, proper subgroups $H$ and
$K$, with $\Order{H} < \Order{K}$. We have thus $\Order{B_{i}'} =
\Order{K} > \Order{B_{0}} = \Order{H}$ for all $i$. If $h \in K$, so
that $h \bbeta^{-1} \in B_{0}'$, we may modify $\beta$ by a diagonal
permutation, so that $0 \bbeta = 1$ still holds, but $h \bbeta^{-1}
\notin B_{0}$, as $\Order{B_{0}} < \Order{B_{0}'}$. We have thus $0
(\balpha \circ \bbeta^{-1}) = 0$ and $x (\balpha \circ \bbeta^{-1})
\notin B_{0}$, as requested. If $h \notin K$, so that $h \bbeta^{-1}
\in B_{i}'$, for some $i \ne 0$, we may modify $\beta$ by a regular
permutation on $B_{i}'$, so that $h \bbeta^{-1} \notin B_{0}$, as
$\Order{B_{0}} < \Order{B_{i}'}$. Here, too, we have $0 (\balpha \circ
\bbeta^{-1}) = 0$ and $x (\balpha \circ
\bbeta^{-1}) \notin B_{0}$.
If $G$ is elementary abelian, of order $p^{2}$, let $H$ and $K$ be any
two nontrivial, proper subgroups. Here we have $B_{i} = B_{i}'$ for all
$i$. As $H \cap K = 1$, and $h \ne 1$, we have $h \notin K$, so that
$h \bbeta^{-1} \notin B_{0}$.
We have thus proved $\Span{\Eh}$ to be $2$-transitive in all
cases.
\subsection{Completion of the proof}
Suppose first that the order $n$ of $G$ is even, and let $H$ be a
subgroup of $G$ of order $2$.
With respect to \HG, a nontrivial
regular permutation on a given block will be a transposition. Since
$\Span{\Eh}$ is $2$-transitive, it follows that $\Span{\Eh}$ contains
all transpositions, and thus $\Span{\Eh} = \Sym(n)$.
Note that in this case, and with this choice of $H$, the permutations
of Section~\ref{sec:blocks} clearly generate the wreath product $\Sym(2)
\wreath \Sym(n/2)$. This is well-known to be a maximal subgroup of
$\Sym(n)$. We have not used this fact here, but our proof of
$2$-transitivity could be read to mean that $\Span{\Eh}$ contains
properly this wreath product.
When $n$ is odd, we begin with showing that $\Span{\Eh}$ contains a
$3$-cycle.
Start with a
diagonal permutation $\sigma$ which is the transposition $(a b)$ on
each block (see the observation at the end of Section~\ref{sec:blocks}). Fix any block $B$.
The regular permutation on the block $B$ induced by a suitable element
of $H$ will be a $p$-cycle of the form $\pi = (a b c \dots)$, for some
$c$. Conjugate $\sigma$ by $\pi$ to get a permutation $\sigma^{\pi}
= \pi^{-1} \sigma \pi$
which is the transposition $(a b)$ on all blocks, except that on block
$B$ it will be $(a b)^{(a b c
\dots)} = (b c)$. We obtain that the product $\sigma^{\pi} \cdot
\sigma$ is the identity on all blocks, except on block $B$,
where it is the $3$-cycle $(b c) (a b) = (a b c)$.
We might now appeal to an observation of C.~Jordan (\cite{Jor},
\cite[Section~5.1, Fact~1]{Cam}) to the effect that a primitive
group that contains a $3$-cycle is either the alternating or the
symmetric group. We then conclude by observing that $\Eh$ contains
an odd permutation, and thus $\Span{\Eh} = \Sym(n)$. This follows
from~\cite[Theorem~5.4]{AlgebraicProperties}: a blockwise permutation
that exchanges just two blocks will be the product of an odd number
$p$ of transpositions.
For completeness, however, we give the short argument (for the case of
$2$-transitive groups) that shows that $\Span{\Eh}$ contains all
$3$-cycles, and thus contains $\Alt(n)$. Let $a, b, c$ be any three
distinct elements of $\I_{n}$. Since $\Span{\Eh}$ contains a
$3$-cycle, and it is $2$-transitive, there are $d, e \in \I_{n}
\setminus \Set{a, b, c}$ such that $(b a d), (b c e) \in
\Span{\Eh}$. If $d = e$, then $(b c d) (b a d)^{-1} = (a b c) \in
\Span{\Eh}$. If $d \ne e$, then $(b a d)^{(b c e)} = (c a d) \in
\Span{\Eh}$, and we argue as in the previous case.
\subsection{A remark on the choice of $\alpha$}
\label{subsec:alpha}
Concerning the definition of $\Eh = \Eh_{\alpha}$ given
in~\eqref{eq:Eh}, we may note that all the transformations
$\balpha \circ \bbeta^{-1}$, that we have considered in this Section
and the previous one, can be taken with respect to a fixed \etls\
$\alpha$, chosen once and for all with respect to \HG, where the
choice of $H$ depends on the properties of the group, as we have just
seen, and we take (just for simplicity) $\alpha_{1}(0) =
\alpha_{2}(0) = 1$.
\section{The case of the cyclic group of order $p^{2}$}
\label{sec:psquare}
If $G = \Span{a}$ is a cyclic group of order $p^{2}$, where $p$ is a
prime, the second part of the argument of
Subsection~\ref{subsec:2-trans} does not work, as $G$ has a unique
nontrivial, proper subgroup $H = \Span{a^{p}}$. In fact, since $\Eh$
consists of the (imprimitive) permutations of
Section~\ref{sec:blocks}, $\Span{\Eh}$ is not $2$-transitive here.
In this case we have the following
\begin{prop}\label{prop:c-dv}
Let $G$ be a cyclic group of order $p^{2}$, where $p$ is a
prime. Then:
\begin{enumerate}
\item
the group $\Span{\Eh}$
is a proper, imprimitive subgroup of $\Sym(p^{2})$;
\item given an \etls\ $\alpha$, there exists a logarithmic signature
$\gamma$ such that
\begin{equation*}
\Span{\Eh \cup \Set{\balpha \circ \bgamma^{-1}}}
=
\Sym(p^{2}).
\end{equation*}
\end{enumerate}
\end{prop}
(The \etls\ $\alpha$ is taken with respect to the only possible choice
\HG.) We refer to ~\cite{NewApproaches} for the general definition of
\emph{logarithmic signatures}. (These are called \emph{group bases} in
\cite{Hor, Impl}.) In the special case of the cyclic group $G$ of
order $p^{2}$ we are considering here, a logarithmic signature is a
pair of (injective) maps $\gamma_{1}, \gamma_{2} : \I_{p} \to G$, so
that each element of $G$ can be written (uniquely) as
$\gamma_{1}(x_{1}) \cdot
\gamma_{2}(x_{2})$, for $x_{1}, x_{2} \in \I_{p}$. As in the case of
an \etls, we
obtain a bijection $\bgamma : \I_{p^{2}} \to G$ as $(x_{2} + p x_{1})
\bgamma = \gamma_{1}(x_{1}) \cdot
\gamma_{2}(x_{2})$.
We choose
$\alpha_{1}(x_{1}) = a^{p x_{1}}$ and $\alpha_{1}(x_{2}) = a^{x_{2}}$,
so that $x \balpha = a^{x}$. Here $B_{0}$ is the set of multiples of
$p$ in $\I_{p^{2}}$. Then we take the logarithmic signature
$\gamma$ defined by $\gamma_{1}(x_{1}) = a^{x_{1}}$ and
$\gamma_{1}(x_{2}) = a^{p x_{2}}$. One sees that $0 (\balpha
\circ \bgamma^{-1}) = 0$, and $p (\balpha
\circ \bgamma^{-1}) = 1$, so that $\balpha
\circ \bgamma^{-1}$ fixes $0$, and takes $p \in B_{0}$ to an element
$1 \notin B_{0}$, as requested.
This yields that the group $\Span{\Eh \cup \Set{\balpha \circ
\bgamma^{-1}}}$ is $2$-transitive; the rest of the proof follows
as in Section~\ref{sec:proof}.
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<?xml version="1.0" encoding="utf-8"?>
<LinearLayout xmlns:android="http://schemas.android.com/apk/res/android"
android:orientation="vertical"
android:layout_width="match_parent"
android:layout_height="match_parent">
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android:id="@+id/checkListCompletedOn"
android:layout_gravity="center"
android:layout_width="wrap_content"
android:layout_height="wrap_content" />
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260Wp solar photovoltaic module of REC brand, model 260PEB with black aluminium frame. High quality long life with reliable power output. The REC Peak Energy Series combines compliance with the highest standards of quality with elegant design and performance.
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"redpajama_set_name": "RedPajamaC4"
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Q: Parse strings with different formats of decimal numbers to float Is there in python a simple way to parse multiple different formats of possible decimal numbers into floats? Following formats could be used (usually amounts):
111 -> 111.00
111.00 -> 111.00
111,00 -> 111.00
11,111.00 -> 11111.00
11.111,00 -> 11111.00
11,111,111.00 -> 11111111.00
11.111.111,00 -> 11111111.00
111.111 -> 111111.00
111,111 -> 111111.00
111.111,00 -> 111111.00
111,111.00 -> 111111.00
At the moment I can only think of looking if there are different special characters ("," and ".") and then look which one is the decimal separator and so on. But that would be a lot of if/else I guess. I also tried locale.atof but the problem is, that I don't know which format the string is. I'm wondering if there is an easier and cleaner way.
A: This isn't the most efficient way to do it, but using regular-expression, we can split the variable into their different parts, and count the "last one" of them, to decide how to do the translation.
import re
values = """111
111.00
111,00
11,111.00
11.111,00
11,111,111.00
11.111.111,00
111.111
111,111
111.111,00
111,111.00"""
values = [i for i in values.split("\n")]
for value in values:
value = re.split(r"\.|,", value.strip())
if len(value[-1]) <=2 and len(value) > 1:
newVal = float("".join(value[:-1]) + "." + value[-1])
else:
newVal =float("".join(value))
print(newVal)
Output:
111.0
111.0
111.0
11111.0
11111.0
11111111.0
11111111.0
111111.0
111111.0
111111.0
111111.0
EDIT:
I didn't sanitize my inputs, the first version included some whitespace, throwing the translation into a loop.
| {
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Q: Django: Moving from XAMPP to Django questions I've worked with XAMPP, WAMPP, MAMPP, etc and am starting to look at Django.
A majority of the work we do is very CMS orientated; although we've been told not to use third-party CMS' (mainly because of user's find them hard to use, and other issues), I've found that I can code a very simple CMS using Cake, CodeIgniter or one of the other PHP frameworks.
And yet, I'm getting increasingly frustrated with the amount of coding I need to do just to get something up and running, and I've been told that Django is a good Python framework to use. It also seems to get a lot of buzz from reddit.
I have some concerns and queries about moving from XAMPP to Django.
1) Security
Any web app should be coded defensively. Over the past few years we've seen a movement towards protecting against XSS, SQL injections, Cross site forgeries, session fixation, session hi-jacking, cookie hi-jacking; the amount of security one needs can be overwhelming.
What things does Django do to prevent/limit XSS, SQL injections, Javascript injections, and santizing input; one normally associates with securing PHP web apps? Is it something I need to worry about, or does Django do all this stuff out of the box.
2) What goes in the /www/ public folder?
In a manual I read it said not to put manage.py or the other .py stuff in the main webroot, so this means I put everything outside of the webroot; so what goes in there?
Do I put the /templates/ directory inside the webroot? How does the server know what to run?
3) Can I still use .htaccess on Django projects? I am familiar with Apache and often use it to do routing, or blocking off bad bots, but will using .htaccess still work?
4) Cronjobs
Do cronjobs still work with Python/Django projects?
5) Running Third party perl/other scripts
In PHP you can use other libraries such as the curl library, ffmpeg, ImageMagik as well as many others; can I still use these libraries with Python/Django?
6) Admin screen
Django gives you an out-of-the-box admin screen; is this only for development purposes or can it put live? I am concerned about any the security of the admin screen.
7) Integration with Discuss, Facebook, Twitter, OpenID, captcha, etc.
There are libraries in PHP that help integrate DisQuss, Facebook, Twitter; but is it relatively easy to do an integration with these and other third party apps?
8) E-commerce, SSL
Are there many e-commerce sites that use Django? I've seen a lot of CMS/Blog type software but not many e-commerce sites. By which I mean, shopping card, Protx/Paypal or Worldpay integration.
That's another thing; there are sandboxes for Protx, Paypal, Worldpay etc for PHP -- but are there any for Django?
9) Is it worth it?
Is it worth moving to Django from an XAMPP background? Will it really make things faster, or is that just marketing hype?
Thanks.
A: *
*Security. The Django core team are very security-conscious, and have taken great care to make things like SQL injection impossible. The next version, 1.2, includes a whole new cross-site request forgery protection library. Obviously, you still need to be aware of these when developing your application, but Django does a lot to help you.
*What goes under /www/public: Nothing. Django doesn't work via the normal Apache serving mechanism: it hooks into (preferably) mod_wsgi, which needs a single file which then tells it to run the rest of the code. The templates can go anywhere, and are pointed to by your Django settings file, but again aren't served directly by Apache.
*.htaccess: You don't really need it, because of point 2: you're not serving things in a filesystem hierarchy. The best way to do it is to set up vhosts and manage things that way.
*Cron jobs: Absolutely. Django is just Python, and you can easily run Python scripts via cron. Django allows you to set up custom command scripts which initialise the ORM and give you access to anything you would need.
*Libraries: Again, because Django is Python, you get access to the huge amount of Python libraries that are out there. For curl, Python has urllib; for ImageMagick, it has PIL; and no doubt there are equivalents of ffmpeg too.
*Admin: Again, security has been thought of from the beginning. Opinions differ as to whether you should use the admin only for your expert users, or customise it and allow access for all users; I've had a lot of success using it as the basis for my custom CMS interfaces.
*Facebook, etc: Yes, there are libraries for all of these.
*E-commerce: There is a whole e-commerce project, Satchmo, written in Django. Libraries exist to interface with all the payment providers.
*Is it worth it? Only you can tell. My experience working alongside a range of developers who have moved from PHP is that they've enjoyed the experience and became much more productive.
A: On SQL Injections: Django uses an ORM, which takes care of SQL injection protection, and you will rarely write you own SQL. If you do, just follow the instructions on how to pass parameters to raw queries and prevent SQL Injections.
There is an entire chapter on the django book about security that should answer all your questions.
On what goes into /www/: anything that is not code? The concern is to not put the python code there.
On .htaccess: Yes, it should still work (for any non Django resources as Daniel points out).
On cronjobs: what do you mean?
On Libraries: Python - the language you will use with Django - is rich in libraries that probably provide the same functionality you are used to. This is a key point: you will need to learn Python well to benefit the most from Django.
On the admin interface: This is actually the thing that will probably help you the most, judging from your question. They are customizable (within some limits) and they really give the staff (it is not intended for public users, but for staff users) the basics of CRUD for your database models. It is a time saver. You might need to write your own templates for advanced functionality, but for most simple CRUD aimed at staff (which is usually the point of a CMS) it is very useful and easy to set up.
On integration: Check Pinax for a group of applications that provide extra functionality. There is a rich and diverse universe of integration solutions out there. It is not unusual to find questions here in SO about django + facebook and others.
On E-commerce: Check Satchmo out.
Is it worth it: Now, I have no experience with XAMPP. I know that I like Python better than both Perl and PHP (and Java, for that matter). I know that as a framework Django is simpler to use, faster to deploy than anything I used before.
My suggestion is the age old: go build a simple project and make up your own mind. You are the only one in position to decide if Django is the framework for you.
An older question on SO discusses some Django limitations. My answer to that might be helpful to you too.
A: I recently moved to developing any new projects in Django, coming from a PHP background. Here are my thoughts on your questions.
1) Security
Strings sent to templates is escaped by default, which takes care of most of that. Since you're using an ORM, SQL injection shouldn't be an issue unless you build raw queries for some reason.
2) What goes in the /www/ public folder?
Django doesn't use a file hierarchy for URLs like a typical PHP setup. The server knows what to run from your urls.py and settings.py pointer to the template folder.
3) Can I still use .htaccess on Django projects? I am familiar with Apache and often use it to do routing, or blocking off bad bots, but will using .htaccess still work?
As noted above, it works for static content just the same. For dynamic pages, you'd want to implement some other form of authentication or redirection for clients you want to block, as far as I know.
4) Cronjobs
There's no reason why you can't use cron for whatever, as you still have a normal Linux system.
5) Running Third party perl/other scripts
You'll want to use the Python versions of those libraries, of course. For instance
FFMpeg
PythonMagick
I replaced most of my need for Curl with the built-in urllib and urrlib2 libraries, but there is also PyCurl if you need it.
6) Admin screen
The Admin screen is intended to be used by your own admins, i.e. site staff. It may be possible to do so, but it's not supposed to be the scaffolding on which you build your public facing project.
7) Integration with Discuss, Facebook, Twitter, OpenID, captcha, etc.
There are a lot of people out there using Python and Django, and I haven't had any problem finding libraries. In my experience there is a bit less support for something than PHP, but what is there is often higher quality.
8) E-commerce, SSL
I haven't tried payment integration, so I can't say. Not sure about the other sites, but the Paypal Sandbox is run by Paypal, isn't it? I don't think it's related to what you're using on the server, so sure, you can access it like normal.
9) Is it worth it? Is it worth moving to Django from an XAMPP background? Will it really make things faster, or is that just marketing hype?
I moved to Django because Python is truly a more compelling language than PHP. Will it make things faster? I'm not sure what the advantages in that respect would be for Django vs.the PHP MVC frameworks. There are no magic bullets.
You do have to keep in mind that you're not just learning a new framework, but also a new language. There will be a bit of a learning curve if you've never used Python before. but I've found both Python and Django to be fairly easy to learn. The clean design of the language is fantastic and Django is veryt well designed, too. I do feel that it's boosting my productivity. I've found snippets for or articles about most everything I need to do in Django as I've been learning, so adapting has been pretty simple.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,730 |
Q: pthread_create segfault (buffered reader example) I have write a simple solution to the buffered reader problem for my OS class, but after a few successful producer threads, I get a segfault. Output, bt and code below:
Output:
Producer 1 exiting
Producer 2 exiting
Producer 3 exiting
Segmentation fault (core dumped)
Thread BT (using GDB thread apply all where):
Thread 2 (Thread 0x7ffff77f6700 (LWP 8310)):
#0 0x0000000000000000 in ?? ()
#1 0x00007ffff7bc4182 in start_thread (arg=0x7ffff77f6700)
at pthread_create.c:312
#2 0x00007ffff78f147d in clone ()
at ../sysdeps/unix/sysv/linux/x86_64/clone.S:111
Thread 1 (Thread 0x7ffff7fd3740 (LWP 8299)):
#0 0x00007ffff78eba27 in mprotect () at ../sysdeps/unix/syscall-template.S:81
#1 0x00007ffff7bc4f21 in allocate_stack (stack=<synthetic pointer>,
pdp=<synthetic pointer>, attr=0x7fffffffde20) at allocatestack.c:650
#2 __pthread_create_2_1 (newthread=0x6021e0, attr=<optimized out>,
start_routine=0x0, arg=0x0) at pthread_create.c:500
#3 0x00000000004009cf in start_producer () at 6-2.c:75
#4 0x00000000004007e9 in main () at 6-2.c:29
Code:
#include <pthread.h>
#include <semaphore.h>
#include <stdio.h>
#include <stdlib.h>
void init();
void start_producer();
void start_consumer();
void *produce();
void *comsume();
int buffer_count;
int max_buffers;
int producer_count;
sem_t *mutex;
sem_t *full;
sem_t *empty;
int main() {
init();
int i = 0;
for(i = 0; i < 3; i++) {
start_producer();
}
for(i = 0; i < 3; i++) {
start_consumer();
}
return 0;
}
void init() {
buffer_count = 0;
max_buffers = 3;
producer_count = 0;
mutex = malloc(sizeof(sem_t));
full = malloc(sizeof(sem_t));
empty = malloc(sizeof(sem_t));
sem_init(mutex, 0, 1);
sem_init(full, 0, 0);
sem_init(empty, 0, max_buffers);
}
void *produce() {
sem_wait(empty);
sem_wait(mutex);
producer_count++;
printf("Producer %d exiting\n", producer_count);
sem_post(full);
sem_post(mutex);
return 0;
}
void *consume() {
sem_wait(full);
sem_wait(mutex);
printf("Consuming produced value: %d\n", producer_count);
producer_count--;
sem_post(empty);
sem_post(full);
return 0;
}
void start_producer() {
pthread_t *thread = malloc(sizeof(pthread_t));
if(pthread_create(thread, NULL, produce(), NULL) != 0)
printf("\tError creating producer thread.\n");
}
void start_consumer() {
pthread_t *thread = malloc(sizeof(pthread_t));
if(pthread_create(thread, NULL, consume(), NULL) != 0)
printf("\tError creating consumer thread.\n");
}
I understand that this is probably a newb issue. I'm having a hard time debugging this. Thanks in advance for your help.
A: You passed NULL to the third argument of pthread_create, which have a big chance to cause Segmentation Fault.
Try these lines
if(pthread_create(thread, NULL, produce, NULL) != 0)
if(pthread_create(thread, NULL, consume, NULL) != 0)
instead of
if(pthread_create(thread, NULL, produce(), NULL) != 0)
if(pthread_create(thread, NULL, consume(), NULL) != 0)
(don't call produce and consume, pass their pointer instead)
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,914 |
WellCare, the managed care company in which Georgia Medicaid recipients are automatically enrolled, got about 4 billion from taxpayers, but that, apparently, may not have been enough. If you want to know why taking on insurance and managed care companies is part of what must happen if we hope to make progress on health care,, read the following about the raids on Wellpoint's Florida HQ. Greed knows no bounds.
WellCare Faces Lawsuits, Calls To End Enrollment
By CAROL GENTRY, The Tampa Tribune
10/30/07: State Halts Expansion Of Troubled WellCare
<http://www.tbo.com/news/metro/MGBFD6P5F8F.html>
10/27/07: Investors Continue WellCare Exodus
<http://www.tbo.com/news/money/MGBIFBSF98F.html>
10/26/07: WellCare Stock Plunges In Wake Of Raid
<http://www2.tbo.com/content/2007/oct/26/na-wellcare-stock-plunges-in-wa
ke-of-raid/>
10/25/07: FBI Agents Raid WellCare
<http://www.tbo.com/news/nationworld/MGBRPJ6R68F.html>
TAMPA - As its share price continued to slide, WellCare Health Plans
dealt Tuesday with a flurry of other problems, led by a halt to its expansion plans in Florida.
It was the sixth day since a coordinated raid by federal and state agents on the Tampa headquarters of the insurer, which has 2.3 million
members in its drug and HMO-style plans - all of them Medicare and Medicaid beneficiaries.
None of the agencies involved in the investigation has said what they
were looking for that day, or why. Neither has the company. But that
doesn't halt speculation, or the rush to the courthouse.
On Tuesday, a New Jersey investor filed suit in Tampa - at least the
fourth federal case so far - that accuses the officers and directors of
falsely inflating financial results to promote a run-up in the stock
price, and then cashing in on their stock options to become
millionaires.
The suit, Rosky v. Farha, names every current member of the board except
former Sen. Bob Graham. He has been on the board only a short time and was not at last week's quarterly meeting, which was interrupted by 200 agents from the FBI, state Medicaid fraud unit and Medicare inspector
>general's office. Todd Farha is WellCare's chairman and chief executive
officer.
WellCare filed a form with the Securities and Exchange Commission on
Tuesday afternoon saying that the investor who sued lacks standing to
pursue the case. The statement did not address the accusations of
financial misstatements or unjust enrichment.
A more immediate problem for WellCare cropped up Monday night when the
Agency for Health Care Administration said it was placing a hold on the
company's request for expansion in the state, pending the outcome of the
investigation. AHCA provided details on the expansion plans Tuesday, releasing letters that outlined the company's requests.
WellCare had asked to begin enrolling Medicaid beneficiaries in an additional 11 counties in Florida.
Already the largest Medicaid contractor in Florida, WellCare
subsidiaries Staywell and Healthease had planned to begin enrolling low-income people in Citrus, Lake and Hendry counties on Dec. 1 and in Hernando and Sumter counties Jan. 1. Healthease planned to move into
five counties in North Florida and Indian River County on the Atlantic
coast in coming months.
Currently WellCare has Medicaid permission to operate in 34 counties in
WellCare also sought permission to expand its enrollment in Duval
County, where the state is conducting a Medicaid Reform project, from a maximum of 3,500 to 6,000 people. That action, too, is on hold.
In announcing the moratorium on WellCare's expansion Monday night, AHCA
Secretary Andrew Agwunobi said in a release that he was taking 'precautionary measures.' He asked that beneficiaries and health care
providers call 1-888-419-3456 if they had concerns about any Medicaid
managed-care plan.
The release also said the agency's inspector general and Medicaid director would review the state's managed-care contracting policies to
see whether they needed strengthening. AHCA spokesman Doc Kokol said
Tuesday that this, too, was merely a precaution, that there was no reason to assume that the questions about WellCare had anything to do with its contract.
Meanwhile, about 2,000 miles north of the company's base, the leader of a consumer group called on federal Medicare officials to stop
automatically assigning low-income beneficiaries into WellCare drug and
HMO plans across the country until the unspecified allegations of financial impropriety are resolved.
'This is something that's pressing for both beneficiaries and
taxpayers,' said Judith Stein of Willimantic, Conn., executive director of the non-profit Washington-based group Center for Medicare Advocacy. WellCare received more than $4 billion from taxpayers last year.
The people about whom Stein is concerned are low-income Medicare beneficiaries who are assigned to a drug plan when they don't select one
on their own. Of the 1.1 million WellCare members from Medicare, about 452,000 arrived at the company through auto-enrollment, according to the Centers for Medicare and Medicaid Services. CMS calls the process 'auto-facilitation.'
About 38,000 of the auto-enrolled Medicare beneficiaries live in Florida, according to CMS data. Florida's Medicaid program also uses auto-enrollment for many in the state program for low-income families.
Stein says it doesn't make sense for CMS to add more money to the hundreds of millions of dollars it sends the company each month while
Its own inspector general and others are investigating how the money is used.
CMS had no immediate response to Stein's request.
Both Florida and Georgia Medicaid programs - WellCare's two largest
customers - also auto-enroll. A Florida consumer group, Community Health
Action Information Network, said Medicaid should halt auto-assignment pending the outcome of the investigation, but spokesmen for both state Medicaid plans said they have no plans to do so.
On Tuesday, the company's stock continued its slide, falling $6.58, or
23 percent, to $22.04. The shares had reached above $120 last week but
plummeted after the Oct. 24 raid by federal and state officials.
Reporter Carol Gentry can be reached at cgentry@tampatrib.com or (813)
>259-7624.
Where were you guys when i was battling these legalized, behemoth bad guys in 2002 & 2004?
I'm having trouble with wellcare in georgia right now. I'm a pregnant benificiary, and they actually DENIED my prenantal vitamins. When my doctor appealed their decision, they denied it again. Then, I developed chronic heartburn. My doctor tried to prescribe something for it, due to the fact I was constantly vomiting, and had tried everything over the counter, and nothing was helping. Once again, they denied it. Now, my iron has dropped to a dangerous level, and my doctor prescribed an iron supplement, which was once again denied. I've filed three seperate grievances, and nothing has been done. I work full time and pay taxes so this service is available to me when needed, so much for "managed health". They are just trying to keep me sick so they don't have to spend money. I've only gained 14 pounds with this pregnancy (and i'm less than 5 weeks from my due date), and they are guessing that my baby will only weigh between 4-5 pounds. Now here's the question...Is Wellcare going to deny services to my baby when he is born???...something to think about. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,523 |
Ancistargis matsunagaensis är en ringmaskart som först beskrevs av Kitamori 1960. Ancistargis matsunagaensis ingår i släktet Ancistargis och familjen Pilargidae. Inga underarter finns listade i Catalogue of Life.
Källor
Havsborstmaskar
matsunagaensis | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,798 |
These are some real problems experienced by WD2500JD-75HBB0 and Western Digital hard drives in general that we see in our lab. We are not affiliated in any way with hard drive manufacturers. All the information below is based solely on our experience and we do not make any claims regarding reliability of the specific model. We see mostly failed drives in our lab and therefore we don't have complete statistics.
The model number is Western Digital(WD2500JD-75HBB0). This drive was removed from an external drive and the actual drive enclosure was wrapped in some sort of bizarre EMI cage. I think - at this point - the problem is a stuck head. Originally, the HD would try to servo a couple times and then quit; in retrospect, the reserve area was probably damaged.
If you experience any of the symptoms described above with your Western Digital WD2500JD-75HBB0 please feel free to contact us to get upfront quote on data recovery from your failed drive. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,803 |
We automatically start feeling relaxed when we enter into a massage or spa center. We get an ambiance specially created to bring the relaxation that every massage center creates to set a perfect mood with a unique setting. A massage therapy amid the soothing surrounding will lift your mood for sure as you can enjoy lots of benefits with the complete relaxation.
People choose a massage therapy to pamper their skin and body when they feel stressed or exhausted. A person who is too occupied with their everyday tasks which are quite stressful. A massage therapy is very helpful to relieve the muscle tension.
What is a massage therapy?
According to the experts, a massage therapy is a normal form of rubbing, pressing and manipulating your skin, muscles, tendons, and ligaments. A therapist uses his fingers and hands which may relax your mind and body. It establishes a connection between mind and body so that you can feel energized and rejuvenated. Also, it is known that massage treats a number of psychological issues such as stress, depression, hyperactivity and post-traumatic stress.
Hot Stone Massage: Hot stone massage therapy is ideal for everyone who wants to improve their well bring in a pleasing environment. In this process, warm stones are placed strategically on your back which increases the penetrating power to treat sore and stiff muscles. The hot stones which are commonly used are Basalt Stones. These stones provide a profound relaxation effect and also, helpful for you to relax your muscle. The heat causes blood vessels to improve the circulation as well as releases accumulated toxins.
Swedish Massage Therapy: This is the most common type of massage therapy where the therapists use long smooth strokes. Along with the firm strokes, kneading and circular movements on superficial layers of muscle with the help of a lotion will be very helpful in toning your muscle. It is very gentle and relaxing which is good for the beginners who never tried any massage therapy.
Deep Tissue Massage: It is used extensively to release chronic tension patterns in muscles as well as toxins. This is the best massage for people under a lot of stress and plans a massage therapy on a regular basis. Using a deep pressure on spammed and knotted muscles, fibrous adhesion and trigger points, it brings a profound relaxation. With the specific kneading and gliding strokes and soothing, it gives you a relaxed feel.
Aromatherapy Massage: In aromatherapy massage a number of essential oils and natura ingredients are used to relax your mind, makes you stress-free and balance your overall physical stress. Lavender is one of the most important essential oils which are used in aromatherapy massage to release your stress and supports your emotional component.
Thai Massage: This massage type used to energize your body by using the gentle pressure on the specific points of your body. It includes compressions and stretches into a proper sequence of the body posture. It works just like Yoga which works more ideally than other practices to reduce stress and improves flexibility of your body.
A practice known as massage therapy works by providing a great relief to your muscles as well as other parts of your body so that you can easily experience a pain-relief in your muscle tension.
1234 19th St NW Suite 600 Washington, DC 20036. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,520 |
Q: Dúvida com validação do conteúdo jquery; Tenho um javascript que deveria executar apenas quando o conteúdo value do edit for diferente de null ou vazio, mais o mesmo está executando sempre.
<script type="text/javascript">
function AdicionarClasseMenuSelecionado(){
var menuatual = document.getElementById('MenuSelecionado').value;
var abaatual = document.getElementById('AbaSelecionado').value;
alert(menuatual);
alert(abaatual);
//remove todos os active do cabecalho
$("ul.nav.nav-tabs li").removeClass("active");
//remove todos os active do content
$(".tab-content div").removeClass("active in");
//adiciona o active que retornou
$("#"+menuatual).addClass("active in");
$("#"+abaatual).addClass("active");
};
$(document).ready(function(){
var menuatual = document.getElementById('MenuSelecionado').value;
if (menuatual != null && menuatual != undefined) {
AdicionarClasseMenuSelecionado();
} else {
alert("não foi enviado nenhum menu");
}
});
</script>
Html:
<input name="MenuSelecionado" type="hidden" value="null" id="MenuSelecionado">
<input name="AbaSelecionado" type="hidden" value="null" id="AbaSelecionado">
A: Veja os valores abaixo:
// false
var menuatual = 0; // zero é false
// true
var menuatual = 1; // número diferente de zero não é false
// false
var menuatual; // undefined é false porque não tem valor atribuído
// true
var menuatual = "0"; // string (mesmo o zero) não é false
// false
var menuatual = ""; // vazio é false, mas não é null nem undefined
// true
var menuatual = " "; // espaço é string, não é false
Analisando a sua comparação:
menuatual != null && menuatual != undefined
Você está dizendo que menuatual não pode ser null e também não pode ser undefined. Então se for vazio (menuatual == '') irá validar, porque vazio não é null nem undefined.
O que você precisa fazer é apenas verificar se a variável menuatual é true simplesmente colocando apenas ela no if:
if(menuatual){
// menuatual é true
// ou seja, não é vazio, não é null, não é undefined
// e não é o número zero (0 é diferente de "0")
}else{
// menuatual é false
}
Ou seja, você espera que a variável não seja undefined, null, vazia ou o número 0.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,070 |
The Protestant cemetery in Augsburg () on Haunstetter road in Hochfeld district of Augsburg was established in 1534 by the City of Augsburg. The cemetery is still in operation and used for burials. It is currently the oldest cemetery in Augsburg.
History and description
The Protestant Cemetery was established in 1534 by the City of Augsburg. Since the Peace of Westphalia in 1648 which ended the Thirty Years' War, the cemetery is owned by the Protestant parishes of the city of Augsburg; St. Anna, St. James, St. Ulrich and Holy Cross churches. In 1700, the administration building was built. The cemetery chapel was built in 1825 by Johann Michael Voit. In addition to the chapel, the morgue building was built in 1837.
In the cemetery, there are numerous grave monuments dating back to 17th century with elaborate tombs of classicism and the Gothic Revival architecture.
A special feature of the cemetery is its collection of old grave books with burial registers dating back to 1658, which have survived until today.
Notables burials
Notables buried include:
Elias Holl (1573–1646), architect of early German Baroque architecture
Anna Barbara von Stetten (1754–1805), philanthropist
Karl Albert Gollwitzer (1839–1917), architect
Gallery
Source and references
External links
Pictures of Augsburg Protestant Cemetery
Cemeteries in Germany
Protestant Reformed cemeteries
Augsburg
1534 establishments in the Holy Roman Empire
Lutheran cemeteries in Germany | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,224 |
<?php
require 'conexao.php';
// Recebe o termo de pesquisa se existir
$termo = (isset($_GET['termo'])) ? $_GET['termo'] : '';
// Verifica se o termo de pesquisa está vazio, se estiver executa uma consulta completa
if (empty($termo)):
$conexao = conexao::getInstance();
$sql = 'SELECT idAluno, nomeAluno, sexoAluno, data_Aluno, enderecoAluno, rgAluno, cpfAluno, estadoCivil, qtdeFilhos, idadeFilhos, telAluno, celAluno, emailAluno, cursoAluno, local_insc, status, foto FROM tab_alunos';
$stm = $conexao->prepare($sql);
$stm->execute();
$geral = $stm->fetchAll(PDO::FETCH_OBJ);
else:
// Executa uma consulta baseada no termo de pesquisa passado como parâmetro
$conexao = conexao::getInstance();
$sql = 'SELECT idAluno, nomeAluno, sexoAluno, data_Aluno, enderecoAluno, rgAluno, cpfAluno, estadoCivil, qtdeFilhos, idadeFilhos, telAluno, celAluno, emailAluno, cursoAluno, local_insc, status, foto FROM tab_alunos WHERE nomeAluno LIKE :nomeAluno OR cursoAluno LIKE :cursoAluno';
$stm = $conexao->prepare($sql);
$stm->bindValue(':nomeAluno', $termo.'%');
$stm->bindValue(':cursoAluno', $termo.'%');
$stm->execute();
$geral = $stm->fetchAll(PDO::FETCH_OBJ);
endif;
?>
<!DOCTYPE html>
<html>
<head>
<meta charset="utf-8">
<title>Fundação Stickel - Cadastro</title>
<link rel="stylesheet" type="text/css" href="css/bootstrap.min.css">
<link rel="stylesheet" type="text/css" href="css/custom.css">
</head>
<body>
<div class='container'>
<fieldset>
<!-- Cabeçalho da Listagem -->
<legend><h1>Sistema de Cadastro</h1></legend>
<!-- Formulário de Pesquisa -->
<form action="" method="get" id='form-contato' class="form-horizontal col-md-10">
<label class="col-md-2 control-label" for="termo">Pesquisar</label>
<div class='col-md-7'>
<input type="text" class="form-control" id="termo" name="termo" placeholder="Infome o Nome ou E-mail">
</div>
<button type="submit" class="btn btn-danger">Pesquisar</button>
<a href='cadastro_artista.php' class="btn btn-danger">Ver Todos</a>
</form>
<!-- Link para página de cadastro -->
</div>
<article class="container">
<hr>
<a href='aluno.php' class="btn btn-danger">Cadastrar o Aluno</a>
<a href='index.php' class="btn btn-danger">Voltar</a>
<div class='clearfix'></div>
</article>
<?php if(!empty($geral)):?>
<div class="table-responsive">
<!-- Tabela de geral -->
<table class="table table-hover">
<tr class='active'>
<th><center>Foto</center></th>
<th><center>Nome</center></th>
<th><center>Sexo</center></th>
<th><center>Nascimento</center></th>
<th><center>Endereço</center></th>
<th><center>RG</center></th>
<th><center>CPF</center></th>
<th><center>Estado Cívil</center></th>
<th><center>Qtde. Filhos</center></th>
<th><center>Idade</center></th>
<th><center>Telefone</center></th>
<th><center>Celular</center></th>
<th><center>E-mail</center></th>
<th><center>Curso</center></th>
<th><center>Inscrição</center></th>
<th><center>Status</center></th>
<th><center>Ação</center></th>
</tr>
<?php foreach($geral as $geral):?>
<tr>
<td><img src='fotos/<?=$geral->foto?>' height='100' width='100'></td>
<td><?=$geral->nomeAluno?></center></td>
<td><?=$geral->sexoAluno?></center></td>
<td><center><?=$geral->data_Aluno?></center></td>
<td><center><?=$geral->enderecoAluno?></center></td>
<td><center><?=$geral->rgAluno?></center></td>
<td><center><?=$geral->cpfAluno?></center></td>
<td><center><?=$geral->estadoCivil?></center></td>
<td><center><?=$geral->qtdeFilhos?></center></td>
<td><center><?=$geral->idadeFilhos?></center></td>
<td><center><?=$geral->telAluno?></center></td>
<td><center><?=$geral->celAluno?></center></td>
<td><center><?=$geral->emailAluno?></center></td>
<td><center><?=$geral->cursoAluno?></center></td>
<td><center><?=$geral->local_insc?></center></td>
<td><center><?=$geral->status?></center></td>
<td>
<a href='editar_aluno.php?idAluno=<?=$geral->idAluno?>' class="btn btn-danger">Editar</a>
<br>
<a href="javascript:if(confirm('Tem certeza que deseja excluir ?')){location= 'deletar_aluno.php?idAluno=<?=$geral->idAluno?>';}">Excluir</a>
</td>
</tr>
<?php endforeach;?>
</table>
</div>
<?php else: ?>
<!-- Mensagem caso não exista geral ou não encontrado -->
<h3 class="text-center text-primary">Não existem Alunos cadastrados!</h3>
<?php endif; ?>
</fieldset>
</div>
<script type="text/javascript" src="js/custom.js"></script>
</body>
</html> | {
"redpajama_set_name": "RedPajamaGithub"
} | 5,990 |
BioFresh's latest newsletter is out just in time for Christmas. You will find a range of topics and news covered in the newsletter from BioFresh's new look to featured science .
It's been a busy few months for the BioFresh team. As we head into our final stage of funding, we've had a makeover, held and participated in workshops on endangered species and key biodiversity areas; had our latest scientific research published in respected journals; and done work on freshwater biodiversity science-policy interfaces, including planning an exciting joint science policy symposium for freshwater life.
Although the BioFresh Project is coming to a close in 2014, much of BioFresh's work will be carried forward. Klement Tocker, the head of BioFresh, explains that there are "strong commitments by several partners to continue the development of the project's core elements such as the portal, the meta database, the atlas, and the blog." "I am convinced that BioFresh will serve as a nucleus in forming a global freshwater biodiversity network," he said. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,886 |
module HasVimeoVideo
class Railtie < Rails::Railtie
initializer 'has_vimeo_video.model_additions' do
ActiveSupport.on_load :active_record do
extend ModelAdditions
end
end
end
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,264 |
{"url":"https:\/\/trac.fact-project.org\/changeset\/19863","text":"# Changeset 19863\n\nIgnore:\nTimestamp:\nNov 7, 2019, 8:53:36 AM (11 months ago)\nMessage:\nUpdated\nFile:\n1 edited\n\n### Legend:\n\nUnmodified\n r10183 Installation Instructions ************************* Copyright (C) 1994, 1995, 1996, 1999, 2000, 2001, 2002, 2004, 2005, 2006, 2007, 2008, 2009 Free Software Foundation, Inc. Copying and distribution of this file, with or without modification, are permitted in any medium without royalty provided the copyright notice and this notice are preserved.\u00a0 This file is offered as-is, without warranty of any kind. Basic Installation ================== Briefly, the shell commands .\/configure; make; make install' should configure, build, and install this package.\u00a0 The following more-detailed instructions are generic; see the README' file for instructions specific to this package.\u00a0 Some packages provide this INSTALL' file but do not implement all of the features documented below.\u00a0 The lack of an optional feature in a given package is not necessarily a bug.\u00a0 More recommendations for GNU packages can be found in *note Makefile Conventions: (standards)Makefile Conventions. The configure' shell script attempts to guess correct values for various system-dependent variables used during compilation.\u00a0 It uses those values to create a Makefile' in each directory of the package. It may also create one or more .h' files containing system-dependent definitions.\u00a0 Finally, it creates a shell script config.status' that you can run in the future to recreate the current configuration, and a file config.log' containing compiler output (useful mainly for debugging configure'). It can also use an optional file (typically called config.cache' and enabled with --cache-file=config.cache' or simply -C') that saves the results of its tests to speed up reconfiguring.\u00a0 Caching is disabled by default to prevent problems with accidental use of stale cache files. If you need to do unusual things to compile the package, please try to figure out how configure' could check whether to do them, and mail diffs or instructions to the address given in the README' so they can be considered for the next release.\u00a0 If you are using the cache, and at some point config.cache' contains results you don't want to keep, you may remove or edit it. The file configure.ac' (or configure.in') is used to create configure' by a program called autoconf'.\u00a0 You need configure.ac' if you want to change it or regenerate configure' using a newer version of autoconf'. The simplest way to compile this package is: 1. cd' to the directory containing the package's source code and type .\/configure' to configure the package for your system. Running configure' might take a while.\u00a0 While running, it prints some messages telling which features it is checking for. 2. Type make' to compile the package. 3. Optionally, type make check' to run any self-tests that come with the package, generally using the just-built uninstalled binaries. 4. Type make install' to install the programs and any data files and documentation.\u00a0 When installing into a prefix owned by root, it is recommended that the package be configured and built as a regular user, and only the make install' phase executed with root privileges. 5. Optionally, type make installcheck' to repeat any self-tests, but this time using the binaries in their final installed location. This target does not install anything.\u00a0 Running this target as a regular user, particularly if the prior make install' required root privileges, verifies that the installation completed correctly. 6. You can remove the program binaries and object files from the source code directory by typing make clean'.\u00a0 To also remove the files that configure' created (so you can compile the package for a different kind of computer), type make distclean'.\u00a0 There is also a make maintainer-clean' target, but that is intended mainly for the package's developers.\u00a0 If you use it, you may have to get all sorts of other programs in order to regenerate files that came with the distribution. 7. Often, you can also type make uninstall' to remove the installed files again.\u00a0 In practice, not all packages have tested that uninstallation works correctly, even though it is required by the GNU Coding Standards. 8. Some packages, particularly those that use Automake, provide make distcheck', which can by used by developers to test that all other targets like make install' and make uninstall' work correctly. This target is generally not run by end users. Compilers and Options ===================== Some systems require unusual options for compilation or linking that the configure' script does not know about.\u00a0 Run .\/configure --help' for details on some of the pertinent environment variables. You can give configure' initial values for configuration parameters by setting variables in the command line or in the environment.\u00a0 Here is an example: .\/configure CC=c99 CFLAGS=-g LIBS=-lposix *Note Defining Variables::, for more details. Compiling For Multiple Architectures ==================================== You can compile the package for more than one kind of computer at the same time, by placing the object files for each architecture in their own directory.\u00a0 To do this, you can use GNU make'.\u00a0 cd' to the directory where you want the object files and executables to go and run the configure' script.\u00a0 configure' automatically checks for the source code in the directory that configure' is in and in ..'.\u00a0 This is known as a \"VPATH\" build. With a non-GNU make', it is safer to compile the package for one architecture at a time in the source code directory.\u00a0 After you have installed the package for one architecture, use make distclean' before reconfiguring for another architecture. On MacOS X 10.5 and later systems, you can create libraries and executables that work on multiple system types--known as \"fat\" or \"universal\" binaries--by specifying multiple -arch' options to the compiler but only a single -arch' option to the preprocessor.\u00a0 Like this: .\/configure CC=\"gcc -arch i386 -arch x86_64 -arch ppc -arch ppc64\" \\ CXX=\"g++ -arch i386 -arch x86_64 -arch ppc -arch ppc64\" \\ CPP=\"gcc -E\" CXXCPP=\"g++ -E\" This is not guaranteed to produce working output in all cases, you may have to build one architecture at a time and combine the results using the lipo' tool if you have problems. Installation Names ================== By default, make install' installs the package's commands under \/usr\/local\/bin', include files under \/usr\/local\/include', etc.\u00a0 You can specify an installation prefix other than \/usr\/local' by giving configure' the option --prefix=PREFIX', where PREFIX must be an absolute file name. You can specify separate installation prefixes for architecture-specific files and architecture-independent files.\u00a0 If you pass the option --exec-prefix=PREFIX' to configure', the package uses PREFIX as the prefix for installing programs and libraries. Documentation and other data files still use the regular prefix. In addition, if you use an unusual directory layout you can give options like --bindir=DIR' to specify different values for particular kinds of files.\u00a0 Run configure --help' for a list of the directories you can set and what kinds of files go in them.\u00a0 In general, the default for these options is expressed in terms of ${prefix}', so that specifying just --prefix' will affect all of the other directory specifications that were not explicitly provided. The most portable way to affect installation locations is to pass the correct locations to configure'; however, many packages provide one or both of the following shortcuts of passing variable assignments to the make install' command line to change installation locations without having to reconfigure or recompile. The first method involves providing an override variable for each affected directory. For example, make install prefix=\/alternate\/directory' will choose an alternate location for all directory configuration variables that were expressed in terms of ${prefix}'.\u00a0 Any directories that were specified during configure', but not in terms of ${prefix}', must each be overridden at install time for the entire installation to be relocated. The approach of makefile variable overrides for each directory variable is required by the GNU Coding Standards, and ideally causes no recompilation. However, some platforms have known limitations with the semantics of shared libraries that end up requiring recompilation when using this method, particularly noticeable in packages that use GNU Libtool. The second method involves providing the DESTDIR' variable. For example, make install DESTDIR=\/alternate\/directory' will prepend \/alternate\/directory' before all installation names. The approach of DESTDIR' overrides is not required by the GNU Coding Standards, and does not work on platforms that have drive letters. On the other hand, it does better at avoiding recompilation issues, and works well even when some directory options were not specified in terms of ${prefix}' at configure' time. Optional Features ================= If the package supports it, you can cause programs to be installed with an extra prefix or suffix on their names by giving configure' the option --program-prefix=PREFIX' or --program-suffix=SUFFIX'. Some packages pay attention to --enable-FEATURE' options to configure', where FEATURE indicates an optional part of the package. They may also pay attention to --with-PACKAGE' options, where PACKAGE is something like gnu-as' or x' (for the X Window System).\u00a0 The README' should mention any --enable-' and --with-' options that the package recognizes. For packages that use the X Window System, configure' can usually find the X include and library files automatically, but if it doesn't, you can use the configure' options --x-includes=DIR' and --x-libraries=DIR' to specify their locations. Some packages offer the ability to configure how verbose the execution of make' will be.\u00a0 For these packages, running .\/configure --enable-silent-rules' sets the default to minimal output, which can be overridden with make V=1'; while running .\/configure --disable-silent-rules' sets the default to verbose, which can be overridden with make V=0'. Particular systems ================== On HP-UX, the default C compiler is not ANSI C compatible.\u00a0 If GNU CC is not installed, it is recommended to use the following options in order to use an ANSI C compiler: .\/configure CC=\"cc -Ae -D_XOPEN_SOURCE=500\" and if that doesn't work, install pre-built binaries of GCC for HP-UX. On OSF\/1 a.k.a. Tru64, some versions of the default C compiler cannot parse its ' header file.\u00a0 The option -nodtk' can be used as a workaround.\u00a0 If GNU CC is not installed, it is therefore recommended to try .\/configure CC=\"cc\" and if that doesn't work, try .\/configure CC=\"cc -nodtk\" On Solaris, don't put \/usr\/ucb' early in your PATH'.\u00a0 This directory contains several dysfunctional programs; working variants of these programs are available in \/usr\/bin'.\u00a0 So, if you need \/usr\/ucb' in your PATH', put it _after_ \/usr\/bin'. On Haiku, software installed for all users goes in \/boot\/common', not \/usr\/local'.\u00a0 It is recommended to use the following options: .\/configure --prefix=\/boot\/common Specifying the System Type ========================== There may be some features configure' cannot figure out automatically, but needs to determine by the type of machine the package will run on.\u00a0 Usually, assuming the package is built to be run on the _same_ architectures, configure' can figure that out, but if it prints a message saying it cannot guess the machine type, give it the --build=TYPE' option.\u00a0 TYPE can either be a short name for the system type, such as sun4', or a canonical name which has the form: CPU-COMPANY-SYSTEM where SYSTEM can have one of these forms: OS KERNEL-OS See the file config.sub' for the possible values of each field.\u00a0 If config.sub' isn't included in this package, then this package doesn't need to know the machine type. If you are _building_ compiler tools for cross-compiling, you should use the option --target=TYPE' to select the type of system they will produce code for. If you want to _use_ a cross compiler, that generates code for a platform different from the build platform, you should specify the \"host\" platform (i.e., that on which the generated programs will eventually be run) with --host=TYPE'. Sharing Defaults ================ If you want to set default values for configure' scripts to share, you can create a site shell script called config.site' that gives default values for variables like CC', cache_file', and prefix'. configure' looks for PREFIX\/share\/config.site' if it exists, then PREFIX\/etc\/config.site' if it exists.\u00a0 Or, you can set the CONFIG_SITE' environment variable to the location of the site script. A warning: not all configure' scripts look for a site script. Defining Variables ================== Variables not defined in a site shell script can be set in the environment passed to configure'.\u00a0 However, some packages may run configure again during the build, and the customized values of these variables may be lost.\u00a0 In order to avoid this problem, you should set them in the configure' command line, using VAR=value'.\u00a0 For example: .\/configure CC=\/usr\/local2\/bin\/gcc causes the specified gcc' to be used as the C compiler (unless it is overridden in the site shell script). Unfortunately, this technique does not work for CONFIG_SHELL' due to an Autoconf bug.\u00a0 Until the bug is fixed you can use this workaround: CONFIG_SHELL=\/bin\/bash \/bin\/bash .\/configure CONFIG_SHELL=\/bin\/bash configure' Invocation ====================== configure' recognizes the following options to control how it operates. --help' -h' Print a summary of all of the options to configure', and exit. --help=short' --help=recursive' Print a summary of the options unique to this package's configure', and exit.\u00a0 The short' variant lists options used only in the top level, while the recursive' variant lists options also present in any nested packages. --version' -V' Print the version of Autoconf used to generate the configure' script, and exit. --cache-file=FILE' Enable the cache: use and save the results of the tests in FILE, traditionally config.cache'.\u00a0 FILE defaults to \/dev\/null' to disable caching. --config-cache' -C' Alias for --cache-file=config.cache'. --quiet' --silent' -q' Do not print messages saying which checks are being made.\u00a0 To suppress all normal output, redirect it to \/dev\/null' (any error messages will still be shown). --srcdir=DIR' Look for the package's source code in directory DIR.\u00a0 Usually configure' can determine that directory automatically. --prefix=DIR' Use DIR as the installation prefix.\u00a0 *note Installation Names:: for more details, including other options available for fine-tuning the installation locations. --no-create' -n' Run the configure checks, but stop before creating any output files. configure' also accepts some other, not widely useful, options.\u00a0 Run configure --help' for more details. Please follow the instructions at https:\/\/trac.fact-project.org\/wiki\/InstallingFACT++","date":"2020-09-28 12:20:35","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.43880680203437805, \"perplexity\": 4644.926391210105}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-40\/segments\/1600401600771.78\/warc\/CC-MAIN-20200928104328-20200928134328-00429.warc.gz\"}"} | null | null |
{"url":"https:\/\/eprint.iacr.org\/2009\/249","text":"Cryptology ePrint Archive: Report 2009\/249\n\nPseudo-randomness and partial information in symbolic security analysis\n\nDaniele Micciancio\n\nAbstract: We prove computational soundness results for cryptographic expressions with pseudo-random keys, as used, for example, in the design and analysis of secure multicast key distribution protocols. In particular, we establish a symbolic notion of independence (for pseudo-random keys) that exactly matches the standard computational security definition (namely, indistinguishability from the uniform distribution) for pseudo-random generators. As a conceptual contribution, we initiate the study of partial information in the context of computationally sound symbolic security analysis. Specifically, we show that (within our admittedly simple, but hopefully evocative setting) partial information can be taken into account in the symbolic model, in a computationally sound way, by simply annotating each key with a label which specifies that the key is either known, unknown, or partially known, without further details about the amount and type of partial information.\n\nCategory \/ Keywords: foundations \/ Computational soundness, formal methods for security, pseudo-random generators, partial information, greatest fix-points","date":"2018-01-22 20:23:06","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8560701012611389, \"perplexity\": 2133.7976693645737}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-05\/segments\/1516084891539.71\/warc\/CC-MAIN-20180122193259-20180122213259-00107.warc.gz\"}"} | null | null |
The IEEE NTUA Student Branch receives the IEEE Regional Exemplary Student Branch Award 2021
The purpose of this award is to provide public recognition of exemplary IEEE Student Branch operations.
Paper by ECE-NTUA Ph.D. candidates V. Alimisis and C.Dimas, Diploma Student G. Gennis and Prof. P. P. Sotiriadis received the Best Paper Award (1st Place) in the IEEE ICM 2021
An Analog Bayesian Classifier Implementation, for Thyroid Disease Detection, Based on a Low-Power, Current-Mode Gaussian Function Circuit
The project FLEXITRANSTORE nominated for the "Good Practice of the Year" Award 2021
Technological Innovation & System Integration
Professor Emeritus Vasilis Maglaris receives a Lifetime Achievement Award
Professor Emeritus Vasilis Maglaris received a Lifetime Achievement Award for his contributions to the evolution of the telecommunications sector.
Gender Equality Plan of the School of Electrical and Computer Engineering of the NTUA
The School was supported during the development of the plan by the CALIPER - Gender Equality in STEM Research project (Horizon 2020).
smarty4covid: BIOSIM join forces with AHEPA and launches the first clinical pilot involving hospitalized COVID-19 patients with the aim to identify novel biomarkers of infection and disease progression
Smarty4covid aspires to discover novel biomarkers with high prognostic value regarding the onset and the progress of COVID-19.
Smart RUE of ECE-NTUA
"Smart RUE" team and the energy transition
HOLOBALANCE project
Personalised coaching for well-being and care of people as they age
ECE-NTUA decorated the Christmas Tree
Merry Christmas and Best Wishes for a Happy New Year | {
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News Villarreal
Soldado: Return better than imagined
By andrew - 16 September 2015, 17:20
Roberto Soldado admits he didn't expect his return to La Liga to go as well as it has, having starred for Villarreal.
The striker moved to El Madrigal after an unhappy spell with Tottenham Hotspur, and was named player of the month by one of the Yellow Submarine's sponsors.
"My return is better than I could have imagined. I really wanted to return to Spain and especially to a club that has everything it needs to do well," he said, AS reports.
"The welcome I received was second to none and we have started well, which in sport is the most important thing.
"We know we have a lot of potential in attack and defence if we work as a team. The new players are trying to get to know their teammates on the pitch.
"For now we've been effective and that's because the team is creating chances. We have to continue on that line. The team will keep scoring goals and that's important.
"It's always nice to play in Europe and we have a big squad with a lot of quality.
"It's important that all the players get minutes and are ready to compete. What everyone wants is to play a lot of games. We have to seize the chances we get.
"Hopefully Villarreal can go as far as possible [in the Europa League]. It's what we're working for."
Tags Roberto Soldado
War of words between Ronald Koeman and Roberto Soldado
Roberto Soldado on Los Galacticos: "It was amazing to share a dressing room with them"
Morning headlines: Pedri sweetens Messi's record-breaking turn while Soldado remembers Los Galacticos | {
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} | 8,640 |
{"url":"https:\/\/astronomy.stackexchange.com\/questions\/24520\/modern-astrophysics","text":"Modern Astrophysics [closed]\n\nI was wondering whether it has proven whether the universe is expanding and if so, what is its ratio, one more question that whether it has an effect for the centrifugal forces, i.e gravity and the weight of the planets and far- away stars. Many thanks.\n\n\u2022 I don't understand much of this. What do you mean by \"ratio\"? What do you mean by \"centrifugal forces ie gravity\"? Gravity isn't a centrifugal force. By \"weight\" do you mean \"mass\"? How do you think expansion might affect mass? You have tagged \"star\" and \"milky-way\" How is this question about stars or the milky-way? Don't comment; please edit. \u2013\u00a0James K Jan 13 '18 at 21:04\n\u2022 Do you mean \"what is its rate\", perhaps? \u2013\u00a0pela Jan 13 '18 at 22:24\n\u2022 If I understand your question, the answer is that expansion doesn't affect gravity or other factors, because expansion of space doesn't apply over short distances. Space expands, objects in space aren't affected, it's the space between objects that expands - and, again, only over very large distances. Space doesn't expand between the Earth and the Sun, for example at least, not in any meaningful way. It certainly doesn't expand within the Earth. \u2013\u00a0userLTK Jan 13 '18 at 23:06\n\u2022 The speed of the expension is the Universe is measured by the so-named Hubble-constant. Its value is around $70 \\frac{km}{s}$ for every 100 Megaparsec. Thus, anything 100 MPc away, is receding with $70 \\frac{km}{s}$ from us. Currently there is no evidence that gravity would change in time. Of course things more far away attracts eachother lesser. \u2013\u00a0peterh - Reinstate Monica Jan 14 '18 at 1:09\n\u2022 It's accepted now that the universe is expanding and this Wikipedia page discusses the reason and evidence. \u2013\u00a0StephenG Jan 14 '18 at 1:36","date":"2020-08-05 11:09:29","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.34859758615493774, \"perplexity\": 762.9834776296137}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-34\/segments\/1596439735939.26\/warc\/CC-MAIN-20200805094821-20200805124821-00355.warc.gz\"}"} | null | null |
\section{Introduction}
\label{intro}
Learning to Recognize Musical Genre from Audio is a challenge track of \textit{The Web Conference 2018}. The main goal of the challenge is to predict musical genres of unknown audio segments correctly, by utilizing the FMA dataset ~\cite{benzi2016fma} as a training set. The challenge therefore focuses on a classification task.
In machine learning, many classification tasks, such as visual object recognition, consider objective and clearly separable classes. In contrast, music genres consider subjective, human-attributed labels. These may be inter-correlated (e.g.\ a \emph{rock} song may also be considered \emph{pop}, many \emph{classical} works are also \emph{instrumental}) and dependent of a user's context (e.g., a \emph{French rock} song is not \emph{International} to a French listener). Generally, no universal genre taxonomy exists, and even the definition of `genre' itself is problematic: what is usually understood as `genre' in Music Information Retrieval would rather be characterized as `style' in Musicology~\cite{liem12-dagstuhl}. This makes genre classification a challenging problem. In our work, considering the given labels in the challenge, we consider a musical genre to be a category that consists of songs sharing certain aspects of musical characteristics.
Commonly, music tracks are released with explicit mentioning of titles and artists. The identity of the artist does not suffer from semantic taxonomy problems, and can thus be considered as a more objective label than the genre label. At the same time, songs from the same artist tend to share prominent musical characteristics. Considering that an artist is commonly mapped into one or multiple specific genres, but not the whole universe of possible genres, and that the other way around, sets of artists can be seen as exemplars for certain music genres, the musical characteristics that identify an artist may also be key features of certain musical genres.
Therefore, it will be beneficial to exploit artist-related information in a genre classification task. At the same time, learning a direct mapping from artist identity to genre label would not be practical. First of all, for an unknown audio segment for which a genre classification should be performed, the artist label may also not be available. Secondly, artist labels may not always be informative to a system, especially when an artist is newly introduced, so no previous history on the artist exists. Finally, an artist may have been active in multiple genres at once, but not be equally representative for all these genres. Given such constraints, we wish to employ a learning framework which only requires artist labels at training time, but not at prediction time, and that will allow for the inclusion of newly introduced artists, for whom not much extra information is available beyond their songs.
In this work, we therefore present a multi-task transfer framework for using artist labels to improve a genre classification model. Assuming that artist labels are given for each track in the training set, these labels are used as side information, allowing a model to learn the mapping between audio and artists, while capturing patterns that might as well be useful for genre prediction.
It has been shown that music representations learned from raw artist labels can effectively transfer to other music-related tasks~\cite{park2017representation}. However, learning more than thousands of artists as individual classes is not efficient for at least two reasons:
\begin{itemize}
\item{Due to data sparsity, only a few tracks are assigned per class;}
\item{Despite the uniqueness of each artist, it can be beneficial to group them into clusters of similar artists, avoiding learning bottlenecks caused by large numbers of classes.}
\end{itemize}
To overcome these potential problems, we therefore apply a label pre-processing step, obtaining Artist Group Factors (AGF) as learning targets, rather than individual artist identities.
Finally, we train Deep Convolutional Neural Networks (DCNNs) employing different learning setups, ranging from targeting genre and various types of AGFs with individual networks, to employing a shared architecture as introduced in multiple previous Multi-Task Learning (MTL) works~\cite{caruana1998multitask, bengio2013representation, liu2015multi, bingel2017identifying, li2014heterogeneous, zhang2016deep, zhang2014facial,kim2018one}.
In the remainder of this paper, we first discuss an initial data exploration leading to our choice for AGFs (Section \ref{motivation}). Subsequently, we will give a detailed description of the proposed approach (Section \ref{method}), followed by a discussion of experimental settings (Section \ref{exp}). Finally, we will present our results (Section \ref{res}), followed by a short discussion and conclusion (Section \ref{disc}).
\section{Initial data exploration}
\label{motivation}
In the beginning of the challenge, we first explored the training data, and investigated a conventional data-driven approach using a DCNN for music genre classification, with genre labels as targets.
First of all, we had some concerns about the reliability of the genre annotations. As they were provided by users who uploaded the content, the users did not have access to a single genre taxonomy and unified annotation strategy. Thus, user-contributed annotations are expected to show more variability than annotations by experts. Furthermore, the dataset included 25,000 tracks from 5,152 unique albums. For 5,028 out of these 5,152 albums, genre annotations were made at the album level. While all tracks in an album can belong to a single genre, this is not always true. Indeed, we could discover examples of the case in which different tracks on the same album would belong to different genres, as well as multiple misannotations. Given these reliability issues, it is not guaranteed that by targeting these annotations only, generalized model performance for genre classification can be achieved.
To this end, while we will consider performance for direct (main top-)genre labels as targets (which we will denote as learning task category \texttt{g} in the remainder of this paper), in order to obtain more generalizable results obtained on more objective and consistent labeling data, we propose a multi-task transfer framework, introducing an Artist Group (AG) prediction task targeting AGFs.
\section{Methodology}
\label{method}
\subsection{Artist Group Factors}
\label{method:agf}
The main idea of extracting AGFs is to cluster artists based on meaningful feature sets that allow for aggregation at (and beyond) the artist level. For instance, one can collect genre labels from songs belonging to each artist, and then construct a Bag-of-Word (BoW) artist-level feature vector. Each dimension of the vector represents a genre, with the magnitude of the vector indicating genre frequency among a song collection. Alternatively, a BoW feature vector can be constructed by counting latent `terms' belonging to each artist, which can be obtained by a dictionary learned from song-level or frame-level features through K-means clustering~\cite{lloyd1982least} or the Sparse Coding~\cite{coates2011importance} method.
Once artist-level BoW feature vectors are constructed, standard clustering methods such as K-Means, or more sophisticated topic modeling algorithms such as Latent Dirichlet Allocation (LDA)~\cite{blei2003latent} can be applied to find a small number of latent groups of artists: the AGFs for this particular feature set. This 2-step cascading pipeline is illustrated in Figure \ref{fig:agf}.
In this work, we exploit four feature sets, which reflect different levels of musical and acoustical aspects of songs. From these feature sets, we obtain artist-level BoW vectors. Subsequently, LDA is applied to transform artist-level BoW vectors into dedicated AGF representations for the particular feature set. We will both consider these artist group prediction tasks and the main genre classification task within our learning framework: an overview summary is given in Table~\ref{tab:agf}.
\begin{figure*}
\includegraphics[width=0.8\textwidth]{agf_pipeline_long}
\caption{Artist group factor extraction pipeline.}
\label{fig:agf}
\end{figure*}
\subsubsection{MFCCs}
\label{method:agf:mfcc}
Mel-Frequency Cepstral Coefficients (MFCCs), which are known to be efficient low-level descriptors for timbre analysis, were used as features of the artist grouping. The coefficients are initially calculated for short-time audio frames. Considering the coefficients over all audio frames of tracks for all artists, we build an universal dictionary of features using K-Means clustering. AGFs resulting from this feature set will belong to learning task category \texttt{m}.
\subsubsection{dMFCCs}
\label{method:agf:dmfcc}
Along with MFCCs, we also use time-deltas of MFCCs (first-order differences between subsequent frames), to consider the temporal dynamics of the timbre for the artist grouping. AGFs resulting from this feature set will be denoted by \texttt{d}.
\subsubsection{Essentia}
\label{method:agf:essentia}
We use song-level feature vectors from Essentia~\cite{bogdanov2013essentia}, which is a music feature extraction library. It extracts descriptors ranging from low-level features, such as statistics of spectral characteristics, to high-level features, including danceability~\cite{herrera2005detrended} or semantic features learned from the data. After filtering descriptor entries that include missing values or errors, we obtained a 4374-dimensional feature vector per track. Before training a dictionary, we apply quantile normalization: a rank-based normalization process that transforms the distribution of the given features to follow a target distribution~\cite{doi:10.1198/016214501753381814}, which we set to be a normal distribution in this case. AGFs resulting from this feature set will belong to learning task category \texttt{e}.
\subsubsection{Subgenres}
\label{method:agf:subg}
We also use the 150 genre labels, including sub-genres, as a pre-defined dictionary for semantic description. For these, we directly build artist-level BoW vectors by aggregating all the genre labels from tracks by an artist. AGFs resulting from this feature set will belong to learning task category \texttt{s}.
\begin{table}[]
\centering
\caption{Details of Learning Targets}
\label{tab:agf}
\begin{tabular}{lcccc}
id & Category & Source & Dictionary & Dimension \\ \hline\hline
\texttt{g} & Main & Genre & N / A & 16 \\ \hline
\texttt{m} & \multirow{4}{*}{AGF} & MFCC & \multirow{3}{*}{K-means} & 25 \\
\texttt{d} & & dMFCC & & 25 \\
\texttt{e} & & Essentia~\cite{bogdanov2013essentia} & & 4374 \\
\texttt{s} & & Subgenre & N / A & 150 \\ \hline
\end{tabular}
\end{table}
\begin{table}[]
\centering
\caption{Network Architectures for Encoder $f$}
\label{tab:dcnn}
\begin{tabular}{ll}
Layers & Output shape \\
\hline\hline
Input layer & $128\times43\times1$ \\
\hline
Conv $5\times5$, ELU & $128\times43\times16$ \\
MaxPooling $2\times1$ & $64\times43\times16$ \\
\hline
Conv $3\times3$, BN, ELU & $64\times43\times32$ \\
MaxPooling $2\times2$ & $32\times21\times32$ \\
Dropout (0.1) & $32\times21\times32$ \\
\hline
Conv $3\times3$, ELU & $32\times21\times64$ \\
MaxPooling $2\times2$ & $16\times10\times64$ \\
\hline
Conv $3\times3$, BN, ELU & $16\times10\times64$ \\
MaxPooling $2\times2$ & $8\times5\times64$ \\
Dropout (0.1) & $8\times5\times64$ \\
\hline
Conv $3\times3$, ELU & $8\times5\times128$ \\
MaxPooling $2\times2$ & $4\times2\times128$ \\
\hline
Conv $3\times3$, ELU & $4\times2\times256$ \\
\hline
Conv $1\times1$, BN, ELU & $4\times2\times256$ \\
\hline
GlobalAveragePooling, BN & 256 \\
\hline
Dense, BN, ELU & 256 \\
Dropout (0.5) & 256 \\
\hline
Output layer 16 or 40 & 16 or 40 \\
\end{tabular}
\end{table}
\subsection{Network Architectures}
\label{method:arch}
The architecture of the proposed system can be divided into two parts, as shown in Figure \ref{fig:transfer}. We first train multiple DCNNs, targeting the various categories of learning targets (genres or various AGFs). Subsequently, transfer takes place: a multilayer perceptron (MLP) for the final genre classification is trained, utilizing features that were derived from the previously trained DCNNs.
\subsubsection{DCNN}
\label{method:DCNN}
We adapted DCNN models to obtain transferable features for genre classification (Table \ref{tab:dcnn}). The input size of the input layer is $128\times$43, which is the size of a spectrogram with 128 mel bins and 43 samples (1 second of audio). After the input layer, there are seven convolutional layers followed by a max-pooling layer, except for the last two layers. The first convolutional layer has $5\times5$ kernels and the last convolutional layer has $1\times$1 kernels. Except for those two layers, all convolutional layers have $3\times$3 kernels. Outputs of the last convolutional layer are subsampled by global-average-pooling. Finally, they are connected to two dense layers for predicting AGF clusters or genres. Batch normalization~\cite{ioffe2015batch} and dropouts~\cite{srivastava2014dropout} are sparsely used to prevent overfitting. Exponential Linear Unit (ELU)~\cite{clevert2015fast} is used as an activation function for the convolutional layers and Softmax is used for the output layer.
\subsubsection{Shared Architecture}
\label{method:shared_arch}
Considering that lower layers of DCNNs usually capture lower-level features such as edges from images or spectrograms, we hypothesized that sharing lower layers among the various DCNNs can be effective under the scenario where multiple learning sources are available. With this approach, one can expect that it not only ensures sufficient specialization on task-specific upper layers, but also benefits from regularization effects on lower layers\cite{kim2018one}. Joint learning of multiple tasks with shared layers can prevent the shared layer to overfit for a specific task, instead learning underlying factors that have commonalities required across tasks~\cite{caruana1998multitask,liu2015representation}.
Throughout the experiment, we used the shared architecture that shares only the first convolutional block. It consists of the first convolutional and the max-pooling layer. For brevity, for the remainder of the paper, we use Single-Task Nets (STNs) and an Multi-Task Net (MTN) to refer to the non-shared networks and shared networks respectively.
\subsubsection{Transfer method}
\label{method:transfer}
The proposed system learns and predicts a genre of an input spectrogram by transferring pre-trained features from Section~\ref{method:DCNN}. We trained an MLP with a single hidden layer; the size of the hidden layer was 1024. ELU non-linearity was used for the hidden layer and Softmax was used for the output layer. Dropouts of 50\% were applied for the input layer and a hidden layer.
Note that for both the feature learning phase and the transfer learning phase, we keep using a segment-wise learning approach. Only at the final inference step, we aggregate all the segment-level predictions, by taking the average of each segment's predicted probability for the genres.
\begin{figure}
\centering
\includegraphics[height=0.3\textheight]{transfer_strategy}
\caption{Illustration for the transfer learning scenario. Dotted lines indicate the setup for the multilayer perceptron for performing final genre classification.}
\label{fig:transfer}
\end{figure}
\subsubsection{Training}
\label{method:train}
At training time, we iteratively update the model parameters with the mini-batch stochastic gradient descent method using the Adam algorithm~\cite{kingma2014adam}. For data augmentation, we randomly crop 1-second excerpts from the entire track included in the mini-batch. We use 64 samples per batch and set the learning rate to 0.001 across the experiments.
For comparison between methods, experiments are run with a fixed number of epochs. We set 1000 epochs for an MTN and 200 for STNs. Since we took a similar stochastic update algorithm to~\cite{liu2015multi} for the shared architecture, for the number of updates for task-specific layers in a shared network, the number of epochs used for training non-shared networks should be multiplied with the number of involved learning tasks. For the transfer learning phase, we also set the number of epochs to train the MLP to 50.
\subsection{Pre-processing}
\label{method:preproc}
We use mel spectrograms as the input representation for the neural networks. We extract 128-dimensional mel spectra for audio frames of 46ms, with 50\% overlap with adjacent frames. To enhance lower-intensity levels of input mel spectrograms at higher frequencies, we take dB-scale log amplitudes of each mel spectrum.
\subsection{Implementation Details}
\label{method:imple}
The experiments were run on GPU-accelerated hardware and software environments. We used Lasagne~\cite{sander_dieleman_2015_27878}, Theano~\cite{2016arXiv160502688short} and Keras~\cite{chollet2015} as main experimental frameworks\footnote{The main code for the experiment can be found in \url{https://github.com/eldrin/Lasagne-MultiTaskLearning}}. We used a number of different GPUs, including NVIDIA GRID-K2, NVIDIA GTX 1070, NVIDIA TITAN X.
\section{Experiments}
\label{exp}
To investigate the effectiveness of various types of AGFs for transfer learning, we trained all 31 possible combinations of given learning tasks, including AGFs (\texttt{m}, \texttt{d}, \texttt{e}, \texttt{s}) and main top-genre labels (\texttt{g}). For each run, to investigate the optimal feature architecture, we tested both shared networks and separate networks for each learning task. This leads to a total number of 62 cases, including all the combinations of learning tasks per network architecture.
However, in all cases in which multiple tasks are considered, the networks have a larger number of parameters compared to the case in which a network focuses on a single task. With a subsequent experiment, we therefore tried to verify the effect of more parameters and larger networks vs. \ the effect of using more tasks. To this end, we train wide Single Task Networks (wSTNs), targeting only genre, but having an equal number of parameters to the MTNs/STNs targeting multiple tasks. Finally, with respect to the number of tasks involved, we compare the best performance of MTNs/STNs to the performance of wSTNs with the same number of parameters.
As for the AGFs using song-level or frame-level features, we trained K-means algorithms employing 2048 clusters. We observed that lower numbers of clusters (e.g.\ 1024) can cause artists with few tracks to get a zero vector as artist-level BoW representation, due to data sparsity. Throughout the experiments, we used a fixed number of latent artist groups, set to 40.
Finally, for the internal evaluation, we divided the given training dataset employing a stratified random 85/15 split.
\section{Results}
\label{res}
\subsection{Multiple Learning Tasks in STN vs.\ MTN}
\label{res:stl}
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{effect_of_n}
\caption{Average performance for the number of tasks involved in feature learning}
\label{fig:effect_of_n}
\end{figure}
\begin{table}[]
\centering
\caption{Comparison of the average performance with or without the main task}
\label{tab:is_main}
\begin{tabular}{lllll}
& \multicolumn{2}{c}{LogLoss} & \multicolumn{2}{c}{F1} \\ \cline{2-5}
& \multicolumn{1}{c}{STN} & \multicolumn{1}{c}{MTN} & \multicolumn{1}{c}{STN} & \multicolumn{1}{c}{MTN} \\ \hline\hline
without \texttt{g} & 1.0079 & 0.9618 & 0.4932 & 0.5168 \\
with \texttt{g} & \textbf{0.8540} & \textbf{0.8486} & \textbf{0.6154} & \textbf{0.6155} \\ \hline
\end{tabular}
\end{table}
In general, we observe that the number of learning tasks has a positive effect on both performance metrics. As shown in Table~\ref{tab:is_main}, it also is found that cases in which the main top-genre classification are included yield better results in comparison to other combinations of tasks.
Considering STN vs.\ MTN, on the log loss metric, MTN shows better results, but in the case of the f1-measure, the opposite is shown. Generally, considering the number of learning tasks and absolute magnitude of differences, the difference observed between the two methods cannot be deemed significant; more experiments with additional datasets and multiple splits would be needed to assess whether statistically significant differences between STN vs.\ MTN approaches can be obtained.
For both STN and MTN, the best performance we achieved uses all the learning tasks, as shown in the last row of Table \ref{tab:main_res}.
\begin{table}[]
\centering
\caption{The performance of various combinations of AGFs and the top-level main genre target as a feature learning task.}
\label{tab:main_res}
\begin{tabular}{lllll}
& \multicolumn{2}{c}{STN} & \multicolumn{2}{c}{MTN} \\ \cline{2-5}
& LogLoss & F1 & LogLoss & F1 \\ \hline\hline
\texttt{g} & 0.8891 & 0.5963 & \multirow{5}{*}{N/A} & \multirow{5}{*}{N/A} \\
\texttt{m} & 1.1812 & 0.3581 & & \\
\texttt{d} & 1.0987 & 0.3967 & & \\
\texttt{e} & 1.2542 & 0.3437 & & \\
\texttt{s } & 0.9404 & 0.5218 & & \\
\texttt{gs} & 0.8606 & 0.6114 & 0.8578 & 0.6190 \\
\texttt{ge} & 0.8811 & 0.5953 & 0.8792 & 0.5996 \\
\texttt{gd} & 0.8845 & 0.5898 & 0.8803 & 0.5955 \\
\texttt{gm} & 0.8874 & 0.5957 & 0.8813 & 0.6037 \\
\texttt{se} & 0.9124 & 0.5537 & 0.9079 & 0.5502 \\
\texttt{sd} & 0.9191 & 0.5601 & 0.9146 & 0.5412 \\
\texttt{sm} & 0.9260 & 0.5581 & 0.9283 & 0.5458 \\
\texttt{ed } & 1.0557 & 0.4433 & 1.0422 & 0.4399 \\
\texttt{em} & 1.1186 & 0.4244 & 1.1060 & 0.4376 \\
\texttt{dm} & 1.0583 & 0.4373 & 1.0704 & 0.4280 \\
\texttt{gse} & 0.8361 & 0.6255 & 0.8335 & 0.6277 \\
\texttt{gsd} & 0.8579 & 0.6280 & 0.8519 & 0.6150 \\
\texttt{gsm} & 0.8486 & 0.6289 & 0.8541 & 0.6153 \\
\texttt{ged} & 0.8528 & 0.6051 & 0.8601 & 0.6067 \\
\texttt{gem} & 0.8645 & 0.5988 & 0.8701 & 0.6056 \\
\texttt{gdm} & 0.8773 & 0.5985 & 0.8845 & 0.5941 \\
\texttt{sed} & 0.8965 & 0.5818 & 0.8867 & 0.5640 \\
\texttt{sem} & 0.9104 & 0.5834 & 0.8889 & 0.5668 \\
\texttt{sdm} & 0.9211 & 0.5629 & 0.9109 & 0.5572 \\
\texttt{edm} & 1.0359 & 0.4879 & 1.0365 & 0.4675 \\
\texttt{gsed} & 0.8211 & 0.6343 & 0.8132 & 0.6328 \\
\texttt{gsem} & 0.8264 & 0.6352 & 0.8172 & 0.6284 \\
\texttt{gsdm} & 0.8407 & 0.6379 & 0.8288 & 0.6170 \\
\texttt{gedm} & 0.8466 & 0.6053 & 0.8450 & 0.6152 \\
\texttt{sedm } & 0.8906 & 0.5856 & 0.8875 & 0.5870 \\
\texttt{gsedm} & \textbf{0.7894} & \textbf{0.6599} & \textbf{0.7727} & \textbf{0.6571} \\ \hline
\end{tabular}
\end{table}
\subsection{Networks for Multiple Learning Tasks vs.\ Large Network on a Single Task}
\label{res:fat_vs_transfer}
We also compared the performance between the best STNs and MTNs for a given number of learning tasks, versus the performance of a wSTN that has equal model capability to these multi-task setups in terms of parameters and architecture, but only is trained on direct main top-genre classification. The corresponding results are shown in Table \ref{tab:multi_single}. It can be seen that MTN representations yield better performance on the log loss metric when all 5 learning tasks (all AGFs and the main top-genre) are used, although at the same time, wSTN performs better when considering the f1-measure for the case in which 2 learning tasks are used. In other cases, differences between the setups appear marginal; further experiments would be needed to assess whether STNs/MTNs will give significant performance boosts in case a larger set of tasks would be considered.
\begin{table}[]
\centering
\caption{Comparison between wSTN (single genre classification task) and STN/MTN setups (multiple tasks) learning setups. The reported performances of STN and MTN consider the task combinations for which the best performance was obtained, given the mentioned number $N$ of tasks.}
\label{tab:multi_single}
\begin{tabular}{lllllll}
& \multicolumn{3}{c}{LogLoss} & \multicolumn{3}{c}{F1} \\ \cline{2-7}
N & wSTN & STN & MTN & wSTN & STN & MTN \\ \hline\hline
2 & 0.8688 & 0.8606 & 0.8578 & 0.6071 & 0.6114 & 0.6190 \\
3 & 0.8546 & 0.8361 & 0.8335 & \textbf{0.6629} & 0.6289 & 0.6277 \\
4 & 0.8278 & 0.8211 & 0.8132 & 0.6451 & 0.6352 & 0.6328 \\
5 & 0.8290 & 0.7893 & \textbf{0.7727} & 0.6528 & 0.6599 & 0.6571 \\ \hline
\end{tabular}
\end{table}
\section{Discussion \& Conclusion}
\label{disc}
In this work, we proposed including several categories of low-rank AGFs, expressing artist-level information, into the task of classifying music genre based on musical audio. Our experimental results support the hypothesis that by targeting different categories of AGFs, deep networks can learn features from musical audio that can meaningfully support genre classification. The inclusion of multiple parallel learning tasks considering different AGF categories, and the inclusion of both genre- and AGF-based tasks in a multi-task setup, also both seem beneficial, although further work will need to be done to assess whether observed effects are truly significant. For this, other datasets will have to be included for training and testing; furthermore, alternative cluster algorithms and clustering parameters should be investigated to achieve the most robust AGF-based features.
\begin{acks}
This work was carried out on the Dutch national e-infrastructure with the support of
SURF Cooperative. And this work is partially supported by the Maria de Maeztu Programme (MDM-2015-0502). We further acknowledge the computing support of Kakao Corporation.
\end{acks}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,089 |
Early in June, Adorama announced its Flashpoint Auto Stand, available in 7-foot and 9-foot versions for $59.95 and $79.95. And shortly after, the company sent us the 9-footer for review.
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Consequently we've awarded it all four photo corners. It makes our life just a bit easier -- and you can't say that about many things these days. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,927 |
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\section{Introduction}\label{introduction}
Recently, in \cite{ DarbyHagerRao11,DarbyHagerRao10, FrancolinHagerRao13,
GargHagerRao11b, GargHagerRao11a, GargHagerRao10a, PattersonHagerRao14},
a class of methods was developed for solving optimal control problems
using collocation at either Gauss or Radau quadrature points.
In \cite{HagerHouRao15b} and \cite{HagerHouRao15c} an
exponential convergence rate is established for these schemes.
The analysis is based on a bound for the inverse of a linearized operator
associated with the discretized problem, and an
estimate for the residual one gets when substituting the solution to the
continuous problem into the discretized problem.
This paper focuses on the estimation of the residual.
We show that the residual in the sup-norm is bounded by the sup-norm distance
between the derivative of the solution to the continuous problem and
the derivative of the interpolant of the solution.
By Markov's inequality $\cite{Markov1916}$,
this distance can be bounded in terms of the Lebesgue
constant for the point set and the error in best polynomial approximation.
A classic result of Jackson \cite{jackson} gives an estimate for
the error in best approximation.
The Lebesgue constant that we need to analyze corresponds to the
roots of a Jacobi polynomial on $(-1, +1)$
augmented by either $\tau = +1$ or $\tau = -1$.
The effects of the added endpoints were analyzed by
V\'{e}rtesi in \cite{Vertesi81}.
For either the Gauss quadrature points
on $(-1, +1)$ augmented by $\tau = +1$ or the Radau quadrature points on
$(-1, +1]$ or on $[-1, +1)$, the bound given in \cite[Thm. 2.1]{Vertesi81}
for the Lebesgue constants is $O(\log (N) \sqrt{N})$,
where $N$ is the number of quadrature points.
We sharpen this bound to $O(\sqrt{N})$.
To motivate the relevance of the Lebesgue constant to collocation methods,
let us consider the scalar first-order differential equation
\begin{equation} \label{de}
\dot{x}(\tau)=f\left(x(\tau)\right), \quad \tau \in [-1, +1],
\quad x(-1) = x_0,
\end{equation}
where $f : \mathbb{R}\rightarrow\mathbb{R}$.
In a collocation scheme for (\ref{de}),
the solution $x$ to the differential equation
(\ref{de}) is approximated by a polynomial $x$
that is required to satisfy the differential
equation at the collocation points.
Let us consider a scheme based on collocation at the Gauss quadrature
points $-1 < \tau_1 < \tau_2 < \ldots < \tau_N < +1$, the roots of the
Legendre polynomial of degree $N$.
In addition, we introduce the noncollocated point $\tau_0 = -1$.
The discretized problem is to find $x \in \C{P}_{N}$,
the space of polynomials of degree at most $N$, such that
\begin{equation}\label{collocated}
\dot{x}(\tau_k) = f(x(\tau_k)), \quad 1 \le k \le N,
\quad x(-1) = x_0.
\end{equation}
A polynomial of degree at most $N$ is uniquely specified by
$N+1$ parameters such as its coefficients.
The $N$ collocation equations and the boundary condition in (\ref{collocated})
yield $N+1$ equations for the polynomial.
The convergence of a solution of the collocated problem (\ref{collocated})
to a solution of the continuous problem (\ref{de})
ultimately depends on how accurately a polynomial interpolant of a
continuous solution satisfies the discrete equations (\ref{collocated}).
The Lagrange interpolation polynomials for the point set
$\{\tau_0, \tau_1, \ldots , \tau_N\}$ are defined by
\begin{equation}\label{lag}
L_i(\tau)=\prod_{\substack{j=0\\ j\neq i}}^N\frac{\tau-\tau_j}
{\tau_i-\tau_j}, \quad 0 \le i \le N.
\end{equation}
The interpolant $x^N$ of a solution $x$ to (\ref{de}) is given by
\[
x^N (\tau) = \sum_{j=0}^N x (\tau_j) L_j(\tau).
\]
The residual in (\ref{collocated})
associated with a solution of (\ref{de}) is the vector with components
\begin{equation}\label{res}
r_0 = x^N(-1) - x_0, \quad r_k = \dot{x}^N(\tau_k) - f(x^N(\tau_k)), \quad
1 \le k \le N.
\end{equation}
For the Gauss scheme,
$r_0 = 0$ since $x$ satisfies the boundary condition in (\ref{de}).
The potentially nonzero components of the residual are $r_k$, $1 \le k \le N$.
As we show in Section~\ref{residual}, the residual can be bounded
in terms of a Lebesgue constant and the error in best approximation for $x$ and
its derivative.
The Lebesgue constant $\Lambda_N$ relative to the point set
$\{\tau_0, \tau_1, \ldots , \tau_N\}$ is defined by
\begin{equation}\label{ln1}
\Lambda_N=\max \left\{
\sum_{j=0}^N\left|L_j(\tau)\right|: \tau\in[-1,1] \right\} .
\end{equation}
The article \cite{Brutman97} of Brutman gives a comprehensive survey on the
analysis of Lebesgue constants, while the book \cite{Mastroianni08}
of Mastroianni and Milovanovi\'{c} covers more recent results.
The paper is organized as follows.
In Section~\ref{residual}, we show how the Lebesgue constant enters
into the residual associated with the discretized problem (\ref{collocated}).
Section~\ref{szego} summarizes results of Szeg\H{o} used in the analysis.
Section~\ref{gauss+} analyzes the Lebesgue constant for the
Gauss quadrature points augmented by $\tau = -1$,
while Section~\ref{radau+} analyzes Radau quadrature points.
Finally, Section~\ref{tight} examines the tightness of the estimates
for the Lebesgue constants.
{\bf Notation.}
$\mathcal{P}_N$ denotes the space of polynomials of degree at most $N$
and $\|\cdot\|$ denotes the sup-norm on the interval $[-1, +1]$.
The Jacobi polynomial $P_N^{(\alpha, \beta)}(\tau)$,
$N \ge 1$, is an $N$-th degree polynomial, and for fixed $\alpha > -1$ and
$\beta > -1$, the polynomials are orthogonal on the interval $[-1, +1]$
relative to the weight function $(1-\tau)^\alpha(1+\tau)^\beta$.
$P_N$ stands for the Jacobi polynomial $P_N^{(0,0)}$, or equivalently,
the Legendre polynomial of degree $N$.
\section{Analysis of the residual}
\label{residual}
As discussed in the introduction,
a key step in the convergence analysis of collocation schemes
is the estimation of the residual defined in (\ref{res}).
The convergence of a discrete solution to the
solution of the continuous problem ultimately depends on
how quickly the residual approaches 0 as $N$ tends to infinity;
for example, see Theorem~3.1 in \cite{DontchevHager97},
Proposition~5.1 in \cite{Hager99c}, or Theorem~2.1 in \cite{Hager02b}.
Since a solution $x$ of (\ref{de}) satisfies the differential equation
on the interval $[-1, +1]$, it follows that
$\dot{x}(\tau_k) = f(x(\tau_k))$, $1 \le k \le N$.
Hence, the potentially nonzero components of the residual can be expressed
$r_k = \dot{x}^N (\tau_k) - \dot{x}(\tau_k)$, $1 \le k \le N$.
In other words, the size of the residual depends on the difference between
the derivative of the interpolating polynomial at the collocation
points and the derivative of the continuous solution at the collocation points.
Hence, let us consider the general problem of estimating the
difference between the derivative of an interpolating polynomial on the
point set $\tau_0 < \tau_1 < \ldots < \tau_N$ contained in $[-1, +1]$
and the derivative of the original function.
\smallskip
\begin{proposition}\label{L1}
If $x$ is continuously differentiable on $[-1, +1]$, then
\begin{eqnarray}
\left\|\dot{x}-\dot{x}^N\right\|
&\le& \left(1+2N^2\right)
\inf_{q \in \mathcal{P}_{N}}\left\|\dot{x}-\dot{q}\right\| \nonumber \\
&& \quad + N^2(1+\Lambda_N)
\inf_{p \in \mathcal{P}_{N}}\left\|x-p\right\|
\label{diffy}
\end{eqnarray}
where $x^N \in \C{P}_N$ satisfies $x^N(\tau_k) = x(\tau_k)$,
$0 \le k \le N$, and $\Lambda_N$ is the Lebesgue constant relative to
the point set $\{ \tau_0, \tau_1, \ldots, \tau_N \}$.
\end{proposition}
\begin{proof}
Given $p \in \mathcal{P}_N$, the triangle inequality gives
\begin{equation}
\left\|\dot{x}-\dot{x}^N\right\|\leq \|\dot{x}-\dot{p}\|+\left\|\dot{p}
-\dot{x}^N\right\|.\label{dify}
\end{equation}
By Markov's inequality $\cite{Markov1916}$, we have
\begin{eqnarray}
\left\|\dot{p}-\dot{x}^N\right\|
&\leq& N^2 \left\|p-x^N\right\|=N^2 \left\|\sum_{i=0}^N(p(\tau_i)-x(\tau_i))
L_i(\tau)\right\|\nonumber \\
&\leq & N^2 \Lambda_N\max_{0\leq i\leq N}|p(\tau_i)-x(\tau_i)|
\le N^2\Lambda_N \|p-x\|. \label{qminusy}
\end{eqnarray}
Let $q \in \C{P}_{N}$ with $q(-1) = x(-1)$.
Again, by the triangle and Markov inequalities, we have
\begin{eqnarray}
\|\dot{x}-\dot{p}\| &\le& \|\dot{x} - \dot{q} \| + \|\dot{q} - \dot{p}\| \le
\|\dot{x} - \dot{q} \| + N^2 \|q - p\| \nonumber \\
&\le&
\|\dot{x} - \dot{q} \| + N^2 (\|q - x\| + \|x - p\|). \label{h71}
\end{eqnarray}
By the fundamental theorem of calculus,
\begin{equation}\notag
\left|q(t)-x(t)\right|=\left|\int_{-1}^{t}
\left(\dot{q}(s)-\dot{x}(s)\right)ds\right|\leq \int_{-1}^{t}
\left|\dot{q}(s)-\dot{x}(s)\right|ds\leq 2\|\dot{q}-\dot{x}\|.
\end{equation}
We combine this with (\ref{h71}) to obtain
\begin{equation}\label{h72}
\|\dot{x}-\dot{p}\| \le (1 + 2N^2) \|\dot{x} - \dot{q} \| + N^2 \|x - p\| .
\end{equation}
To complete the proof, combine (\ref{dify}), (\ref{qminusy}), and (\ref{h72})
and exploit the fact that
\[
\left\{\dot{q}: q(-1) = x(-1), \;\; q \in \C{P}_N \right\} =
\left\{\dot{q}: q \in \C{P}_N \right\}.
\]
\end{proof}
An estimate for the right side of \eqref{diffy} follows from results
on best uniform approximation by polynomials, which
originate from work of Jackson \cite{jackson}.
For example, the following result employs an estimate from Rivlin's
book \cite{Rivlin1969}.
\begin{lemma}\label{L2}
If $x$ has $m$ derivatives on $[-1, +1]$ and $N > m$, then
\begin{equation}\label{jackson}
\inf_{p\in \mathcal{P}_N}\|x-p\|\leq
\left( \frac{12}{m+1} \right) \left( \frac{6e}{N} \right)^m
\|x^{(m)}\|,
\end{equation}
where $x^{(m)}$ denotes the $m$-th derivative of $x$.
\end{lemma}
\begin{proof}
It is shown in \cite[Thm. 1.5]{Rivlin1969} that
\begin{equation}\label{yp}
\inf_{p\in \mathcal{P}_N}\left\|x-p\right\|\leq
\left( \frac{6}{m+1} \right) \left( \frac{6e}{N} \right)^m
\omega_m \left(\frac{1}{N-m}\right),
\end{equation}
where $\omega_m$ is the modulus of continuity of $x^{(m)}$.
By the definition of the modulus of continuity, we have
\[
\omega_m\left(\frac{1}{N-m}\right)=\sup\left\{\left|x^{(m)}(\tau_1)
-x^{(m)}(\tau_2)\right|: {\tau_1, \tau_2 \in[-1,1], |\tau_1-\tau_2|
\leq
\frac{1}{N-m}}\right\}.
\]
Since
\[
|x^{(m)}(\tau_1)-x^{(m)}(\tau_2) |\leq 2
\|x^{(m)}\| ,
\]
(\ref{jackson}) follows from (\ref{yp}).
\end{proof}
If $\Lambda_N = O(N)$ and $m \ge 4$, then
Proposition~\ref{L1} and Lemma~\ref{L2} imply that the components
of the residual approach zero as $N$ tends to infinity.
Moreover, if $x$ is infinitely differentiable and
there exists a constant $c$ such that $\|x^{(m)}\| \le c^m$,
then we take $m = N-1$ in Lemma~\ref{L2} to obtain
\[
\inf_{p\in \mathcal{P}_N}\|x-p\|\leq
\left( \frac{2}{ec} \right) \left( \frac{6ec}{N} \right)^N.
\]
Hence, the convergence is extremely fast due to the $1/N^N$ factor.
\section{Some results of Szeg\H{o}}
\label{szego}
We now summarize several results developed by Szeg\H{o} in \cite{Szego1939}
for Jacobi polynomials that are used in the analysis.
The page and equation numbers that follow refer to the 2003 edition
of Szeg\H{o}'s book published by the American Mathematical Society.
First, at the bottom of page 338, Szeg\H{o} makes the following observation:
\smallskip
\begin{theorem}\label{jacobi}
The Lebesgue constant for the roots of the Jacobi polynomial
$P_N^{(\alpha, \beta)}(\tau)$ is $O(N^{0.5+\gamma})$
if $\gamma := \max(\alpha, \beta) > -1/2$,
while it is $O(\log N)$ if $\gamma \le-1/2$.
\end{theorem}
\smallskip
For the Gauss quadrature points, $\alpha = \beta = 0$, $\gamma = 0$,
and $\Lambda_N = O(\sqrt{N})$.
The result that we state as Theorem~\ref{jacobi}
is based on a number of additional properties of Jacobi polynomials
which are useful in our analysis.
The following identity is a direct consequence of the Rodrigues formula
\cite[p. 67]{Szego1939} for $P_N^{(\alpha,\beta)}$.
\smallskip
\begin{proposition}\label{flip}
For any $\alpha$ and $\beta \in \mathbb{R}$, we have
\begin{equation}\label{eq8}
P_N^{(\alpha, \beta)}(\tau)=(-1)^NP_N^{(\beta, \alpha)}(-\tau)
\quad \mbox{for all } \tau \in [-1, +1].
\end{equation}
\end{proposition}
\smallskip
The following proposition provides some bounds for Jacobi polynomials.
\smallskip
\begin{proposition}\label{pro1}
For any $\alpha$ and $\beta \in \mathbb{R}$
and any fixed constant $c_1 > 0$,
we have
\[
P_N^{(\alpha,\beta)}(\cos\theta)=\left\{
\begin{array}{clcccl}
O\left(N^\alpha\right) &\mbox{if } \theta \in
[&0&,& c_1N^{-1} &],\\[.05in]
\theta^{-\alpha-0.5}O\left(N^{-1/2}\right)
&\mbox{if } \theta \in [ &c_1N^{-1} &, & \pi/2 &],\\[.05in]
(\pi-\theta)^{-\beta-0.5}O\left(N^{-1/2}\right)
&\mbox{if } \theta \in [&\pi/2&,& \pi- c_1N^{-1}&],\\[.05in]
O\left(N^\beta\right) &\mbox{if } \theta \in [&\pi- c_1N^{-1}&,& \pi&].
\end{array}
\right.
\]
\end{proposition}
\smallskip
\begin{proof}
The bounds for $\theta \in [0, cN^{-1}]$ and for
$\theta \in [cN^{-1}, \pi/2]$ appear in \cite[(7.32.5)]{Szego1939}.
If $\theta \in \left[\pi/2, \pi\right]$, then
$\pi-\theta \in \left[0, \pi/2 \right]$ and by \eqref{eq8},
\begin{equation}\label{h1}
P_N^{(\alpha, \beta)}(\cos \theta)=P_N^{(\alpha, \beta)}(-\cos(\pi- \theta))
=(-1)^NP_N^{(\beta, \alpha)}(\cos(\pi- \theta)).
\end{equation}
Hence, for $\theta \in [\pi/2, \pi]$,
the first two estimates in the proposition applied to the right
side of (\ref{h1}) yield the last two estimates.
\end{proof}
The next proposition provides an estimate for the derivative of a
Jacobi polynomial at a zero.
\smallskip
\begin{proposition}\label{pro2}
If $\alpha>-1$ and $\beta>-1$, then there exist constants
$\gamma_2 \ge \gamma_1 > 0$, depending only on $\alpha$ and $\beta$, such that
\[
\gamma_1 i^{-\beta - 1.5} N^{\beta + 2} \le
\left|\dot{P}_N^{(\alpha, \beta)}(\tau_i)\right| \le
\gamma_2 i^{-\beta - 1.5} N^{\beta + 2}
\]
whenever $\tau_i \le 0$ where
$\tau_1 < \tau_2 < \ldots < \tau_N$ are the zeros of $P_N^{(\alpha, \beta)}$
(the smallest zero is indexed first).
Moreover, if $\theta_i \in [0, \pi]$ is defined by
$\cos \theta_i = \tau_i$, then there exist constants
$\gamma_4 \ge \gamma_3 > 0$, depending only on $\alpha$ and $\beta$, such that
\begin{equation}\label{h9}
\gamma_3 \sqrt{N} (\pi - \theta_i)^{-\beta - 1.5} \le
\left|\dot{P}_N^{(\alpha, \beta)}(\tau_i)\right| \le
\gamma_4 \sqrt{N} (\pi -\theta_i)^{-\beta - 1.5}
\end{equation}
whenever $\theta_i \in [\pi/2, \pi]$.
\end{proposition}
\smallskip
\begin{proof}
In \cite[(8.9.2)]{Szego1939}, it is shown that there exist
$\gamma_2 \ge \gamma_1 > 0$, depending only on $\alpha$ and $\beta$, such that
\begin{equation}\label{h7}
\gamma_1 i^{-\beta - 1.5} N^{\beta + 2} \le
\left|\dot{P}_N^{(\beta, \alpha)}(\sigma_i)\right| \le
\gamma_2 i^{-\beta - 1.5} N^{\beta + 2}
\end{equation}
whenever $\sigma_i \ge 0$ where
$\sigma_1 > \sigma_2 > \ldots > \sigma_N$ are the zeros of
$P_N^{(\beta, \alpha)}$ (the largest zero is indexed first).
By Proposition~\ref{flip}, $\tau_i$ is a zero of $P_N^{(\alpha,\beta)}$
if and only if $-\tau_i$ is a zero of $P_N^{(\beta,\alpha)}$.
Hence, the zeros of $P_N^{(\beta,\alpha)}$ are
$-\tau_1 > -\tau_{2} > \ldots > -\tau_N$.
Moreover,
\begin{equation}\label{h7.5}
\dot{P}_N^{(\alpha,\beta)}(\tau) = \pm
\dot{P}_N^{(\beta,\alpha)}(-\tau).
\end{equation}
The bound given in the proposition for
$|\dot{P}_N^{(\alpha,\beta)}(\tau_i)|$ with $\tau_i \le 0$ is exactly the
bound (\ref{h7}) for
$|\dot{P}_N^{(\beta,\alpha)}(\sigma_i)|$ with $\sigma_i \ge 0$.
It is shown in \cite[(8.9.7)]{Szego1939}, that there exist constants
$\gamma_4 \ge \gamma_3 > 0$, depending only on $\alpha$ and $\beta$, such that
\begin{equation}\label{h8}
\gamma_3 \sqrt{N} \phi_i^{-\beta - 1.5} \le
\left|\dot{P}_N^{(\beta, \alpha)}(\sigma_i)\right| \le
\gamma_4 \sqrt{N} \phi_i^{-\beta - 1.5}
\end{equation}
whenever $\phi_i \in [0, \pi/2]$ where $\cos \phi_i = \sigma_i$.
Since $\cos \phi_i = \sigma_i = -\tau_i = \cos (\pi - \theta_i)$,
it follows that $\phi_i = \pi - \theta_i$, and
(\ref{h7.5}) and (\ref{h8}) yield (\ref{h9}).
\end{proof}
\section{Lebesgue constant for Gauss quadrature points augmented by $-1$}
\label{gauss+}
In this section we estimate the Lebesgue constant for
the Gauss quadrature points augmented by $\tau_0 = -1$.
Due to the symmetry of the Gauss quadrature points, the same
estimate holds when the Gauss quadrature points are augmented by $+1$
instead of $-1$.
The Gauss quadrature points are the zeros of the Jacobi polynomial
$P_N^{(0, 0)}(\tau)$, which is abbreviated as $P_N(\tau)$.
By Theorem~\ref{jacobi}, the Lebesgue constant for the Gauss
quadrature points themselves is $O(\sqrt{N})$.
The effect of adding the point $\tau_0 = -1$ to the Gauss quadrature
points is not immediately clear due to the new factor $(1 + \tau_i)$
in the denominator of the Lagrange polynomials;
this factor can approach 0 since roots of $P_N$
approach $-1$ as $N$ tends to infinity.
Nonetheless, with a careful grouping of terms,
Szeg\H{o}'s bound in Theorem~\ref{jacobi}
for the Gauss quadrature points can be extended to handle the new
point $\tau_0 = -1$.
\smallskip
\begin{theorem}\label{gausstheom}
The Lebesgue constant for the point set consisting of the Gauss
quadrature points $-1 < \tau_1 < \tau_2 < \ldots < \tau_N < +1$
$($the zeros of $P_N)$ augmented with $\tau_0 = -1$ is $O(\sqrt{N})$.
\end{theorem}
\smallskip
\begin{proof}
Define
\[l(\tau)=(\tau-\tau_1)(\tau-\tau_2)\dots (\tau-\tau_N),
\quad \mbox{and}\quad L(\tau)=(\tau+1)l(\tau).
\]
The derivative of $L(\tau)$ at $\tau_i$ is
\[
\dot{L}(\tau_i)=l(\tau_i)+(\tau_i+1)\dot{l}(\tau_i)=\left\{
\begin{array}{cl}\displaystyle
l(-1), & i = 0, \\[.1in]
(\tau_i+1)\dot{l}(\tau_i), &i> 0.
\end{array}
\right.
\]
Hence, the Lagrange polynomials $L_i(\tau)$ associated with the
point set $\{\tau_0 , \tau_1, \ldots, \tau_N\}$ can be expressed as
\begin{equation}\label{Li}
L_i(\tau)=\frac{L(\tau)}{\dot{L}(\tau_i)(\tau-\tau_i)}=\left\{
\begin{array}{cl}
l(\tau)/l(-1), &i=0, \\[.1in]
\displaystyle\frac{L(\tau)}{(\tau_i+1)\dot{l}(\tau_i)(\tau-\tau_i)},
& i> 0.
\end{array}
\right.
\end{equation}
Since $P_N$ is a multiple of $l$ (it has the same zeros), it follows that
\[
L_i(\tau)=\left\{
\begin{array}{cl}
P_N(\tau)/P_N(-1), &i=0,\\[.1in]
\displaystyle
\frac{(\tau+1)P_N(\tau)}{(\tau_i+1)\dot{P}_N(\tau_i)(\tau-\tau_i)},
&i > 0.
\end{array}
\right.
\]
By \cite[(7.21.1)]{Szego1939},
$|P_N(\tau)| \le 1$ for all $\tau \in [-1, +1]$, and by
\cite[(4.1.4)]{Szego1939}, $|P_N(-1)| = (-1)^N$.
We conclude that $|L_0 (\tau)| \le 1$ for all $\tau \in [-1, +1]$.
Hence, the proof is complete if
\begin{equation}\label{h3}
\max \left\{ \sum_{i=1}^N|L_i(\tau)| : \tau \in[-1, 1] \right\} = O(\sqrt{N}) .
\end{equation}
For any $\tau \in [-1, +1]$, the integers $i \in [1, N]$ are partitioned
into the four disjoint sets
\begin{eqnarray*}
\C{I}_1 &=& \{ i \in [1,N]: \tau_i \ge 0 \}, \\
\C{I}_2 &=& \{ i \in [1,N]: -1 < \tau_i < 0, \; \tau_i > \tau \}, \\
\C{I}_3 &=& \{ i \in [1,N]: -1 < \tau_i < 0, \; \tau_i \le \tau, \;
\tau - \tau_i \le \tau_i + 1 \}, \\
\C{I}_4 &=& \{ i \in [1,N]: -1 < \tau_i < 0, \; \tau_i \le \tau, \;
\tau - \tau_i > \tau_i + 1 \}.
\end{eqnarray*}
Let $\C{I}_{123}$ denote $\C{I}_1 \cup \C{I}_2 \cup \C{I}_3$.
Observe that for any $i \in \C{I}_{123}$ and $\tau \in [-1, +1]$,
$(\tau+1)/(\tau_i + 1) \le 2$.
Consequently, for all $i \in \C{I}_{123}$,
\[
|L_i (\tau)| =
\left| \frac{(\tau+1)P_N(\tau)}{(\tau_i+1)\dot{P}_N(\tau_i)(\tau-\tau_i)}
\right| \le
\frac{2|P_N(\tau)|}{|\dot{P}_N(\tau_i)(\tau-\tau_i)|} .
\]
This bound together with Theorem~\ref{jacobi} imply that
\[
\sum_{i \in \C{I}_{123}} |L_i(\tau)| \le
\sum_{i \in \C{I}_{123}}
\frac{2|P_N(\tau)|}{|\dot{P}_N(\tau_i)(\tau-\tau_i)|} \le
2 \sum_{i=1}^N
\frac{|P_N(\tau)|}{|\dot{P}_N(\tau_i)(\tau-\tau_i)|} = O(\sqrt{N})
\]
since the terms in the final sum are the Lagrange
polynomials for the Gauss quadrature points.
To complete the proof, we need to analyze the terms in (\ref{h3})
associated with the indices in $\C{I}_4$.
These terms are more difficult to analyze since $\tau_i + 1$
in the denominator of $L_i$ could approach 0 while $\tau +1$ in
the numerator remains bounded away from 0.
For $i \in \C{I}_4$, we have
\[
\tau + 1 = (\tau - \tau_i) + (\tau_i + 1) \le 2 (\tau - \tau_i)
\]
since $\tau - \tau_i > \tau_i + 1$.
Hence,
\[
|L_i (\tau)| \le \frac{2|P_N(\tau)|}{|(\tau_i + 1) \dot{P}_N (\tau_i)|} \le
\frac{2}{|(\tau_i + 1) \dot{P}_N (\tau_i)|}
\]
since $|P_N(\tau)| \le 1$ for all $\tau \in [-1, +1]$
by \cite[(7.21.1)]{Szego1939}.
It follows that
\begin{equation}\label{h6}
\sum_{i \in \C{I}_{4}} |L_i(\tau)| \le
\sum_{i \in \C{I}_{4}}
\frac{2}{|(\tau_i + 1) \dot{P}_N (\tau_i)|} \le
\sum_{-1 < \tau_i < 0 }
\frac{2}{|(\tau_i + 1) \dot{P}_N (\tau_i)|} .
\end{equation}
Given $\theta \in [\pi/2, \pi]$, define $\phi = \pi - \theta$.
Observe that
\[
\left|\frac{\phi^2}{1+\cos \theta}\right|
=\frac{\phi^2}{2\cos^2(\theta/2)}
=\frac{2(\phi/2)^2}{\sin^2 (\phi/2)}
\leq \max_{x\in [0, \pi/4]}
\frac{2x^2}{\sin^2 x} =\frac{\pi^2}{4}.
\]
Hence, for $\theta \in [\pi/2, \pi]$, we have
\begin{equation}\label{h5}
1 + \cos \theta \ge \left( \frac{4}{\pi^2} \right) \phi^2 =
\frac{4}{\pi^2} (\pi - \theta)^2 .
\end{equation}
By the bounds \cite[(6.21.5)]{Szego1939} for the roots of the
Jacobi polynomial $P_N^{(\alpha, \beta)}$ when
$\alpha$ and $\beta \in [-0.5, +0.5]$, it follows that
\begin{equation}\label{*}
\left(\frac{2i-1}{2N+1}\right) \pi \leq \pi-\theta_i
\leq \left(\frac{2i}{2N+1}\right) \pi, \quad
1 \le i \le N,
\end{equation}
where $\cos \theta_i = \tau_i$.
This implies the lower bound
\begin{equation}\label{h4}
\pi - \theta_i \ge
\left(\frac{2i-1}{2N+1}\right) \pi \ge
\left( \frac{i}{3N} \right)\pi > \frac{i}{N} .
\end{equation}
We combine (\ref{h5}) and (\ref{h4}) to obtain
\begin{equation}\label{eq3}
1+\tau_i \ge \frac{4}{\pi^2}(\pi-\theta_i)^2\geq\frac{4}{\pi^2}
\left(\frac{i}{N}\right)^2.
\end{equation}
By Proposition~\ref{pro2},
\[
|\dot{P}_N(\cos \theta_i )| \ge \gamma_1 i^{-1.5} N^2.
\]
This lower bound for the derivative and the lower bound (\ref{eq3}) for
the root imply that
\[
\frac{1}{(1+\tau_i)|\dot{P}_N(\tau_{i})|} \le
\left( \frac{\pi^2}{4 \gamma_1} \right) i^{-1/2} .
\]
Hence, we obtain the following bound for the $\C{I}_4$ sum in (\ref{h6}):
\[
\sum_{-1<\tau_i<0}\frac{2}{(1+\tau_i)|\dot{P}_N(\tau_{i})|} \le
\left( \frac{\pi^2}{2 \gamma_1} \right)
\sum_{i = 1}^N i^{-1/2} \le
\left( \frac{\pi^2}{2 \gamma_1} \right)
\int_0^N i^{-1/2} di = O(\sqrt{N}) .
\]
This bound inserted in (\ref{h6}) completes the proof.
\end{proof}
\section{Lebesgue constants for the Radau quadrature points}
\label{radau+}
Next, we estimate the Lebesgue constant for the Radau quadrature scheme.
There are two versions of the Radau quadrature points depending on whether
$\tau_1 = -1$ or $\tau_N = +1$.
Since these two schemes have quadrature points that are the
negatives of one another, the Lebesgue constants are the same.
The analysis is carried out for the case $\tau_N = +1$.
In this case, the Radau quadrature points are the $N-1$ roots of
$P_{N-1}^{(1,0)}$ augmented by $\tau_N = 1$.
Szeg\H{o} shows that the Lebesgue constant for the roots of
$P_{N-1}^{(1,0)}$ is $O(N^{3/2})$.
We show that when the quadrature point $\tau_N = 1$ is included,
the Lebesgue constant drops to $O(\sqrt{N})$.
The analysis requires an estimate for the location of the zeros of
$P_{N-1}^{(1,0)}$.
Our estimate is based on some relatively recent results on
interlacing properties for the zeros of Jacobi polynomials obtained by
Driver, Jordaan, and Mbuyi in \cite{DriverJordaanMbuyi2008}.
Let $\tau_i'$ and $\tau_i''$, $i\geq 1$, be zeros of
$P_{N-1}$ and $P_{N}$ respectively, arranged in increasing order.
Applying \cite[Thm. 2.2]{DriverJordaanMbuyi2008}, we have
\[
\tau_i'' < \tau_i < \tau_{i}' ,
\]
$i = 1, 2, \ldots, N-1$, where
$-1 < \tau_1 < \tau_2 < \ldots < \tau_{N-1} < +1$ are the zeros of
$P_{N-1}^{(1,0)}$.
Let $\theta_i \in [0, \pi]$ be defined by $\cos \theta_i = \tau_i$.
By the estimate (\ref{*}) for the zeros of $P_N$, it follows that
the zeros of $P_{N-1}^{(1,0)}$ have the property that
\begin{equation}\label{zeros}
\left( \frac{2i-1}{2N-1} \right) \pi < \theta_{N-i} <
\left( \frac{2(i+1)}{2N+1} \right) \pi, \quad
1 \le i \le N-1.
\end{equation}
When $i$ is replaced by $N-i$, these bounds become
\begin{equation}\label{zeros*}
\left( \frac{2i-1}{2N+1} \right) \pi < \pi - \theta_{i} <
\left( \frac{2i}{2N-1} \right) \pi, \quad
1 \le i \le N-1.
\end{equation}
Together, (\ref{zeros}) and (\ref{zeros*}) imply that
\begin{equation}\label{phibounds}
\pi - \theta_{i} > i/N \quad \mbox{and} \quad \theta_{N-i} > i/N,
\quad 1 \le i \le N-1;
\end{equation}
moreover, taking into account both the upper and lower bounds, we have
\begin{eqnarray}
\theta_i - \theta_{i+1} &<& \left( \frac{4(i+N)+2N+1}{4N^2 - 1}\right) \pi
\le \left( \frac{10N - 7}{4N^2 - 1} \right) \pi \nonumber \\
&<& \left( \frac{5(2N - 1)}{4N^2 - 1} \right) \pi < \frac{2.5\pi}{N},
\quad 1 \le i \le N-2.
\label{separation}
\end{eqnarray}
Thus, the interlacing properties for the zeros leads to explicit
bounds for the separation of the zeros; for comparison,
Theorem~8.9.1 in \cite{Szego1939} yields $\theta_i - \theta_{i+1} = O(1)/N$,
while (\ref{separation}) yields an explicit constant $2.5\pi$.
These estimates for the zeros of $P_{N-1}^{(1,0)}$ are used
to derive the following result.
\smallskip
\begin{theorem}\label{radau}
The Lebesgue constant for the Radau quadrature points
\[
-1 < \tau_1 < \tau_2 < \ldots < \tau_N = 1
\]
$($the zeros of $P_{N-1}^{(1,0)}$ augmented by $\tau_N = +1)$ is $O(\sqrt{N})$.
\end{theorem}
\smallskip
\begin{proof}
The Lagrange interpolating polynomials $R_i$, $1 \le i \le N$,
associated with the Radau quadrature points are given by
\[
R_i(\tau)= \left( \frac{1-\tau}{1-\tau_i} \right)
\prod_{\substack{j=1\\ j\neq i}}^{N-1}\frac{\tau-\tau_j}
{\tau_i-\tau_j}, \quad 1 \le i \le N-1, \quad
R_N(\tau) =
\prod_{\substack{j=1}}^{N-1}\frac{\tau-\tau_j}
{1-\tau_j}.
\]
Similar to (\ref{Li}), the $R_i$ can be expressed
\begin{equation}\label{Ri}
R_i(\tau)=\left\{
\begin{array}{cl}
\displaystyle\frac{(1-\tau)P_{N-1}^{(1,0)}(\tau)}
{(1-\tau_i)\dot{P}_{N-1}^{(1,0)}(\tau_i)(\tau-\tau_i)},
&i < N, \\[.20in]
\displaystyle{\frac{P_{N-1}^{(1,0)}(\tau)}{P_{N-1}^{(1,0)}(1)}}.
&i=N.\\
\end{array}
\right.
\end{equation}
By \cite[(4.1.1)]{Szego1939} and \cite[(7.32.2)]{Szego1939}, we have
\begin{equation}\label{h22}
P_{N-1}^{(1,0)}(1)= N \quad \mbox{and} \quad
|P_{N-1}^{(1,0)}(\tau)|\le N \mbox{ for all } \tau \in [-1, +1] .
\end{equation}
We conclude that $|R_N (\tau)| \le 1$ for all $\tau \in [-1, +1]$.
Hence, the proof is complete if
\begin{equation}\label{h10}
\max \left\{ \sum_{i=1}^{N-1}|R_i(\tau)| : \tau \in [-1, +1] \right\}
=O(\sqrt{N}) .
\end{equation}
Let $\delta > 0$ be a small constant.
Technically, any $\delta$ satisfying $0 < \delta < 1/2$ is
small enough for the analysis.
Szeg\H{o} establishes the following bounds when analyzing the
Lebesgue constants associated with the roots of Jacobi polynomials:
\begin{equation}\label{radaulebesgue}
\sum_{i = 1}^N \left| \frac{P_{N}^{(1,0)}(\tau)}
{\dot{P}_{N}^{(1,0)}(\tau_i)(\tau-\tau_i)} \right| =
\left\{
\begin{array}{ll}
O(\sqrt{N}) & \mbox{if } \tau \in [-1, \delta-1], \\
O(\log N) & \mbox{if } \tau \in [\delta-1 , 1 - \delta], \\
O(N^{3/2}) & \mbox{if } \tau \in [1 - \delta, 1].
\end{array} \right.
\end{equation}
Szeg\H{o} considers the general Jacobi polynomials
$P_N^{(\alpha, \beta)}$ on pages 336--338 of \cite{Szego1939},
while here we only state the results
corresponding to $\alpha = 1$ and $\beta = 0$.
We first show that (\ref{h10}) holds when $\tau \in [-1, 1-\delta]$.
Observe that $(1 - \tau)/(1-\tau_i) \le 4/\delta$
when $\tau_i \le 1 - \delta/2$ and $\tau \in [-1, +1]$.
It follows from (\ref{radaulebesgue}) that
\begin{eqnarray}
\sum_{\tau_i \le 1-\delta/2} |R_i(\tau)| &\le& \left( \frac{4}{\delta} \right)
\sum_{\tau_i \le 1-\delta/2}
\left| \frac{P_{N-1}^{(1,0)}(\tau)}
{\dot{P}_{N-1}^{(1,0)}(\tau_i)(\tau-\tau_i)} \right| \nonumber \\
&=& \left\{ \begin{array}{ll}
O(\sqrt{N}), & \tau \in [-1, \delta-1], \\
O(\log N), & \tau \in [\delta-1, 1 - \delta] .
\end{array} \right. \label{h11}
\end{eqnarray}
When $\tau_i > 1-\delta/2$ and $\tau \in [-1, +1-\delta]$, we have
$|\tau - \tau_i| \ge \delta/2$; hence,
\begin{equation}\label{h12}
\sum_{1 > \tau_i > 1-\delta/2} |R_i(\tau)| \le \left( \frac{4}{\delta} \right)
\sum_{1 > \tau_i > 1-\delta/2}
\left| \frac{P_{N-1}^{(1,0)}(\tau)}
{(\tau_i - 1)\dot{P}_{N-1}^{(1,0)}(\tau_i)} \right| .
\end{equation}
We have the following bounds for the factors on the right side of (\ref{h12}):
\begin{itemize}
\item[(a)]
By Proposition~\ref{pro1},
$|P_{N-1}^{(1,0)} (\tau)| = O(1)$ if $\tau \in [-1, \delta - 1]$ and
$|P_{N-1}^{(1,0)} (\tau)| = O(N^{-1/2})$ if $\tau \in [\delta - 1, 1-\delta]$.
\item[(b)]
By (\ref{h8}),
$|\dot{P}_{N-1}^{(1, 0)}(\tau_i)| \ge
\gamma_3 \theta_i^{-5/2} \sqrt{N-1}$, where $\cos \theta_i = \tau_i \ge 0$.
\item[(c)]
By a Taylor expansion around $\theta = 0$,
\begin{equation}\label{1-cos}
\theta^2/4 \le 1 - \cos \theta \le \theta^2/2, \quad \theta \in [0, \pi/2].
\end{equation}
\end{itemize}
By (b) and the lower bound in (c) at $\theta = \theta_i$, we have
\begin{equation}\label{h100}
(1-\tau_i) |\dot{P}_{N-1}^{(1,0)}(\tau_i)| \ge
0.25 \gamma_3 \theta_i^{-1/2} \sqrt{N-1} .
\end{equation}
We combine this with (a) and (\ref{h12}) to obtain
\[
\sum_{1 > \tau_i > 1-\delta/2} |R_i(\tau)| =
\left\{ \begin{array}{lll}
O(N^{-1/2})\displaystyle\sum_{i = 1}^N \sqrt{\theta_i} &= O(\sqrt{N}),
& \tau \in [-1, \delta - 1], \\
O(N^{-1})\displaystyle\sum_{i = 1}^N \sqrt{\theta_i} &= O(1),
& \tau \in [\delta-1, 1-\delta] ,
\end{array} \right.
\]
since $\theta_i \in [0, \pi]$.
This establishes (\ref{h11}) for all $\tau \in [-1, 1-\delta]$.
To complete the proof of (\ref{h10}), we need to consider
$\tau \in (1-\delta, 1]$.
The analysis becomes more complex since
Szeg\H{o}'s estimate (\ref{radaulebesgue}) is $O(N^{3/2})$ in this region,
while we are trying to establish a much smaller bound in (\ref{h10});
in fact, the bound in this region is $O(\log N)$ as we will show.
For the numerator of $R_i (\tau)$ and
$\tau \in (1-\delta, 1]$,
Proposition~\ref{pro1} and (\ref{1-cos}) yield
\begin{eqnarray}
(1-\tau) |P_{N-1}^{(1,0)} (\tau)| &=&
(1-\cos \theta)|P_{N-1}^{(1,0)}(\cos \theta)| =
\left\{ \begin{array}{ll}
\theta^2 O(N), & \theta \in [0, 1/N], \\
\theta^{1/2} O(N^{-1/2}), & \theta \in [1/N, \pi/2],
\end{array} \right. \nonumber \\
&=&
O(N^{-1/2}). \label{h14}
\end{eqnarray}
Given $\tau \in (1-\delta, 1]$,
let us first focus on those $i$ in (\ref{h10}) for which
$|\tau-\tau_i| \ge \delta$.
In this case, $\tau_i \le 1-\delta$ or
$1-\tau_i \ge \delta$, and (\ref{h14}) gives
\begin{eqnarray}
\sum_{|\tau-\tau_i|\ge\delta}|R_i(\tau)| &=&
\sum_{|\tau-\tau_i|\ge\delta}
\left| \frac{(\tau-1)P_{N-1}^{(1,0)}(\tau)}
{(\tau_i-1)\dot{P}_{N-1}^{(1,0)}(\tau_i)(\tau-\tau_i)} \right| \nonumber \\
&\le& \frac{O(N^{-1/2})}{\delta^2} \sum_{|\tau-\tau_i|\ge\delta}
\frac{1}{|\dot{P}_{N-1}^{(1,0)} (\tau_i)|}. \label{h50}
\end{eqnarray}
The lower bounds (\ref{h9}) and (\ref{h8}) imply that
\begin{equation}\label{h51}
\sum_{|\tau-\tau_i|\ge\delta}|R_i(\tau)| =
O(N^{-1}) \sum_{\tau_i \ge 0} \theta_i^{5/2}
+
O(N^{-1}) \sum_{\tau_i < 0} |\pi - \theta_i|^{3/2} = O(1),
\end{equation}
since the terms in the sums are uniformly bounded and there are at
most $N$ terms.
The next step in the proof of (\ref{h10}) for $\tau \in (1-\delta, 1]$
is to consider those terms corresponding to $|\tau-\tau_i|<\delta$.
Since $\delta$ is small, it follows that both $\tau$ and $\tau_i$ are
near 1, and consequently, $\theta$ and $\theta_i$
are small and nonnegative,
where $\cos \theta = \tau$ and $\cos \theta_i = \tau_i$.
In particular, $0 \le \theta_i \le \pi/2$.
In this case where $\tau_i$ is near $\tau$,
it is important to take into account the fact that
$\tau - \tau_i$ is a divisor of the numerator $P_{N-1}^{(1,0)}(\tau)$.
To begin, we combine the lower bound in (\ref{h8}) and the bounds in
(\ref{1-cos}) to obtain
\begin{equation}\label{h19}
\frac{(1-\tau)}
{(1-\tau_i)|\dot{P}_{N-1}^{(1,0)}(\tau_i)|} \le
\frac{2\theta^2}{\theta_i^2 (\gamma_3 \sqrt{N}) \theta_i^{-5/2}} =
O(N^{-1/2}) \theta^2\sqrt{\theta_i} .
\end{equation}
It follows from (\ref{Ri}) that
\begin{equation}\label{h15}
|R_i(\tau)|= O(N^{-1/2}) \theta^2\sqrt{\theta_i}
\left| \frac{P_{N-1}^{(1,0)}(\tau)}{\tau-\tau_i} \right| .
\end{equation}
The mean value theorem and the formula
\cite[(4.21.7)]{Szego1939} for the derivative of
$P_{N-1}^{(\alpha, \beta)}(\tau)$ in terms of
$P_{N-2}^{(\alpha+1, \beta+1)}(\tau)$ gives the identity
\begin{equation} \label{h17}
\left|\frac{P_{N-1}^{(1,0)}(\tau)}{\tau-\tau_i}\right|
= \left|\frac{P_{N-1}^{(1,0)}(\tau)-P_{N-1}^{(1,0)}
(\tau_i)}{\tau-\tau_i}\right|
=\left(\frac{N+1}{2}\right)\left|P_{N-2}^{(2,1)}(\cos\eta_i)\right|,
\end{equation}
where $\eta_i$ lies between $\theta$ and $\theta_i$.
Together, (\ref{h15}) and (\ref{h17}) imply that
\begin{equation}\label{h20}
|R_i(\tau)| = O(N^{1/2}) \theta^2 \sqrt{\theta_i}
\left|P_{N-2}^{(2,1)}(\cos\eta_i)\right|.
\end{equation}
The estimate (\ref{h20}) is useful when $\tau_i$ is near $\tau$.
When $\tau_i$ is not near $\tau$, we proceed as follows.
Use the identity
\[
\cos\alpha-\cos\beta
=-2\sin\displaystyle{\frac{(\alpha+\beta)}{2}}
\sin\displaystyle{\frac{(\alpha-\beta)}{2}},
\]
to deduce that
\begin{equation}\label{h70}
|\tau - \tau_i| = |\cos \theta - \cos \theta_i| \ge
\frac{2}{\pi^2} \left| \theta^2 - \theta_i^2 \right|
\end{equation}
when $|\theta + \theta_i| \le \pi$, which is satisfied since
both $\theta$ and $\theta_i$ are near 0.
Exploiting this inequality in (\ref{h15}) yields
\begin{equation}\label{h21}
|R_i(\tau)|= O(N^{-1/2}) \theta^2\sqrt{\theta_i}
\left| \frac{P_{N-1}^{(1,0)}(\tau)}{ \theta^2 - \theta_i^2} \right| .
\end{equation}
Recall, that we now need to analyze the interval $\tau \in [1-\delta, 1]$
and those $i$ for which $|\tau-\tau_i| < \delta$.
Our analysis works with the variable $\theta \in [0, \pi/2]$,
where $\cos \theta = \tau$.
The interval $\theta \in [0, \pi/2]$ corresponds to
$\tau \in [0,1]$ which covers the target interval $[1-\delta, 1]$ when
$\delta$ is small.
By \cite[(7.32.2)]{Szego1939}, we have
\[
|P_{N-2}^{(2,1)}(\cos\eta_i)| \le N(N-1)/2 .
\]
If $\theta \in [0, c/N]$, where $c$ is a fixed constant independent of $N$,
then it follows from
(\ref{h20}) that $|R_i(\tau)| = O(N^{1/2}) \sqrt{\theta_i}$.
Moreover, if $\theta_i \le 2 \theta \le 2c/N$, then
$|R_i (\tau)| = O(1)$.
By the bounds (\ref{phibounds}),
the number of roots that
satisfy $\theta_{N-i} \le 2c/N$ is at most $2c$, independent of $N$.
On the other hand, if $\theta_i > 2 \theta$, then $\theta< \theta_i/2$ and
\[
\left|\theta_i^2-\theta^2\right|=\theta_i^2-\theta^2\geq
(3/4) \theta_i^2 .
\]
With this substitution in (\ref{h21}), we have
\[
|R_i(\tau)|= O(N^{-1/2}) \theta^2\theta_i^{-3/2}
\left| {P_{N-1}^{(1,0)}(\tau)} \right| .
\]
By (\ref{h22}), $| P_{N-1}^{(1,0)}(\tau)| \le N$.
Hence, if $\theta \in [0, c/N]$, then by (\ref{phibounds}), we have
\begin{eqnarray*}
\sum_{|\tau-\tau_i| < \delta} |R_i(\tau)| &=&
O(N^{-3/2}) \sum_{|\tau-\tau_i| < \delta} \theta_i^{-3/2} =
O(N^{-3/2}) \sum_{i=1}^{N-1} \theta_i^{-3/2} \\
&=& O(N^{-3/2}) \sum_{i=1}^{N-1} \theta_{N-i}^{-3/2} =
O(1) \sum_{i=1}^{N-1} i^{-3/2} = O(1),
\end{eqnarray*}
for all $\theta \in [0, c/N]$.
Finally, suppose that $\theta \in [c/N, \pi/2]$.
By (\ref{separation}) the separation between adjacent zeros
$\theta_i$ and $\theta_{i+1}$ is at most $2.5\pi/N$.
Hence, if $\theta_i$ is within $k$ zeros of $\theta$, then
$\eta_i \ge \theta - \gamma N^{-1}$, $\gamma = 2.5\pi k$.
Here $k \ge 2$ is an arbitrary fixed integer.
By Proposition~\ref{pro1}, we have
\[
\left|P_{N-2}^{(2,1)}(\cos\eta_i)\right| =
(\theta- \gamma N^{-1})^{-5/2}O(N^{-1/2}) .
\]
Choose $c > 2\gamma$.
If $\theta \in [c/N, \pi/2]$,
then $\theta/2 \ge c/(2N) \ge \gamma/N$.
Hence, $\theta- \gamma /N \ge \theta/2$ and
\[
\left|P_{N-2}^{(2,1)}(\cos\eta_i)\right| =
(\theta/2)^{-5/2}O(N^{-1/2}) =
\theta^{-5/2}O(N^{-1/2}) .
\]
Combine this with (\ref{h20}) to obtain
\[
|R_i(\tau)| = O(1) \sqrt{\theta_i/\theta} .
\]
when $\theta \in [c/N, \pi/2]$ and $\theta_i$ is within $k$
zeros of $\theta$.
If $\theta_i \le \theta$, then $R_i (\tau) = O(1)$.
If $\theta_i > \theta$ and $\theta_i$ is within $k$ zeros of $\theta$,
then $\theta_i - \theta \le \gamma/N$, and
\[
\theta_i/\theta \le (\theta + \gamma/N)/\theta \le 1 + \gamma/c
\]
when $\theta \in [cN, \pi/2]$.
Thus $|R_i (\tau)| = O(1)$ when $\theta \in [cN, \pi/2]$ and
$\theta_i$ is within $k$ zeros of $\theta$.
This analysis of $R_i$ when $\theta_i$ is close to $\theta$ needs to
be complemented with an analysis of $R_i$ when $\theta_i$ is not
close to $\theta$ and $\theta \in [c/N, \pi/2]$.
For $\theta$ in this interval, Proposition~\ref{pro1} yields
$|P_{N-1}^{(1,0)} (\cos \theta)| = \theta^{-3/2}O(N^{-1/2})$.
By (\ref{h21}), we have
\begin{equation}\label{h23}
|R_i (\tau)| = O(N^{-1}) \frac{\sqrt{\theta} \sqrt{\theta_i}}
{|\theta^2 - \theta_i^2|} .
\end{equation}
If $\theta \ge 2\theta_i$, then
$\theta^2 - \theta_i^2 \ge (3/4)\theta^2$ and
\[
|R_i (\tau)| = O(N^{-1}) \theta^{-3/2} \theta_i^{1/2} .
\]
By (\ref{zeros*}), we have
\[
|R_{N-i} (\tau)| = O((N\theta)^{-3/2}) \sqrt{i+1} .
\]
Recall that we are focusing on those $i$ for which $\theta_{N-i} \le \theta/2$.
The lower bound $\theta_{N-i} \ge i/N$ from (\ref{phibounds}) implies that
$i \le N\theta/2$ whenever $\theta_{N-i} \le \theta/2$.
Hence, the set of $i$
satisfying $i \le N\theta$ is a superset of the $i$ that we need to consider,
and we have
\begin{eqnarray*}
\sum_{\theta_i \le \theta/2} |R_i (\tau)| &=&
\sum_{\theta_{N-i} \le \theta/2} |R_{N-i} (\tau)| =
O((N\theta)^{-3/2}) \sum_{i \le N\theta} \sqrt{i+1} \\
&=& O((N\theta)^{-3/2}) (N\theta + 1)^{3/2} = O(1) .
\end{eqnarray*}
On the other hand, if $\theta < 2 \theta_i$, then we have
\[
\frac{\sqrt{\theta} \sqrt{\theta_i}}
{|\theta^2 - \theta_i^2|} =
\frac{\sqrt{\theta} \sqrt{\theta_i}}
{|(\theta - \theta_i)(\theta + \theta_i)|} \le
\frac{\sqrt{\theta} \sqrt{\theta_i}}
{|(\theta - \theta_i)\theta_i|} \le
\frac{\sqrt{2}}
{|\theta - \theta_i|} .
\]
Combine this with (\ref{h23}) to obtain
\[
|R_i (\tau)| = \frac{O(1)}{|N\theta - N\theta_i|} .
\]
Earlier we showed that
$|R_i (\tau)| = O(1)$ for those $i$ where the associated $\theta_i$
is within $k$ zeros of $\theta$.
When $\theta_i$ is more than $k$ zeros away from $\theta$,
we exploit the estimate (\ref{zeros}) for the zeros to deduce that
$|N\theta - N\theta_i|$ behaves like an arithmetic sequence of natural numbers.
Hence, the sum of the $|R_i (\tau)|$ over these natural numbers,
where we avoid the singularity, is bounded by a multiple of $\log N$.
This completes the proof.
\end{proof}
\section{Tightness of estimates}\label{numerical}
\label{tight}
At the bottom of page 110 in \cite{Vertesi81}, V\'{e}rtesi states
some lower bounds for the Lebesgue function.
In the case of the Gauss quadrature points augmented by $\tau_{N+1} = +1$
and the Radau quadrature points with $\tau_N = +1$, the associated
Lebesgue function is of order $\sqrt{N}$ at
$\tau = (\tau_1 + \tau_{2})/2$,
the midpoint between the two smallest quadrature points.
It follows that the $O(\sqrt{N})$ estimates for the Lebesgue constant are tight.
To study the tightness of the estimates,
the Lebesgue constants were evaluated numerically and
fit by curves of the form $a \sqrt{N} + b$, $10 \le N \le 100$
(see Figures~\ref{graphgauss}--\ref{graphradau}).
A fast and accurate method for evaluating the Gauss quadrature points,
which could be extended to the Radau quadrature points,
is given by Hale and Townsend in \cite{HaleTownsend13}.
Figure~\ref{graphgauss}--\ref{graphradau} indicate that
a curve of the form $a \sqrt{N} + b$ is a good fit to the Lebesgue constant.
Another Lebesgue constant which enters into the analysis of
the Radau collocation schemes studied in \cite{HagerHouRao15c} is the
Lebesgue constant for the Radau quadrature points on $(-1, +1]$
augmented by $\tau_0 = -1$.
As given by V\'{e}rtesi in \cite[Thm. 2.1]{Vertesi81}, the
Lebesgue constant is $O(\log n)$.
Trefethen \cite{Trefethen13} points out that the Lebesgue constant
on any point set has the lower bound
\[
\Lambda_N \ge \left( \frac{2}{\pi} \right) \log (N) + 0.52125\ldots ,
\]
due to Erd\H{o}s \cite{Erdos61} and Brutman \cite{Brutman78}.
For comparison, Figure~\ref{graphradau-1} plots this lower bound
along with the computed Lebesgue constant.
When the number of interpolation points range between 10 and 100,
the Lebesgue constant for the Radau quadrature
points augmented by the point $-1$ differs from the smallest possible Lebesgue
constant by between 0.70 and 0.84.
\begin{figure}
\centering
\includegraphics[scale=.4]{gauss.eps}
\caption{Least squares approximation to the Lebesgue constant for
the point set corresponding to the Gauss quadrature points augmented by $-1$
using curves of the form $a\sqrt{N}+b$}
\label{graphgauss}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=.4]{radau.eps}
\caption{Least squares approximation to the Lebesgue constant for
the point set corresponding to the Radau quadrature points using curves of the
form $a\sqrt{N}+b$}
\label{graphradau}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=.4]{radau1.eps}
\caption{Least squares approximation to the Lebesgue constant for
the point set corresponding to the Radau quadrature points on
$(-1, +1]$ augmented by $-1$ using curves of the form $a\log N +b$}
\label{graphradau-1}
\end{figure}
\section{Conclusions}
\label{conclusions}
In Gauss and Radau collocation methods for unconstrained control problems
\cite{HagerHouRao15b, HagerHouRao15c},
the error in the solution to the discrete problem is bounded by the
residual for the solution to the continuous problem inserted in the
discrete equations.
In Section~\ref{residual}, we observe that the residual in the sup-norm
is bounded by the distance between the derivative of the continuous
solution interpolant and the derivative of the continuous solution.
Proposition~\ref{L1} bounds this distance in terms of the error in
best approximation and the Lebesgue constant for the point set.
We show that the Lebesgue constant for the point sets associated with
the Gauss and Radau collocation methods is $O(\sqrt{N})$, and
by the plots of Section~\ref{numerical}, the Lebesgue constants are
closely fit by curves of the form $a\sqrt{N}+b$.
\section*{Acknowledgments}
Special thanks to Lloyd N. Trefethen for pointing out Brutman's paper
\cite{Brutman97} and for providing a copy when we had trouble locating
the journal.
Also, we thank a reviewer for pointing out the book \cite{Mastroianni08}
which contains newer results as well as additional references.
\bibliographystyle{siam}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,832 |
La Chiesa cattolica in Mali è parte della Chiesa cattolica universale in comunione con il vescovo di Roma, il papa.
Storia
La Chiesa cattolica inizia a essere presente in Mali tra il 1876 ed il 1881, quando il cardinale Lavigerie invia alcuni missionari a Timbuctù, che però vengono uccisi. Nello stesso periodo i padri Spiritani, dal Senegal, inviano alcune spedizioni missionarie, che riescono a fondare la prima parrocchia in Mali nel settembre 1888: è l'inizio dell'evangelizzazione del Paese, che si estende con l'arrivo nel 1895 dei padri Bianchi. Il territorio appartiene all'immenso vicariato apostolico del Sahara, da cui si distacca nel 1921 quello di Bamako (che diventa arcidiocesi nel 1955). È del 1942 la creazione della prima prefettura apostolica a Gao. Nel 1988 è celebrato il primo centenario della presenza della Chiesa cattolica in Mali, e due anni dopo papa Giovanni Paolo II compie la visita pastorale alla Chiesa cattolica del Mali.
Organizzazione territoriale
La Chiesa cattolica è presente sul territorio con 1 sede sede metropolitana e 5 diocesi suffraganee:
Arcidiocesi di Bamako
Diocesi di Kayes
Diocesi di Mopti
Diocesi di San
Diocesi di Sikasso
Diocesi di Ségou
Statistiche
Alla fine del 2004 la Chiesa cattolica in Mali contava:
37 parrocchie;
130 preti;
259 suore religiose;
59 istituti scolastici;
28 istituti di beneficenza.
La popolazione cattolica ammontava a 241.921 cristiani, pari all'1,79% della popolazione.
Nunziatura apostolica
La delegazione apostolica dell'Africa Occidentale mutò il proprio nome in delegazione apostolica in Mali e Mauritania il 21 maggio 1973 con il breve Ex quo divino di papa Paolo VI. Il 3 giugno 1980, con l'instaurazione di rapporti diplomatici tra la Santa Sede e il Mali, nacque la nunziatura apostolica del Mali in forza del breve Vigilem curam di papa Giovanni Paolo II. Sede attuale della nunziatura è Conakry in Guinea; fino al 2007 i delegati ed i nunzi apostolici risiedevano a Dakar in Senegal.
Delegati apostolici
Giovanni Mariani, arcivescovo titolare di Missua (17 ottobre 1973 - 11 gennaio 1975 nominato nunzio apostolico in Venezuela)
Luigi Barbarito, arcivescovo titolare di Fiorentino (5 aprile 1975 - 10 giugno 1978 nominato pro-nunzio apostolico in Australia)
Luigi Dossena, arcivescovo titolare di Carpi (24 ottobre 1978 - 3 giugno 1980 nominato pro-nunzio apostolico)
Pro-Nunzi apostolici
Luigi Dossena, arcivescovo titolare di Carpi (3 giugno 1980 - 30 dicembre 1985 nominato nunzio apostolico in Perù)
Pablo Puente Buces, arcivescovo titolare di Macri (12 marzo 1986 - 31 luglio 1989 nominato nunzio apostolico in Libano)
Antonio Maria Vegliò, arcivescovo titolare di Eclano (21 ottobre 1989 - dicembre 1994 nominato nunzio apostolico)
Nunzi apostolici
Antonio Maria Vegliò, arcivescovo titolare di Eclano (dicembre 1994 - 2 ottobre 1997 nominato nunzio apostolico in Libano e Kuwait e delegato apostolico nella Penisola Arabica)
Jean-Paul Aimé Gobel, arcivescovo titolare di Galazia in Campania (6 dicembre 1997 - 31 ottobre 2001 nominato nunzio apostolico in Nicaragua)
Giuseppe Pinto, arcivescovo titolare di Anglona (5 febbraio 2002 - 6 dicembre 2007 nominato nunzio apostolico in Cile)
Martin Krebs, arcivescovo titolare di Taborenta (8 settembre 2008 - 8 maggio 2013 nominato nunzio apostolico in Nuova Zelanda, nelle Isole Cook, nelle Kiribati, in Palau e negli Stati Federati di Micronesia e delegato apostolico nell'Oceano Pacifico)
Santo Rocco Gangemi, arcivescovo titolare di Umbriatico (5 febbraio 2014 - 25 maggio 2018 nominato nunzio apostolico in El Salvador)
Tymon Tytus Chmielecki, arcivescovo titolare di Tre Taverne (26 marzo 2019 - ? dimesso)
Jean-Sylvain Emien Mambé, arcivescovo titolare di Potenza Picena, dal 2 febbraio 2022
Conferenza episcopale
L'episcopato locale è riunito nella Conferenza episcopale del Mali (Conférence Episcopale du Mali, CEM), istituita nel 1970.
La CEM è membro della Regional Episcopal Conference of West Africa (RECOWA) e del Symposium of Episcopal Conferences of Africa and Madagascar (SECAM).
Elenco dei presidenti della Conferenza episcopale:
Luc Auguste Sangaré, arcivescovo di Bamako (1970 - 1987)
Jean-Baptiste Maria Cissé, vescovo di Sikasso (1987 - 1996)
Jean-Gabriel Diarra, vescovo di San (1996 - aprile 2009)
Jean-Baptiste Tiama, vescovo di Sikasso (aprile 2009 - marzo 2017)
Jonas Dembélé, vescovo di Kayes, dal marzo 2017
Elenco dei segretari generali della Conferenza episcopale:
Presbitero Edmond Dembélé, dal 2012
Bibliografia
Guida delle missioni cattoliche 2005, a cura della Congregatio pro gentium evangelizatione, Roma, Urbaniana University Press, 2005
Voci correlate
Cristianesimo in Mali
Altri progetti
Collegamenti esterni
Breve storia delle diocesi del Mali
Sito ufficiale della Conferenza Episcopale del Mali
La Chiesa cattolica in Mali sul sito di Gcatholic
La Chiesa cattolica in Mali sul sito di Catholic Hierarchy
Sito della Communauté des étudiants croyants del Mali
Sito della Caritas del Mali
Breve Ex quo divino, AAS 65 (1973), p. 627 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,419 |
\section{Introduction}
Given Banach spaces $\mathcal{X}$ and $\mathcal{Y}$
and Banach space operators $A\in L(\mathcal{X})$ and $B\in L(\mathcal{Y}).$
Let $L_{A}\in L(L(\mathcal{X}))$ and $R_{B}\in L(L(\mathcal{Y}))$ be the left and the right
multiplication operators, respectively, and denote by $\delta_{A,B}\in L(L(\mathcal{Y},\mathcal{X}))$
the generalized derivation $\delta_{A,B}(X)=(L_{A}-R_{B})(X)=AX-XB$. The problem of transferring spectral properties from
$A$ and $B$ to $L_A,$ $R_B$, $L_A R_B$ and $\delta_{A,B}$ was studied by numerous mathematicians,
see \cite{Bao, BD, BA, CH, D2, D9, H2, LB, LBA} and the references therein.
The main objective of this paper is to study the
problem of transferring the left polaroid property and its strong version finitely left polaroid property,
from $A$ and $B^{*}$ to $\delta_{A,B}.$
After section 2 where several basic definitions and facts will be recalled,
we will prove that if $A$ is left polaroid and satisfies property $(\mathcal{P}_l)$ and $B$ is right polaroid and
satisfy property $(\mathcal{P}_r),$ then $\delta_{A,B}$ is left polaroid.
Also, we prove that $A$ is finitely left polaroid and $B$ is finitely right polaroid
if and only if $\delta_{A,B}$ is finitely left polaroid.
In the fourth section, we give necessary and sufficient conditions for $\delta_{A,B}$ to satisfy generalized a-Weyl's theorem.
In the last section we apply results obtained previously. If $\mathcal{X}=H$ and $\mathcal{Y}=K$ are Hilbert spaces,
we prove that if $A\in L(H)$ and $B\in L(K)$ are
completely totally hereditarily normaloid operators, then
$f(\delta_{A,B})$ satisfies generalized a-Weyl's theorem, for every analytic function $f$ defined on a neighborhood of
$\sigma(\delta_{A,B})$ which is non constant on each of the components of its domain. This generalize results obtained
in \cite{BA, CH, D2, Du8, LB, LBA}.
\section{Notation and terminology}
Unless otherwise stated, from now on $\mathcal{X}$ (similarly, $\mathcal{Y}$)
shall denote a complex Banach space and $L(\mathcal{X})$ (similarly, $L(\mathcal{Y})$)
the algebra of all bounded linear maps defined on and with values in $\mathcal{X}$
(respectively, $\mathcal{Y}$).
Given $T\in L(\mathcal{X}),$ $N(T)$ and $R(T)$ will stand for the null space and the range of $T$
respectively. Recall that $T\in L(\mathcal{X})$ is said to be bounded below, if $N(T)=\{0\}$
and $R(T)$ is closed. Denote the approximate point spectrum of $T$ by
\begin{equation*}
\sigma_{a}(T)=\{\lambda\in\mathbb{C}: T-\lambda I\,\,\,\text{is not bounded below}\}.
\end{equation*}
Let \begin{equation*}\sigma_{s}(T)=\{\lambda\in\mathbb{C}: T-\lambda I\,\,\,\text{is not surjective}\}\end{equation*}
denote the surjective spectrum of $T.$
In addition, $\mathcal{X}^*$ will denote the dual space of $\mathcal{X},$ and if
$T\in \mathcal{X},$ then $T^* \in L(\mathcal{X}^{*})$ will stand for the adjoint map
of $T.$ Clearly, $\sigma_{a}(T^*)=\sigma_{s}(T)$ and
$\sigma_{a}(T) \cup \sigma_{s}(T)=\sigma(T),$ the spectrum of $T.$ Recall that the ascent $asc(T)$ of an operator $T$ is defined by $asc(T)=\inf\{n\in\mathbb{N}:N(T^{n})=N(T^{n+1})\}$ and the descent
$dsc(T)=\inf\{n\in\mathbb{N}:R(T^{n})=R(T^{n+1})\}$, with $\inf\emptyset=\infty.$ It is well known that if $asc(T)$ and $dsc(T)$ are both finite, then they are equal.\\
\indent A complex number
$\lambda\in\sigma_{a}(T)$ (respectively, $\lambda\in\sigma_{s}(T)$) is left pole (respectively, right pole) of order $d$ of $T\in L(\mathcal{X})$ if $asc(T-\lambda I)=d<\infty$ and $R((T-\lambda I)^{d+1})$ is closed (respectively, $dsc(T-\lambda I)=d<\infty$ and $R((T-\lambda I)^{d})$ is closed).
We say that $T$ is left polar (respectively, right polar) of order d at a point $\lambda\in \sigma_a(T)$
(respectively, $\lambda \in \sigma_s(T)$) if $\lambda$ is a left pole of $T$ (respectively, right pole of $T$) of order d.
Now, $T$ is left polaroid (respectively, right polaroid) if $T$ is left polar (respectively, right polar ) at
every $\lambda \in iso\sigma_a(T)$ (respectively, $\lambda \in iso\sigma_s(T)$), where
$iso\mathcal{K}$ is the set of all isolated points of $\mathcal{K}$ for $\mathcal{K} \subseteq\mathbb{C}.$
In \cite{BD} a left polar operator $T\in L(\mathcal{X})$ of order $d(\lambda)$ at $\lambda \in \sigma_a(T),$
satisfies property $(\mathcal{P}_l)$ if the closed subspace $N((T-\lambda)^{d(\lambda)})+R(T-\lambda)$
is complemented in $\mathcal{X}$ for every $\lambda \in iso\sigma_a(T).$
Dually, a right polar operator $T\in L(\mathcal{X})$ of order $d(\lambda)$ at $\lambda \in \sigma_s(T),$
satisfies property $(\mathcal{P}_r)$ if the closed subspace
$N(T-\lambda) \cap R((T-\lambda)^{d(\lambda)})$ is complemented in $\mathcal{X}$ for every $\lambda \in iso\sigma_s(T).$
If $\mathcal{X}=H$
is a Hilbert space, then every left polar (respectively, right polar) operator $T\in L(H)$ of order $d(\lambda)$ at
$\lambda \in iso\sigma_a(T)$ (respectively, $\lambda \in iso\sigma_s(T)$)
satisfies property $(\mathcal{P}_l)$ (respectively, $(\mathcal{P}_r)).$
On the other hand, it is known that $T\in L(\mathcal{X})$ is right polaroid if and only if $T^{*}$ is left polaroid
and $T$ is polaroid if it is both left and right polaroid, whenever
$iso \sigma(T)= iso\sigma_a(T)\cup iso\sigma_s(T).$\\
\indent Recall that $T\in L(\mathcal{X})$ is said to be a Fredholm operator, if both $\alpha(T)=dimN(T)$ and
$\beta(T)=dim \mathcal{X} \setminus R(T)$ are finite dimensional, in which case its index is given by
$ind(T)=\alpha(T)-\beta(T).$ If $ R(T)$ is closed and $\alpha(T)$ is finite (respectively, $\beta(T)$ is finite),
then $T \in L(\mathcal{X})$ is said to be upper semi-Fredholm (respectively, lower semi-Fredholm)
while if $\alpha(T)$ and $ \beta(T)$ are both finite and equal, so the index is zero and $T$ is said to be Weyl operator.
These classes of opertaors generate the Fredholm spectrum, the upper semi-Fredholm spectrum, the lower semi-Fredholm spectrum
and the Weyl spectrum of $T\in L(\mathcal{X})$ which will be denoted by $\sigma_{e}(T),$
$\sigma_{SF_{+}}(T),$ $\sigma_{SF_{-}}(T)$ and $\sigma_{W}(T),$ respectively.
The Weyl essential approximate point spectrum and the Browder essential approximate point spectrum of $T\in L(\mathcal{X})$ are the sets
\begin{equation*}
\sigma_{aw}(T)=\{\lambda\in\sigma_{a}(T): \lambda \in \sigma_{SF_{+}}(T)\,\, or \,\, 0<ind(T-\lambda I)\}
\end{equation*}
and
\begin{equation*}
\sigma_{ab}(T)=\{\lambda\in\sigma_{a}(T): \lambda\in\sigma_{aw}(T)\,\,\text{
or}\,\,\ asc(T-\lambda I)=\infty\}.
\end{equation*}
It is clear that
\begin{equation*}
\sigma_{SF_{+}}(T)\subseteq\sigma_{aw}(T)\subseteq\sigma_{ab}(T)\subseteq\sigma_{a}(T).
\end{equation*}
\indent For $T\in L(\mathcal{X})$ and a nonnegative integer $n$ define $T_{n}$ to be the restriction of
$T$ to $R(T^{n})$ viewed as a map from $R(T^{n})$ into $R(T^{n})$. If for some integer $n$
the range space $R(T^{n})$ is closed and the induced operator $T_{n}\in L(R(T^{n}))$
is Fredholm, then $T$ will be said B-Fredholm.
In a similar way, if $T_{n}$ is upper semi-Fredholm (respectively, lower semi-Fredholm )
operator, then $T$ is called upper semi B-Fredholm (respectively, lower semi B-Fredholm).
In this case the index of $T$ is defined as the index of the semi-Fredholm operator $T_{n}$, see \cite{B1}.
$T\in L(\mathcal{X})$ is semi B-Fredholm if $T$ is upper semi B-Fredholm or lower semi B-Fredholm.
Let
\begin{equation*}
\Phi_{SBF}(\mathcal{X})=\{T\in L(\mathcal{X}) : T\,\,\,\text
is semi B-Fredholm}\,\,\},
\end{equation*}
\begin{equation*}
\Phi_{SBF_{+}^{-}}(\mathcal{X})=\{T\in\Phi_{SBF}(\mathcal{X}): T\,\,\,\text
is upper semi B-Fredholm with}\,\,ind(T)\leq0\}
\end{equation*}
and
\begin{equation*}
\Phi_{SBF_{-}^{+}}(\mathcal{X})=\{T\in\Phi_{SBF}(\mathcal{X}): T\,\,\,\text
is lower semi B-Fredholm with}\,\,ind(T) \geq 0\}.
\end{equation*}
Then the upper semi B-Weyl and lower semi B-Weyl spectrum of $T$ are the sets
\begin{equation*}
\sigma_{UBW}(T)=\{\lambda\in\sigma_{a}(T): T-\lambda I\not\in\Phi_{SBF_{+}^{-}}(\mathcal{X})\}
\end{equation*}
and
\begin{equation*}
\sigma_{LBW}(T)=\{\lambda\in\sigma_{a}(T): T-\lambda I\not\in\Phi_{SBF_{-}^{+}}(\mathcal{X})\},
\end{equation*}
respectively.
$T\in L(\mathcal{X})$ will be said B-Weyl, if $T$ is both upper and lower semi B-Weyl (equivalently $T$ is B-Fredholm operator of index zero).
The B-Weyl spectrum $\sigma_{BW}(T)$ of $T$ is defined by
\begin{equation*}
\sigma_{BW}(T)=\{\lambda\in\mathbb{C}: T-\lambda I\,\,\,\text
is not B-Weyl operator}\}.
\end{equation*}
Let $\Pi^{l}(T)$ denote the set of left pole of $T\in L(\mathcal{X})$.
\begin{equation*}
\Pi^{l}(T)=\{\lambda\in\sigma_{a}(T): asc(T-\lambda I)=d<\infty\,\,\,\text{and}\,\ R((T-\lambda I)^{d+1})\,\,\text{is closed}\}.
\end{equation*}
A strong version of the left polaroid property says that
$T\in L(\mathcal{X})$ is finitely left polaroid (respectively, finitely right polaroid)
if and only if every $\lambda\in iso\sigma_{a}(T)$
( respectively, $\lambda\in iso\sigma_{s}(T)$)
is left pole of $T$ and $\alpha(T-\lambda I)<\infty$ (respectively, right pole of $T$
and $\beta(T-\lambda I)<\infty$). Let $\Pi_{0}^{l}(T)$ (respectively, $\Pi_{0}^{r}(T)$)
denote the set of finite left poles (respectively, the set of finite right poles ) of $T.$
Then $T\in L(\mathcal{X})$ is finitely left polaroid (respectively, finitely right polaroid) if and only if
$iso \sigma_a(T)=\Pi_{0}^{l}(T)$ (respectively, $iso \sigma_a(T)=\Pi_{0}^{r}(T)$).\\
\indent For $T\in L(\mathcal{X})$ define
$$ \Delta (T)= \{ n \in \mathbb{N} : m \geq n, m\in \mathbb{N} \Rightarrow R(T^n) \cap N(T) \subseteq R(T^m) \cap N(T) \}.$$
The degree of stable iteration is defined as $dis(T)=\inf \Delta (T) $ if $\Delta (T)\neq \emptyset,$ while
$dis(T)=\infty$ if $\Delta(T)=\emptyset.$ $T\in L(\mathcal{X})$
is said to be quasi-Fredholm of degree d, if there exists $d\in \mathbb{N} $ such that
$dis(T)=d,$ $R(T^n)$ is a closed subspace of $\mathcal{X}$ for each $n\geq d$ and $R(T)+N(T^n)$ is a closed subspace
of $\mathcal{X}.$ An operator $T\in L(\mathcal{X})$ is said to be semi-regular,
if $R(T)$ is closed and $N(T^{n})\subseteq R(T^{m})$ for all $m,n \in \mathbb{N} .$\\
\indent An important property in local spectral theory is the single valued extension property.
An operator $T\in L(\mathcal{X})$ is said to have the single valued extension property
at $\lambda_{0}\in\mathbb{C}$ (abbreviated SVEP at $\lambda_{0}$), if for every open
disc $\mathbb{D}$ centered at $\lambda_{0}$, the only analytic function $f:\mathbb{D}\rightarrow \mathcal{X}$
which satisfies the equation $(T-\lambda I)f(\lambda)=0$ for all $\lambda\in\mathbb{D}$
is the function $f\equiv0$. An operator $T\in L(\mathcal{X})$ is said to have SVEP if $T$ has SVEP at every $\lambda\in\mathbb{C}.$\\
Furthermore, for $T\in L(\mathcal{X})$ the quasi-nilpotent part of $T$ is defined by
\begin{equation*}
H_{0}(T)=\{x\in \mathcal{X}: \lim_{n\rightarrow\infty}\|T^{n}(\mathcal{X})\|^{\frac{1}{n}}=0\}.
\end{equation*}
It is easily seen that $N(T^{n})\subset H_{0}(T)$ for every $n\in\mathbb{N}$.
The analytic core of an operator $T\in L(\mathcal{X})$ is the subspace $K(T)$ defined as the set of all $x\in \mathcal{X}$ such that there exists a constant $c>0$ and a sequence of elements $x_{n}\in \mathcal{X}$ such that $x_{0}=x$, $Tx_{n}=x_{n-1},$ and $\|x_{n}\|\leq c^{n}\|x\|$ for all $n\in\mathbb{N}$, the spaces $K(T)$ are hyperinvariant under $T$ and satisfy $K(T)\subset R(T^{n})$, for every $n\in\mathbb{N}$ and $T(K(T))=K(T),$ see \cite{P1} for information on $H_{0}(T)$ and $K(T)$.
\section{Left polaroid generalized derivation}
We begin this section by recalling some results concerning spectra of generalized derivations.\\
Let $\mathcal{X}$ and $\mathcal{Y}$ be two Banach spaces and consider
$A\in L(\mathcal{X})$ and $B\in L(\mathcal{Y}).$ Let
$\delta_{A,B} \in L(L(\mathcal{Y},\mathcal{X}))$ the generalized derivation induced by $A$
and $B,$ i.e., $$\delta_{A,B}(X)=(L_{A}-R_{B})(X)=AX-XB \mbox{ where } X\in L(\mathcal{Y},\mathcal{X}).$$
According to \cite[Theorem 3.5.1]{LN}, we have that $$\sigma_{a}(\delta_{A,B})=\sigma_{a}(A)-\sigma_{s}(B).$$
and it is not difficult to conclude that
$$iso \sigma_a (\delta_{A,B})= (iso\sigma_a (A)- iso\sigma_a (B^*))\setminus acc\sigma_a(\delta_{A,B}).$$
The following results concerning upper semi Fredholm spectrum and Browder essential approximate point spectrum
of generalized derivation was proved in \cite{BA, LZ}. It will be used in the sequel.
\begin{lemma} \label{propostion1}
Let $\mathcal{X}$ and $\mathcal{Y}$ be two Banach spaces and consider $A\in L(\mathcal{X})$
and $B\in L(\mathcal{Y}),$ then the following statements hold.
\begin{enumerate}
\item[i)] $\sigma_{SF_{+}}(\delta_{A,B})=(\sigma_{SF_{+}}(A)-\sigma_{s}(B))\cup(\sigma_{a}(A)-\sigma_{SF_{-}}(B)).$
\item[ii)] $\sigma_{ab}(\delta_{A,B})=(\sigma_{ab}(A)-\sigma_{s}(B))\cup(\sigma_{a}(A)-\sigma_{ab}(B^{*})).$
\end{enumerate}
\end{lemma}
The following lemma concerning the Weyl essential approximate point spectrum of generalized derivation
will be used in the sequel.
\begin{lemma}\label{lema2}
Let $\mathcal{X}$ and $\mathcal{Y}$ be two Banach spaces and consider $A\in L(\mathcal{X})$
and $B\in L(\mathcal{Y}),$ then
$$\sigma_{aw}(\delta_{A,B})\subseteq(\sigma_{aw}(A)-\sigma_{s}(B))\cup(\sigma_{a}(A)-\sigma_{aw}(B^{*})).$$
\end{lemma}
\begin{proof}
Let $\lambda\not\in(\sigma_{aw}(A)-\sigma_{s}(B))\cup(\sigma_{a}(A)-\sigma_{aw}(B^{*})).$
If $\mu_{i}\in\sigma_{a}(A)$ and $\nu_{i}\in\sigma_{s}(B)$ are such that $\lambda=\mu_{i}-\nu_{i},$
then $\mu_{i}\not\in\sigma_{SF_{+}}(A)$ and $\nu_{i}\not\in\sigma_{SF_{-}}(B),$ hence from
statement i) of Lemma \ref{propostion1} $\lambda\not\in\sigma_{SF_{+}}(\delta_{A,B})$.
Now, we will prove that $$ind(\delta_{A,B}-\lambda I)\leq0.$$
Suppose to the contrary that $ind(\delta_{A,B}-\lambda I)>0$, then $\lambda\not\in\sigma_{e}(\delta_{A,B}).$ It follows from \cite[Corollary 3.4]{E}
that $$\lambda=\mu_{i}-\nu_{i}\,\,\,\ (1\leq i\leq n),$$ where
$\mu_{i}\in iso\sigma(A)$ for $1\leq i\leq m$ and $\nu_{i}\in iso\sigma(B),$ for
$m+1\leq i\leq n$. We have that
$ind(\delta_{A,B}-\lambda I)$ is equal to
$$\sum_{j=m+1}^{n}dim H_{0}(B-\nu_{j})ind(A-\mu_{j})-\sum_{k=1}^{m}dim H_{0}(A-\mu_{k})ind(B-\nu_{k}).$$
Since $\mu_{i}\in iso\sigma(A),$ for $1\leq i\leq m$ and
$\nu_{i}\in iso\sigma(B),$ for $m+1\leq i\leq n$, it follows that
$dim H_{0}(A-\mu_{j})$ is finite, for $1\leq j\leq m$ and $dim H_{0}(B-\nu_{k})$ is finite,
for $m+1\leq k\leq n$ and we have also $ind(A-\mu_{i})\leq0$ and $ind(B-\nu_{j})\geq0.$
Thus $ind(\delta_{A,B}-\lambda I)\leq0.$ This a contradiction.
Hence $\lambda\not\in\sigma_{aw}(\delta_{A,B}).$
\end{proof}
\indent According to \cite{BD} left polaroid operator (respectively, right polaroid ) operator satisfies property
$(\mathcal{P}_{l}),$ (respectively, $(\mathcal{P}_{r})$), if it is left polar at every
$\lambda \in iso\sigma_a(T)$ (respectively, right polar at every $\lambda \in iso\sigma_s(T)$
which satisfies property $(\mathcal{P}_{l}),$(respectively property $(\mathcal{P}_{r}).$\\
\indent The Following Lemma is the dual version of \cite[Lemma 3.1]{BD}.
\begin{lemma}\label{lemmaD}
Let $\mathcal{X}$ a Banach space.
If $T\in L(\mathcal{X})$ is right polaroid and satisfies property $(\mathcal{P}_{r}),$
then for every $\lambda \in iso\sigma_s(T)$ there exists T-invariant closed
subspaces $N_1$ and $N_2$ such that
$\mathcal{X}=N_1 \oplus N_2,$ $(T-\lambda)|_{N_1}$ is nilpotent of order $d(\lambda)$ and
$(T-\lambda I)|_{N_2}$ is surjective, where $d(\lambda)$ is the order of the right pole at $\lambda.$
Moreover, $K(T-\lambda I)=R((T- \lambda I)^{d(\lambda)}).$
\end{lemma}
\begin{proof}
From the hypothesis $T-\lambda$ is quasi-Fredholm of degree $d(\lambda)$
and the closed subspaces $N((T-\lambda I)^{d(\lambda)})+R(T-\lambda)$ is complemented in
$\mathcal {X}.$ Since $T\in L(\mathcal{X})$ is right polaroid and satisfies property $(\mathcal{P}_{r}),$
then $N(T-\lambda) \cap R((T-\lambda)^{d(\lambda)})$ is complemented in $\mathcal {X}.$
From \cite[ Theorem 5]{Mu}, there exists T-invariant closed subspaces $N_1$ and $N_2$
such that $\mathcal{X}=N_1 \oplus N_2,$ $(T-\lambda)|_{N_1}$ is nilpotent of order $d(\lambda)$ and
$(T-\lambda I)|_{N_2}$ is semi-regular.
Since $dsc(T-\lambda I)=d(\lambda),$ then
the semi-regular operator $(T-\lambda I)|_{N_2}$ is surjective. Since
$K(T-\lambda I)=K((T-\lambda I)|{N_1}) \oplus K((T-\lambda I)|{N_2})= 0 \oplus N_2=N_2,$ then
we can conclude from \cite[Theorem 2.7]{P2} that $K(T-\lambda I)=R((T- \lambda I)^{d(\lambda)}.$
\end{proof}
Next follows the main result of this section.
\begin{theorem}\label{theorem1}
Let $\mathcal{X}$ and $\mathcal{Y}$ be two Banach spaces and let $A\in L(\mathcal{X})$
be left polaroid and $B\in L(\mathcal{Y})$ be right polaroid.
If $A$ satisfies property $(\mathcal{P}_l)$ and $B$ satisfy property $(\mathcal{P}_r),$ then
\begin{center}$\delta_{A,B}$ is left polaroid.\end{center}
\end{theorem}
\begin{proof}
Let $\lambda\in iso\sigma_{a}(\delta_{A,B}),$ then
there exist $\mu\in\sigma_{a}(A)$ and $\nu\in\sigma_{s}(B)$ such that $\lambda=\mu-\nu$,
it follows from statement that $\mu\in iso\sigma_{a}(A)$ and $\nu\in iso\sigma_{s}(B)=iso\sigma_{a}(B^{*}).$
Since $A$ is left polaroid, then there exist $A$-invariant closed subspaces
$M_{1}$ and $M_{2}$ such that $\mathcal{X}=M_1 \oplus M_2,$
$(A- \mu I)|_{M_{1}}= A_{1}- \mu I|_{M_1}$ is nilpotent of order $d_1$ where
where $d_1=d(\mu)$ is the order of the left pole of $A$ at $\mu$
and that $(A- \mu I)|_{M_2}=A_{2}- \mu I|_{M_2}$ is bounded below.
Also, since $B$ is right polaroid,
then there exists $B$-invariant closed subspaces $N_1$ and $N_2$ such that
$\mathcal{Y}=N_1 \oplus N_2,$ $(B-\nu)|_{N_1}=B_{1}-\nu I |_{N_1}$ is nilpotent of order $d_2$
where $d_2=d(\nu)$ is the order of the right pole of $B$ at $\nu$ and $(B-\nu I)|_{N_2}=B_{2}-\nu I|_{N_2}$ is surjective.
Let $d=d_{1}+d_{2}$
and $X\in L(N_{1}\oplus N_{2},M_{1}\oplus M_{2})$ have the representation $X=[X_{kl}]_{k,l=1}^{2}$.
We will prove that $asc(\delta_{A,B}-\lambda I)$ is finite.
We have
\begin{eqnarray*}
(\delta_{A_1,B_1}-\lambda I)^{d+1}&=&(L_{A_1-\mu I}-R_{B_1-\nu I})^{d+1} \\
&=&\sum_{k=0}^{d+1}(-1)^{d+1-k}\left(
\begin{array}{c}
d+1 \\
k\\
\end{array}
\right)(L_{A_1-\mu I})^{k}(R_{B_1-\nu I})^{d+1-k}=0,
\end{eqnarray*} that is
$\delta_{A_{1},B_{1}}-\lambda I$ is nilpotent of order $d,$ and hence $asc(\delta_{A_{1},B_{1}}-\lambda I)<\infty .$\\
On the other hand,
\begin{eqnarray*}
(\delta_{A_{1},B_{2}}-\lambda I)^{d+1}(X_{12})&=&\sum_{k=0}^{d_{1}-1}(-1)^{d_{1}-1-k}\left(
\begin{array}{c}
d_{1}-1 \\
k\\
\end{array}
\right)(L_{A_{1}-\mu I})^{k}(R_{B_{2}-\nu I})^{d_{1}-1-k}(X_{12})\\
&=&(\delta_{A_{1},B_{2}}-\lambda I)^{d}(X_{12}),
\end{eqnarray*}similarly we get
$(\delta_{A_{2},B_{1}}-\lambda I)^{d+1}(X_{21})=(\delta_{A_{2},B_{1}}-\lambda I)^{d}(X_{21}).$ Thus
$$asc(\delta_{A_{1},B_{2}}-\lambda I)<\infty \mbox{ and } asc(\delta_{A_{2},B_{1}}-\lambda I)<\infty .$$
Now, we will prove that $0\not\in\sigma_{a}(\delta_{A_{2}-\mu I|_{M_2},B_{2}-\nu I|_{N_2}}).$
For this, it suffices to prove that $\sigma_{a}(A_{2}-\mu I|_{M_2} ) \cap \sigma_{s}(B_{2}-\nu I|_{N_2})= \emptyset.$
Suppose that there exists a complex number $\alpha $ such that
$\alpha \in \sigma_{a}(A_{2}-\mu I|_{M_2}) \cap \sigma_{s}(B_{2}-\nu I|_{N_2}),$ then
$\alpha \in\sigma_{a}(A_{2}-\mu I|_{M_2})$ and $ \alpha \in\sigma_{s}(B_{2}-\nu I|_{N_2}),$
from \cite[Theorem 2.48]{P1}, $0\in\sigma_{a}(A_{2}-(\mu +\alpha)I|_{M_2} )$ and $0 \in\sigma_{s}(B_{2}-(\nu + \alpha)I|_{N_2}).$
Since $(\mu +\alpha)$ is isolated in the approximate point spectrum of $A$ and $(\nu +\alpha)$ is isolated in the surjective spectrum of $B,$
then by the hypothesis $A$ is left polaroid which satisfies property $(\mathcal{P}_l)$
and $B$ is right polaroid which satisfies property $(\mathcal{P}_r),$ we conclude that
$$(A- (\mu +\alpha )I)|_{M_2}=A_{2}- (\mu + \alpha)I|_{M_2}$$ is bounded below and
$$(B-(\nu+\alpha)I)|_{N_2}=B_{2}-(\nu +\alpha)I|_{N_2}$$ is surjective. That is
$$0 \notin \sigma_{a}(A_{2}-(\mu +\alpha)I|_{M_2}) \mbox{ and } 0 \notin \sigma_{s}(B_{2}-(\nu + \alpha)I|_{N_2}).$$
This is a contradiction, hence $0\not\in\sigma_{a}(\delta_{A_{2}-\mu I|_{M_2},B_{2}-\nu I|_{N_2}}).$
Finally, $$asc(\delta_{A_2,B_2}-\lambda I)\leq d<\infty,$$ and hence, $$asc(\delta_{A,B}-\lambda I)\leq d<\infty.$$
Now,we prove that $(\delta_{A,B}-\lambda I)^{d+1}(L(\mathcal{Y},\mathcal{X}))$ is closed. Since
$0\not\in\sigma_{a}(\delta_{A_{2},B_{2}}-\lambda I),$ then from \cite[Lemma 1.1]{AA}
$(\delta_{A_{2},B_{2}}-\lambda I)^{d+1}(L(N_{2},M_{2})$ is closed.
We have that $\delta_{A_{1},B_{1}}-\lambda I$ is nilpotent of order $d,$ then by \cite[Theorem 2.7]{OBO}
it follows that $$(\delta_{A_{1},B_{1}}-\lambda I)^{d+1}(L(N_{1},M_{1})) \mbox{ is closed }.$$
From the fact that $0\not\in\sigma_{a}(\delta_{A_{i},B_{j}}-\lambda I)$ and \cite[Lemma 1.1]{AA},
it follows that $$(\delta_{A_{i},B_{j}}-\lambda I)^{d+1}(L(N_{j},M_{i}) \mbox{
is closed for }1\leq i,j\leq2 \mbox{ and }i\neq j,$$ consequently
$(\delta_{A,B}-\lambda I)^{d+1}(L(\mathcal{X},\mathcal{Y}))$ is closed. Hence
$\lambda$ is a left pole of $\delta_{A,B}$ which means that $\delta_{A,B}$ is left polaroid.
\end{proof}
In the case of Hilbert spaces, we have the following corollary.
\begin{corollary}
Let $H$ and $K$ be Hilbert spaces and let $A\in L(H)$ and $B \in L(K).$
If $A$ and $B^*$ are left polaroid, then
\begin{center}$\delta_{A,B}$ is left polaroid.\end{center}
\end{corollary}
In the following Theorem, we characterize finitely left polaroid generalized derivation.
\begin{theorem}
Let $\mathcal{X}$ and $\mathcal{Y}$ be two Banach spaces and let $A\in L(\mathcal{X})$
and $B\in L(\mathcal{Y})$. Then
$A$ and $B^*$ are finitely left polaroid operators if and only if
\begin{center}$\delta_{A,B}$ is finitely left polaroid.\end{center}
\end{theorem}
\begin{proof}
Let $\lambda\in iso\sigma_{a}(\delta_{A,B}),$ then there exist
$\mu\in\sigma_{a}(A)$ and $\nu\in\sigma_{s}(B)$ such that
$\lambda=\mu-\nu,$ hence we have
$\mu\in iso\sigma_{a}(A)$ and $\nu\in iso\sigma_{s}(B)=iso\sigma_{a}(B^{*}).$
Suppose that $A$ and $B^{*}$ are finitely left polaroid, then from \cite[Corollary 2.2]{R1} we have
that $\mu\not\in\sigma_{ab}(A)$ and $\nu\not\in\sigma_{ab}(B^{*})$, applying statement ii) of Lemma \ref{propostion1},
we get $\lambda\not\in\sigma_{ab}(\delta_{A,B}),$
hence by \cite[Corollary 2.2]{R1} $\delta_{A,B}$ is finitely left polaroid.
Conversely, suppose that $\delta_{A,B}$ is finitely left polaroid and prove that
$A$ and $B^{*}$ are finitely left polaroid.
For this, let $\mu\in iso\sigma_{a}(A)$ and $\nu\in iso\sigma_{a}(B^{*}),$ then
$\lambda=\mu-\nu \in iso\sigma_{a}(\delta_{A,B}).$ Since
$\delta_{A,B}$ is finitely left polaroid, then by \cite[Corollary 2.2]{R1}
$\lambda=\mu - \nu \not\in\sigma_{ab}(\delta_{A,B}),$ and hence by statement ii) of Lemma \ref{propostion1}
$\mu\notin\sigma_{ab}(A)$ and $\nu\notin\sigma_{ab}(B^{*}).$ We conclude from
\cite[Corollary 2.2]{R1} that $A$ and $B^{*}$ are finitely left polaroid.
\end{proof}
\section{Consequences on Weyl's type theorem}
For $T\in L(\mathcal{X})$, let $E^{a}(T)=\{\lambda\in iso\sigma_{a}(T): 0<\alpha(T-\lambda I)\}$ and
$E_{0}^{a}(T)=\{\lambda\in E^{a}(T): \alpha(T-\lambda I)<\infty\}.$
Recall that $T$ is said to satisfy a-Browder's theorem (respectively, generalized a-Browder's theorem)
if $\sigma_{a}(T)\backslash\sigma_{aw}(T)=\Pi_{0}^{l}(T)$ (respectively, $\sigma_{a}(T)\backslash\sigma_{UBW}(T)=\Pi^{l}(T)$).
From \cite[Theorem 2.2]{AZ6} we have $T$ satisfies a-Browder's theorem if and only if
$T$ satisfies generalized a-Browder's theorem.
$T$ is said to satisfy a-Weyl's theorem (respectively, generalized a-Weyl's theorem) if
$\sigma_{a}(T)\backslash\sigma_{aw}(T)=E_{0}^{a}(T)$
(respectively, $\sigma_{a}(T)\backslash\sigma_{UBW}(T)=E^{a}(T)$).\\
For $T\in L(\mathcal{X})$, let $E(T)=\{\lambda\in iso\sigma(T): 0<\alpha(T-\lambda I)\}$ and
$E_{0}(T)=\{\lambda\in E(T): \alpha(T-\lambda I)<\infty\}.$ Recall that
$T$ is said to satisfy Weyl's theorem (respectively, generalized Weyl's theorem) if
$\sigma(T)\backslash\sigma_{W}(T)=E_{0}(T)$ (respectively, $\sigma(T)\backslash\sigma_{BW}(T)=E(T)$).
We know that $T$ satisfies generalized a-Weyl's theorem implies that
$T$ satisfies a-Weyl's theorem and this implies that $T$ satisfies Weyl's theorem.
Next generalized a-Weyl's theorem for $\delta_{A,B}$ will be studied.
\begin{theorem}\label{thmanswer}
Let $\mathcal{X}$ and $\mathcal{Y}$ be two Banach spaces and let $A\in L(\mathcal{X})$
and $B\in L(\mathcal{Y})$. Suppose that $A$ and $B^{*}$ satisfy a-Browder's theorem.
If $A$ is left polaroid and satisfies property $(\mathcal{P}_l)$ and $B$ is right polaroid
and satisfies $(\mathcal{P}_r)$, then the following assertions are equivalent.
\begin{itemize}
\item[i)] $\delta_{A,B}$ satisfies generalized a-Weyl's theorem.
\item[ii)] $\sigma_{aw}(\delta_{A,B})=(\sigma_{aw}(A)-\sigma_{s}(B))\cup(\sigma_{a}(A)-\sigma_{aw}(B^{*})).$
\end{itemize}
\end{theorem}
\begin{proof} If $A$ and $B^{*}$ satisfy a-Browder theorem, then they satisfy generalized a-Browder theorem,
by \cite[Theorem 4.2]{BA} it follows that $\delta_{A,B}$ satisfies generalized a-Browder's theorem if and only if $\sigma_{aw}(\delta_{A,B})=(\sigma_{aw}(A)-\sigma_{s}(B))\cup(\sigma_{a}(A)-\sigma_{aw}(B^{*})).$ That is $\sigma_{a}(\delta_{A,B})\backslash\sigma_{UBW}(\delta_{A,B})=\Pi^{l}(\delta_{A,B}).$
Since $A$ is left polaroid and $B$ is right polaroid,
then from Theorem \ref{theorem1} $\delta_{A,B}$ is left polaroid,
consequently $\Pi^{l}(\delta_{A,B})=E^{a}(\delta_{A,B}).$
Thus $\delta_{A,B}$ satisfies generalized a-Weyl's theorem.
The reverse implication is obvious from the implication $\delta_{A,B}$ satisfies generalized a-Weyl's theorem implies
$\delta_{A,B}$ satisfies generalized a-Browder's theorem
\end{proof}
In the case of Hilbert spaces operators, we have the following corollaries.
\begin{corollary}
Let $H$ and $K$ be two Hilbert spaces and let $A\in L(H)$
and $B\in L(K)$. Suppose that $A$ and $B^{*}$ satisfy a-Browder's theorem.
If $A$ is left polaroid and $B$ is right polaroid, then the following assertions are equivalent.
\begin{itemize}
\item[i)] $\delta_{A,B}$ satisfies generalized a-Weyl's theorem.
\item[ii)] $\sigma_{aw}(\delta_{A,B})=(\sigma_{aw}(A)-\sigma_{s}(B))\cup(\sigma_{a}(A)-\overline{\sigma_{aw}(B^{*})}).$
\end{itemize}
\end{corollary}
\begin{corollary}\label{coro}
Let $\mathcal{X}$ and $\mathcal{Y}$ be two Banach spaces and let $A\in L(\mathcal{X})$
and $B\in L(\mathcal{Y})$. Suppose that $A$ and $B^{*}$ satisfy a-Browder's theorem.
If $A$ is left polaroid and satisfies property $(\mathcal{P}_l)$ and
$B$ is right polaroid and satisfies property $(\mathcal{P}_r)$, then the following assertions are equivalent.
\begin{itemize}
\item[i)] $\delta_{A,B}$ has SVEP at $\lambda\not\in\sigma_{UBW}(\delta_{A,B}).$
\item[ii)] $\delta_{A,B}$ satisfies a-Browder's theorem.
\item[iii)] $\delta_{A,B}$ satisfies a-Weyl's theorem.
\item[iv)] $\delta_{A,B}$ satisfies generalized a-Weyl's theorem.
\item[v)] $\sigma_{aw}(\delta_{A,B})=(\sigma_{aw}(A)-\sigma_{s}(B))\cup(\sigma_{a}(A)-\sigma_{aw}(B^{*})).$
\end{itemize}
\end{corollary}
\begin{proof}
$(i)\Leftrightarrow(ii)$ follows from \cite[Theorem 2.1]{AZ8}, $(iii)\Leftrightarrow(iv)$ follows from \cite[Theorem 3.7]{AA} and $(iv)\Leftrightarrow(v)$ follows from Theorem \ref{thmanswer}.
\end{proof}
In the following result, we give sufficient conditions for $\delta_{A,B}$ to satisfy
a-Browder's theorem.
\begin{theorem}\label{theo}
Let $\mathcal{X}$ and $\mathcal{Y}$ be two Banach spaces and let $A\in L(\mathcal{X})$
and $B\in L(\mathcal{Y})$.
If $A$ has SVEP on the complement of $\sigma_{SF_{+}}(A)$ and
$B$ has SVEP on the complement of $\sigma_{SF_{-}}(B)$, then
\begin{center}
$\delta_{A,B}$ satisfies a-Browder's theorem.
\end{center}
\end{theorem}
\begin{proof}
Let $\lambda\in\sigma_{a}(\delta_{A,B})\backslash\sigma_{aw}(\delta_{A,B}),$ then $\lambda\in\sigma_{a}(\delta_{A,B})\backslash\sigma_{SF_{+}}(\delta_{A,B}),$ from statement i) of Lemma \ref{propostion1} there exist $\mu\in\sigma_{a}(A)\backslash\sigma_{SF_{+}}(A)$ and $\nu\in\sigma_{s}(B)\backslash\sigma_{SF_{-}}(B)$ such that $\lambda=\mu-\nu$. Since $A$ has SVEP at $\mu\not\in\sigma_{SF_{+}}(A)$ and $B$ has SVEP at $\nu\not\in\sigma_{SF_{-}}(B)$, it follows from \cite[Corollary 2.2]{R1} that $\mu\not\in\sigma_{ab}(A)$ and $\nu\not\in\sigma_{ab}(B^{*})$, applying
statement ii) of Lemma \ref{propostion1} we get $\lambda\not\in\sigma_{ab}(\delta_{A,B})$ this is equivalent to $\lambda\in\Pi_{0}^{l}(\delta_{A,B}).$ Hence $\delta_{A,B}$ satisfy a-Browder's theorem.
\end{proof}
\section{Application}
A Banach space operator $T\in L(\mathcal{X})$ is said to be hereditary normaloid, $T\in\mathcal{HN}$, if every part of $T$ (i.e., the restriction of $T$ to each of its invariant subspaces) is normaloid (i.e., $\|T\|$ equals the spectral radius $r(T)$), $T\in\mathcal{HN}$ is totally hereditarily normaloid $\mathcal{THN}$ if also the inverse of every invertible part of $T$ is normaloid and $T$ is completely totally hereditarily normaloid
(abbreviated $T\in\mathcal{CHN}$), if either $T\in\mathcal{THN}$ or $T-\lambda I\in\mathcal{HN}$ for every complex number $\lambda.$ The class $\mathcal{CHN}$ is large. In particular, let $H$ a Hilbert space and $T\in L(H)$ a Hilbert space operator. If $T$ is hyponormal ($T^{*}T\geq TT^{*}$) or p-hyponormal ($(T^{*}T)^{p})\geq(TT^{*})^{p}$) for some ($0<p\leq1$) or w-hyponormal $((|T^{*}|^{\frac{1}{2}}|T||T^{*}|^{\frac{1}{2}})^{\frac{1}{2}}\geq|T^{*}|),$ then $T$ is in $\mathcal{THN}.$ Again totaly *-paranormal operators ($\|(T-\lambda I)^{*}x\|^{2}\leq\|(T-\lambda I)x\|^{2}$ for every unit vector $x$) are $\mathcal{HN}$ operator and paranormal operators ($\|Tx\|^{2}\leq\|T^{2}x\|\|x\|,$ for all unit vector $x$) are $\mathcal{THN}$ operators.
It is proved in \cite{D2} that if $A,B^{*}\in L(H)$ are hyponormal, then generalized Weyl's theorem holds for $f(\delta_{A,B})$ for every $f\in\mathcal{H}(\sigma(\delta_{A,B})),$ where $\mathcal{H}(\sigma(\delta_{A,B}))$ is the set of all analytic functions defined on a neighborhood of $\sigma(\delta_{A,B}),$ this result was extended to log-hyponormal or p-hyponormal operators in \cite{Du8} and \cite{LB}. Also in \cite{CH} and \cite{LBA} it is shown that if $A,B^{*}\in L(H)$ are w-hyponormal operators, then Weyl's theorem holds for $f(\delta_{A,B})$ for every $f\in\mathcal{H}(\sigma(\delta_{A,B})).$
Let $\mathcal{H}_{c}(\sigma(T))$ denote the space of all analytic functions defined on a neighborhood of $\sigma(T)$ which is non constant on each of the components of its domain. In the next results we can give more.
\begin{theorem}\label{theorem4}
Suppose that $A,B\in L(H)$ are $\mathcal{CHN}$ operators, then \begin{center}
$\delta_{A,B}$ satisfies a-Browder's theorem.
\end{center}
\end{theorem}
\begin{proof}
Since $A$ and $B$ are $\mathcal{CHN}$ operators, it follows from \cite[Corollary 2.10]{D5} that $A$ has SVEP on the complement of $\sigma_{SF_{+}}(A)$ and $B$ has SVEP on the complement of $\sigma_{SF_{-}}(B),$ then by Theorem \ref{theo} a-Browder's theorem holds for $\delta_{A,B}$.
\end{proof}
\begin{corollary}\label{corolla} Suppose that $A,B\in L(H)$ are $\mathcal{CHN}$ operators, then the following assertions are equivalent.
\begin{itemize}
\item[i)] $\delta_{A,B}$ has SVEP at $\lambda\not\in\sigma_{UBW}(\delta_{A,B})$
\item[ii)] $\delta_{A,B}$ satisfies a-Browder's theorem.
\item[iii)] $\delta_{A,B}$ satisfies a-Weyl's theorem.
\item[iv)] $\delta_{A,B}$ satisfies generalized a-Weyl's theorem.
\item[v)] $\sigma_{aw}(\delta_{A,B})=(\sigma_{aw}(A)-\sigma_{s}(B))\cup(\sigma_{a}(A)-\overline{\sigma_{aw}(B^{*})}).$
\end{itemize}
\end{corollary}
\begin{proof}
Since $A$ and $B$ are $\mathcal{CHN}$ operators, it follows from \cite[Corollary 2.15]{D5} that $A$, $B$, $A^{*}$ and $B^{*}$ satisfy a-Browder's theorem.
By \cite[Proposition 2.1]{D5}, we conclude that $A$ and $B^{*}$ are left polaroid. Now,
then the equivalences follows from Corollary \ref{coro}.
\end{proof}
\begin{corollary} Suppose that $A,B\in L(H)$ are $\mathcal{CHN}$ operators, then \begin{center}
$f(\delta_{A,B})$ satisfies generalized a-Browder's theorem, for every $f\in\mathcal{H}_{c}(\sigma(\delta_{A,B}))$.\end{center}
\end{corollary}
\begin{proof}
By Corollary \ref{corolla} and \cite[Corollary 3.5]{D7}, we get generalized a-Browder's theorem holds for $f(\delta_{A,B})$.
\end{proof}
\begin{corollary} Suppose that $A,B\in L(H)$ are $\mathcal{CHN}$ operators, then \begin{center}
$f(\delta_{A,B})$ satisfies generalized a-Weyl's theorem, for every $f\in\mathcal{H}_{c}(\sigma(\delta_{A,B}))$.\end{center}
\end{corollary}
\begin{proof}
By \cite[Proposition 2.1]{D5} and Theorem \ref{theorem1},
we get $\delta_{A,B}$ is left polaroid and from Corollary \ref{corolla} we have
$\delta_{A,B}$ satisfies generalized a-Weyl's theorem,
apply \cite[Theorem 3.14]{D7} we get generalized a-Weyl's's theorem holds for $f(\delta_{A,B})$.
\end{proof}
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press releases /
KPN & Notificare pushing it to a partnership
Rotterdam, the Netherlands, June 18 2019
We are happy to announce a new partnership with KPN. Notificare is a leading Contextual Marketing Automation Platform that allows marketers to send context-relevant and interactive messages to their customers. With this partnership, KPN clients can benefit from the most relevant contextual messages Notificare offers. Integration with the Notificare platform ensures that both mobile app data and website data become insightful and actionable.
Rich and interactive messages for web and mobile
In recent years Notificare has evolved into a powerful Marketing Automation platform where rich and interactive messages play a central role. By using Location Based Marketing, the relevance for the end customer quickly increases. The platform uses a user-friendly dashboard to set up campaigns, segment, and instantly view the campaign results. All the platform functionalities are also accessible through the robust API. As of today, this API is also available in the KPN Store to make it even easier to use and combine with other APIs.
KPN API Store
APIs have proven to play an essential role in developing new propositions for consumer and business to business markets. With the KPN API Store, KPN provides developers, innovators, and product owners access to building blocks to develop and enrich their own propositions.
"In the past year, the KPN API Store has grown with easy-to-use and flexible building blocks for fast developing. With Notificare, there is a strategic, business and technical fit. As a result, the first app with Notificare is already in the App Stores, and we are looking forward to many to come." Says Wouter van Schaik, Lead KPN API Store.
"We believe in the combination and integration of the various related business solutions. The API Store is a good starting point to enter the different APIs. In the coming year, we will be releasing more of our services via the KPN platform. So that users can enjoy, for example, the loyalty solutions for retailers that we offer." Says Robert Leefmans, CEO of Notificare
Notificare in the API Marketplace
Notificare will help KPN's clients move up the innovation curve, transform processes and increase customer satisfaction thanks to reliable and smart notification APIs. The first Notificare API can be tested for free in the KPN API Store.
More about the KPN API Store and the integration of Notificare. https://developer.kpn.com/marketplace/push-notificare
About Notificare
Notificare is a leading Marketing Automation Platform that empowers marketers to send context-relevant and interactive messages to their customers. Headquartered in Rotterdam, Netherlands, founded in 2012 and ISO/IEC 27001:2013 certified. More information: https://notificare.com/contact
About KPN
KPN is a leading telecommunications and IT provider and market leader in the Netherlands. With fixed and mobile networks for telephony, data and television, KPN serves customers at home, at work, and abroad. KPN focuses on both private customers and business users, from small to large. In addition, KPN offers telecom providers access to its widespread networks. | {
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«Пи́терборо Юна́йтед» (полное название — Футбольный клуб «Питерборо Юнайтед»; , ) — английский профессиональный футбольный клуб из города Питерборо, графство Кембриджшир, Восточная Англия. Основан в 1934 году.
В настоящее время выступает в Лиге 1, третьем по значимости дивизионе в системе футбольных лиг Англии.
Клубное прозвище — «пош» (The Posh). Основные противники клуба — «Кембридж Юнайтед», «Нортгемптон Таун» и «МК Донс».
Титулы
Третий дивизион Футбольной лиги:
1991/92
Четвёртый дивизион Футбольной лиги:
1960/61, 1973/74
Трофей Футбольной лиги:
2013/14
Рекорды
В марте 2011 года клуб объявил о начале продаж сезонных абонементов стоимостью 15 тысяч фунтов — почти в семь раз дороже, чем самый дорогой абонемент в Премьер-лиге. Помимо удовольствия лицезреть любимую команду с самых лучших мест, обладатели эксклюзивных абонементов, если таковые, конечно, найдутся, будут получать в неограниченном количестве бесплатную еду и напитки, смогут общаться с игроками команды, а также будут регулярными гостями одного из директоров клуба на выездных матчах.
Кроме того, «Питерборо» предложил и 50 «пожизненных» абонементов чуть «подешевле» — по 12 тысяч фунтов. Они рассчитаны на 75 лет и включают в себя кубковые и товарищеские матчи, а в случае смерти его обладателя могут быть переданы членам семьи. Можно их продать и третьей стороне.
Текущий состав
Известные игроки
Андре Боукод
Габриэль Закуани
Дэвид Симэн
Джимми Буллард
Джордж Бойд
Джейк Ливермор
Дуайт Гейл
Зэт Найт
Мэттью Этерингтон
Саймон Дэвис
Сейдо Берахино
Тэрри Блай
Примечания
Ссылки
Официальный сайт клуба
База данных «Питерборо Юнайтед»
Футбольные клубы Англии
Футбольные клубы Кембриджшира
ФК «Питерборо Юнайтед» | {
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Q: How can I change my xpath dynamically in Selenium WebDriver? In my application id is changing dynamically,and name is not given to all elements.
Now I want to apply dynamic way of searching "x path" in different Division in HTML.
Whenever I refresh page , database value can be added or removed from page.
So is there any way for taking dynamic path of one element??
A: You can use custom XPath or CSSPath.
This path have some condition like "OR", "AND", etc.
You can also use some functions like contains(), text(), not(), etc.
References :
http://www.w3schools.com/xpath/xpath_functions.asp
http://www.w3schools.com/cssref/css_selectors.asp
| {
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When it's time to really relax and spoil yourself, Central West Florida has just what you need.In addition to day spas sprinkled throughout the area, several resorts offer spa treatments to their guests as well as to the public. What's more, to enhance your experience, several spas feature water-inspired treatments. After all, you are in Florida.
With a list of spa packages that goes on and on, you're sure to find the right combination at Pia Esthetics Day Spa in St. Petersburg. Check out the two-and-a-half-hour Beauty Break (massage and pedi), or go for the four-plus-hour Extravaganza (body sugar scrub, chocolate cake mani-pedi and deep pore rescue facial).
When Hernando de Soto reached the shores of Old Tampa Bay in 1539, he thought he'd found the legendary Fountain of Youth that Ponce De León had missed. Since its founding in 1925, the historic Safety Harbor Resort and Spa is "where natural healing mineral waters flow" under the grounds. Today, the 50,000-square-foot spa and fitness center, which Spa Finder Magazine consistently recognizes among its top 10 U.S. spas, features everything you need for that youthful glow. Spend a day and choose from more than 50 spa and salon treatments.
In Tampa's SoHo neighborhood, Spa Evangeline takes organic to a new level with fresh fruit scrubs, buttery lotions and botanical oils infused with handpicked herbs. Each treatment is customized to meet your needs, and each ingredient is carefully selected to awaken your senses and refresh your soul. You can even create your own aromatherapy blend to use during your treatment and take home to enjoy afterward.
You'll feel instantly relaxed as you cozy into Amy's Day Spa, housed in a bungalow, also in Tampa's SoHo neighborhood. The Signature Zen Journey will take you through five hours of pampering, including a 90-minute customized massage, facial and a mani-pedi. Feel those cares melting away?
It really is all about you at All About You Salon & Day Spa in Pasco County's Zephyrhills, where the team is ready to pamper you from head to toe with treatments ranging from massages and facials to manis and pedis, and much more.
Following a day in the springs in Crystal River, relax at Spa Bleu, an Aveda spa that has nearly everything you could desire, from one-of-a-kind treatments to a full day of pampering filled with manicures, pedicures, massages, facials, Vichy showers and more.
Set amidst 900 acres of rolling hills, pine trees and four golf courses, Indaba–The Spa at Innisbrook, a Salamander Golf & Spa Resort in Palm Harbor, defines tranquility. Inside the separate dressing rooms are lounge areas, saunas/steam rooms and hot tubs. State-of-the-art treatment rooms evoke instant relaxation, making it simple to ease into your treatment. For a change of pace, try the Copperhead Golf Ball Massage, a terrific take on a hot-stone massage.
The 12,000-square-foot Spa at Sandpearl on Clearwater Beach is a relaxing retreat from the start. From the whirlpool and steam rooms to treatment rooms and couples' suites complete with Swiss showers, you won't want to leave once you step foot inside. And that's before the treatments even begin! For a real treat, try the Ocean Memory Ritual, an organic algae-based remedy that renders the skin smooth and firm. It's little wonder that the Spa at Sandpearl is rated among the "Top Spas in North America" by the readers of Condé Nast Traveler.
South of Clearwater Beach is Spa Oceana at the Don CeSar Hotel St. Pete Beach. Five treatments on the spa menu are exclusive to the Don CeSar, including Spa Oceana Splendor, Oceana Aromatherapy Massage, Beach Glow Body Treatment, Sea of Life Facial and the Floral Blossom Manicure & Pedicure. For a truly decadent spa experience, choose the Spa Oceana Sunset Bliss Massage—the most expensive and exclusive couples massage in the world at $3,500 per couple. Start with a little relaxation in a rooftop cabana and delicacies prepared by the hotel's executive chef and a bottle of Cristal, followed by a sunset, seaside couples massage that will melt the world away.
Just down the beach is Sandava Spa at the Hyatt Regency Clearwater Beach Resort and Spa. With a focus on individually crafted experiences customized with fresh, organic ingredients, Sandava individualizes your spa time to specifically suit your needs. For instance, the Sun Repair Soother is perfect after a few days under the rays, or opt for one of the spa's signature treatments, such as the Sun-Warmed Shell Massage, Florida Orange Scrub or the Blueberry Bliss Facial. Sandava has seasonal offerings, too, so be sure to check the menu for even more choices when you make your appointment.
Spoil yourself and get treated like royalty at one of the many spas in Central West Florida. | {
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{"url":"https:\/\/www.studysmarter.us\/textbooks\/physics\/physics-for-scientists-and-engineers-a-strategic-approach-with-modern-physics-4th\/the-magnetic-field\/q-12-what-are-the-magnetic-fields-at-points-a-to-c-in-fig-ur\/","text":"Suggested languages for you:\n\nQ. 12\n\nExpert-verified\nFound in: Page 830\n\n### Physics for Scientists and Engineers: A Strategic Approach with Modern Physics\n\nBook edition 4th\nAuthor(s) Randall D. Knight\nPages 1240 pages\nISBN 9780133942651\n\n# What are the magnetic fields at points a to c in FIG URE EX29.12? Give your answers as vectors.\n\nAt a.\n\nAt b.\n\nAT C.\n\nSee the step by step solution\n\n## Step 1: Given information\n\nWe have given,\n\nthe diagram shown the current carrying wires.\n\nWe have to find the magnetic field at a and c.\n\n## Step 2: Simplify\n\nThe magnetic field due to current carrying wire at a distance is\n\nLet us find the magnetic field due to upper wire at point a.\n\nThe distance between the point a and wires are,\n\nthen,\n\nSince the distance of point a from both the wire is same. so magnetic fields will be same but the direction will opposite.\n\nThen total magnetic field will be\n\n## Step 3: Simplify\n\nFor magnetic field at b, Since the distance is perpendicular.\n\nthen Total magnetic field at b\n\nFor magnetic field at c, we find it similarly as we find for at a.\n\nSince the distance of point c from both the wire is same. so magnetic fields will be same but the direction will opposite.\n\nThen total magnetic field will be","date":"2022-11-28 17:32:33","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8091695308685303, \"perplexity\": 1618.8989880820347}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-49\/segments\/1669446710534.53\/warc\/CC-MAIN-20221128171516-20221128201516-00768.warc.gz\"}"} | null | null |
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<string name="alert_dialog_ok">Okay</string>
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<string name="welcome">Welcome</string>
<string name="login">Login</string>
<string name="password">Password</string>
<string name="email">Email</string>
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<string name="title_work_with_pressure">Work with pressure</string>
<string name="title_family">Family</string>
<string name="title_settings">Settings</string>
<string name="title_logout">Logout</string>
<string name="navigation_drawer_open">Navigation drawer open</string>
<string name="navigation_drawer_close">Navigation drawer close</string>
<string name="welcome_home_dear">Welcome home dear!</string>
<string name="now_its_time_to_look_up_your_family">Now, Its time to look up your family</string>
<string name="time_to_tune_your_settings">Time to tune your settings…</string>
<string name="work_starts_to_do_its_work_explodes_in_peace">Work starts to do its work\nExplodes in peace</string>
<string name="logout">Logout</string>
<string name="logout_msg">Are you sure want to logout?</string>
<string name="ninety_nine_plus">99+</string>
<string name="current_location">current location</string>
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Trinity (Play Inc.) Title Wave Character Generator Manual, Volume 7, Version 2.1 ©2000.
Trinity (Play Inc.) User Guide Volume 1, Version 2.1 ©2000.
GVG Horizon Series Routing System HX-64 Parts List and Drawings. In hard Binder.
Service manual for Ikegami HC-240A 3 CCD Camera, in original hard binder, Excellent condition.
Maintenance manual for NEC SP-3 CCD Camera, VG condition.
Sony BKU-905 Time Code Generator/Reader Operation and Maintenance Manual1st Edition (Revised 1) For the BVU-950 Umatic.
Sony BVE-800 Operation and Maintenance Manual 1st Edition (Revised 9). Good condition, some marks and wear to cover.
Sony Operation Manual For the BVE900 editing system. Covers all aspects of setup and operation. Torn covers, internal Good to VG.
Sony BVE-9000 Editing Control System Operation Manual. Covers BVE-9000 System setup.
Sony BVS-3100 Operation Manual,1st Ed Rev 2.
Sony Operation Manual covers BVS-3100, 3200, 3200C and PAL Versions.
Sony BVV-5 Operation Manual. pn 3-718-094-04 (4). 1991.
Sony BVW-40 Operation & Maintenance Manual Vol 1, 1st Ed.
Sony Betacam BVW-40 Manual, paper cover, Very good condition.
Sony DME Switcher Operating Instructions, for Model DFS-300 switcher, some stains to cover, overall VG condition. | {
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Радинь (або Радин, Радзинь-Подляскі, давніше Козій Ринок, ) — місто в східній Польщі. Адміністративний центр Радинського повіту Люблінського воєводства.
Положення
Місто розташоване на південному Підляшші, до 1946 року на українсько-польському етнічному пограниччі.
Історія
1468 року вперше згадується православна церква в місті.
У часи входження до складу Російської імперії був центром Радинського повіту Сідлецької губернії.
За даними етнографічної експедиції 1869—1870 років під керівництвом Павла Чубинського, у місті переважно проживали польськомовні римо-католики, меншою мірою — греко-католики, які також розмовляли польською.
У 1917—1918 роках у місті діяла українська школа, заснована 19 листопада 1917 року, у якій навчалося 50 учнів, учитель — М. Тульчий.
У міжвоєнні 1918—1939 роки польська влада в рамках великої акції Ревіндикації перетворила місцеву православну церкву на римо-католицький костел.
Розпорядженням міністра внутрішніх справ 14 квітня 1934 року розширено територію міста — передані з ґміни Біла Радинського повіту: землі села Надвітнє і фільварку Надвітнє, землі фільварку Губернія, цегельня, урочище Рабштин фільварку Біла, палац з парком і луками, села Козиринок Старий і Козиринок Новий, землі фільварку Кути, землі шпиталю св. Кунегунди, цвинтар і землі католицької і колишньої православної парафій.
Під час Другої світової війни в Радині діяв відділ Українського Допомогового Комітету.
Населення
Демографічна структура станом на 31 березня 2011 року:
Відомі особи
Фіалек Іпполит (1875—1936) — український радянський політичний діяч, член Української Центральної Ради.
Кароль Ліпінський (1790—1861) — польський скрипаль, композитор і педагог, один з найвідоміших скрипалів світу усіх часів.
Примітки
Література
Міста Люблінського воєводства
1468 у Європі | {
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Q: PHP for iPhone Native Can I compile PHP to a native iPhone app?
A: No. You can, though, make web apps that look an awful lot like they're iPhone apps.
A: Take a look at Titanium Mobile.
I've used their desktop developer, which can use PHP but not the mobile version
A: Perhaps a bit of elaboration. You cannot "compile" PHP. It is a scripting language. Yes you can compress, obfuscate, and even turn it into an Apache Module etc. but it is an interpreted language so at some point will need the/a php interpreter and a web server. So to get it to run on any hardware platform mean having the php engine and/or an Apache server to be there.
So to make any self contained app you need to have a programming language that will compile and produce an executable file that is supported on the target hardware or is self supporting. But essentially produces a bundle containing all the resources required to run that app whether it is an iPhone, Palm Pre, Mac or PC. For the iPhone Objective-c is used to program, connect to the libraries and frameworks and produce the required bundle to install on the iPhone. You can use Mono via Monotouch to do the development work but it then compiles to the required bundle for installation.
If you want to use PHP then build a web based app that will be accessible via Mobile Safari on the iPhone. To make the app render well you will need quite a bit of HTML, JavaScript and CSS but as has been mentioned there are frameworks to ease this pain such as iUi, iWebkit, jQTouch etc.
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RELX Survey: AI Adoption Accelerates during COVID-19 Pandemic; 68% of US Businesses Increased Investment in AI Technologies
Use of AI technologies reached 81%, up 33 percentage points since 2018
AI investment and adoption accelerated during COVID-19 pandemic
86% of survey respondents believe that ethical considerations are a strategic priority in the design and implementation of their AI systems
AI is more likely to be used to increase efficiencies and worker productivity than to replace labor
US competitiveness in AI on the world stage remains a concern
NEW YORK--(BUSINESS WIRE)--The adoption of new artificial intelligence (AI) technologies and further investment in existing AI technologies accelerated during the COVID-19 pandemic, according to a study released today by RELX, a global provider of information-based analytics and decision tools for professional and business customers. The study also reveals that overall implementation of AI technologies across the business landscape increased for the third consecutive year.
The 2020 RELX Emerging Tech Executive Report marks the third edition of the survey and provides a three-year overview of AI adoption. It features insights from business leaders across eight industries (government, healthcare, insurance, legal, science/medical, banking and agriculture) and covers AI's impact on businesses' success, the future of work, global competition, ethics, and the global COVID-19 response. More than 1,000 U.S. senior executives were surveyed.
COVID-19 Drove AI Technology Investment and Adoption
COVID-19 is the most pressing issue facing US executives today as it reshapes and disrupts industries across the US. The majority of respondents (68%) increased their investment in AI technologies during the COVID-19 pandemic with 48% investing in new AI technologies and 46% investing further in AI technologies already in use at their companies. Similarly, 63% of business leaders polled report that AI technologies had a positive impact on their business's ability to stay resilient in the face of the pandemic.
For many respondents, the COVID-19 response effort underscored the importance of AI, with 77% agreeing that these technologies helped slow the spread of the virus and almost eight in ten (79%) responding that countries should share AI technology resources in light of the pandemic.
"Businesses' response to COVID-19 has confirmed the view of US business leaders that artificial intelligence has the power to create smarter, more agile and profitable businesses," said Vijay Raghavan, Executive Director of the Chief Technology Officer Forum at RELX. "Businesses face more complex challenges every day and AI technologies have become a mission-critical resource in adapting to, if not overcoming, these types of unforeseen obstacles and staying resilient."
The Use of Artificial Intelligence has Increased Across Nearly all Sectors Polled
Artificial intelligence is transforming industries and changing competitive dynamics. Modern organizations are increasingly implementing AI technologies to uncover new efficiencies and inform business decisions.
While AI is now driving market reinvention, in 2018, less than half (48%) of business leaders responded that their companies used these technologies. The 2020 RELX Emerging Tech Executive Report shows that this number has reached 81%, up from 72% in 2019. The use of AI is viewed as a key competitive differentiator and 60% of those who say their company utilizes these technologies have increased their data scientist and technologist headcount to support these technologies while 60% say that they've increased the areas of their business touched by AI.
AI is a value add for businesses, but there are hurdles that need to be cleared on the path to emerging technology adoption. The leading reasons for companies not using AI are budget constraints (44%) and lack of technical expertise (39%). More than half of all respondents agree that in order to increase investment in AI, data quality and availability needs to improve (57%), and there needs to be a better understanding of the legal and regulatory implications (53%) for adopting these technologies.
Ethical AI is a Priority and a Competitive Advantage
The rapid adoption and implementation of AI needs to be balanced with ethical considerations not only for greater social good, but also because it is a differentiator in a crowded landscape. Almost nine in ten (89%) business leaders believe that ethical standards in the development and use of emerging technologies lead to a competitive advantage for businesses. A similar number (86%) report that ethics considerations have been made a strategic priority in the design and implementation of their AI systems.
Only 5% of respondents believe that no regulations are needed for AI technologies. While this represents a small number of business leaders, there are differences in opinions as to how AI should be regulated. Most (68%) business leaders believe emerging technologies should be regulated at a national level, while 60% believe they should be regulated at an international level and 45% believe they should be regulated at a state level.
International Competition Remains a Concern for US Businesses
The study shows a clear perceived link between technological supremacy and economic growth. While the vast majority (90%) of business leaders see the US as the leader in artificial intelligence, up 10 percentage points from the year prior, 82% say that they are concerned with the possibility of other countries surpassing the US in terms of AI technology development and implementation. This sentiment has increased every year since the survey began in 2018, reflecting an ongoing concern that foreign competitors could overtake US businesses.
Among business leaders who share these concerns, half (50%) state that their business would be negatively impacted if other countries passed the US in AI expertise. Respondents believe the keys to promoting AI development and implementation domestically are rolling out programs to help employees stay competitive as AI becomes further ingrained in the business world (59%), increasing research and development funding (58%) and providing training opportunities for employees (58%). Training is on the rise, as 75% of companies offer training on AI technologies, up from 62% in 2019 and 46% in 2018.
"A trend we've seen over the last three years is that AI implementation consistently outpaces training," said Raghavan. "Companies that do not dedicate the necessary resources to training existing employees on new AI technologies risk leaving growth opportunities on the table and using biased or otherwise flawed systems to make and enforce major decisions."
Key three-year trends include:
Artificial intelligence (AI) technologies are being utilized by my business
I am somewhat or very concerned about other countries being more advanced than the US in artificial intelligence technology development and implementation
I believe US companies should invest in the future artificial intelligence workforce through educational initiatives such as university partnerships
My company currently offers training on artificial intelligence technologies
The US government should develop programs to help employees stay competitive as AI becomes more integrated in the business world
The US government should leave the promotion of AI technologies to the private sector
A summary of the findings, with a breakdown of data relating to each of the eight industries surveyed can be found here.
Infographics can be found here.
With Ipsos, RELX surveyed 1,014 adults in the United States between the ages of 30 – 74. To qualify, respondents had to be employed full-time, have a household income of at least $50,000, work at a company with more than 50 employees, and currently be a business executive or business decision maker/leader at their company. Respondents also had to be employed in one of eight industries featured in this report to qualify, and they had to either use AI technology at their business or be aware of it. The qualifications were consistent to those used in 2019 though new definitions were included in this survey to describe each industry.
About RELX
RELX is a global provider of information-based analytics and decision tools for professional and business customers. The Group serves customers in more than 180 countries and has offices in about 40 countries. It employs over 33,000 people, of whom almost half are in North America. The shares of RELX PLC, the parent company, are traded on the London, Amsterdam and New York Stock Exchanges using the following ticker symbols: London: REL; Amsterdam: REN; New York: RELX. The total market capitalization is approximately £33.7bn, €37bn, $43.6bn.
Ali Donzanti
Allison+Partners
+ 1 646.428.0627
relx@allisonpr.com | {
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84 THE AIR. Freedom and the Super-lens O N Sunday, July 12th, a televi- SIon camera equipped with a lens of heroic magnifying power was placed in the center-field bleach- ers of the Boston baseball park and pointed straight at the catcher's mitt, four hundred and twenty feet away. This action produced two results, in the following order: (1) a splendid view of certain phases of a Red Sox-Yankee game, and (2) censorshIp. On Mon- day, the 13 th, the hasehall commission- er, Ford C. Frick, asked the National Broadcasting Company, which owns the super-lens, to take its gadget out of baseball and keep it out. N.B.C. promptly agreed to do so. The news- papers reported some of the details of this quick and friendly work of sup- pression, hut none of them mentioned what seems to me to be a keJ factor in the case-the principle of so-called free- dom of the air. Freedom of the air is a broad and deeply mysterious idea. To date, neither Congress nor the Federal Communica- tions Commission has done much to ex- plain what it means. I've heard it defined by theorists as the right of audiences throughout the countr) to receive the best that broadcasters can provide in the way of service and tech- nique, without restriction or discrimination For one rea- son and another, the baseball industry has found itself pecul- iarly involved in problems of aerial freedom. Network tele- casting of major-league ball games on weekends is thought by many people to be destroy- ing mInor-league baseball. Minor-league leaders have de- scnbed the practice as canni- balism. Mr. Fnck has said that he opposes such telecasting (which makes a great deal of money for the major-league magnates, who pay most of his salary) hut is afraid that any restriction of it would violate freedom of the air. Suppression of N.B.C.'s super-lens would also cem, on the face of it, to violate freedom 0 f the air, but Mr. Frick, at the behest of a few major-league magnates, \.... ... .\' 'Iì -:: ) ( \ 1';1 \ .:: 1 · >If . ,:î has asked for its suppressIon. The situa- tion is confused, and so, obviously, is the Commissioner. As his industry's agent-but a God-fearing, law-fearing agent-Mr. Frick has for some time been begging the government to spell out baseball's duties In the television field. Philosophically and legally, this good man has implied, he doesn't know where he stands. The same thing IS true, of course, of everyone else connected with television, but no one else appears to care. The affair of the big lens in Boston was one more chapter In a lone- ly moral struggle. As I've said here before, watching a baseball game on television is like watching one through a small selection of knotholes. N.B.C.'s operation in cen- ter field opened up a new and better knothole The super-lens IS known in the trade as a Spacemaster. Sam \\T eller mIght have called it a "patent double- million-magnifyin' gas microscope of hextra power." The newspapers have called it an eighty-inch lens. Actually, according to engineers I know, it is formed b} the union of a standard lens with a fifteen-power eyepiece multi- '" ( ">i \" "" " \; f "",,* "'ø' Â ..P, :. ../ "-< < -<. C^ '\ .,... \.. .... , '- " .... dULY 2. 5. I 9 59 plier. Thanks to the combination, this Cyclopean eye has approximately the power of an eighty-inch lens. In Boston, by looking at the catcher from in front, it gave the audience a fine, novel view of the strIke zone, the behavior of pitched balls, and the finger signals transmitted by the catcher to the pitcher. (Normal- ly, these signals are invisible to every- one but the pitcher, the second baseman, and the shortstop. On most teams, the general rule, whIch can be vaned if there is a runner spying from second base, is one finger for a fast ball, two fingers for a curve, and a wiggling of all five fingers for a slow ball, or change-of-pace. ) Dunng the game in question, the center-field camera with the big lens was by no means the only one in action. Other cameras, behind the plate and in the grandstand, offered other, more familiar views from time to time. But it was the center-field view, unquestionably, that pleased the tele- vision audience most. Within certain severe hmlts, it gave the TV VIewer an advantage over every spectator in the ballpark. That last point is significant. When Mr. Frick asked N.B.C to repeal the Spacemaster, he said, by way of ex- planation that use of the lens "could cause all kinds of trouble." He dIdn't .. ! '\ -t " .- * t . *' '\ " "' .ý , , .. , ....., j<., 1; t",. ' . .,: .... "!< 't .. , *^-'. :0 . , ,"l- .;, J A". ,:0 , ., 4 ^ ,t\' ,-' :!" ((All men are not created equal. I'm vastly superior to you." | {
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The Arabic language is one of the major languages of the world. If you are learning the Arabic language watching the language spoken among people is a great way to increase your vocabulary comprehension and can help you understand better how the language is expressed. Within our selection of language DVDs we have American films enjoyable by both children and adults translated into Arabic that will enable you to absorb the Arabic language as you continue to learn the language. If you already know Arabic well, these DVDs will provide an excellent opportunity to enjoy these movies in a language that you understand. Combining these films with Arabic Books and Arabic Software is the best way to round out you or your children's educational experience in learning the Arabic language. So select the Arabic language learning products now to find out more specific details of what we offer.
The twenty minute Arabic dvd exposes children to animals from their environment in the Arab world.
Collection of 32 popular children's Arabic rhymes from all over the Arab world.
A playful and interactive introduction to Arabic numbers 1-10.
Little Pim Arabic dvds for babies, toddlers and preschoolers.
he Arabic phonic alphabet is easy to learn once you know the sounds!
Ali Azoua - 2000 Moroccan film directed by Nabil Ayouch.
2007 film directed by Erin Kolirin and the winner of over 35 international film awards.
A romantic comedy centered around the lives of five Lebanese women living in Beirut.
2001 film directed by Mehdi Charef.
2002 film directed by Elia Suleiman.
2005 documentary directed by Haydar Daffar on life in Baghdad.
1947 film directed by Ahmed Badrakhan.
A coming-of-age comedy/drama set in Tunisia.
2005 film directed by Joana Hadjithomas and Khalil Joreige.
A look at terrorism in Algeria through the eyes of Rachida, a teacher.
2005 Arabic film directed by Hany Abu-Assad.
2002 Arabic film directed by Raja Amari.
2007 Lebanese film directed by Philippe Aractingi.
French and Arabic 2008 film directed by Karin Albou.
1987 film directed by Michel Kheleifi.
Three stories set among the Bedouin of Jahalin in the hills of the Judean desert. | {
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Take your personal ministry to the Next Level by helping StudyLight build churches and supporting pastors in Uganda.
Gill's Exposition of the Whole Bible
Gospels Only
Golden Chain Commentary
Lightfoot's Commentary
Ryle's Exposiory Thougths
Fourfold Gospel
Gospels Compared
Lapide's Commentary
And it came to pass on the second sabbath day after the first,.... Or "second first sabbath", concerning which interpreters are greatly divided. Some think, that it was either the seventh day of the feast of unleavened bread, or the eighth day of the feast of tabernacles. Others, that it was the sabbath which fell that year on the day of Pentecost; and that as there were three grand festivals among the Jews, the feasts of passover, Pentecost, and tabernacles; so when the sabbath day fell on the feast of the passover, it was called the first prime sabbath, when on the feast of Pentecost, it was called the second prime sabbath, and when on the feast of tabernacles, the third prime sabbath. Others have been of opinion, that as the Jews had two beginnings of their year, the one on civil accounts in Tisri, the other on ecclesiastical accounts in Nisan; so the first sabbath in Tisri was called the first first sabbath, and that in Nisan, which was this, the second first sabbath: but what seems most likely is, that this sabbath was, as it may be rendered, "the first sabbath after the second"; that is, the first sabbath after the second day of the passover, when the sheaf of the firstfruits was offered, and harvest might be begun; which suits well with ears of corn being ripe at this time, which the disciples rubbed. So the Jews reckoned the seven weeks from thence to Pentecost by sabbaths; the first after the second day they called the second first, or the first after the second day; the second they called the second second; and the third was named the second third; and so on, the second fourth, the second fifth, the second sixth, and second seventh, which brought on Pentecost, when the harvest was ended. So in the Jewish liturgies, there are collects for the first sabbath after the passover, and for the second sabbath after the passover, and so on to the sabbath before Pentecost. The eastern versions, Syriac, Arabic, Persic, and Ethiopic, not knowing what should be meant by it, have only rendered it, "on the sabbath day", as in Mt. 12:1. :-.
That he went through the corn fields; that is, Jesus, as the Syriac, Persic, and Ethiopic versions:
and his disciples plucked the ears of corn, and did eat, rubbing them in their hands: after they had plucked them they rubbed them in their hands to get clean off the husk or beard, that were on them, and then ate the grains. And as plucking of the ears of corn was forbidden on a sabbath day, :-, so was rubbing them; though if they were rubbed before, the chaff might be blown off from them in the hand, and eat on the sabbath day: the rule is this l;
"he that rubs ears of corn on the evening of the sabbath, (i.e. on the sixth day,) may blow them from hand to hand on the morrow, and eat''
But the disciples both plucked them, and rubbed them, and blew away the chaff from them on the sabbath day, and therefore were complained of by the Pharisees.
l T. Bab. Betza, fol. 12. 2. & 13. 2. Vid. Maimon. Hilch. Sabbat, c. 21. sect. 14. 17.
And certain of the Pharisees said unto them,.... Unto the disciples. The Evangelists Matthew and Mark say, that they said this to Jesus: no doubt but they said it to both, first to one, and then to the other; probably last of all to Christ, who returned an answer to it:
why do ye that which is not lawful on the sabbath day? as to pluck ears of corn, and rub them, and eat them; :-
And Jesus answering them, said,.... For they brought the charge against the disciples to him, being desirous to know what he would say, and that they might have something to accuse him of; and who, at once, took up the cause of his disciples, and vindicated them, by observing what David did, when he, and his men were an hungry; how that he went into the tabernacle, and took the showbread, and ate of it, and gave it to his men, who also ate of it; which, according to the law, was only allowed to priests; and by taking notice of another instance, which this evangelist does not relate; namely, how on the sabbath days the priests, by doing various servile works, profaned the sabbath day, and yet were not charged with any blame;
:-.
And he said unto them,.... He adds this at the close of the instances he gave, at the end of his vindication of his disciples, and discourse with the Pharisees, as a full answer to their cavils;
that the son of man is Lord also of the sabbath; and may do what he will, and suffer his disciples to do whatever he pleases on that day;
And it came to pass also on another sabbath,.... Whether the following sabbath, or some time after, is not certain,
that he entered into the synagogue. The Arabic version reads, "into their synagogue", as in Matthew 12:9 the synagogue of the Jews; in what place, whether at Capernaum, or some other city of Galilee, is not so clear:
and taught; explained the Scriptures to the people, and instructed them in the doctrines of the Gospel:
and there was a man whose right hand was withered; who was in the synagogue, and one of his hearers; Matthew 12:9- :
And the Scribes and Pharisees watched him,....
whether he would heal on the sabbath day: there being such an object before him:
that they might find an accusation against him; as they had before against his disciples.
But he knew their thoughts,.... Being the omniscient God; though they had said nothing of their intentions, he knew what they designed, should he heal the man with his withered hand, as they expected he would:
and said to the man which had the withered hand, rise up, and stand forth in the midst. The Syriac and Persic versions add, "of the synagogue", and which is the true sense; :-
and he arose and stood forth; he rose up from his seat, and stood up in the midst of the synagogue, and of the people, that he might be seen of all.
Then said Jesus unto them,.... The Scribes and Pharisees, who were watching him, and whose thoughts, and the reasonings of their minds, purposes, and intentions, he full well knew:
I will ask you one thing; or question, as they had asked him one before; Matthew 12:10
is it lawful on the sabbath days to do good, or to do evil? to save life, or to destroy it?
Matthew 12:10- : to which may be added, that to save life on the sabbath day was agreeable to their own canons: there were many things which they allowed might be done on the sabbath day, when life was in danger, which otherwise were not lawful; Matthew 12:10- :.
And looking round about upon them all,.... The Scribes and Pharisees, and the rest of the people in the synagogue;
he said to the man; who had the withered hand,
stretch forth thy hand, and he did so; he stretched it out, as the Syriac and Persic versions render it, which he was not able to do before:
and his hand was restored well as the other; the phrase, "well as the other", is left out in one copy, and in the Vulgate Latin version; and so is the word "well" in the Syriac and Arabic versions; and the word "immediately" is added in the Ethiopic version. And certain it is, that his withered hand was restored sound and well as the other, directly.
And they were filed with madness,.... Both at the cure, and because they could not answer him; nor properly fix a charge upon him, or accuse him before the people, without bringing their resentments on them:
and communed one with another what they might do with Jesus: this they did after they came out of the synagogue, and when with the Herodians, as in :-.
And it came to pass in those days,.... When Christ was teaching by the lake of Gennesaret, or in one or other of the cities of Galilee near that place:
that he went out; of the synagogue and city where he had been:
into a mountain to pray; for the sake of solitude, and which lay near the sea of Tiberias; :-.
and continued all night in prayer to God; or "with" God, as the Ethiopic version renders it; or "in the prayer of God" as the phrase may be literally rendered; not in a prayer of God's making; though the Jews m sometimes speak of the prayer of God, and give us a form of it: but either this respects the object of his prayer; it was made to God, as our translation suggests; or the nature, matter, and manner of it: it was a divine prayer, it regarded divine things, and was put up in a very fervent manner, and with great vehemence; so the coals of love or jealousy are said to be "coals of fire, which hath שלהבת יה, the flame of Jehovah"; that is as we render it, "a most vehement flame", Song of Solomon 8:6 In like manner, "prayer of God" is a most vehement prayer; strong cries sent up to God with great eagerness and importunity, fervency, and devotion; and such was Christ's prayer, and in which he continued all night: unless by the prayer of God should be meant, as is thought by many, an house of prayer to God, in which Christ lodged all night, and spent it in prayer to God in it. Certain it is, the Jews had their "proseuchre", or prayer houses. Philo the Jew n often speaks of them, and so does Josephus o; and there seems to be mention made of them in the Talmudic writings: when R. Jochanan ben Zaccai came to Vespasian, in his camp before Jerusalem, Vespasian asked him, what he should give him? he replied p,
"I desire nothing of thee but this "Jabneh", (a famous university,) that I may teach in it the disciples, and fix in it תפלה, "an oratory", or "prayer house", and do in it, all the commandments said in the law.''
And in another place q,
"R. Judah says, that Samuel said it is free for a man to make water within four cubits, של תפילה, which I should choose to render, "of the proseucha", or "prayer house":''
though the Gemarists afterwards, and so the gloss seem to explain it of the time after prayer, in which a man should wait before he evacuates, even as long as he might go the length of four cubits. Juvenal r has reference to one of these oratories, when he says, "in qua te qucero proseucha?" and in one of these, it is very likely, Christ was in prayer all night long; for by the sea side, and by the side of rivers, these oratories were used to be; Acts 16:13.
m T. Bab. Beracot, fol. 7. 1. Bereshit Rabba, sect. 56, fol. 50. 2. n De Vita Mosis, l. 3. p. 685. in Flaccum, p. 971, 972, 982. leg. ad Caium. p. 1011, 1012, 1013, 1014, 1016, 1040, 1043. o In Vita. p Abot R. Nathan, c. 4. fol. 2. 4. q T. Bab. Megilia, fol. 27. 2. r Satyr. 3. l. 295.
And when it was day,.... Or morning; having spent the whole night in prayer to God, no doubt for his disciples, whom he was about to send forth as his apostles, to preach his Gospel, and work miracles, and for their success therein:
he called unto him his disciples; the whole company of them, as in Luke 6:17 all that were his followers, and professed to believe in him, or as many as he pleased; see Mark 3:13.
And of them he chose twelve; and ordained them, and sent them out to preach, heal sicknesses, and cast out devils:
whom he also named apostles; or "messengers", from their being sent by him on such important business; and their names are as follow.
Simon, whom he also named Peter,.... Which signifies a rock, or stone, as Cephas also does, see John 1:42 from his constancy, steadfastness, and solidity:
and Andrew his brother; who was called at the same time with him, and were brethren, both in nature and grace:
James and John: the two sons of Zebedee, who were called next:
Philip and Bartholomew; the latter of these is by some thought to be Nathanael.
Matthew and Thomas,.... The first of these was a publican, and who also was called Levi; and the latter had besides the name of Didymus, and was he that was so unbelieving of Christ's resurrection:
James the son of Alphaeus; sometimes called James the less, and the brother of our Lord: and
Simon called Zelotes; or the Canaanite;
And Judas the brother of James,.... Of that James, that was the son of Alphaeus; though the Syriac and Arabic versions call him "the son of James", very wrongly: this Judas was also called Thaddaeus and Lebbaeus, and is the writer of the epistle that bears his name:
and Judas Iscariot, which also was the traitor; both his surname and his character are mentioned, to distinguish him from the other Judas: it is easy to observe, that these twelve are mentioned by pairs, or couples, and so they were sent out, two by two; see Mark 6:7 as were also the seventy disciples afterwards; see Luke 10:1 There seems to be an allusion to the pairs and couples of the Jewish fathers and doctors, who in their succession are thus paired: Jose ben Joezer, and Joseph ben Jochauan; Joshua ben Perachia, and Nathan the Arbelite; Simeon ben Shetach, and Judah ben Tabai; Shemain and Abtalion; the two sons of Bethira, whose names were Judah and Joshua; Hillell and Shammai s: all before Christ's time.
s Pirke Abot, c. 1.
And he came down with them,.... With the twelve apostles, from the top of the mountain, where he had been praying all night, and where he had been that morning, ordaining, and giving instructions to the twelve he had chosen:
and stood in the plain; in a lower part of the mountain, in a plain place on it; which was large, and capable of holding a great number of people; for it was still upon the mount, that Christ taught his disciples, and said many of the things hereafter mentioned in this chapter; see Matthew 5:1.
And the company of his disciples: not only the twelve, but the large number out of which he had chosen twelve;
and a great multitude of people; who were hearers of him, and attendants on him, and who had a great esteem for him, though they were not as yet of the number of his disciples; who came
out of all Judea, and Jerusalem, and from the sea coast of Tyre and Sidon: drawn from these several parts by the fame of him, some for one thing, and some another; some of
which came to hear him: to hear him preach, and that they might know what manner of doctrine he taught: and others of them,
to be healed of their diseases; their bodily diseases, and some came perhaps for both.
And they that were vexed with unclean spirits,.... Were possessed with devils, and sadly tormented and afflicted by them:
and they were healed: both such that had bodily diseases, and were under diabolical possessions.
And the whole multitude sought to touch him,.... That is, the multitude of those that were sick and possessed; for they were persuaded, and they found it true by experience, that if they could but touch any part of his body, or his garments, they should be cured of their diseases:
for there went virtue out of him; in great abundance, as water from a fountain; without his speaking a word, or using any gesture, such as laying his hands on them:
and they were healed; in this secret and private way, of whatsoever disease they were afflicted with.
And he lifted up his eyes on his disciples,.... Either the whole company of them, or rather the twelve apostles, whom he saw coming to him, and fixing his eyes on them, he sat,
and said; what follows, with many other things recorded by Matthew:
blessed be ye poor; not only in the things of this world, having left all for Christ, but poor in Spirit, as in Matthew 5:3,
Matthew 5:3- ::
for yours is the kingdom of God; or heaven, so in Matthew 5:3.
Blessed are ye that hunger now,.... Not only suffer hunger and thirst in a literal sense, in this present life, but who have hunger and thirst in a spiritual sense, after righteousness and eternal life, as in Matthew 5:6 where it is also said as here:
for ye shall be filled: with righteousness and life;
Matthew 5:6- :.
blessed are ye that weep now; under afflictions and pressures of life, and mourn for sin, their own, and others:
for ye shall laugh; be filled with spiritual joy and pleasure, and be comforted with the consolations of the Spirit;
Blessed are ye when men shall hate you,.... For the sake of Christ, and his Gospel:
and when they shall separate you from their company; either from civil conversation with them, as if they were Gentiles and uncircumcised persons; or from their religious assemblies, and so may have respect to that sort of excommunication in use, among the Jews, called נדוי or "separation": by which persons were not only excluded from the congregation, but from all civil society and commerce: such a person might not sit nearer to another than four cubits, and this continued for thirty days; and if not discharged then, he continued thirty more t:
and shall reproach you: as heretics, apostates, and enemies to the law of Moses, as the Jews did reproach the Christians;
and cast out your name as evil; or "as of evil men": as the Syriac and Arabic versions render it: this may have respect to the greater sorts of excommunication, used among them, called "Shammatha" and "Cherem", by which a person was accursed, and devoted to destruction; so that our Lord's meaning is, that the should be esteemed and treated as the worst of men, and stigmatized in the vilest manner they were capable of:
for the son of man's sake; not for any immorality committed by them, but only for professing and, preaching that the Messiah was come in the flesh, and that Jesus of Nazareth was he; and that he who was the son of man, according to his human nature, was, the Son of God according to his divine nature.
t Vid. Maimon. Talmud Tora, c. 7. sect. 4, 5, 6.
Rejoice ye in that day,.... When they should be hated, discarded, reproached, and anathematized: and leap for joy; as if the greatest honour and happiness imaginable had been conferred on them; and as persons do, when in the greatest rapture:
for behold, your reward is great in heaven, for in the like manner did their fathers unto the prophets;
But woe unto you that are rich,.... Not in worldly riches and substance, for some of these have been, and are happy persons in a spiritual sense; and at most, it can only mean such, who trust in their riches, and place their, happiness in them; but it chiefly regards such, as are rich in their own opinion, and stand in need of nothing; who place their confidence in their own righteousness, and do not apply to Christ, in whom alone are durable riches and righteousness:
for ye have received your consolation; which they take from their own works, and a very unstable and short lived one it is; for while they are crying Peace, Peace, to themselves, from their own services, sudden destruction comes upon them, and all their comforts vanish away: for there is no true solid comfort but in Christ, and in his righteousness; that administers consolation now, and lays a foundation for everlasting comfort hereafter.
Woe unto you that are full,.... Not so much with the plenty and affluence of the things of this life, as of themselves, and their own righteousness, and so with conceit, vanity, and pride, and have no appetite for spiritual things, nor do they hunger and thirst after Christ, and the grace that is in him:
for ye shall hunger; not that they shall truly and spiritually desire an interest in Christ, and his righteousness, or heaven and eternal life hereafter; but they shall be in starving and famishing circumstances; and whilst the saints are feeding upon the joys and glories of the other world, compared to a banquet, they shall be without, and have no share in these things; Isaiah 65:13.
Woe unto you that laugh now; at sin, rejoice in iniquity, make a mock at it, instead of mourning for it; or that glory in themselves, and in their righteousness, and rejoice in their boastings:
for ye shall mourn and weep; shall be cast into outer darkness, where are weeping, waiting, and gnashing of teeth; and for all the fire they have kindled, and sparks they have encompassed themselves with, and danced in and about, this they shall have at the hand of God, they shall lie down in sorrow, and ever continue in it.
Woe unto you when all men shall speak well of you!.... The word "all", is left out in the Vulgate Latin, Syriac, Arabic; Persic: and Ethiopic versions, and is wanting in many copies, though it is in the Alexandrian copy; and the meaning is, it looks ill in persons, when the men of the world, wicked men, all of them, or the greater part of them, applaud and commend them; for this can never be, if they are truly religious persons, and are faithful to their principles, and upright in their practices; and do not connive at, or comply with the errors and evil ways of wicked men; for it is no bad sign, to have the good word of good men, and therefore these must be excepted, and the passage must be limited to bad men;
for so did their fathers to the false prophets; they spoke well of them, and heaped favours, riches, and honours upon them, that they might prophesy unto them things; 1 Kings 22:6, smooth things and deceit.
But I say unto you which hear,.... The Ethiopic version adds "me", and the generality of interpreters understand the passage of the hearers of Christ, as distinct from the disciples, or together with them, and of the better sort of them; and of such as had ears to hear, and who heard with a desire of understanding, and of putting into practice what they heard; but I rather think it regards the hearers of the Scribes and Pharisees, then present, who had heard and received the traditions of the elders, to which the following rules of Christ are opposed; and to each of which, with others in Matthew, these words are prefixed;
ye have heard that it was said by them of old time--but I say unto you,.... Matthew 5:21 with which compare this phrase, and the sense will appear to be this; to you that hear day by day, the traditions of the elders urged upon you, and the false glosses the Scribes and Pharisees put upon the word of God; in opposition to them, I say to you what follows:
love your enemies; whereas you have heard them say, hate your enemies, keep enmity in your hearts to them, and revenge yourselves on them:
do good to them that hate you; whereas you have heard it said, that you should only do good to your friends, and should keep anger in your bosoms to such who hate you, and do you an injury;
Matthew 5:21- : Matthew 5:21- :
Bless them that curse you;.... In common discourse, or anathematize you in their synagogues:
and pray for them which despitefully use you: so Christ himself did;
And unto him that smiteth thee on the one cheek,.... The right cheek,
offer also the other; the left cheek, by turning it to him, that he may smite that likewise, if he thinks fit: by which proverbial expression, Christ teaches patience in bearing injuries and affronts, and not to seek private revenge; but rather, suffer more, than indulge such a temper; and for the same purpose is what follows urged:
and him that taketh away thy cloak, forbid not to take thy coat also: the phrase is inverted in Matthew;
Mt 5:40.
And give to every man that asketh,....
And of him that taketh away thy goods; not by force, but by consent, having either lent them, or sold them to him: for if they were taken away by force, the person so taking them was to be deemed a thief and a robber, and to be treated as such; but one that takes them by agreement, and is not able to make a return of them, or to give a valuable consideration for them, of such an one ask them not again: do not exact or demand them, but give him a release, as the law requires, in Deuteronomy 15:2 which seems to be respected here; and where the same word is used by the Septuagint, as here.
And as ye would that men should do to you,.... In matters of justice and beneficence were they in your case, and you in theirs;
do ye also to them likewise: a golden rule this, agreeably to the light of nature, and divine revelation, and is the sum and substance of the law and prophets; :-.
For if ye love them which love you, what thank have ye?.... Or, "what grace have ye?" this is no fruit, nor evidence of grace, nor any exercise of the true grace of love; nor is it any favour conferred upon the object loved, which deserves the respect shown, nor can any reward be expected for such treatment: and thus it is expressed in Matthew, "what reward have ye?" and the Arabic version renders it so here:
for sinners also love those who love them: men that are destitute of the grace of God, profligate sinners, even the worst of them, such as publicans, do this; :-.
And if ye do good to them which do good to you,.... As one good turn deserves another:
what thank have ye? what grace or goodness is there in such an action? what glory or merit is there in it?
for sinners also do even the same: wherefore no man should conclude himself a righteous man, or better than sinners, on such an account: this is to be found among the worst of men, and is natural to them, unless they are brutes indeed, to be kind to such as are kind to them. And yet, this was the whole of the doctrine of the Jews about doing good to men: for so they say u,
"an Israelite is obliged to do good to an Israelite his companion, and to lend without usury: this is kindness and goodness, and a greater good it is than a gift; for many men are ashamed to take a gift, and are not ashamed to take a loan: but not so an Israelite to a Gentile; for he is not bound to do good, or show kindness to him, or to lend him his money freely; for many of them hate the Israelites; but it must be owned, that if a Gentile does a kindness, or good, to an Israelite; the Israelite is also bound to show kindness to him, and do him good.''
In direct opposition to such narrow sentiments does our Lord deliver himself in this, and the following verses.
u Kimchi in Psal. xv. apud Huls. Theolog. Jud. par. 1. p. 420.
And if ye lend to them of whom ye hope to receive,.... The same again, as from their brethren the Jews; or usury, as from the Gentiles:
what thank have ye? and yet they looked upon this, in the first instance of it, as a very great kindness, and act of goodness, as appears from the above citation:
for sinners also lend to sinners, to receive as much again; or "what is equal", and answerable to what they have lent them; that is, the same, or what is equivalent to it.
But love ye your enemies,.... As before urged in Luke 6:27
and do good and lend; not to your friends only, but to your enemies;
hoping for nothing again; either principal or interest, despairing of seeing either; lending to such persons, from whom, in all appearance, it is never to be expected again. The Persic version renders it, "that ye may not cause any to despair": and the Syriac version, "that ye may not cut off", or "cause to cease the hope of men"; and the Arabic version, "that ye do not deceive the hope of any" that is, by sending such away, without lending to them, who come big with expectations of succeeding:
and your reward shall be great: God will bless you in your worldly substance here, and will not forget your beneficence hereafter:
and ye shall be the children of the Highest: that is of God; one of whose names is עליון "the Most High"; Psalms 82:6 the meaning is, that such who from principles of grace, and with right views do such acts of kindness and beneficence to their fellow creatures and Christians, shall be, made manifest, and declared to be the children of God; since they will appear to be born of him, and made partakers of the divine nature, and bear a resemblance to him, by their imitating him:
for he is kind to the unthankful and to the evil; by causing his sun to rise, and his rain to fall on them, as on the righteous and the good; for as Jews w observe,
"there is no difference with him, whether on the right hand or the left; for he is gracious, and does good, even to the ungodly.''
And elsewhere they say x, that
"he does good, and feeds the righteous and the ungodly.''
w R. Abraham ben Dior in Sepher Jetzira, p. 19. x Zohar in Exod. fol. 69. 2, 3.
Be ye therefore merciful,.... Tenderhearted, kind, beneficent to all men, friends and foes:
as your Father also is merciful; that is your Father which is in heaven; who is good to all, and his tender mercies are over all his works: nothing is more common in Zohar y, and the Talmud z than to express the Divine Being by no other name, than "the Merciful"; אמר
רחמנא, "the Merciful said" so, and so; that is, God: and so the Arabians generally begin their books and chapters with these words, "in the name of God, exceeding merciful", or "the merciful commiserator": a saying much like to this in the text, is the Targum of Jonathan, on
Leviticus 22:28.
"O my people, the children of "Israel, as your father",
רחמן, "is merciful" in heaven, so be ye merciful on earth.''
y Zohar in Lev. fol. 2. 2. & 9. 4. & 20. 1. & 22. 1. z T. Bab. Moed Katon, fol. 15. 2.
Judge not, and ye shall not be judged,....
Condemn not, and ye shall not be condemned; censure not men's persons, and judge not their state, or adjudge them to condemnation, for every offence in practice, or because they differ in principle, lest you should be treated in like manner by others; and especially, lest you should fall under the righteous censure, judgment, and condemnation of God:
forgive; offences and trespasses committed against you, bear with, and pass by injuries and affronts:
and ye shall be forgiven; of God;
Give, and it shall be given unto you,.... Give liberally of your worldly substance to indigent persons, as you have an opportunity, according to your ability, and as cases require: and it shall be returned again to great advantage; with great recompense, either in temporals or spirituals, or both:
good measure, pressed down, and shaken together, and running over, shall men give into your bosom. The allusion is to dry measure among the Jews, for to liquids, the terms used will not agree; and which, though right and full, which is here called good measure, they thrust and pressed to make it hold more; and shook it also for the same purpose, and then heaped it up as much as they could, till it fell over: of all these methods used in measuring, we have instances in their writings; which may serve to illustrate this passage: it is said of a one, that
"he measured, במדה כתושה, "with measure pressed down"; and therefore they measured to him, with measure pressed down.''
Some of their measures they heaped, and some they did not: they say b;
"all the measures which were in the sanctuary, נגדושות "were heaped", except the high priest's, and his heap was contained in it.''
And elsewhere they observe c that
"there were two decimaries (or tithing vessels) in the sanctuary, one was גדוש, "heaped", and the other was מחוק, "stricken": with that which was heaped they measured all the fine flour for the meat offerings, and with the stricken, that which was for the cakes of the high priest.''
With respect to this distinction of measures, they say it is a tradition of the Rabbins d, that they do not "strike" in the place where
"they "heap", nor heap in the place where, they strike.''
Between these two measures there was another, which was full measure and just, and right, without heaping or striking e, R. Papa inquired, whether the handful
"(of sweet incense the high priest took on the day of atonement) which is spoken of Leviticus 16:12 was of "stricken" or "heaped" measure; R. Abba said to R. Ase, come, hear, the handful spoken of, is neither of stricken nor heaped measure, אלא טפופות, "but of equal measure";''
sufficiently full, and no more. Dr. Lightfoot reads it, מצופות, "flowing over"; by what authority I cannot say; though the gloss says, the word signifies,
"flowing over, by reason of its height,''
But flowing or running over measure, was the same with that which was heaped, as appears from the following instance f:
"all those that המשפיעין במדה גסה "cause to abound", or run over with the great "measure", it is lawful for them to sell that, of which it is doubted whether it has been tithed or not; and these are they, that "cause to run over", or "heap" with the great measure, as corn factors and fruiterers.''
Who buy corn and fruits to sell again, and which they buy by the large measure, and fill it up, add unto it, and heap it up; and so get more than what is properly due unto them, as the commentators observe g: would you know the quantity of the heap, or that which ran over, or the difference between even measure, and that which was heaped, learn, it from hence: in 1 Kings 7:26 it is said, the molten sea held two thousand baths, and in 2 Chronicles 4:5 three thousand baths; which difficulty the Jewish writers solve this way, by observing, that the former text is to be understood of liquid measure, and the latter of dry measure, which was heaped: hence says R. Abai, we learn that, גודשא תלתא הוי, "the heap is the third part" of the measure h: now to this superabundant measure, Christ here refers; and signifies, that a large compensation should be made to such, who give liberally and generously to needy persons; that as they abounded in their acts of beneficence, so an overflowing plenty of good things should be returned to them: and when he says, that this should be "given into their bosom", he alludes to the long and large garments the Jews wore, into which they were capable of receiving large lapfuls of good things: the words may be read impersonally, "shall be given into your bosom"; or if personally, they may be understood of God, angels, and men, in different senses: the phrase "shaken together", is not in the Syriac and Persic versions: "for with the same measure that ye mete withal, it shall be measured to you again"; a common proverb with the Jews:
2 Chronicles 4:5- :.
a T. Bab. Yebamot, fol. 107. 2. T. Hieros. Yebamot, fol. 13. 3. b Misn. Menachot, c. 9. sect. 5. c T. Bab. Menachot, fol. 37. 1, 2. d T. Bab. Bava Bathra, fol. 89. 1. e T. Bab. Yoma, fol. 48. 1. f Misn. Demai, c. 2. sect. 4. g Maimon. & Bartenora in ib. h T. Bab. Erubin, fol. 14. 2. Vid Targum, Jarchi, Kimchi, & R. Levi ben Getshorn, in 1 Kings vii. 26. Bemidbar Rabba, sect. 11. fol. 204. 3.
And he spake a parable unto them,.... The Vulgate Latin reads, "he spake also a parable unto them"; besides what he said; and the Arabic version renders it, "another similitude", parable, or proverb, distinct from the comparisons, allusions, and proverbial expressions in the preceding verses. Though it should be observed, that these words were not spoken at the same time, nor on the mount, as the foregoing were; but this, and what follow, are a collection of various expressions of Christ at different times, some delivered on the mount, and others elsewhere; unless it should be rather thought, that these proverbs and sentences were repeated at different places and times, which is not improbable:
can the blind lead the blind? they may do so, as the blind Scribes and Pharisees led the blind people of the Jews, which is what our Lord intends; but if they do, as they did,
shall they not both fall into the ditch? yes, verily, what else can be expected? :-.
The disciple is not above his master,.... Or "more excellent", as the Syriac, Arabic, and Persic versions render it; that is, in learning and knowledge; if the master is ignorant, the scholar will be so too; and thus it is with teachers, and their people under their care; if the leaders are blind and ignorant, those under their instructions will remain so likewise. These words are an illustration of the preceding parable, and are used to another purpose here than in Matthew 10:24.
Matthew 10:24- :
but every one that is perfect shall be as his master. The Vulgate Latin reads it, "every one shall be perfect if he is as his master"; that is, if his master is a man of general learning, and a complete scholar, if he is like him, he will be so too: the Persic version renders it, "every disciple that desires perfection shall be as his master": whoever is ambitious of being a thorough scholar, and is diligent and industrious, by all ways and means, to obtain such a character, shall be even as good an one as his master, under whom he learns, and better he cannot well expect to be; and this is sufficient; and so the Ethiopic version renders it, "is it not enough that every one be as his master?" agreeably to Matthew 10:25
Maimonides i has an expression much like this:
"he that learns, shall not be greater than he of whom he learns, but shall be, כמותו, "as he".''
Christ, in this last clause, seems to design his own disciples, who, when perfect in knowledge, which is not to be expected in this state, unless in a comparative sense, will be like himself.
i Misn. Bava Kama, c. 2. sect. 5.
And why beholdest thou the mote that is in thy brother's eye,.... A lesser sin in comparison of others; for all sins are not alike, as the Stoics asserted: and though none are to be countenanced and indulged, yet some are not so severely to be animadverted upon as others, the nature, occasions, circumstances, and aggravations considered; for no man is perfect, or wholly free from sin; nor are the words preceding to be understood of such a perfection; for which reason perhaps these words, with what follow, are mentioned:
but perceivest not the beam that is in thine own eye? meaning a greater sin, such are guilty of, who are inquisitive searchers into the faults of others, and severe animadverters on them; and yet are blind to their own iniquities, and take no notice of them. These proverbial expressions were delivered by Christ on the mount, and are the same with those in Matthew 7:3.
Either how canst thou say to thy brother,.... Guilty of the lesser sin;
brother, let me pull out the mote that is in thine eye; that is, suffer me to reprove thee for thy sin: the word "brother" is omitted in the Cambridge copy of Beza's, and in the Persic version; nor is it in Matthew; but in the Syriac and Ethiopic versions it is read, "my brother"; pretending great affection and sincerity:
when thou thyself beholdest not the beam that is in thine own eye? that is, takest no notice of, and dost not refrain from a greater iniquity continued in:
thou hypocrite; as such an one must be, that bears hard upon his brother, and severely censures him for a small crime, when he indulges in himself a far more abominable sin:
cast out first the beam out of thine own eye, and then shalt thou see clearly to pull out the mote that is in thy brother's eye: the sense is, that a man should first reform himself, and then others.
For a good tree bringeth not forth corrupt fruit,.... The particle, "for" is left out in the Syriac, Arabic, Persic, and Ethiopic versions; and so it is in Beza's ancient copy: nor do these words stand in close connection with the preceding in Matthew's Gospel, though they may be very well considered as an illustration of them; for as that cannot be called a good tree, which brings forth bad fruit; so such men cannot be accounted good men, let them make ever so large pretensions to such a character, who are very busy in espying, discovering, and censuring the faults of their brethren; when they take no notice of, nor refrain from, nor relinquish their own. These words, with what follow in this, and the next verse, and the similes in them, are used by our Lord in Matthew, on account of false prophets or teachers; where he suggests, that as good and faithful ministers of the Gospel cannot, and do, not bring forth, and publish corrupt notions, and false doctrines, usually and knowingly; even usual, nor can it be, that a good tree should bring forth corrupt fruit; so,
neither doth a corrupt tree bring forth good fruit; or men of corrupt minds deliver good and sound doctrine, or the wholesome words of our Lord Jesus Christ: but here they seem to be applicable to other persons, even true believers and hypocrites: the former are comparable to good trees, and are called trees of righteousness, which being planted by the river of the love of God, and rooted in Christ, and filled with the fruits of righteousness by him, do not bring forth the evil fruit of sin, as the common and constant course of their lives and conversations; for that they never commit sin, or are entirely without it, cannot be said; but sin is not their usual and common practice, or they do not live in sin: and the latter, hypocrites, who pretend to a great deal of religion, and have none that is true and real, these are comparable to corrupt trees; which, though they may make a fair show, yet do not bring forth good fruit, or perform works of righteousness which are truly such; what they do have only the appearance of good works, and are not properly so;
For every tree is known by its own fruit,.... Good and bad preachers are known by their doctrines, the one being agreeable, the other disagreeable to the word of God; and good and bad men are known by their lives and conversations: the grace of God revealed to good men, and wrought in them, teaches them to live soberly, righteously, and godly; a holy life is the fruit of grace, and an evidence of it; and the wickedness that is in the heart of unregenerate men, and even the hypocrisy of formal professors, will show themselves in the common and ordinary course of their conversations:
for of thorns men do not gather figs, nor of a bramble bush gather they grapes; nor can they be expected from them: and no more can an unregenerate man perform good works, or bring forth: fruits of righteousness acceptable unto God; for these require a knowledge of his will, obedience to it, a principle of grace, love to God, faith in Christ, and a view to the glory of God; all which are wanting in such a person.
A good man out of the good treasure of his heart,.... This, because of its suitableness and agreement with what goes before, is placed by Luke here; though, according to Matthew, it was spoken at another time and place, unless it should be a repetition there;
for of the abundance of the heart his mouth speaketh. The Vulgate Latin, Arabic, Ethiopic, Syriac, and Persic versions, leave out the word "his"; and the two latter read "lips", instead of "mouth";
And why call ye me Lord, Lord,.... Or, "my Lord, my Lord", as the Syriac version renders it; acknowledging, in words, his government over them; claiming an interest in him, and making use of his name and authority:
and do not the things which I say; or "command"; and therefore such words in their mouths would be of no use to them, since they neither did his Father's will, which he taught them, nor observed his commands and ordinances which he enjoined them; and therefore should not enter into the kingdom of heaven, nor be owned by him another day, but should be bid to depart from him; :-.
Whosoever cometh to me,.... To be a disciple and follower:
and heareth my sayings, and doth them;
I will show you to whom he is like; or "to what thing he is like"; so the Syriac and Arabic versions; though what follows seems better to agree with person than thing.
He is like a man which built an house,.... That is, intended to build one, having drawn the scheme of it in his mind, and provided materials, and fixed upon the spot of ground:
and digged deep, and laid the foundation on a rock; that is, he dug deep in the earth, till he came at a rock, and there, and then, he laid the foundation of his house; in which he acted the part of a wise man, as he is called in Matthew: so a sensible sinner, desirous of building his soul, and the salvation of it, on a sure bottom, digs deep into the Scriptures, diligently searches them, till he finds out the scheme of salvation by Christ; which lies deep in God's counsel and covenant, was ordained before the world began, and was hid in God till revealed in the Gospel: and finding Christ to be the rock of ages, in whom is everlasting strength, and the foundation which God has laid, nor is there another; he makes use of him as such, and builds the hope of his eternal salvation on him:
and when the flood arose; an inundation, a multitude of waters, the swelling of the sea; or rather "when it was tide", as the word here used signifies k:
the stream beat vehemently upon the house; or the river, up which the tide came, dashed and broke against it; by which may be signified the temptations of Satan, the persecutions of the world, the corruptions of men's hearts, and the errors and heresies of false teachers:
and could not shake it; as none of these can so shake as to move a soul, thus built on Christ, off of him the foundation:
for it was founded upon a rock;
k Vid. Rivinum de Venilia Salacia, &c. p. 681, 632.
But he that heareth, and doth not,.... Hears Christ's sayings externally, but does not obey his commands:
is like a man that without a foundation built upon the earth: that is, without digging for a foundation, built his house upon the surface of the earth; "upon the dust of it", as the Syriac version renders it; or, "upon the sand", as Matthew says: "against which the stream did beat vehemently, and immediately it fell, and the ruin of that house was great";
The New John Gill's Exposition of the Entire Bible Modernised and adapted for the computer by Larry Pierce of Online Bible. All Rights Reserved, Larry Pierce, Winterbourne, Ontario.
A printed copy of this work can be ordered from: The Baptist Standard Bearer, 1 Iron Oaks Dr, Paris, AR, 72855
Gill, John. "Commentary on Luke 6". "Gill's Exposition of the Entire Bible". https://www.studylight.org/commentaries/eng/geb/luke-6.html. 1999. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,700 |
#import "NSObject.h"
@class ACAccount, NSDictionary, NSURL, SLTwitterRequest;
@interface TWRequest : NSObject
{
SLTwitterRequest *_request;
}
- (void)performRequestWithHandler:(id)arg1;
- (id)signedURLRequest;
- (void)addMultiPartData:(id)arg1 withName:(id)arg2 type:(id)arg3;
@property(retain, nonatomic) ACAccount *account;
@property(readonly, nonatomic) NSDictionary *parameters;
@property(readonly, nonatomic) NSURL *URL;
@property(readonly, nonatomic) long long requestMethod;
- (void)dealloc;
- (id)initWithURL:(id)arg1 parameters:(id)arg2 requestMethod:(long long)arg3;
@end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,119 |
A Publication Scheme is a guide to the classes of information that the council publishes, or intends to publish routinely. Section 19 of the Freedom of Information Act 2000 requires every public authority to adopt and maintain a Publication Scheme which has been approved by the Information Commissioner, and to publish information in accordance with the scheme.
The idea of the Scheme is to let everyone know what information will be automatically or routinely published and/or made available by the Council.
The term 'published' is broad and is not limited to information produced in paper form. As far as the Freedom of Information Act is concerned, information made publicly available is published.
Information provided on our website is therefore part of the Publication Scheme, in addition to printed documents.
How we make decisions - Policy proposals and decisions, decision making processes, internal criteria and procedures, consultations.
Our policy and procedures - Current written protocols for delivering our functions and responsibilities.
The services we offer - Advice and guidance, booklets and leaflets, transactions and media releases, a description of the services offered.
These types of information are available on our website, under the Transparency web pages or on the relevant service area pages; or by contacting the relevant council department. If the information is not easily identifiable, please contact the council by email to information@iow.gov.uk for assistance.
The purpose of the scheme is to make the maximum amount of information available at the minimum cost and inconvenience. Charges made for routinely published information will be justified, transparent and kept to a minimum.
Material published and accessed on the website will be provided free of charge.
where they are legally authorised, justified and in accordance with a published schedule of fees which is readily available to the public.
If a charge is to be made, confirmation of the payment due will be given in advance. Payment may also be required in advance.
pro-actively publish or otherwise make available as a matter of routine, information (including environmental information) which is held by the council and falls within the classifications.
specify the information held by the Council and which falls within the classifications. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,167 |
categories > Comic Talk and General Discussion * > We need YOUR stuff for Quackcast 19 and 20!
Quackcast 19 is being recorded on Thursday this week (the 17th).
The theme is techniques and materials , there'll also be a bit about Digital art VS Traditional art.
All we need are short scripts or a short recording about two minutes in length on the subject of what materials you prefer to use for your comics and WHY?
-Whether that's pen and inks on bristleboard, pens on lined paper, sprites in MS paint, or Photoshop with a tablet or whatever, it doesn't matter.
Please submit it Via PQ to either Skoolmunkee or myself before Thursday the 17th!
Quackcast 20 is being recorded on Thursday next week (the 24th).
The theme is What You've Learned - by making comics!
All we need are short scripts or a short recording about two minutes in length on the subject of, well, what you've learned by making comics.
- Something about your improvement in art or writing? Technical or software skills? Lessons learned about yourself, your process, etc? Something about 'the business' or the world at large? Etc.
Please submit it Via PQ to either Skoolmunkee or myself before Thursday the 24th!
Then where did quackcast #18 go?
IT'S THE TWILIGHT ZONE I TELL YOU!
1337, there's a 5-day delay between recording and posting. We recorded 18 on Thursday, it will get posted Tuesday. It's in the limbo-zone.
Audio clips should be in mp3 format (192 kbps if possible), and can be uploaded to yousentit or something and then give the download link to us.
Oz, I had to edit your post, sorry. :] You got the week wrong!
I changed the description to 2 minutes.
I would love to hear you rap it. DO IT!
This sounds like it could be neat. Sadly, I have plans with family coming up. :/ I wouldn't have the time. Maybe on the next one, though.
So I just gotta record me discussing something for two minutes? And I PQ the admin the recording off of Audacity on mp3 or something? Seems fair enough.
Yeah, it could even be just a couple sentences of advice or something. It doesn't have to be a big ol Thing.
Where is the new one any way?
And yeah i will be sending you some stuff soon ish.
It was late getting up but it's there now.
We've got something from elektro already! That's right, elektro is better than all of you.
That's right, elektro is better than all of you.
He Does have an awesome voice.
Could you post a reminder of where you want it uploaded.
I'll probably write one small script for one of those podcast, depending on if I have the time to do it for the 17th or more likely the 24th.
i'll give it a go. i'd love to give ya a soundbite if that' what yer lookin for with the MP3's or whatever. but i cant do that without sending a vid file i dont think, so ill work on a script. whatever's better for you guys.
The easiest place is probably yousendit. :] Then PQ one of us with the link.
You could try downloading Audacity. It's completely free.
Please submit it Via PQ to either skoolmunkee or ozoneocean before Thursday the 24th!
Quackcast 21 is being recorded on Thursday the 31st.
The theme is Speeding Up The Process!
All we need are short scripts or a short recording no more than two minutes in length on the subject of how to make comics faster!
Planning, writing, drawing, processing, webpage, etc is all game. Most especially we'd like tips or advice from people who've had some stuff work for them!
Please submit it Via PQ to either Skoolmunkee or ozoneocean before Thursday the 30th!
For quackcast 19, we got 1 script, 3 recordings about the topic, and 1 recording promoting a good resource thread! We'd like to have around that many every time because it's great to include other voices and knowledge!
We've got two recordings for QC 20 so far, it would be great to have a bit more to play (or read)!
I should have something ready for #21 for Speeding Up the Process. :) I haven't used my mic since… like… xD ….. *blows it off* …Ah, what the heck. :D I'll send it soon. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,652 |
Idrottsmärke är ett utmärkelsetecken som erhölls vid avläggandet av vissa kontrollerade idrottsprov.
I Sverige märks:
Riksidrottsförbundets idrottsmärke, instiftat 1907 för män och 1916 för kvinnor, senare flera gånger reviderat. Finns i brons, silver och guld, för prov i ett flertal grenar under kontroll av två av distriktsstyrelsen godkända kontrollanter.
Skolidrottsmärket, instiftat 1919 för pojkar och 1923 för flickor, finns i järn, brons, silver och guld. Skånes gympnastikförbund tog 1930 fram ett eget skolidrottsmärke, där simprov inte ingick.
Skidlöparmärket, instiftat 1916 för män, 1921 för kvinnor och 1923 för ungdom
Skridskomärket instiftat 1928
Cykelmärket instiftat 1926
Simmärken, äldst kandidatmärket från 1907 (ändrat flera gånger), varpå magistermärket i järn, brons silver och guld följde 1932.
Paddlarmärket instiftat 1918
Fäktarmärket instiftat 1932
Militära idrottsmärket, instiftat 1914 och sedan 1941 kallat Fältidrottsmärket
Landstormens idrottsmärke, instiftat 1918
Källor
Sport | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,543 |
Q: Terraform Azurerm error: linuxConfiguration.ssh.publicKeys.path is invalid I am using Terraform v0.12.24 with provider.azurerm v2.2.0
I get this error below when i try to create scaleset VMs:
Error: compute.VirtualMachineScaleSetsClient#CreateOrUpdate: Failure
sending request: StatusCode=400 -- Original Error:
Code="InvalidParameter" Message="The value of parameter
linuxConfiguration.ssh.publicKeys.path is invalid."
Target="linuxConfiguration.ssh.publicKeys.path"
on scaleset.tf line 1, in resource
"azurerm_virtual_machine_scale_set" "demo": 1: resource
"azurerm_virtual_machine_scale_set" "demo" {
I am using Windows 10 for the terraform configuration. My os_profile_linux_config is as below:
storage_profile_image_reference {
publisher = "Canonical"
offer = "UbuntuServer"
sku = "18.04-LTS"
version = "latest"
}
os_profile_linux_config {
disable_password_authentication = true
ssh_keys {
key_data = file("C:/Users/jack/Documents/key/id_rsa.pub")
path = "C:/Users/jack/Documents/key"
}
}
First, i have tried two different key pairs. one is created by puttygen and the other by ssh-keygen with git bash. I had the same error with both of them.
Do you have any idea?
A: For your issue, I think you misunderstand the property path of the ssh_key, it shows here:
ssh_keys - (Optional) Specifies a collection of path and key_data to
be placed on the virtual machine.
It's the path inside the VM that you want to create, not the path of the machine that you execute the Terraform. And also:
Note: Please note that the only allowed path is
/home//.ssh/authorized_keys due to a limitation of Azure.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,701 |
CITN
HomeAccountingTop 6 Richest Accountants In The World
Top 6 Richest Accountants In The World
Onyema Donald
Naturally, there are some benefits attached to managing other people's money – one of them the knowledge of how to build up a personal fortune of your own. If a celebrity salary is on your agenda, then the list of six accountants who have hit the big time will inspire you.
1. Phil Knight, co-founder of Nike. Net worth: $25.1bn
You probably know Phil Knight better as the co-founder of Nike, but he is a certified public Accountant. After graduating from business school, Knight visited Japan where he door-stepped the boss of the Onitsuko Tiger running shoe brand and secured US distribution rights. Spotting an opportunity, he teamed up with his former track running coach and launched the business that became Nike in 1964. Nike is now worth about $86.2bn.
Kumar Mangalam Birla
2. Kumar Mangalam Birla, chairman of Aditya Birla Group. Net worth: $8.3bn
Kumar Mangalam Birla is a member of a prominent Indian business family, he studied accountancy and business, but nothing could prepare him for the loss of his father at the young age of 28 – and the huge task of taking over one of India's biggest business groups. But apprehension about Birla's ability to fill his successful father's boots turned out to be unfounded: he has successfully increased revenues 20-fold over the past 20 years to over $40bn.
Denise Coates
3. Denise Coates, director of Bet365. Net worth: $2.9bn
Denise Coates is an English businesswoman, she started out as a humble cashier in her family's bookmaking company, Provincial Racing. She trained as an accountant after university, but showed her true talent when she took charge of the business and transformed the family's fortunes. In 2001, Coates started the popular online betting firm Bet365.com with a bank loan. The business is now one of the world's largest online gambling companies, operating in almost 200 countries and serving millions of customers across the world. That's one gamble that sure paid off for Denise!
4. Arthur Blank, co-founder of The Home Depot. Net worth: $2.6bn
Born in the 1940s in New York, Arthur Blank studied for a degree in accounting and business administration. He moved into the retail sector, where he rose to become a vice president of finance. When both he and his friend and CEO Bernard Marcus were fired in 1978, the pair decided to pool their expertise and set up their own business. They called it Home Depot – and The Home Depot brand is now worth a cool $149.2bn.
Paul Coulson
5. Paul Coulson, chairman of Ardagh Group. Net worth: $1.5bn
A native of Dublin, Paul Coulson is known as one of the richest people in Ireland, and he certainly seems to have been bestowed with an immeasurable amount of that famous Irish luck. Starting out at PricewaterhouseCoopers in London, Coulson founded finance company Yeoman, before making the switch over to the glass industry in 1998. Running glass-bottling company Ardagh, Coulson propelled the firm to worldwide success; the once small firm is now worth an estimated $3.34bn.
Brian Souter and Ann Gloag
6. Sir Brian Souter and Ann Gloag, founders of Stagecoach Group. Net worth: $1.49bn
Scottish businessman Sir Brian Souter and his sister Ann Gloag were the children of a bus driver, but little did they know that the mode of public transport would line their pockets for their lifetimes. Souter graduated from university with a degree in accountancy and economics, and went on to found international transport firm Stagecoach Group with his sister in 1980. The company has since become a big name around the world, with operations in the UK, US and Canada. No sign of sibling rivalry there!
10 Top Universities To Study Accounting In UK
Accounting As The Language of Business
Exam Preparation Tips For Accounting Students And Professionals
Factors To Consider When Buying Accounting Software
Relationship Between Accounting And Other Disciplines
The Old And New Name Of Accounting Terms
ICAN March 2020 Diet Professional Exam Fees And Timetable
How To Check ICAN Professional Exam Result
List Of ICAN Accredited Study Centers In Lagos State With Their Contacts
How to Generate ICAN e-Voucher 12 Digits Pin
ICAN Exemption Guidelines, Requirements And Fees | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,447 |
package org.locationtech.geowave.test.basic;
import java.io.IOException;
import java.math.BigDecimal;
import java.math.BigInteger;
import java.util.Arrays;
import java.util.Calendar;
import java.util.Date;
import java.util.HashMap;
import java.util.Map;
import org.geotools.feature.AttributeTypeBuilder;
import org.geotools.feature.simple.SimpleFeatureBuilder;
import org.geotools.feature.simple.SimpleFeatureTypeBuilder;
import org.junit.AfterClass;
import org.junit.Assert;
import org.junit.BeforeClass;
import org.junit.Test;
import org.junit.runner.RunWith;
import org.locationtech.geowave.adapter.vector.FeatureDataAdapter;
import org.locationtech.geowave.core.geotime.store.query.ExplicitSpatialQuery;
import org.locationtech.geowave.core.geotime.util.GeometryUtils;
import org.locationtech.geowave.core.store.CloseableIterator;
import org.locationtech.geowave.core.store.api.QueryBuilder;
import org.locationtech.geowave.core.store.api.Writer;
import org.locationtech.geowave.core.store.cli.store.DataStorePluginOptions;
import org.locationtech.geowave.core.store.query.constraints.QueryConstraints;
import org.locationtech.geowave.test.GeoWaveITRunner;
import org.locationtech.geowave.test.TestUtils;
import org.locationtech.geowave.test.annotation.GeoWaveTestStore;
import org.locationtech.geowave.test.annotation.GeoWaveTestStore.GeoWaveStoreType;
import org.locationtech.jts.geom.Coordinate;
import org.locationtech.jts.geom.Geometry;
import org.opengis.feature.Property;
import org.opengis.feature.simple.SimpleFeature;
import org.opengis.feature.simple.SimpleFeatureType;
import org.slf4j.Logger;
import org.slf4j.LoggerFactory;
@RunWith(GeoWaveITRunner.class)
public class GeoWaveVectorSerializationIT extends AbstractGeoWaveIT {
private static final Logger LOGGER = LoggerFactory.getLogger(GeoWaveVectorSerializationIT.class);
@GeoWaveTestStore(
value = {
GeoWaveStoreType.ACCUMULO,
GeoWaveStoreType.BIGTABLE,
GeoWaveStoreType.CASSANDRA,
GeoWaveStoreType.HBASE,
GeoWaveStoreType.DYNAMODB,
GeoWaveStoreType.KUDU,
GeoWaveStoreType.REDIS,
GeoWaveStoreType.ROCKSDB,
GeoWaveStoreType.FILESYSTEM})
protected DataStorePluginOptions dataStore;
private static long startMillis;
@Override
protected DataStorePluginOptions getDataStorePluginOptions() {
return dataStore;
}
@BeforeClass
public static void reportTestStart() {
startMillis = System.currentTimeMillis();
LOGGER.warn("-----------------------------------------");
LOGGER.warn("* *");
LOGGER.warn("* RUNNING GeoWaveVectorSerializationIT *");
LOGGER.warn("* *");
LOGGER.warn("-----------------------------------------");
}
@AfterClass
public static void reportTestFinish() {
LOGGER.warn("-----------------------------------------");
LOGGER.warn("* *");
LOGGER.warn("* FINISHED GeoWaveVectorSerializationIT *");
LOGGER.warn(
"* "
+ ((System.currentTimeMillis() - startMillis) / 1000)
+ "s elapsed. *");
LOGGER.warn("* *");
LOGGER.warn("-----------------------------------------");
}
@Test
public void testFeatureSerialization() throws IOException {
final Map<Class, Object> args = new HashMap<>();
args.put(
Geometry.class,
GeometryUtils.GEOMETRY_FACTORY.createPoint(new Coordinate(123.4, 567.8)).buffer(1));
args.put(Integer.class, 23);
args.put(Long.class, 473874387l);
args.put(Boolean.class, Boolean.TRUE);
args.put(Byte.class, (byte) 0xa);
args.put(Short.class, Short.valueOf("2"));
args.put(Float.class, 34.23434f);
args.put(Double.class, 85.3498394839d);
args.put(byte[].class, new byte[] {(byte) 1, (byte) 2, (byte) 3});
args.put(Byte[].class, new Byte[] {(byte) 4, (byte) 5, (byte) 6});
args.put(Date.class, new Date(8675309l));
args.put(BigInteger.class, BigInteger.valueOf(893489348343423l));
args.put(BigDecimal.class, new BigDecimal("939384.93840238409237483617837483"));
args.put(Calendar.class, Calendar.getInstance());
args.put(
String.class,
"This is my string. There are many like it, but this one is mine.\n"
+ "My string is my best friend. It is my life. I must master it as I must master my life.");
args.put(long[].class, new long[] {12345l, 6789l, 1011l, 1213111111111111l});
args.put(int[].class, new int[] {-55, -44, -33, -934839, 55});
args.put(double[].class, new double[] {1.125d, 2.25d});
args.put(float[].class, new float[] {1.5f, 1.75f});
args.put(short[].class, new short[] {(short) 8, (short) 9, (short) 10});
final SimpleFeatureTypeBuilder builder = new SimpleFeatureTypeBuilder();
final AttributeTypeBuilder ab = new AttributeTypeBuilder();
builder.setName("featureserializationtest");
for (final Map.Entry<Class, Object> arg : args.entrySet()) {
builder.add(
ab.binding(arg.getKey()).nillable(false).buildDescriptor(
arg.getKey().getName().toString()));
}
final SimpleFeatureType serTestType = builder.buildFeatureType();
final SimpleFeatureBuilder serBuilder = new SimpleFeatureBuilder(serTestType);
final FeatureDataAdapter serAdapter = new FeatureDataAdapter(serTestType);
for (final Map.Entry<Class, Object> arg : args.entrySet()) {
serBuilder.set(arg.getKey().getName(), arg.getValue());
}
final org.locationtech.geowave.core.store.api.DataStore geowaveStore =
dataStore.createDataStore();
final SimpleFeature sf = serBuilder.buildFeature("343");
geowaveStore.addType(serAdapter, TestUtils.DEFAULT_SPATIAL_INDEX);
try (Writer writer = geowaveStore.createWriter(serAdapter.getTypeName())) {
writer.write(sf);
}
final QueryConstraints q =
new ExplicitSpatialQuery(((Geometry) args.get(Geometry.class)).buffer(0.5d));
try (final CloseableIterator<?> iter =
geowaveStore.query(QueryBuilder.newBuilder().constraints(q).build())) {
boolean foundFeat = false;
while (iter.hasNext()) {
final Object maybeFeat = iter.next();
Assert.assertTrue(
"Iterator should return simple feature in this test",
maybeFeat instanceof SimpleFeature);
foundFeat = true;
final SimpleFeature isFeat = (SimpleFeature) maybeFeat;
for (final Property p : isFeat.getProperties()) {
final Object before = args.get(p.getType().getBinding());
final Object after = isFeat.getAttribute(p.getType().getName().toString());
if (before instanceof double[]) {
Assert.assertTrue(Arrays.equals((double[]) before, (double[]) after));
} else if (before instanceof boolean[]) {
final boolean[] b = (boolean[]) before;
final boolean[] a = (boolean[]) after;
Assert.assertTrue(a.length == b.length);
for (int i = 0; i < b.length; i++) {
Assert.assertTrue(b[i] == a[i]);
}
} else if (before instanceof byte[]) {
Assert.assertArrayEquals((byte[]) before, (byte[]) after);
} else if (before instanceof char[]) {
Assert.assertArrayEquals((char[]) before, (char[]) after);
} else if (before instanceof float[]) {
Assert.assertTrue(Arrays.equals((float[]) before, (float[]) after));
} else if (before instanceof int[]) {
Assert.assertArrayEquals((int[]) before, (int[]) after);
} else if (before instanceof long[]) {
Assert.assertArrayEquals((long[]) before, (long[]) after);
} else if (before instanceof short[]) {
Assert.assertArrayEquals((short[]) before, (short[]) after);
} else if (before.getClass().isArray()) {
Assert.assertArrayEquals(
returnArray(p.getType().getBinding(), before),
returnArray(p.getType().getBinding(), after));
} else if (before instanceof Geometry) {
Assert.assertTrue(((Geometry) before).equalsExact((Geometry) after, 1e-7));
} else {
Assert.assertTrue(before.equals(after));
}
}
}
Assert.assertTrue("One feature should be found", foundFeat);
}
TestUtils.deleteAll(dataStore);
}
public <T> T[] returnArray(final Class<T> clazz, final Object o) {
return (T[]) o;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,247 |
<?php
namespace Dgafka\AuthorizationSecurity\Domain\User\Identity;
use Dgafka\AuthorizationSecurity\Domain\User\User;
/**
* Class IdentityUser - User for Identity Based Access Control
*
* @package Dgafka\AuthorizationSecurity\Domain\User
* @author Dariusz Gafka <dgafka.mail@gmail.com>
*/
final class IdentityUser extends User
{
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,219 |
Q: ricci flow on surfaces In Hamiltons paper "Ricci flow on surfaces" there is an estimate on $|\nabla R|^2$ which shows that $|\nabla R|^2 \leq C_1 \exp{\frac{rt}{2}}$ for some constant $C_1$.
Actually for any solution of the Ricci flow on a surface $|\nabla R|^2$ evolves by $$\frac{\partial}{\partial t}|\nabla R|^2 = \Delta|\nabla R|^2 - 2|\nabla\nabla R|^2 + (4R-3r)|\nabla R|^2.$$
Using the estimate $|R-r|\leq C \exp{rt}$, the above equation implies $$\frac{\partial}{\partial t}|\nabla R|^2 \leq \Delta|\nabla R|^2 - 2|\nabla\nabla R|^2 + (r+4C\exp{rt})|\nabla R|^2.$$
My problem is in the next step where it is stated that taking $t>0$ large enough one can show that $$\frac{\partial}{\partial t}|\nabla R|^2 \leq \Delta|\nabla R|^2 + \frac{r}{2}|\nabla R|^2.$$ After that it is well understood that we can apply maximum principle to get the above estimate. Please anyone help me to understand the last arguement. This case is for $r<0$.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,960 |
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"In his well-researched and convincingly argued book, Robert Dienesch has demonstrated clearly that the American spy satellite program, rather than being a knee-jerk reaction to the launching of _Sputnik_ by the Soviet Union in 1957, was instead the culmination of years of effort by the Truman and Eisenhower administrations."
—Galen Perras, associate professor of history at the University of Ottawa and author of _Franklin Roosevelt and the Origins of the Canadian-American Security Alliance, 1933–1945: Necessary but Not Necessary Enough_
"Dienesch combines an explication of high-level policy formulation with technical details about reconnaissance satellite development. He penetrates the secrecy that surrounded America's first military satellite program, WS-117L, to assess both its contributions and disappointments."
—Rick W. Sturdevant, deputy director of history, Air Force Space Command
# Eyeing the Red Storm
# Eyeing the Red Storm
# Eisenhower and the First Attempt to Build a Spy Satellite
Robert M. Dienesch
University of Nebraska Press | Lincoln & London
© 2016 by Robert M. Dienesch
Cover design by the University of Nebraska Press
Author photo courtesy of Jennifer Mattinson
All rights reserved
Library of Congress Cataloging-in-Publication Data
Dienesch, Robert M.
Eyeing the red storm: Eisenhower and the first attempt to build a spy satellite / Robert M. Dienesch.
pages cm
Includes bibliographical references and index.
ISBN 978-0-8032-5572-2 (cloth: alk. paper)
ISBN 978-0-8032-8675-7 (epub)
ISBN 978-0-8032-8676-4 (mobi)
ISBN 978-0-8032-8677-1 (pdf)
1. Space surveillance—United States—History—20th century. 2. Artificial satellites—United States—History—20th century. 3. Astronautics, Military—United States—History—20th century. 4. Military surveillance—United States—History—20th century. 5. Eisenhower, Dwight D. (Dwight David), 1890–1969. 6. Cold War—History. I. Title. II. Title: Eisenhower and the first attempt to build a spy satellite.
UG1523.D54 2016
358'.88—dc23
2015028635
The publisher does not have any control over and does not assume any responsibility for author or third-party websites or their content.
I dedicate this book to four people who have meant the world to me.
It is dedicated to the memory of my grandparents: Michael and Susanna Dienesch and Christa Laudenbach; unfortunately they did not live to see the completion of this work. I never knew if they really understood what I was working on; I do know that they loved me and that my world is a little less without them.
The fourth person is my wife, Jennifer Mattinson. Before you came into my life, I was lost and without direction. You have brought back the light and magic in my life. I dedicate this book to you, my cheering squad, my rock, my love.
# Contents
Acknowledgments
Introduction: Filling in the Gap
List of Abbreviations
Part 1. Eisenhower's Delicate Balance
1. Truman and Eisenhower on the Cold War (1945–55)
2. Eisenhower and Defense: Three Challenges, Three Responses (1953–56)
3. Eisenhower and Satellite Reconnaissance: Three Projects (1954–58)
Part 2. WS-117L
4. Origins: RAND and Satellite Reconnaissance (1945–54)
5. WS-117L: Two Stages (1954–57)
6. Satellite Photography, Film Return, and the Birth of CORONA (1957–58)
7. SENTRY/SAMOS, MIDAS, and the Dissolution of WS-117L (1958–60)
Epilogue: WS-117L in Perspective
Appendix: Historiography of Eisenhower and Space Reconnaissance
Notes
Bibliography
Index
# Acknowledgments
Inevitably during the research and writing of a work such as this, a large number of people have given me assistance. It is impossible to give credit to each of them in turn, so I beg their forgiveness for not including a detailed list. However, certain individuals and organizations do stand out for special thanks.
As with every research project a great deal of archive work is essential. I have been lucky that the archival staff I worked with were exceptional. The knowledge of the archivists at the Dwight D. Eisenhower Archive in Abilene, Kansas, and their willingness to assist made my time there not just productive but a splendid experience. I look forward to having the opportunity to work with them again. My thanks to the archivists at both the National Archives and Records Administration in College Park, Maryland, and the George Washington University Space Policy Institute Archives. Both staffs were extremely helpful and effective.
I also want to thank the staff at the journal _Quest: The History of Spaceflight Quarterly_. Excerpts of my argument on Eisenhower's domestic perspective and his role in the creation of WS-117L were published in their special issue on the fiftieth anniversary of satellite reconnaissance (volume 17, number 3) in 2010. They took a chance on an unknown historian from Canada, and it has been deeply appreciated.
Dr. Marc Milner, thank you for being a good friend and confidant over the years, listening when I needed to talk and giving me lots of encouragement. Dr. David Charters, thank you for taking on the role of thesis advisor and supervising me throughout my research.
Nicholas, Rob, Jeff, Andre, Dave, and the entire circle of friends who have journeyed with me on this project, you kept me sane and on an even keel, gave me useful advice, and kept me laughing. Thanks, guys.
Of course, my family: you deserve a great deal of credit for all of this. Supporting me when I was ready to give up, keeping me on course and working steadily, you have all earned a portion of the credit for this volume.
Finally, one other person stands out. Dwayne Day, you are a scholar and a gentleman. When my research looked about to founder because of a lack of material in the National Archives, you opened up a whole new set of doors for me, which included your own personal document collection. Considering we had only talked on the phone, that was an incredibly generous gesture. I can only pray that I will be able to repeat the favor for another grad student in the future.
# Introduction
# Filling in the Gap
When Americans went to bed on October 3, 1957, little did they realize that the night sky was about to change forever. The only forewarning was a small article in the _New York Times_ , "Soviet Expert Tells West of Test of Rocket." Found at the bottom left corner of the first page on the morning of October 4, the piece would have garnered only passing interest had it not been for the events of the day.
At Tyuratam in Kazakhstan, just minutes before midnight on October 4, a Soviet rocket blasted off. At its top sat the Soviet Union's contribution to the satellite program of the International Geophysical Year (IGY). Weighing only eighty-four kilograms, the polished metal sphere of _Sputnik_ , meaning "traveler," was an oversized radio beacon. Its sole purpose was to orbit the earth and make electronic noise. And what a noise it made! It permanently altered the tranquil night sky. The earth and the moon were no longer alone. A man-made signal pierced the eerie silence of space.
Americans did not immediately grasp what had happened. Traveling at over seventeen thousand miles an hour, the satellite (complete with its booster and protective shroud following in its wake) flew over the United States twice before the U.S. government became aware of it. Radio Moscow first broke the news to the world, playing up the achievement for its propaganda value. In the United States the shock and disbelief were evident. Believing strongly that the nation had to act and that the Americans' IGY counterpart to _Sputnik_ —VANGUARD—would not provide results in the near future, Werner Von Braun, long a vigorous advocate of satellite research and director of the Development Operations Division of the Army Ballistic Missile Agency (ABMA), felt that all he needed was the opportunity to launch a satellite. He saw the Soviet success as a challenge that the United States had to meet immediately.
Most of the rest of the country reacted with shock and fear. The event shook Senate Majority Leader Lyndon B. Johnson (D-TX), who quickly realized its political implications. Following dinner on the evening of the 4th, he led guests at his Texas ranch outside to stare at the dark sky, which was no longer reassuring but now created a sense of dread. The consequences for U.S. national security were very clear. If the Union of Soviet Socialist Republics (USSR) could put a satellite in orbit, the same rocket booster could theoretically deliver a nuclear warhead to an American city. That prospect held enormous potential as a means of challenging Dwight D. Eisenhower's Republican administration (1953–61).
Media reaction was swift. Newspapers featured numerous articles that fueled further apprehension. While families gathered around their radios and televisions to listen to the insistent beeping of the new satellite; the print media sought to explain the event and its implications, while at the same time feeding public anxieties. If the Soviet satellite was heavier than the one the United States would launch, obviously the USSR was further ahead in space technology. By October 6 the press had linked recent cutbacks in U.S. defense expenditures to the USSR's victory in the first leg of what people would soon call "the space race," and the Eisenhower administration, with its goals of balancing budgets and restraining defense estimates, became the scapegoat. On October 8 Democratic senators entered the fray, demanding investigations and action. They accused the administration of complacency and of withholding funds from the satellite effort.
The Soviet's _Sputnik_ advantage evaporated in August 1960, when the United States launched _Discoverer XIV_. Officially a scientific satellite, DISCOVERER was in fact the cover name for CORONA, the world's first functioning spy satellite. In less than three years from its beginning in early 1958, the United States had leaped well ahead of its chief rival in the military use of space for aerial reconnaissance. This gave it a significant edge—one that it maintained and expanded throughout the cold war.
This book seeks to explain how and why the United States surpassed _Sputnik_ in only thirty-four months. It is remarkable that the United States was able to overcome all the technical challenges to orbit a reconnaissance satellite in such a short time. How was such a complex program able to succeed so rapidly? The answer rests with the work done on space programs from 1945 through 1958. The main focus is on the pre-CORONA program known as WS-117L, developed during the early years of the Eisenhower administration, starting in 1953 and ending in 1960–61. Most recent scholarship has concentrated on the triumphant CORONA satellites. While mentioning WS-117L, scholars usually do so in a cursory way (see the appendix). But without knowing fully about WS-117L, we cannot grasp the story of CORONA, and to truly understand CORONA's success it is essential to examine the satellite program as a whole. It is also necessary to situate it within the wider context of Eisenhower's presidency, especially his handling of national security policy.
The WS-117L program was the world's first attempt to develop a spy satellite and as such broke major theoretical and technical barriers. However, we know very little about it, and discussions in the literature are inaccurate and severely limited. Why is there so little insight into WS-117L? Part of the reason rests with the nature of the subject. Secrecy shrouds such programs and reduces the amount of material available to most scholars. In the information vacuum that this created around WS-117L, leaked details from military and other sources, combined with a healthy dose of speculation, led to several works that weave information and myth into a narrative account. Much of the information is contradictory, confusing, and inaccurate, often merging details from later programs with the initial WS-117L.
There is also a lack of material in other sources. The Eisenhower administration started and prosecuted the program. While there is a large body of writing on Eisenhower in the White House that is well developed and demonstrates a growing level of sophistication and research over time, the absence of information on WS-117L is glaring. The reason rests with Eisenhower himself. A firm believer in security and secrecy, he refused to release information on intelligence matters, even when it was to his advantage. Thus he left little material on his role in the program and the reasons for its acceptance.
The problem of figuring out WS-117L becomes more intense when we learn that the one organization that many people assume was most active in spying, the Central Intelligence Agency (CIA), had no real role in the original satellite program. WS-117L was a U.S. Air Force (USAF) program, and the CIA hardly participated at all. It was only in February 1958, in the wake of _Sputnik_ , that this changed with Eisenhower's decision to separate part of the original WS-117L satellite into a separate program named CORONA. The CIA took on the job of developing CORONA with the USAF and only at that point began to play a major role in satellite reconnaissance. Not surprisingly CIA historians have looked mostly at CORONA and are virtually silent about WS-117L. Unfortunately understanding WS-117L is all the more difficult because CORONA was far more successful, producing the first operational spy satellite system. Thus it attracts a great deal of research, in the process obscuring the original program.
This combination of forces has effectively masked the WS-117L program. In particular the fixation on CORONA, as well as lack of information and slow declassification of documents, has created a great many myths. Since WS-117L is the origin point of virtually all military satellite efforts, and satellite reconnaissance proved pivotal in the cold war, an effort to reassess it is essential.
By examining satellite reconnaissance prior to CORONA, this book is unique. It makes a valuable contribution to our understanding not only of military space programs but also of the history of the cold war and of the Eisenhower administration. The examination of WS-117L that I provide here is the first comprehensive study of the program. Any understanding of CORONA is incomplete without such material. The relationship between WS-117L, the cold war, and the Eisenhower presidency is also a vital element in the story. The entire spy satellite episode helped shape U.S. cold war policy.
So as the troubled nation nervously eyes the skies, the American satellite story begins.
# Abbreviations
ABMA: Army Ballistic Missile Agency
ADC: Air Defense Command
AEC: Atomic Energy Commission
AFB: Air Force Base
AFDAP: Air Force Development and Planning
ARDC: Air Research and Development Command
ARPA: Advanced Research Projects Agency
BUORL: Boston University Optical Research Laboratory
CGS: Coordinating Committee on General Sciences, Department of Defense Research and Development
CIA: Central Intelligence Agency
CIG: Central Intelligence Group
CSAGI: Comité Special Anneé Geophysique Internationale 1957–58 (Special Committee for the International Geophysical Year)
DCI: Director of Central Intelligence
DoD: Department of Defense
DPO: Development Planning Objective
FY: fiscal year
GNP: gross national product
ICBM: intercontinental ballistic missile
IGY: International Geophysical Year (1957–58)
IRBM: intermediate range ballistic missile
JCS: Joint Chiefs of Staff
JPL: California Institute of Technology's Jet Propulsion Laboratory
JRDB: Joint Research and Development Board
kw: kilowatt
Mc: megacycles
NARA: National Archives and Records Administration
NATO: North Atlantic Treaty Organization
NIE: National Intelligence Estimate
NSA: National Security Agency
NSC: National Security Council
ODM-SAC: Office of Defense Management–Science Advisory Committee
R&D: research and development
RCA: Radio Corporation of America
RDB: Research and Development Board
SAC: Strategic Air Command
SIGINT: Signals Intelligence
SNIE: Special National Intelligence Estimate
TCP: Technological Capabilities Panel
USAAF: U.S. Army Air Forces
USAF: U.S. Air Force
USN: U.S. Navy
WADC: Wright Air Development Command
WDD: Western Development Division
WS: weapon system
# Part 1
# Eisenhower's Delicate Balance
#
# Truman and Eisenhower on the Cold War (1945–55)
A distasteful but vital necessity.
—President Dwight D. Eisenhower
War . . . always starts with a Pearl Harbor kind of attack. In an atomic war the first attack, no matter how well prepared for it we may be, will really be a disaster.
—Louis Ridenour
Virtually every history of satellite reconnaissance justifies the creation of the program by citing the need for U.S. intelligence on Soviet military capabilities. The argument focuses on the growing atomic threat from the USSR in the early 1950s combined with problems in penetrating Soviet security as the primary drives for the setting up of the WS-117L satellite program. This interpretation is logical. There certainly were serious problems affecting the gathering of intelligence on the Soviet Union from 1945 to 1953, and by the time Dwight Eisenhower took office in January 1953, the lack of information was a growing concern. But was the monitoring of Soviet military developments the sole reason for the WS-117L?
In this chapter I look at the contrasting views of presidents Harry S. Truman (1945–53) and Dwight D. Eisenhower (1953–61) on the threats and challenges that the cold war posed to the United States beginning soon after the end of the Second World War.
## Harry Truman and the Cold War (1945–53)
The United States required intelligence in the light of increasing tensions with the Soviet Union. During Truman's years in the White House the international situation changed dramatically. Truman and his administration expected the wartime U.S.-Soviet relationship to last into the peace, and it startled them to find the peace so short-lived. Joseph Stalin, determined to ensure the safety of eastern and east-central Europe from Western influence, established communist governments there, thus creating a sphere of influence around his vast nation. When the Americans responded by providing assistance (initially financial, but later military) to help contain what they saw as communist ambitions, the result was a cold war that polarized the world and lasted decades. The deterioration of U.S.-Soviet relations and the start of the cold war did not occur overnight but rather emerged slowly through the second half of the 1940s and finally became an _id_ é _e fixe_ in the American psyche during the first three years of the 1950s.
With mounting evidence of a change in Soviet attitudes toward the United States and a growing sense of hostility from the Soviet Union, the Truman administration became increasingly aware of the need for strong intelligence about the threats confronting the country. Pearl Harbor was the best argument for more national intelligence. The most shocking and destructive single experience in American history up to that time, this event traumatized every living adult American. The attack was possible because of a clear intelligence failure, the heart of which was the American inability to monitor Japanese military movements and intentions effectively. Most American information came from code-breaking, especially the top Japanese diplomatic code, PURPLE. However, absence of military data from such signals and the huge volume of traffic left intelligence experts unable to interpret indicators of a possible attack. This inadequacy and the absence of an effective system for coordinating and forwarding information to key U.S. commands climaxed in Pearl Harbor, which propelled the nation into war.
The memory of that fateful day was still very fresh after war's end. Americans often recalled with perfect clarity what they were doing when they heard about the attack and the feelings it created, even years later. The trauma of the event and the suspicion that poor intelligence was probably to blame for it found reinforcement in the numerous investigations into what happened on that day. Starting soon afterward with the Roberts Commission, the military and civilian arms of government conducted eight inquiries. The last, the formal Joint Congressional Committee Investigation (November 1945–July 1946), produced forty volumes of material, including much of the testimony from other investigations. Next to President Kennedy's assassination on November 22, 1963, and the attacks on the World Trade Center in New York on September 11, 2001, no other event in modern American history has had such an impact. It was the fear of another Pearl Harbor that underscored the desire for intelligence and for satellite reconnaissance.
In 1945 the United States was the sole possessor of atomic weapons, but that monopoly proved to be a paper tiger. Although the government was drafting plans to use such devices against the Soviet Union, it had very few bombs with which to do so. Immediately after the war it had no need to increase its atomic stockpile rapidly, as it did not see the Soviets as a threat or believe that they had any such devices. Possessing only conventional and chemical weapons, the Soviet Union could not project its power beyond Europe, let alone attack North America directly. Lacking any bases in countries near the United States and possessing only relatively short-range aircraft, it could not appreciably threaten its chief rival in the near future.
Conversely, because the United States did have nuclear weapons, it thought that dropping a few from long-range bombers would quickly knock out the Soviets if such an eventuality became necessary. Since the heavy bomber was the only feasible means of delivering the weapons, the U.S. Air Force understandably concentrated on preparations for strategic bombing, which became its primary task and evolved into a "bomber mentality" with respect to procurement, training, and intelligence assessment. General Eisenhower noted that proclivity before he entered the White House in 1953, by which time the air force's fixation had become the norm. Authors such as Lawrence Aronsen argue that a major reason for this was the air force's solid belief that the Soviets wanted war and that the bomber was their only means of devastating the United States.
One clear legacy of Pearl Harbor was the assumption that future wars would begin with a surprise attack. Having seen the advantage that surprise gave an attacker and knowing that the Soviets would eventually develop atomic weapons, Americans viewed the example of Pearl Harbor with great unease. The pairing of an unexpected attack and the power of nuclear weapons seemed a nightmarish combination that would paralyze the victim's economy and government. A corollary to this was the notion of a distinct U.S. disadvantage in the new atomic era. The postwar tendency in democratic societies to downplay military preparedness suddenly became a potential danger. The destructiveness of a nuclear strike meant that lack of preparation and neglect of military abilities would bring rapid defeat to a democracy.
Immediately after the war strategic thinkers began to assess the impact of the bomb on warfare and national security. Bernard Brodie, an architect of nuclear deterrence theory and an articulate spokesman for the role of nuclear weapons in peacetime, quickly grasped the weapons' implications. Arguing that their vast power would render any attack devastating, he concluded that the age of defense was over: some bombers would always make it through the defenses. Their destructiveness would prevent the victim's buildup of sizable military forces after the initial assault. Thus the United States had to be constantly ready to wage preemptive war. No weapon system, however "superior," could guarantee strategic superiority. To Brodie the best solution was deterrence. The key to safety was the retention of enough nuclear weapons to convince a potential aggressor of the likelihood of massive retaliation.
An integral component of deterrence was the acquisition of accurate intelligence on the Soviet Union. However, until 1947 the United States had no centralized structure for doing so. The Office of Strategic Services under Gen. William J. Donovan had run wartime intelligence gathering and covert operations. It was unpopular within the administration, however, and Truman disbanded it quickly at war's end.
Increasing tensions with the Soviet Union soon forced the president to rethink his decision and establish a permanent intelligence agency. The first steps in this direction took place on January 22, 1946, a little over one month before former British prime minister Winston Churchill added the phrase _Iron Curtain_ to the Western world's vocabulary. Truman authorized creation of the Central Intelligence Group, or CIG. The CIG was to correlate, evaluate, and disseminate all intelligence relating to national security, and the primary target was the Soviet Union. Although a major step forward, the CIG did not last long. Increasing demands for intelligence, rivalry among the military services, and limited resources prevented it from being totally effective. Moreover the new cold war necessitated an ever-stronger intelligence organization. Thus in 1947 the National Security Act gave the CIG a permanent statutory foundation as the Central Intelligence Agency (CIA).
One of an intelligence community's primary tasks is preparing intelligence estimates. Under the CIG its Central Reports Staff had handled this task, assisting the director in producing these evaluations. In 1947 the Office of Research and Evaluation took on this duty for the new CIA. Accepting Truman's geostrategic vision of national security, the CIG (and the CIA) looked at a variety of factors that affected the United States. "National security" in the broadest sense now involved more than just weapons; it included economic forces, political and ideological threats, and control of resources and industrial infrastructure. Thus the concept became much broader in the Truman era.
The primary focus remained the Soviet Union, although the danger came not just from tanks and atomic weapons but from elements that experts had never really considered before in great detail. Seeing the Soviet regime as hostile in every way, U.S. officials perceived possible threats not just in Soviet military actions but in national and regional instability and weakness in various parts of the world. The Soviets could exploit this situation through political, economic, and psychological means to undermine potential American strength.
As officials were preparing the first estimates, lack of hard intelligence about the Soviet Union quickly became apparent. Initial estimates, such as the report of October 1946, "Soviet Capabilities for the Development and Production of Certain Types of Weapons and Equipment," indicated the problems of predicting the Soviet Union's capabilities. Noting that "any report of this nature is at best educated guesswork," the authors pointed out that "an estimate of capabilities ten years hence obviously cannot be based on evidence, but only on a projection from known facts in the light of past experience and reasonable conjecture." As a foundation for their assessment, the writers relied on their current estimates of Soviet scientific and industrial abilities. They also compared American experience and estimates of capabilities (both current and predicted) with past Soviet capabilities. They also obtained information from former Soviet prisoners—mainly German scientists the Soviets had captured during the war and forced to work for them and who were now going home.
Such estimates were problematic. First, by relying on American experience, they failed to account for the different setup of the Soviet economy: a "command economy" could call on more resources and use different avenues of research and development, thereby speeding development. Assuming that their own country's research and development was normative, American analysts found the Soviet Union to be far behind. Such a conclusion assumed that the Soviets would follow the same steps in the same order and in the same amount of time. This fallacy—"mirror imaging," as the historian Abram Shulsky describes it—involves "assessing or predicting a foreign government's actions by analogy with the actions that the analyst feels he (or his government) would take were he (or it) in a similar position."
Second, this misinterpretation helped to create another problem: underestimation of Soviet capabilities, in which the absence of hard data played a role. Viewing the Soviets as backward, many Americans (including Truman) rejected the idea that the Soviets could compete in highly scientific and technical fields such as atomic energy. Noting the Soviet Union's postwar rebuilding and "limited technological development," the CIG's report argued that the USSR would be incapable of research and development in many advanced areas such as nuclear weapons for at least ten years. Most predictions for the period up to 1950 attributed any Soviet achievements to captured German technology and scientists. The CIG anticipated that by 1948 the Soviets would only develop bombers with performance characteristics similar to the B-29s they had captured and interred in Asia during the war. It suggested that by 1950 they might produce almost 150 aircraft per month.
Central to U.S. national security, of course, was the Soviets' ability to attack the United States directly with atomic weapons. While acknowledging the Soviets' overwhelming conventional strength, particularly in Europe, the CIA did not believe they were willing to risk open war in the face of the U.S. nuclear arsenal (at least in 1948). The Soviet Union would require enough atomic weapons and an effective means of delivery, neither of which it possessed in 1948.
The accepted opinion of the U.S. scientific community in the period 1946–49 was that at worst the Soviet Union was five years from developing its own atomic weapon. Members of the Interim Committee who advised the president on postwar atomic energy concurred. Secretary of War Henry Stimson had set up the group in the spring of 1945, and it consisted of Vannevar Bush, James F. Byrnes, and James B. Conant. The president ignored its prediction of a U.S. monopoly lasting only three to four years, as did many of his key advisors, who wanted to be the only player for a longer period of time.
Gen. Leslie Groves, director of the Manhattan Project, was by May 1945 sure that the United States and Britain had a monopoly on the crucial ingredient: high-grade uranium. Basing his reasoning on a special study of the world's uranium deposits called the Murray Hill Area project, conducted for him by top experts from 1943 to 1945, Groves believed that the Soviet Union lacked uranium and this would keep them at least twenty years behind in their development of nuclear weapons. No one ever challenged the highly secret findings, and Groves's committee had excluded any experts who might have dissented. As a result many U.S. officials thought that their country had gained control over the requisite raw resources. All of these experts and officials seriously underestimated Soviet capabilities.
This type of complacency and overconfidence was common among U.S. officials, including the Joint Chiefs of Staff (JCS). Truman and his administration did not expect a Soviet nuclear device before 1950 or even 1955. Even the CIA did not anticipate an atomic test prior to 1953. In a July 1948 memorandum for the president the CIA made it clear that it based its assessment on American, British, and Canadian experience. The agency found no reason to expect a Soviet weapon before the 1950s. Noting that it was "impossible to determine its exact status or to determine the date scheduled by the Soviets for the completion of their first atomic bomb," the memo added that the Soviets' supply of fissionable material would allow for only between twenty and fifty weapons by 1955, depending on the date of their first atomic test.
The nuclear bomb would be useful only if the Soviets could deliver it to its target. In the CIA's report of September 28, 1948, "Threats to the Security of the United States," the agency stated that it did not believe the Soviets could attack the United States directly except via one-way suicide missions, which they could not launch at a scale sufficient to cripple the United States. The CIA predicted that the Soviets would not present a palpable threat from bombers and possibly by launching short-range missiles from submarines until 1955.
The Truman administration felt safe behind American technological superiority—until 1949. The first Soviet test of an atomic bomb on August 29 of that year shocked the U.S. government, forcing a reversal in attitude about Soviet capabilities. This success antedated even the CIA's earliest prediction, and the threat of a nuclear attack would increase exponentially as Soviet weapon stockpiles grew. In fact by February 1950 the Joint Intelligence Committee (an interservice office that reported directly to the JCS) predicted that the Soviets would expand their atomic arsenal and would attack the United States "at any time they assessed that it was to their advantage to do so." For the first time in American history, the country faced the prospect of a devastating attack that directly threatened its survival.
In April 1950 the CIA completed a comprehensive examination of the Soviet bomb's implications for U.S. security, and the results were not encouraging. It expected the Soviet Union to have approximately one hundred atomic weapons by 1953; the estimate climbed to two hundred for 1954–55. The Soviet version of captured American B-29 bombers (the TU-4, or BULL, bomber, as the West called it) was the expected delivery system. The report predicted that two hundred bombs reaching key targets could decisively cripple the United States.
In June 1950 a further report, "The Effect of the Soviet Possession of Atomic Bombs on the Security of the United States," determined that the Soviet devices placed U.S. security in increasing jeopardy. Predicting a Soviet arsenal of between 70 and 135 warheads by mid-1953, the authors believed that the enemy could inflict critical damage. The greatest threat was "a single surprise attack on the United States and its foreign installations, which could seriously limit U.S. offensive capabilities, possibly to a critical degree." More important, possession of the atomic bomb would greatly strengthen Soviet influence over other states by weakening their resolve to resist communism. Noting that the Soviet Union's basic objective was to establish communism worldwide, the report concluded that the Soviets would probably not start a general war unless they thought an attack by the West was imminent. However, if the balance of power began to shift in their favor, this attitude was likely to change.
In November 1950 a CIA estimate, "Soviet Capabilities and Intentions," reinforced this interpretation. While adopting the same tone as earlier reports, it was more dire, predicting a probable general war sometime between 1950 and 1954 to install communist regimes in the West when the Soviets believed their strength to be at its peak. The bomb would be the major factor in the Soviet Union's estimation of its military power. Estimating that it already had 22 warheads, the CIA now predicted an arsenal of roughly 235 weapons by mid-1954. Equally important, the Soviets would have enough planes and trained personnel to deliver their entire inventory. American deterrence could no longer guarantee Soviet forbearance. The CIA expected that an attack would become more likely should the Soviets decide they could cripple or eliminate the U.S. arsenal in a single, decisive stroke.
By early 1952 the agency had reversed its position. With the shock of the Korean War somewhat dissipating, it reported on January 8, 1952, that the Soviet Union was unlikely to initiate a general conflict even if it believed that it had the advantage, since its leaders preferred using any means short of war. The CIA did not expect the Soviets' development of intercontinental ballistic missiles (ICBMs) to change this situation. So long as they did not make a major technological advance, the threat of their attacking seemed very small; such a breakthrough could have led them to conclude that they could destroy the United States without sustaining effective retaliation. This meant not that nuclear war was impossible but that it was unlikely to be the product of a conscious choice.
The shock of the first Soviet weapon was not the only one for the United States during this period. In September 1949 Chinese communists seized power and within weeks consolidated their control over continental China. Truman had hoped to contain communism within the Soviet Union and Eastern Europe by helping Western Europe and Japan rebuild, so the disappearance of the atomic monopoly and the "loss" of China shook many people in the administration and the public at large. In less than six months the United States went from "winning" the cold war to appearing to be on the edge of losing it. With ties between communist China and the Soviet Union strengthening, Truman and his administration were not ready for the next shock. Without warning, on June 24, 1950, North Korean troops poured across the 38th parallel into "democratic" South Korea. The Korean War drove home the problems facing the United States. Seeming to prove communism's lust to conquer by any means, the surprise invasion demonstrated that the U.S. government lacked solid intelligence concerning the intentions, plans, and capabilities of its greatest rival.
The crux of the U.S. intelligence problem during Truman's administration was how to obtain the necessary information. The United States and the Soviet Union opposed each other politically and philosophically, and their societies worked differently. The United States was a comparatively open society, with freedom built into the Constitution and into the very fabric of everyday life. Freedom of the press, freedom of speech, freedom to travel within the United States, and relatively free access to information made the nation an open target for foreign intelligence agents. In October 1956 President Eisenhower observed that "a Russian can now buy an air ticket in New York and learn about our whole country" and suggested that there was little that anyone could do about it.
In contrast the Soviet Union was a tightly controlled, "closed society"—in intelligence parlance, a "denied area." The Communist Party dominated every aspect of life. It restricted travel and thereby prevented its citizens (and anyone else) from seeing vast areas of the country. Using secret police and forcing people to spy on each other allowed the state to monitor its own people and foreigners. Harsh punishment and a climate of fear meant that few would provide information to U.S. agents or even make contact with them. These same controls restricted the activities of foreign visitors. Tight border security (complete with guards and fences) and the requirement of papers for domestic travel kept out most foreign agents or severely circumscribed their movements.
The United States used every means possible to overcome these obstacles. Following war's end it gained intelligence mainly through indirect means. Captured reconnaissance photographs from the German Luftwaffe and the debriefing of German prisoners, émigrés, and defectors all provided some information, even if it was out of date and covered only a small part of the Soviet Union. Attachés at the U.S. embassy and occasionally tourists—both real and "special" ones (whom the CIA selected)—could provide more timely data, but the paranoid and heavy-handed regime monitored such individuals carefully to keep them from learning anything of real value.
By 1948, in an effort to increase the amount of intelligence available, the United States under Truman's direction—and its allies—began to use camera-equipped aircraft for flights near and occasionally over Soviet territory. They also employed camera-equipped balloons as well as ground-, air-, and ship-based equipment to monitor electronic signals. The culmination of the balloon efforts was the 1956 program GENETRIX, "an Air Force meteorological survey" that used specially designed helium-filled balloons to photograph the Soviet Union in the first attempt to penetrate deep into its territory for intelligence purposes. Most balloons disappeared, and it was difficult to locate the subjects of recovered photos. U.S. analysts had never seen vast areas of the Soviet interior.
Beginning in 1948 the United States also shared with Britain and Canada the task of intercepting Soviet signals. During the 1950s the National Security Agency (NSA) and its Signals Intelligence (SIGINT) and Communication Intelligence products remained the key source of information on the Soviet Union. However, because of excessive secrecy and difficulties in penetrating Soviet codes, SIGINT's value is difficult to assess.
The reconnaissance efforts in this era inside the Eastern Bloc also began to include a human component. In the late 1940s the United States began infiltrating agents into Eastern Europe, equipping them with false papers, money, and transmitters and air-dropping them into Soviet Bloc countries or rural Soviet border areas. A great deal of effort and money went into training these agents and preparing phony documents and histories, but few of these people provided any intelligence. The KGB and its network of informants apprehended most of them when they landed or shortly thereafter. Due to the large-scale failure of the program, all attempts to insert agents ended in 1954.
The first Soviet atomic test in 1949 helped persuade the Truman administration to draft its most important document of the cold war: NSC-68. Finished prior to the Korean War but not formally accepted until after hostilities began, it called for increasing conventional and nuclear forces to counter the escalating Soviet threat. The Soviet test was also the key factor in American resolve to develop hydrogen weapons—a decision that escalated both the cold war and U.S. intelligence problems. The atomic bomb was a threat to U.S. security, albeit a relatively small one. Delivering it required long-range heavy bombers, whose slow speed would give a target nation hours to prepare for an attack, evacuate vulnerable populations, and perhaps intercept and destroy the delivery aircraft.
In 1950 the ICBM as a means of delivering nuclear warheads had been a concept years ahead of its time, but its potential was already clear. Work had begun in the field immediately after the war; however, progress had been very slow. In his December 1945 report, "Towards New Horizons," Dr. Theodore von Karman, a noted mathematician and expert on aeronautical sciences and head of Caltech's Guggenheim Aeronautical Labs in Pasadena, predicted that it would soon become a vital new weapon system. The atomic warheads' excessive weight, and the need for great accuracy to be effective, challenged researchers. The greater power of hydrogen weapons meant that smaller warheads were possible and the need for high accuracy became unnecessary.
## Eisenhower's New Approach (1953–55)
By the time of Eisenhower's election in November 1952, the government was acutely aware of the new reality of the cold war. It lacked reliable and accurate intelligence on the Soviet Union. None of its sources of information had alerted it to the pace of that country's atomic research in the late 1940s. Likewise, none of them knew about the state of Soviet research and development of its own hydrogen bomb, let alone warned of its test in 1953. U.S. intelligence was also unable to determine either the status of the Soviet bomber force (the most likely means of attack in the near future) or whether the Soviets had undertaken large-scale research in rocketry. This situation was coming to a head in January 1953, when Eisenhower became president.
The new chief executive took office worrying about an external military threat. Eisenhower's biographer Stephen Ambrose demonstrates that Ike suffered from the same shock regarding Pearl Harbor as his fellow citizens. It left a permanent mark on people's psyches, a mental sore spot, that made most U.S. leaders during the 1950s obsessive about the threat of another surprise attack. When atomic and hydrogen weapons made such an eventuality more feasible, many anticipated a new December 7 of epic proportions.
Curtis Peebles supports Ambrose's interpretation. In _The Corona Project_ , he argues that the new president faced two overwhelming problems that shaped U.S. policy throughout his years in office: overwhelming fear of a sudden Soviet attack and the inability of the U.S. intelligence establishment to penetrate Soviet secrecy, especially concerning military activity. He highlights Eisenhower's meeting with the President's Scientific Advisory Committee on March 27, 1954, when he explained his fear that modern (i.e., nuclear) weapons and a closed society gave the Soviet Union a major advantage. Many historians take this reasoning as the basis for Ike's decision to support the creation of a system of satellite reconnaissance.
Eisenhower's military experience and knowledge made him well aware of the U.S. situation. Having experienced war and fearing nuclear weapons, he was probably the American most aware of the destructive potential of a surprise attack. Seeing war as "completely stupid and futile," he could find no way to decouple thermonuclear war from a conventional conflict. A "low-intensity" war would spread to include a general armed struggle between the superpowers and the use of nuclear weapons with nightmarish consequences.
Briefings that the new chief executive received strengthened his fears. For example, on May 18, 1953, the Special Evaluation Subcommittee of the National Security Council (NSC) gave him its estimate of the scale of damage that the Soviets could inflict. Expecting attacks on bomber and forward-staging bases and on major population and industrial areas, the report painted a very bleak picture. The Strategic Air Command (SAC) could expect to lose between about one-quarter and one-third of its strength, and the country between one-third (in 1953) and two-thirds (by 1955) of its industrial output. Depending on timing, casualty rates ranged from 9 million people for 1953 to 12.5 million for 1955, half of them either from the immediate blast effects or from radiation exposure. The psychological impact would be unspeakable. In the face of such estimates Eisenhower strongly opposed nuclear war except as a last resort.
For most of its history the U.S. heartland had been far distant from any potential enemy. In both world wars the Atlantic and Pacific oceans provided a great deal of protection. Now new weapons could eliminate the vast American industrial base rather quickly.
Early in his first term Eisenhower spoke about the danger to Americans and the world in no uncertain terms. In his December 1953 "Atoms for Peace" speech to the United Nations, he pointed out that the United States could "inflict terrible losses upon an aggressor," but there was no real defense against nuclear holocaust. Eisenhower continued, "But let no one think that the expenditure of vast sums for weapons and systems of defense can guarantee absolute safety for the cities and citizens of any nation. The awful arithmetic of the atomic bomb does not permit of any such easy solution. Even against the most powerful defense, an aggressor in possession of the effective minimum number of atomic bombs for a surprise attack could probably place a sufficient number of his bombs on the chosen target to cause hideous damage." Eisenhower rejected the notion that the United States and the Soviet Union were "two atomic colossi . . . doomed malevolently to eye each other indefinitely across a trembling world." 47 By proposing a world stockpile of nuclear materials and internationalization of research on harnessing atomic energy for peaceful purposes, he hoped to defuse tension and preserve peace. Unfortunately this effort failed because of Soviet mistrust of American motives.
To Eisenhower, then, the only effective way to prevent nuclear war was increasingly vigorous intelligence gathering to provide warning of an attack and to reveal Soviet military capabilities. His wartime experience had taught him the value of accurate information. When he took office, the scale of U.S. intelligence efforts surprised him. Three sources kept him constantly up to date: daily briefings on security developments, special studies and briefings by scientists and experts at his request, and formal National Intelligence Estimates (NIEs) or Special National Intelligence Estimates (SNIEs) from the CIA for the NSC.
Overall the Office of National Estimates played a pivotal role in intelligence and in the president's and the NSC's deliberations. It provided about fifty NIEs and SNIEs per year, usually at the request of policymakers, synthesizing a vast amount of material into forecasts. NIEs were only as good as the raw data that informed them. These reports directly influenced the U.S. defense posture and procurement as well as foreign policy.
Growing military awareness of a crisis in U.S. intelligence was evident in an air force request of May 1953. That month Brig. Gen. W. M. Burgess, deputy chief of staff for intelligence, USAF, admitted in an official study, "Cost and Effectiveness of the Defense of the United States against Air Attack in 1952–1957," that there was little information available concerning the Soviet order of battle. The problem lay in identifying the types and numbers of Soviet aircraft. The study reported that the turbo-prop Type-31 bomber was the only plane definitely in active service. It also predicted that Soviet versions of the American B-47 and B-52 jet bombers probably existed because the Soviets had near-complete access to American developments in military aviation. Since it assumed that the Soviet aircraft industry was simply copying foreign designs, the document concluded that the industry would follow the American lead and thus lag behind by only a few months. In short, the writers engaged in mirror-imaging.
By February 1954, however, Eisenhower began to doubt seriously that the Soviets could have any significant number of aircraft similar to the B-52, which was still under development in the United States. Basing his judgment on American experience, he believed firmly that the Soviets were not yet able to surmount the huge technical problems that the United States had already addressed.
American concerns were also growing about Soviet research on long-range ballistic missiles. In 1953, lacking sufficient knowledge of Soviet work in this field, the Operations and Planning Group in the USAF Air Defense Command (ADC) turned to the intelligence community. Noting that the United States was preparing to meet a bomber threat, the ADC thought that such defenses aimed at aircraft-delivered bombs would be useless in the face of missile systems. Thus it desperately needed intelligence so that it could make appropriate preparations. It recommended maximum effort to obtain information and establish a factual basis for the evaluation of the status of and program for Soviet development of both long-range missiles and heavy bombers.
Despite ongoing worries over deficiencies in intelligence, the CIA continued issuing reports without possessing solid information. In July 1953, for example, it released an SNIE on the expected Soviet capability of attacking North America. Focusing on the period from mid-1953 to mid-1955, it did not assess the likelihood of such an eventuality but looked at how the Soviet Union _could_ attack _if_ it chose to do so. Expecting that the Soviets would continue to erode American superiority in atomic weapons, it analyzed the principal means of delivery: long-range aircraft. It estimated that as of July 1, 1953, one thousand TU-4s were available for attacks and noted sightings of heavy jet-bomber prototypes in the air. Projecting from American experience, the CIA estimated two hundred heavy jet bombers available by 1955, supplementing the projected two thousand TU-4s and about eighty jet-powered medium bombers.
The report is pivotal because it indicated that in 1953 there was already evidence of the Soviet Union's developing long-range bombers. Also significant, it predicted that Soviet progress in long-range aviation would resemble the American experience. It presumed that Soviet leaders concentrated on heavy bombers and in-flight refueling to permit maximum use of aircraft for North American attacks. But its most important feature was its speculative, mirror-imaging character; its prognostications rested on virtually no hard data. It also discussed the possibility that the Soviet Union was using the work of German scientists and captured missiles to develop long-range guided missiles.
By early 1954, although intelligence efforts impressed Eisenhower, the limited amount of useful information that resulted disappointed him. He found two major problems: reports made no distinction between Soviet capabilities and intentions and seemed lacking in perspective. The briefings and reports did not weigh the Soviets' strengths and aspirations against the Americans' and did not take into account American ability to counter Soviet bombers and retaliate in kind.
These criticisms indicated serious problems. Failure to differentiate between capability and intention was largely a function of the data available. U.S. intelligence could monitor some technical aspects of Soviet capabilities, and diplomats could photograph and count aircraft during parades and aviation-day flyovers. However, the United States lacked detailed knowledge about production rates or capacities. Extrapolating from this limited knowledge, the United States had to estimate likely capabilities of Soviet long-range aircraft, their ranges, their bomb loads, and so forth. Nevertheless capabilities were the easiest factor to estimate.
The real problem lay in determining what the Soviet Union was planning to do. Was it going to build a large bomber fleet, and if so, why? Did it intend to create a bomber force so massive that it could overwhelm U.S. defenses, or was it simply mimicking the U.S. bomber mentality? Were the aircraft seen just prototype aircraft? The challenge faced by U.S. intelligence was how to answer these questions. The United States had no high-level spies inside the Soviet leadership, rendering it almost totally unaware of Soviet decision making and plans. It knew nothing about the Soviet mind-set. American assumptions rested on the interpretations of those people, such as U.S. diplomat George Kennan, who had had significant experience with Soviet leaders. Kennan's "long telegram" of 1946 had defined American views, even though he was an outsider within the USSR. Without access to the inner sanctums, U.S. intelligence could only guess at Soviet intentions.
Furthermore, without direct entrée to Soviet production facilities or airfields, and lacking accurate knowledge of production rates, plans, or decision-making processes, U.S. intelligence could only estimate vaguely the rate of aircraft production and deployment. Military self-interest at home, particularly in the USAF, often skewed interpretation of findings in what one of Eisenhower's advisors called "sales promotion intelligence." Playing up the need to enhance security, the air force used the scarce information available on Soviet bombers to maximize the perceived threat. It pioneered use of the "worst-case scenario" as the normative standard for intelligence reporting. Inflating the danger of course facilitated increases in deterrence requirements and thus budget requests, thus playing into competition for funding within the administration. As late as 1956 Adm. Arthur W. Radford indicated to the president that debate over military programs reaffirmed the necessity for firmer intelligence estimates.
The phenomenon of the "bomber gap" graphically illustrates this problem. In 1954 and 1955 the CIA and the air force were presenting increasingly bleak intelligence. Focusing on long-range bombers and citing studies by Albert Wohlstetter and other experts at the RAND Corporation concerning SAC's vulnerability, they predicted that the Soviet Union would launch a surprise attack when it had sufficient might. Initial estimates offered modest projections of Soviet bomber strength. In April 1954 the TU-16 medium bomber ("Badger") and the new Type-37 four-engine heavy jet bomber ("Bison," first seen in July 1953 on the tarmac of the Ramenskoye test facility) appeared in Moscow flyovers, with between twelve and twenty TU-16s at a distance. The next month photographs of both TU-16s and Type-37s appeared, leading to speculation that the Soviets had roughly forty Bisons in service.
The CIA saw the Bison as a prototype, with expected production to begin in 1956, and concluded that it would not pose a strategic threat until about 1960, but the air force did not agree. By June 1954 estimates of Soviet bomber strength had increased. In NIE 11-5-54 (June 7, 1954) the expansion (and corresponding ability to deliver nuclear weapons) seemed significant enough to change the world's balance of power. The almost simultaneous appearance of Badgers and Bisons was surprising, and so predictions of the Soviet Union's bomber strength escalated further. Predictions were now of 20 Badgers by mid-1954, 120 in mid-1955, and about 600 by 1959.
In 1955, during practice flybys for the May Day parade in Moscow, observers spotted as many as ten Bisons in various formations. American experience suggested to the USAF that between twenty-five and forty could already be in service. Thus the USAF concluded that there was a gap between U.S. and Soviet bomber strengths and it was decidedly in the Soviets' favor. This quickly became a major political issue. Leaking these estimates to the media during testimony before Congress, air force leaders cited the Soviet bombers to help justify more military spending and thereby put a great deal of pressure on the administration. The air force emphasized the worst-case scenario and predicted that the Soviets would maximize the rate of bomber production. U.S. experience suggested that the Soviets would have started with six aircraft per month, so at least thirty were already operational, if the estimate of the size of Soviet production plants was correct and if one assumed a maximum pace for production.
In hindsight there was indeed a gap, but it was in favor of the United States. During the second half of 1955 the United States began producing the first B-52s for training purposes, taking possession of roughly eighteen. By December 1956 it had 1,470 bombers for operations, including forty-five B-52s. The Soviet Union had only forty operational bombers in 1956, half of them Bisons. The flybys of the latter were a ruse, using the same aircraft over and over to confuse U.S. intelligence. Unfortunately this did not become clear until after mid-1956, when American U-2 aircraft were able to fly over most key Soviet bomber fields.
Although all the U.S. military services had access to the same information about Soviet long-range bombers, the air force supported the worst-case predictions: some eighty aircraft already operational and six hundred to seven hundred probably ready by mid-1959. In congressional testimony during 1956 and 1957 the air force maintained that by 1959 Soviet bombers would outnumber U.S. bombers by two to one. The army and the navy, along with the CIA, were very skeptical of these figures. Only the air force saw the situation as absolutely dire; unfortunately its pessimism carried considerable weight with Eisenhower's critics and with supporters of a more powerful military.
By October 1956 SNIEs of Soviet bomber strength had again fallen victim to the air force's most unnerving prognostications. Presuming that Soviet goals included neutralization of U.S. retaliatory capacity, the air force whipped sketchy information into the prediction that the Soviets already had 1,400 bombers in mid-1956. The air force expected this number to grow by mid-1960 to 1,500 bombers, five hundred of them Bisons. The army disagreed strongly on the grounds that the air force lacked compelling evidence, but the intelligence community could not disprove the claims. The obvious solution, from the air force's perspective, was an increase in the main American deterrent: heavy bombers. By emphasizing the threat the air force was attempting to goad Eisenhower into lifting spending restrictions. In fact there was no "bomber gap." By mid-1956 U-2s flying over the Soviet Union revealed to the president that bomber estimates were grossly inaccurate.
Even as the alleged bomber gap dominated U.S. security concerns, the development of the hydrogen bomb (H-bomb) began to render the bomber obsolete. The U.S. government was well aware that the value of atomic weapons was on the decline. In November 1952 a test at Elugelab in the Pacific produced the first hydrogen explosion, with a total yield of 10.5 megatons. The scale of destruction that this test implied startled experts. They had predicted that within fifty square miles of "ground zero" there would be death rates of probably 100 percent and within three hundred square miles massive destruction.
The nature of the cold war had changed yet again. RAND studies soon predicted that only fifty-five hydrogen warheads of a 20-megaton yield could totally destroy fifty of the largest Soviet cities and kill over 35 million people in minutes. The detonation of the largest American thermonuclear weapon on March 1, 1954, yielding 15 megatons, only reinforced the shift. The United States had the advantage in atomic warheads, but the H-bomb would limit the utility of its nuclear arsenal. Like HMS _Dreadnought_ in 1906—an advance in military technology that made all capital-class warships preceding it obsolete—the H-bomb made atomic weapons useful but of lesser importance.
The H-bomb had startling repercussions for the ICBM program. In April 1946 the Convair Corporation had received a contract for $1.4 million from the U.S. Army Air Forces to study what was required for a ballistic missile with a range of up to five thousand miles. The government canceled the resulting MX-774 program in 1948 because of public opposition to defense spending and the seemingly insurmountable technical problems. Reactivated in 1951, the MX-774 program became the ATLAS missile system, but it was years away from becoming operational.
The H-bomb changed this sluggish pace. The Strategic Missiles Evaluation Committee, also known as the Teapot Committee, first met in November 1953 to study the impact of the H-bomb on ICBM development. Working under Dr. John von Neumann, the committee acted quickly, submitting its final report in February 1954. The Teapot report clearly indicated that the new generation of hydrogen weapons exposed the United States to greater danger. The lighter warheads and greater destructive yield would make it easier to surmount the two greatest problems in ICBM development: thrust and guidance. A smaller rocket could launch a lighter warhead and was easier to manufacture and deploy. The greater destructive power lessened the need for accuracy. Instead of having to be within meters of a target for maximum effect, the warhead could detonate several kilometers away and still destroy it. More important, the ICBM was faster, and there were no effective countermeasures or adequate warning systems. The resulting prediction was that it was the ultimate delivery system. Logically the Teapot report pushed for immediate, strong acceleration of ICBM programs under one command.
An apparently increasing Soviet bomber threat was creating apprehension, and the Soviet H-bomb only exacerbated fear of a nuclear Pearl Harbor. The air force and its supporters, including Senator Stuart Symington (D-MO), a staunch critic of the administration's defense efforts, advocated larger budgets and more bombers in response. Looking to make political gains from the threat of Soviet bombers, the senator, among others, used anxiety about a sudden attack to advance the Democratic agenda domestically. In the 1955 report of the air force subcommittee of the Senate Armed Services Committee and in his public statements, Symington blamed the president and his defense policies for the seeming decline in U.S. air power.
By early 1954 Eisenhower had been viewing the far greater bomber threat and the new ICBMs as a frightening reversal of U.S. strategic fortunes. The CIA was the administration's principal intelligence tool, and it would warn of an impending attack and monitor the world situation. But it was now obvious that its intelligence picture was incomplete, with no indication that the situation would improve. During a special NSC meeting on March 31, 1953, the director of the CIA, Allen Dulles, had admitted to the president "shortcomings of a serious nature" in intelligence. A year later NSC paper 5408, on continental defense, acknowledged continuing inadequacies: "In view of the implications of nuclear weapons in the hands of the Soviet Union, greater knowledge of Soviet capabilities and intentions is essential for military and non-military measures to reach maximum effectiveness."
Lacking meaningful intelligence on Soviet military capabilities, knowing the effects of modern war, and anxious that an unexpected attack or a stupid mistake could lead to nuclear war, Eisenhower felt that he was at a major disadvantage when it came to selecting an appropriate level of defense preparedness. Fearing that the situation could only worsen, the president turned for help to the scientists on the Science Advisory Committee, part of the Office of Defense Management. During a meeting with them on March 27, 1954, he raised his concerns about a surprise attack and the problem of penetrating a closed society like the Soviet Union. He asked them to prepare a detailed study of the issues. James Killian, president of MIT, headed the study group. The Killian Commission and its report transformed U.S. intelligence operations, especially with its call for the more active application of science to intelligence gathering. The results of this effort included both the U-2 and satellite reconnaissance.
As I mentioned, most historians see the need for intelligence as the driving force behind development of satellite reconnaissance. Historians such as Curtis Peebles and R. Cargill Hall cite the problem of gathering useful intelligence and the fear of surprise attack to explain the Eisenhower administration's move in that direction. The argument is not without merit. Both the president and the intelligence community wanted more information. The only antidote to growing anxiety about Soviet capabilities and intentions was concrete data from the CIA. The CIA's inability to provide timely and accurate intelligence necessitated a new means of gathering information. For this to happen two other elements were essential: political will and a viable alternative.
The arguments of Peebles, Day, Hall, and others focus, however, only on that aspect of Eisenhower's political agenda. By fixating solely on spy satellites' collection of intelligence, these historians have missed the wider implications of the decision to develop satellite reconnaissance. The will to act was the product of a variety of political forces, not solely the need for intelligence.
#
# Eisenhower and Defense
# Three Challenges, Three Responses (1953–56)
We face a hostile ideology—global in scope, atheistic in character, ruthless in purpose, and insidious in method. Unhappily the danger it poses promises to be of indefinite duration. To meet it successfully, there is called for, not so much the emotional and transitory sacrifices of crisis, but rather those which enable us to carry forward steadily, surely, and without complaint the burdens of a prolonged and complex struggle—with liberty the stake.
—Dwight D. Eisenhower, "Farewell Address"
If he was to accept something as radical as satellite reconnaissance, Eisenhower needed convincing. The potential intelligence from a spy satellite was a strong incentive for its development. However, such a move involved a risky technological leap of faith. After all, until _Sputnik_ orbited in 1957, space flight was only theoretically possible, and few people thought of it as likely in the near future—surely there were other, less challenging solutions to the deficiencies in intelligence.
Why, then, would Eisenhower turn to space-based reconnaissance to address the issue? Satellites in fact promised the president something very valuable: a long-term source of high-quality information on which to base U.S. policy. To understand Eisenhower's decision it is essential to grasp his thinking on defense and national security. Throughout his two terms in the White House, Eisenhower struggled doggedly with three major problems that plagued his administration, problems so systemic that he believed they threatened the country's long-term safety. The first two were products of the cold war itself and constituted his administration's primary policy challenges; he articulated these in an election speech in Pittsburgh on October 28, 1952. For him national security required balancing two massive tasks: defense of the nation's freedom against political and military disaster abroad and protection of the people against economic disaster at home. Hence he saw his first task as protecting the United States from Soviet aggression. He accepted much of Truman's cold war philosophy, including the belief that the Soviet Union was aiming the communist world's formidable power and aggressive policy at undermining the United States. He did not believe that the Soviets would resort to war but thought that the United States had to prepare for a cold war struggle of unpredictable length, probably decades.
His second problem—economic security—emerged out of the first. Facing a significant communist threat, Truman had dramatically increased defense spending; as a result rising taxes, inflation, and debt threatened the economy. Eisenhower realized that this could create far-reaching economic problems, undercutting the nation's well-being. He believed that a strong economy was the cornerstone of security policy and felt strongly that the country had to husband its economic strength, protecting itself by a level of defense spending that it could sustain indefinitely.
The third threat was the military's reaction to the first two. The military establishment and its supporters, both inside and outside of Congress, believed that the United States had to be ready to defeat communism at any time with overwhelming force. Lacking a clear intelligence picture of the Soviet Union, the military reacted by overemphasizing defense. Willing to accept any level of military expenditure, these pressure groups would fundamentally alter the country to save it. Eisenhower, however, believed that massive outlays on defense would help to create a garrison-state mentality that, in an effort to guarantee absolute security, would undermine the very fabric of American democracy. The new president had to prevent overmilitarization without compromising national security. Any understanding of his decisions about satellite reconnaissance has to consider these three challenges—national security, economic security, and pressure from the military—and how he proposed to address them.
## National Security: The Evolving Dilemma
At the start of his presidency Eisenhower faced the strategic situation that Truman left him. Like his immediate predecessor and most of his contemporaries, Eisenhower accepted many basic cold war assumptions: the Soviet regime was heavily armed, totalitarian, and hostile; it would try to expand its sphere of influence by any means possible; and its goal was world domination. He and Truman differed, however, over the immediate physical threat to the United States. Truman thought Soviet aggression would build toward an imminent attack during the mid-1950s. Eisenhower, in contrast, did not consider war inevitable; rather he saw the need to formulate a security policy that the nation could maintain over decades.
Truman formed his views on the Soviets during the tumultuous years immediately following the Second World War. Possessing no real background to help him evaluate the Soviet threat, but having experienced the shock of the news of Pearl Harbor, his administration struggled to adapt to the new cold war. In the initial euphoria over war's end in 1945, U.S. leaders hoped for peaceful coexistence; distance and the atomic monopoly helped them to feel secure. Unfortunately, when Truman was vice president (January–April 1945), President Franklin Roosevelt had kept him in the dark about U.S. agreements with the Soviets about postwar Europe. Expecting the Soviets to allow free elections in Eastern Europe, Truman felt shock at Stalin's insistence on maintaining a strong sphere of influence in that region, especially in Poland. Between the Eastern European issues and the apparent pressure by communist forces in Greece, Turkey, and the rubble of Western Europe, the U.S. president faced what looked like an escalating Soviet threat. Between 1946 and 1948 he attempted to devise a policy that would contain Soviet expansion and allow some measure of peaceful coexistence. NSC 20 (November 24, 1948) encapsulated U.S. policy, calling for a buildup in deterrent power while strengthening other nations to counter communist actions. The 1947 Marshall Plan to rebuild Europe and the Truman Doctrine pledging U.S. support for nations resisting oppression are excellent examples of the latter.
As long as the United States had a monopoly on atomic weapons and the protection of distance, a direct Soviet attack seemed unlikely. But the Soviet atomic bomb of 1949, the "loss" of China to the communists that year, and the sudden start of the Korean War in June 1950 shook the U.S. government. Suddenly the Soviets seemed to be actively strengthening international communism. In response the administration reexamined national security policy and adopted a harder stance toward the Soviet Union. On September 30, 1950, Truman formally adopted the pivotal policy paper NSC 68, much darker and more dire than NSC 20: "The issues that face us are momentous, involving the fulfillment or destruction not only of this republic but of civilization itself." According to the Truman interpretation that informed it, communism was inherently hostile to the United States and advocated a policy that was diametrically opposed to the American political worldview. Armed conflict between the superpowers seemed inevitable. The communist leaders, keen to expand their influence, would use any means available, violent or nonviolent, to subvert noncommunist governments. Truman saw his country as the only one that could arrest this influence.
NSC 20 and NSC 68 differed on the scale of the threat. NSC 68 described Soviet nuclear capabilities as extensive. The authors, especially the principal writer, Paul Nitze, believed that the Soviet Union would attack the United States if it had the military advantage. As long as it believed that it could not attack successfully, the balance of power would preserve peace. Unfortunately Soviet progress was chipping away at the American lead in technology and research in such key areas as nuclear weapons.
Using available information, NSC 68 identified 1954 as the year when the balance would shift to the Soviets and an attack would occur. It insisted that the United States should seek to prevent this outcome by building up its strength, both military and economic, to provide a credible deterrent. It proposed that remaining more powerful than the Soviet Union would allow the Americans to hold onto the balance of power. Therefore it called for massive increases in defense spending and for notable corresponding growth in the deficit and in taxes to cover increased costs for national security.
In the wake of the Korean attacks Truman embraced NSC 68. He authorized a rapid increase in spending in an effort to provide short-term security against the threat of Soviet attack by 1954–55. The annual defense budget went from less than $13 billion (approximately 5 percent of the gross national product, or GNP) in 1950 to $60 billion (18.5 percent of GNP) in the spring of 1951. By 1952, according to Stephen Ambrose, Truman projected more than $50 billion for defense to prepare for the "year of maximum danger." But once the Korean War began to stabilize and a U.S.-Soviet war became almost a nonissue, Truman started to slow the military's expansion while keeping spending on it higher than prewar levels. The result was a series of budgetary deficits and a "feast-to-famine" pattern in defense spending.
Eisenhower thought Truman's handling of the cold war flawed. The core of the problem lay in his predecessor's buildup of American strength to meet the perceived threat of a "year of maximum danger"—the exact opposite of what Eisenhower thought the appropriate response of a democracy. Instead of panicking, a democratic government prepares militarily only on a defensive (i.e., long-term) basis. As Eisenhower noted in his personal diary, "We do not attempt to build up to a D-day because, having no intentions of our own to attack, we must devise and follow a _system that we can carry as long as there appears to be a threat in the world capable of endangering our national safety_." Since the cold war was a long-term confrontation, no single day could hold greater risk for the United States; rather the threat would last, and a sustainable defense was essential for the long run.
Eisenhower considered the notion of a "year of maximum danger" a fallacy in keeping with the traditional U.S. pattern of "boom-and-bust" military preparedness. He found that the idea collapsed in the face of logic and reason. The fixing of such a date was highly subjective and invariably shaped defense spending. When the threat seemed particularly serious, military expenditure grew, as in the wake of Korea and NSC 68, and when the threat appeared to recede or become less severe, it waned. Truman's approach had been disastrous, with the government ordering expensive equipment one year and then delaying or canceling orders the next. The dizzying oscillation of defense spending between 1945 and 1953 did more harm than good, preventing the military from modernizing and maintaining steady capability.
While serving with the Joint Chiefs of Staff (JCS) in 1948 (and indeed as early as 1946), General Eisenhower had pushed for a constant level of spending—on the order of $15 billion per year—to maintain an adequate military. He anticipated that this amount would allow sustainable defense and steady development and modernization. At the same time, a fixed budget would permit (and force) the services to allocate their resources more effectively. In the long run such an arrangement would have helped to prevent inflation and to preserve the strength of the U.S. economy.
During an April 1953 meeting with Republican congressional leaders, the chief executive, sick of criticisms that he would harm the air force by changing spending patterns, lashed out at those people who predicted that the country would be in more danger on any particular day, describing the idea as "pure rot" and adding that he had opposed this delusion all along. "I have always fought the idea of X units by Y date. I am not going to be stampeded by someone coming along with a damn trick formula of 'so much by this date.'" He argued that there was no way to achieve maximum military effectiveness in peacetime. That would require full mobilization of troops, which the United States could do only in war. There was no single perilous moment but rather "an age of danger." The situation required an indefinite defense.
Eisenhower's concern over the long-term nature of the cold war only grew with the presence of nuclear weapons in large numbers. His views on the matter were clear as early as 1946, when in a letter to his boyhood friend Everett "Swede" Hazlett, he described them as a "hellish contrivance." They made war vastly more destructive, spreading its damage worldwide. For Eisenhower atomic weapons made general war obsolete because they had changed the nature of international conflict. Conventional war was no longer feasible because, first and foremost, it would inevitably escalate into a nuclear holocaust. The advantages that came with the possession of nuclear weapons would create an overwhelming temptation to use them at the outset of a war in an effort to gain military advantage. The United States, however, did not have the luxury of employing them first. As a democratic nation it could not attack with such devices but could only keep them in readiness to counterattack and to paralyze the enemy at the start of hostilities.
Following on the heels of this understanding came Eisenhower's second crucial insight into the nature of the cold war: for the first time ever war could devastate the United States. He tried repeatedly to convey the scale and implications of this destruction to his cabinet, the NSC, and members of Congress. In a short off-the-record speech in 1954 during a visit to Quantico, Virginia, he made it clear that there would be no victor in nuclear war. He could not bring himself to call the prospect of wiping out Soviet cities and capacity to wage war a victory. In the wreckage of that nation, where a government and functioning society had existed, there would be only a vast territory of devastation and ruin. The United States would have defeated the Soviet Union, but Eisenhower saw no political gains in creating such a wasteland. His views were clear: "Here would be a great area from the Elbe to Vladivostok and down through Southeast Asia torn up and destroyed without government, without its communications, just an area of starvation and disaster. I ask you what would the civilized world do about it? I repeat there is no victory in any war except through our imaginations, through our dedication and through our work to avoid it." The devastating character of nuclear conflict had transformed the pace and tempo of warfare. This was strongly evident in the president's statements during a February 1955 meeting with leaders from Congress, where he dismissed the idea of sending troops to Europe after an attack. The vaporizing of American cities and infrastructure would rapidly degrade military capacity. The United States would not have the luxury of a protracted offensive buildup like that of 1941–44. The ability to marshal troops and to fight would disappear as moving, supplying, and replacing units (not to mention controlling them) became virtually impossible. Within the first few days both sides would tire themselves out. With destruction of cities from the air, it would be the task of the army to bring order to the chaos. The main role for the U.S. Army was first and foremost a domestic one: reestablishing control within the continental United States. Anyone who argued differently was "just talking through his hat. It couldn't be done and if I tried to do it, you would want to impeach me."
By January 1956 Eisenhower was painting an even bleaker picture of the consequences of a nuclear attack. During an NSC discussion about stockpiling strategic materials in case of war, the president's tone left no one with any illusions. While attempting to describe the possible scale of destruction, Eisenhower lamented, "We were simply unequal to imagining the chaos and destruction which such a war would entail." Rapid victory was out of the question: following the initial nuclear blows, he expected, the nation would be in ruins, with massive casualties and devastation of ports and cities. Although damage would be severe on both sides, the war would not be over. To ensure victory the United States would have to invade the Soviet Union and confirm that it could no longer fight, but repairing its own infrastructure and rebuilding its production to launch such an expedition would take at least three or four years. In that time it would not be able to conduct conventional operations overseas.
In June 1954 Eisenhower set up an evaluation subcommittee to assess Soviet ability to inflict direct damage on the United States. This group—the chairman of the JCS, the director of the CIA, and several other key members of the administration—met periodically throughout his presidency. Its reports estimated Soviet damage under a variety of circumstances. The initial review provided both a worst- and a better-case scenario for a hypothetical attack on the United States at a predefined date. On January 23, 1956, some eleven days after the NSC meeting where the president had argued that it would take Americans years to dig out of the devastation, the subcommittee reported to him. A surprise attack on July 1, 1956, would, it projected, inflict shocking damage and destruction. Approximately 65 percent of the population would require medical care that would no longer be available. With total economic collapse and disintegration of the government, the country would be in a hopeless state. Even a one-month warning made for no sizable improvement. Although the United States expected to inflict three times as many casualties on the Soviet Union, the numbers were staggering to the president.
In December 1956 the subcommittee again met with the NSC, and its projections for an attack in 1959 were even more dire. With both sides dramatically increasing their nuclear stockpiles, there would be massive damage. The United States could expect approximately 40 percent of its people to die and an additional 13 percent to sustain serious injuries and require medical attention at a time when it was scarce, if available at all. American retaliatory capability was the only factor preventing the Soviet Union from emerging as the dominant power within a day of the attack. By 1957 the forecast for an attack in 1960 showed no major changes, except for casualty figures: roughly half the people of both countries would die outright, with equivalent devastation to the structures of society. As we saw, Eisenhower had a dim view of general war as a viable strategy. To Richard L. Simon of the Simon and Schuster publishing house the president clearly let his views be known: "War implies a contest; when you get to the point that contest is no longer involved and the outlook comes close to destruction of the enemy and suicide for ourselves—an outlook that neither side can ignore—then arguments as to the exact amount of available strength as compared to somebody else's are no longer the vital issues." Eisenhower pointed out that both sides would soon have sufficient nuclear strength to bring about total annihilation of one another: "Already we have come to the point where safety cannot be assumed by arms alone. But I repeat that their usefulness becomes concentrated more and more in their characteristics as deterrents than in instruments with which to obtain victory over opponents as in 1945."
He saw the nuclear deterrent as a hopeful sign. Knowing that the United States could not initiate a war, he believed strongly that the Soviet Union would also refrain from doing so. From the start of his administration he argued that anyone—including Soviet leaders—who knew the weapons' destructive power would never use them. Although accident or miscalculation was still possible, the president thought it inconceivable that anyone would launch a nuclear strike. In July 1954 his administration took the position that increasing nuclear capabilities on both sides would make warfare less likely. Since total war meant total destruction, the very presence of nuclear weapons helped to prevent hostilities. By 1958 this stance had evolved: the nuclear deterrent rendered it impossible to imagine nuclear war arising from a conscious choice. If it did happen, it would be the result of an irrational act or accident, not of advanced planning.
The nuclear deterrent had changed the nature of the cold war. While Eisenhower rejected Truman's view that a war with the Soviets was inevitable in the near future, he did accept that the United States needed to contain communism indefinitely. The indefinite nature of the cold war was the real danger. It required the United States to be ready to fight not on a particular day but perpetually if it was to maintain a credible deterrent. The cold war was a long, drawn-out struggle that could span generations, pitting the entire strength of both superpowers against each other. Eisenhower felt that he had to educate the military about this reality.
The key to preserving peace was maintaining a balance of power. Rejecting the temptation to flaunt military strength, Eisenhower did not take a bellicose stand against the Soviets. He spelled out U.S. strategy in a memorandum of February 1953 to the NSC. The United States had to block Soviet expansion, decrease its power and influence, and develop the free world's strength to contain the Soviets politically and geographically. To do this the United States had to establish and sustain, for as long as necessary, a state of limited mobilization for war. This meant maintaining sufficient conventional military strength to deter Soviet aggression, developing the capability to mobilize rapidly in case of war, and backing it up with a nuclear arsenal second to none.
In 1953 the administration conducted an in-depth study of policy alternatives to prepare itself better to wage the cold war. Eisenhower was a big supporter of study groups to provide detailed recommendations in response to specific problems. In May 1953 he formed Project SOLARIUM, consisting of a group of scientists and other experts, to examine three alternative courses of action and report back to the NSC. Each panel looked at a different approach to dealing with the Soviet Union. The first, Task Force A, was chaired by George F. Kennan and included key military and civilian figures, many of whom had helped to create the policy of containment. Since 1948 this policy had called for the preservation of armed strength and a strong economy to deter communist aggression and expansion without increasing the risk of general war. Believing that the inherent contradictions and flaws in the Soviet Union would lead to its collapse, Task Force A focused on continuing containment. It called for a flexible policy to ensure that the United States could adapt to changing situations, but the danger was that the government would fixate exclusively on containing and destroying the Soviet Union. By emphasizing military containment of the Soviet Union, the United States ran the risk of losing its support from the free world, which feared another war. Therefore it had to focus not just on military preparations but also on psychological, economic, and political warfare.
Task Force B included many figures who played a greater role in the Eisenhower administration, including Lt. Gen. James H. Doolittle, Douglas MacArthur Jr. (council for the State Department), and Maj. Gen. James McCormack; it examined a policy position that emphasized a more forceful stance on Soviet expansion. The United States should "complete the line now drawn in the NATO area and the Western Pacific so as to form a continuous line around the Soviet bloc beyond which the U.S. will not permit Soviet or satellite military forces to advance without general war." Thus it had to maintain the military capacity to fight a general war against the Soviet Union for an almost indefinite period. Task Force B felt that this policy, though taking a hard line on Soviet expansion, meshed well with other policy positions and provided optimal flexibility.
Task Force C included Lt. Col. Andrew J. Goodpaster (later a key aide in the administration), Lt. Gen. Lyman L. Lemnitzer (military planner, foreign affairs expert, and future supreme allied commander, Europe), and Frank G. Wisner (deputy director of plans for the CIA). Its report articulated the principle of "roll back." Accepting the other two task forces' approaches as a backdrop for a far more proactive policy, it advocated combining strengthening of the West to resist Soviet expansion (containment) with an aggressive political strategy that included covert, diplomatic, economic, military, and propaganda attacks on the Soviet Union. In the long term this policy would, it hoped, disrupt the Soviet Bloc's control over its territory, accelerate popular resistance, and exploit weaknesses within the Soviet Union—all in the cause of "rolling back" communism.
Following the presentations of the three task forces in July 1953, Eisenhower summarized his understanding of these views to the entire group at Project SOLARIUM. He concluded from the project's deliberations that containment remained the best strategy for fighting the cold war. The only thing worse than losing a global nuclear war, he stressed again, was winning one; another war would destroy individual freedoms—a defining characteristic of American society. Thus U.S. policy must protect the nation and prevent open war. It is significant that the president emphasized the economic costs of defeating the Soviet Union. Looking at the long haul, he made it clear that demanding too much of the economy (and by extension the citizens) could lead to federal interventions in the economy that would erode individual liberty and rights. This was something that he was not willing to contemplate.
SOLARIUM confirmed for Eisenhower that containing communism was the only practical, sustainable strategy. He realized that the United States needed not overwhelming military force but rather a strong long-term deterrent, which nuclear weapons could supply. To be effective the strategy had to involve two elements: integration of atomic weapons into defense planning and development and a clear and straightforward policy for their use, about which the United States would inform the Soviet Union. How, then, to develop a strategy for use of suicidal weapons and yet make it credible enough for opponents to believe?
Eisenhower's solution was "massive retaliation." Secretary of State John Foster Dulles first spoke about it to the Council on Foreign Relations on January 12, 1954. A nuclear deterrent was to be the cornerstone of long-term U.S. security. The strategy played to American strengths. In Eisenhower's words, the United States "cannot be strong enough to go to every spot in the world, where our enemies may use force or the threat of force, and defend those nations." Instead it would rely on a two-step solution. First, local forces, to which the Americans would extend limited aid, were to contain aggression as and where it occurred, and, when necessary, mobile U.S. reserves would join in the effort, along with naval and air support. By not committing massive ground forces, the United States would avoid frittering away scarce resources on hundreds of small battlefields. By building up the noncommunist world, Eisenhower expected to export some of the cost of a large military establishment and also expand the forces opposing communism.
Second, the Americans would expand their own nuclear forces and continental defense. To prevent large-scale Soviet expansion and general war, they would turn their nuclear advantage into an umbrella of protection. The administration made it clear that when it faced such a crisis, it would react by using its greatest strength, long-range bombers carrying nuclear weapons against what it considered the "heart" of the problem: the Soviet Union. A preponderance of atomic weapons countered massive Soviet conventional forces. Knowledge of the devastating consequences of an attack in Europe, or elsewhere, would contain Soviet aggression. For Eisenhower, then, the key to diminishing the risk of a large, general war was maintenance of a credible deterrent in the form of nuclear forces ready to attack the Soviet Union.
## Economic Security: Stabilizing Defense Spending
The president's realization that the cold war could last indefinitely went hand in hand with his awareness that the military threat masked a more subtle danger to the core of American strength: the economy. He firmly believed that his was the best country in the world because of its individual freedoms and the material, intellectual, and spiritual opportunities that it provided for its people. Chief among the advantages it offered was the free-market economy. Eisenhower worried that the high cost of a military buildup might irreparably damage the U.S. economy. By expanding its defenses the nation might actually undercut its long-term ability to maintain them. This would force the federal government to make changes to the very fabric of society in the name of security.
The economy that Eisenhower inherited had experienced some dramatic upheavals since 1945. Truman had managed to avoid a severe postwar depression and had controlled unemployment thereafter, but the task had not been easy. Unemployment had increased dramatically, reaching 2.7 million by March 1946. As well, wholesale dismantling of the military reduced it by 1947 from over 12 million personnel in uniform to 1.5 million. These years also saw a series of clashes with labor. Demanding higher wages, automobile workers, steel workers, and coal miners went out on strike four times before Truman left office. Twice, in an effort to end a strike, the chief executive seized coal mines and the steel industry to restore production. Nonetheless he kept the economy expanding. Between 1946 and 1952 the GNP increased consistently, rising from $209 billion to $346 billion. A great deal of this growth resulted from deficit financing and the Korean War.
For Eisenhower, who promised the nation "security with solvency," a major concern was the oscillation in defense spending under Truman. The defense budget for 1945 stood at $73 billion, or about 77 percent of the total federal budget. In the wake of the war and in response to strong pressure to demobilize, defense spending dropped rapidly. By 1947 expenditures hit an all-time postwar low of just over $9 billion, or about 24 percent of the overall budget. The figure for 1950 was slightly larger, at roughly $14 billion (33 percent of the federal budget). It was not until the Korean War and the adoption of NSC 68 on September 30, 1950, that the amount again began to soar. It jumped to over $33 billion in 1951 (74 percent of the budget) and reached $48.7 billion in 1953. And the defense budget for fiscal year 1955 that Eisenhower inherited was even larger: $46.3 billion, out of $78.6 billion. Truman left an overall national debt of over $275 billion. Thus since 1950 the greatest proportional increase in the national budget had occurred in defense, which by 1953 made up about 67 percent of the total budget.
Eisenhower's views on the economy were fundamentally the same as those of the rest of his party. Like most Republicans he subscribed to the concept of government and economic policy that Seymour Harris has described as traditionally Jeffersonian. Eisenhower believed in thrift, hard work, and minimal government interference in business. He felt that a government created the proper climate for prosperity when it kept taxes and spending low. According to the economic historian Craufurd Goodwin, Eisenhower, like all Republicans, avoided meddling in the economy too extensively because he believed that it was self-regulating. The more the government intervened in it, the greater the problems.
The economy drew a great deal of Eisenhower's attention both before and while he was in the White House. Throughout the 1952 campaign, in a series of speeches, he hammered home its role as the foundation of American strength. In August 1952, addressing the American Legion Convention in New York City, he went to the very heart of the matter: "We must keep America economically strong. Even our great military effort must not break our great competitive system because in the combination of American spiritual, economic and military strength is the cornerstone of a free world." 43 One month later he went further, identifying the economy as a direct target of the nation's enemies. In October 1952 he again emphasized this point when he spoke in Jackson, Michigan, and in New Brunswick, New Jersey. Stating that "a free America must be the cornerstone of any free structure in the world," he argued that its people must maintain "our scientific strength, our productive and industrial strength, our financial strength to keep that economy sound. We must stay solvent." 44
The danger was that the deficit of roughly $9.4 billion that Eisenhower inherited from Truman would lead to increased inflation. His predecessor had also authorized $81 billion in appropriations, and this increase in the nation's debt seemed a portent of economic stagnation. By July 1953 the administration had to borrow more funds to cover its predecessor's bills, raising the national debt to more than $272 billion and forcing the president to ask Congress to increase the debt limit. Eisenhower had good reason to fear inflation: it decreased the dollar's buying power, thus making goods more costly. If the dollar bought less, the government would have to pay more for defense, which would increase the deficit or taxes. This in turn ran the risk of creating an inflationary spiral. Eisenhower believed that the government had to balance the budget so as not to increase the deficit and, where possible, to start paying off accumulated debt.
The largest percentage of the budget went to national security, mainly the Department of Defense and the Atomic Energy Commission (AEC). In FY 1954, Defense, the AEC, and the Mutual Security Program used up about 70 percent of the budget, or roughly $50 billion. Eisenhower worried that the country was living beyond its means, spending more on defense than the economy could support in an effort to gain "perfect" security. In his view controlling defense spending was the key element to balancing the budget. The government had to find a workable balance between maintaining sufficient strength to counter possible Soviet aggression and ensuring economic solvency. Ideally this meant an austere defense budget in the range of $36 billion to $38 billion per year. Eisenhower hoped to accomplish this sort of restraint with a combination of constantly upgraded, modern defenses, strong reserve forces available as necessary, and a vibrant economy. Failing to hold down defense spending would bring about a high risk of inflation and, eventually, severe government controls on the economy and a "garrison state" mentality. Since the armed forces existed to preserve a "way of life," not just property and territory, Eisenhower believed that to change the essence of the American system to defeat the Soviet Union would mean collapse in the long run.
These ideas were not new to Eisenhower. From 1946 to 1948, while serving on the JCS, he had pushed for a constant level of spending. The Truman administration, however, under pressure to demobilize and reconvert the economy to consumer production, was slashing defense spending. In 1948 Eisenhower lamented in his diary that inflation had reached the point that even if the defense budget dropped to a constant $15 billion annually it would not be enough to meet U.S. security commitments. By increasing government spending through deficit financing and by raising taxes, the administration eroded the dollar's value and hurt the economy. As the dollar fell in value, the economy and the American people suffered.
The danger that the cold war posed to the economy was a central feature of Eisenhower's discussions while onboard the USS _Helena_ during his return from Korea in 1952. He proposed a straightforward solution that balanced needs and resources, a middle course between rearmament and a sound economy.
Economic considerations loomed large in NSC meetings, especially in Eisenhower's first year in the White House. His first State of the Union Address, on February 2, 1953, enunciated the issue succinctly: "Our problem is to achieve adequate military strength within the limits of endurable strain upon our economy. To amass military power without regard to our economic capacity would be to defend ourselves against one kind of disaster by inviting another."
In a draft statement of February 1954 to the NSC about continental defense, he elevated the economic issue into an integral part of the struggle to win the cold war: "The survival of the free world depends upon the United States maintaining: (a) sufficient strength, military and non-military, to deter general war, to prevent or counter aggression, and to win a general war if it is forced upon us; and (b) a sound, strong economy, capable of supporting such strength over the long pull and of rapidly and effectively changing to full mobilization." He also clearly laid out his position to senators during a private meeting in April 1956: "If we attempt to match soldier for soldier, weapon with weapon, etc., with the Russians, we will bring on ourselves at the very least economic suicide." He insisted that "a bankrupt America is more the Soviet goal than an America conquered on the battlefield." With defense portions of the U.S. budget so large, overspending had a major impact. The resulting rise in taxes diverted money from the productive private sector to the public purse, which was not productive.
In response the president sought to correct problems in planning and spending for national security to preserve the economy. Foremost he dismissed NSC 68's conception of a "year of maximum danger." In a radio address on May 19, 1953, he spelled out the economic reasons for his rethinking of cold war strategy. He argued that the Soviets wanted the United States to overspend on security in order to precipitate an economic disaster. He called for reason and restraint in defense spending. He thought that the defense program "must, first of all, be one which we can bear for a long—and indefinite—period of time." He would not accept a pattern of "sudden, blind response to a series of fire-alarm emergencies, summoning us to amass forces and material with a speed that is heedless of cost, order and efficiency." Booms in spending on defense usually gave way to smaller budgets when the threat did not materialize; preoccupation with numbers was a hallmark of this sort of unrealistic thinking. Well aware that no fixed number of ships, planes, or guns (let alone dollars) could guarantee security, the president pushed for constant spending that guaranteed national security over an extended period without crippling the economy.
Fluctuations in defense expenditures harmed the military budget and had far-ranging repercussions on the nation as a whole. The largest single component of the annual U.S. budget that the administration could adjust without legislative agreement was defense spending. Its ups and downs affected taxes, the deficit, unemployment rates, inflation, and more. Thus preparing for war by a specific date was more disruptive than beneficial. Spending highs and lows affected the whole economy, and if war did not come by the appointed time, what then? Producing and training a large military force with very expensive equipment would have caused major interruptions to the economy. But once it was clear that war would not come, demobilization would follow, shifting production back to civilian uses and again disrupting the economy while it adjusted back to "normalcy."
Eisenhower relied on his belief that the Soviet Union was not seeking a general war and pushed at every turn to make defense spending sustainable. Military expenditures, unlike consumer goods, massively drained valuable resources. In the domestic market, goods would spur economic growth; military items, however, were expensive and quickly became obsolete. Despite the income that military production generated, the goods that it produced were a drain on the economy. In a speech before the American Society of Newspaper Editors on April 16, 1953, Eisenhower eloquently linked the costs of large-scale defense buildups with their domestic consequences:
Every gun that is made, every warship launched, every rocket fired signifies, in the final sense, a theft from those who hunger and are not fed, those who are cold and not clothed. . . . The cost of one modern heavy bomber is this: a modern brick school in more than 30 cities. It is two electric power plants, each serving a town of 60,000 population. It is two fine, fully equipped hospitals. It is some 50 miles of concrete highway. We pay for a single fighter plane with a half million bushels of wheat. We pay for a single destroyer with new homes that could have housed more than 8,000 people.
Anyone could understand this assessment of the cost of the arms race. At best, large-scale expenditures on defense could help protect what a country had; at worst, they deprived people of basic necessities.
Eisenhower also disagreed with NSC 68's assumption that as much as 20 percent of GNP should go to defense, supported by deficit financing. He thought this ran contrary to common sense. Instead he ordered his administration to start cutting expenditures and balancing the budget. Despite immediate and sharp cuts in spending across the board, Eisenhower was realistic enough to understand that achieving a balanced budget would take time. The administration's first task was to slow the rate of spending of the last Truman budget. This resulted in some confrontations in April 1953 with Senator Robert Taft (R-OH) and other Republican leaders over the failure to cut defense spending substantially. Taft was particularly vocal about the need to reduce taxes and balance the budget. Eisenhower pushed Defense to begin eliminating nondefense costs, for example in administration, procurement, and overhead; eliminating duplication and waste at this level would save large amounts. The administration also began preparing a comprehensive "inventory" of the entire defense establishment to serve as a base line for understanding what the military had and to act as a foundation for future cuts.
The president realized, however, that cutting back on overhead and running Defense on an austere budget would not by itself stabilize defense spending. The administration turned to reorganization of the defense establishment in terms of the new realities of nuclear weapons and the urgency of economy. It started with cutting back on force size, keeping only key units, such as the elements of SAC necessary for retaliation and deterrence, at full strength. All units continued, but most of them with less than full strength—an unpopular move, particularly with the army. However, this was only the beginning. On July 1, 1953, Eisenhower ordered the JCS to reexamine the whole defense posture with an eye to countering long-term Soviet aggression. This scrutiny covered not only strategic concepts but also roles and missions and many other areas, with emphasis on reducing costs. The resulting "New Look" was to maximize every defense dollar to ensure both security and affordability.
Twice the JCS reported back to the president on this matter. On August 27, 1953, it presented its views and findings to the NSC. The four service chiefs had prepared this first look by the military at economy. Concentrating on budgetary issues, the report recommended reductions in forces abroad (especially in Japan, Korea, and central Europe) as cost-cutting measures. The United States would keep foreign bases but reduce the strength of overseas forces, transferring many of their tasks to local troops. The JCS believed that this could save money and ease the overextension in personnel; strategic and tactical nuclear weapons would offset decreased force sizes. The only real opposition to this document came from Gen. Matthew Ridgway, the army's chief of staff. He objected to the removal of forces from overseas bases and did not want the United States to rely solely on the nuclear deterrent and the air force. He felt that recent experience, as in the Korean War, suggested that it would be unwise to reduce the military's limited war capabilities and to rely solely on deterrence. Clearly he was also trying to protect the army's share of the budget.
On October 13, 1953, there was a follow-up discussion of the defense budget for FY 1955. Since neither the Soviet threat nor U.S. security policy had changed and the administration had not issued a clear directive on use of nuclear weapons, Assistant Secretary of Defense Wilfred McNeil and the JCS presented a budget with few major spending cuts. McNeil predicted a defense budget for FY 1955 of $43 billion for combat forces alone; the budget for support elements was still being drafted. There would be probably about 3.5 million people in uniform. The administration's reaction was severe, with Secretary of the Treasury George Humphrey one of the most vocal critics. Predicting that the Department of Defense would spend $48 billion in FY 1954 and $47 billion (including carryover funds) in FY 1955, Humphrey insisted on cuts in the budget. The president reacted just as strongly, pointing out that the need for austerity meant that the military should be reducing its personnel ceilings and force sizes and reexamining its posture. For the military a key element in reducing spending was the extent of reliance on atomic weapons. Eisenhower was emphatic that the military should use these devices at any tactical level if an attack occurred. NSC 162, adopted a few weeks later, normalized their use if war started.
Eisenhower insisted that Defense cut spending by about $4 billion in FY 1954 and by $6.6 billion in FY 1955. When the JCS refused to reduce expenditures, civilian leaders adjusted the budget before it went to Congress on May 7, 1953. The submission called for total spending of $43.2 billion for FY 1954, about $35 billion of it new spending. Overall this budget was below Truman's estimate by $5 billion. The air force lost roughly $4.7 billion, dropping to $13.7 billion, which forced a reduction in force size from 143 air wings to 120, only 114 of them likely to be operational by mid-1954. The navy's allotment fell by $1.7 billion to $9.8 billion. The army's actually went up, from $12.1 billion to $13.7 billion, to cover the final expenses of the Korean War.
Such reductions occurred in virtually every Eisenhower budget. FY 1955 was to see new defense spending fall by $3.5 billion and previous expenditures by $4 billion. The overall defense budget was $37.6 billion (57.3 percent of the national budget). The army lost $4.6 billion and had to reduce its standing forces from twenty divisions to seventeen, for a loss of 317,200 personnel by mid-1955. Facing a decrease of 51,000 personnel owing to a loss of $1.5 billion, the navy cut its strength from 1,126 ships to 1,080. Reflecting Eisenhower's emphasis on massive retaliation, only the air force was able to expand. Rising from 955,000 personnel to 970,000, its wings increased from 115 in number in mid-1954 to 120 by mid-1955, with a projected goal of 137 by 1957. Brushing off protests from the army, the House Appropriations Committee cut the 1955 budget by an additional $1.1 billion, half of this from the army.
This pattern remained through most of Eisenhower's two terms. The only exceptions were FYs 1956 and 1958, when Congress instigated small increases. The defense budget for FY 1961 stood at roughly $41 billion.
The resulting New Look represented an attempt to readjust the U.S. strategic posture from preparation for war by a given date to sustained effort over the long haul. The U.S. advantage in nuclear weapons and strategic bombers became the focal point. By integrating atomic weapons into defense strategy down to the tactical level, and emphasizing their destructive power, the United States attempted to magnify their deterrent value across the full spectrum of operations, but it could now reduce its conventional forces. Although Eisenhower described the New Look as a reallocation of resources that emphasized nuclear deterrence, the budgetary savings were equally important.
Unfortunately for the president, however, atomic and hydrogen weapons, although providing a means of reducing expenditures, were a double-edged sword. Soviet possession of the H-bomb, following their successful test on October 12, 1953, created anxiety within the U.S. defense community and the nation in general. The result was a greater desire for stronger continental defense. The support of the House and the Senate was important, but it was the military that felt the overriding need for greater efforts to prevent a "nuclear" Pearl Harbor. Eisenhower's belief that deterrence was the strongest defense, which sentiment the Defense Department echoed, did not find universal acceptance. More to the point, his initial attempts to reduce the defense budget put considerable pressure on the president.
The New Look was a compromise between containment and financial overreach. Eisenhower predicted that the defense budget would level off by 1959 at about $30 billion—$35 billion per annum—less than Truman's projections but more than the department was spending before the war in Korea. The primary deterrent force consisted of Strategic Air Command's bomber forces, which could deliver large-scale nuclear strikes in accordance with the doctrine of massive retaliation. Conventional forces would become a mobile strategic reserve smaller than what most critics expected. At the same time, the United States encouraged Allied forces such as those in the North Atlantic Treaty Organization to be more active in their own defense, thus allowing greater U.S. reductions. To this end Eisenhower reinstituted spending caps on Defense to limit its share of the budget. In short, by relying on nuclear weapons he hoped to justify a more limited conventional force and save substantially in defense spending.
Eisenhower solved the dual military and economic challenges of the cold war by creating a three-part, "layered" defense against Soviet aggression. First, the nuclear deterrent would contain the Soviet danger indefinitely. Second, the concept of sufficiency dictated U.S. deployment of enough forces (including atomic weapons) to augment local military forces in threatened areas. Nuclear weapons and local forces would provide adequate defense without damaging the U.S. economy. The third layer was the U.S. economy itself.
## Pressure from the Military: Addressing Interservice Rivalry
Eisenhower's identification of the problems and the appropriate response did not result in an immediate solution. Rather they fueled a third major issue that haunted his presidency: chronic interservice rivalry over funding, which the cuts exacerbated. The president estimated in 1957 that he spent about two-thirds of his time fighting pressure from the competing services to increase spending on defense. While he was looking at the "big picture" and the nation's long-term survival, the military focused on its primary mission: defense. The services' secondary, unstated mission—protection and expansion of their share of the budget—directly reinforced their principal task.
Force reductions after war's end in 1945 had generated competition for limited funding. Each military branch played up its own ability to protect national security to ensure its continuing role and its need for more funding. Eisenhower was very familiar with the issue from his time as informal chair of the JCS. Seeing up close the rivalry between the army air forces and the navy over funding, he realized that Defense was causing many of its own problems. The larger its budget slice, the more important a service felt and the more prestige it assumed. The capacity to fulfill missions was not the issue; indeed the services were adept at creatively expanding their missions in an effort to justify more funding. Their tendency to run to Congress and the public with their convictions and complaints just raised the stakes.
The problem became so severe by 1949 that General Eisenhower advocated drastic controls. First, the establishment of "majority rules" within the JCS would mean that all the services would have to come to a joint decision and then carry it out cooperatively. Second, the services had to realize that the president (i.e., Truman) was in charge. According to Eisenhower's personal diary, he felt that only the chief executive could resolve the issue: "I believe the President has to show the iron underneath the pretty gloves, some of our services are forgetting that they have a Commander in Chief. They must be reminded of this, in terms of direct, _unequivocal_ language." Eisenhower feared that the debates would one day erupt into a major confrontation that Washington papers and Congress would hear about, which would probably cause major problems in budget appropriations and in congressional committees. Only the president could prevent loss of control over the process.
By 1949 the problems were too large for Eisenhower to correct. He felt that the members of the JCS so assiduously followed the doctrinal positions of their own services that they were unable to do their jobs, which involved a profound responsibility to the nation. The president and the secretary of defense would have to fix the problem. The future president was also aware that the military was not averse to playing up perceived threats to increase funding.
In January 1952 his intimate knowledge of interservice rivalry led him to think carefully about the Truman budgets that called for $85 billion per year in defense spending. He saw the budget's size as a clear sign that the military was not doing its primary duty. Its obsession with spending and budgets and weapon systems had blinded it to other threats to the American way of life, such as economic stress.
By the time Eisenhower entered the White House, the pattern of interservice rivalry was firmly in place. The new chief executive found within the military and other branches of government a tendency to approach everything from the perspective of bureaucratic self-interest. Looking narrowly at specific programs and special projects, or simply at increasing their share of the governmental pie, they were reluctant to cut favored projects (such as the atomic-powered bomber), scale back on spending, or think about the budget from the perspective of a team working for the nation. In fact the military was willing to use the very democratic process that it was defending against the administration. The services proved adept at public relations. They launched aggressive campaigns in support of particular projects or goals, manipulating the media through Pentagon leaks and press conferences. Pet programs and even entire services thus appeared in the best possible light, occasionally at the expense of the other services, all in an effort to preserve or expand funding. Eisenhower wanted the public "to have a complete faith in the services—that is what he is working for." Unfortunately, because of the rivalry "the American public has lost a large measure of confidence in the services." The competition had become so intense and overt that the public began to question whether the military could do its job.
It was almost impossible for most people to judge the legitimacy of the armed forces' claims. Much of the general population had no real experience with the military and saw no reason to question its demands for new equipment or more forces. The military overcame objections of knowledgeable observers about procurement or budgetary issues by controlling the process of research and development. The first step in developing a new weapon system was demonstration of either a threat or a defined need. The source of the military's information was inevitably military intelligence, and that was a major part of the problem. Deputy Secretary of Defense Roger Keys argued in November 1953 that the services—especially the air force—were notorious for "sales promotion intelligence," or manipulation of intelligence to justify new projects. The practice resulted in large and overcharged programs of research, development, and procurement. The exaggeration implicit in worst-case scenarios accelerated requests for more equipment, personnel, and funding than was strictly necessary. The result was an overload of programs, which, once it started, was difficult to curb or cut.
To help facilitate this the services maintained lobby groups in Washington, whose sole purpose was to press their case for the "needed" program. By 1953 Eisenhower had reports of up to seventy-five officials from the Department of Defense on Capitol Hill as military lobbyists, and he opposed their activities. The lobbyists reinforced the services' direct approaches to Congress. Going to staunch supporters such as Senator Stuart Symington, the air force tried to muster political clout to increase spending. By inflating the Soviet threat and overestimating the "weakening" of American defenses, the armed service lobbies applied considerable pressure for more spending in Congress. This activity disrupted Eisenhower's control over the military. The armed forces in essence was circumventing the normal chain of command.
All four military services complained vocally about the president's budget cuts. By December 1954 Eisenhower was sure that there were too many military personnel who had no clear and useful purpose. He wanted to cut the noncombat people—the "dishwashers and waiters"—without sacrificing combat capability. Initially the most vigorous attacks on the administration's budget came from the army, led by General Ridgway. The emphasis on nuclear weapons over conventional forces gave the air force greater stature in national security, and thus the cuts fell heaviest on the army, and to a lesser extent on the navy. Army force levels dropped from 1.3 million personnel in December 1954 to approximately 1 million by June 1956. As commander in chief Eisenhower had to judge what was necessary for the good of the country. It was up to his administration to establish what was essential for national security—a decision that it based both on the economy and on military strength. Ridgway and the top army brass strongly opposed the cuts and on several occasions raised the issue with the president. Emphasizing the damage to the morale of allies and to the armed forces themselves, Ridgway and Secretary of the Army Robert Stevens articulated what Eisenhower called the "parochial" view of the army; they argued for balanced U.S. military forces and less reliance on nuclear deterrence.
The president's son, John Eisenhower, in 1972 gave an intriguing interpretation of the army's struggles for funding. He argued that his father was harder on the army than necessary to prevent charges of favoritism for his old service. The son was also critical of Admiral Radford (chairman, JCS) for taking advantage of his position to manipulate the situation and of Secretary of Defense Charles Wilson for his inability to control the military establishment. As for interservice rivalry, the younger Eisenhower pointed out that each branch handled relations with the White House differently. The air force, emphasizing its central role in nuclear deterrence, lobbied mostly in Congress, where many strong supporters from the struggle to create an independent air force still sat, and it had no hesitation about taking its position to the public. The navy used its contact with the White House to pass on its views directly to the president. Ironically the army had the least skill in public relations and tended to be more disciplined, so it was unable to fight cutbacks effectively. Thus the air force was best at verbalizing criticism of the budget.
Not surprisingly the president was not happy with the air force. The central issue in contention was the number of aircraft in its arsenal. In 1953, taking its case to Congress and the public, the service claimed that it needed 141 air groups by 1954 in order to meet its responsibilities. The president was furious. In conversations with Republican leaders he attacked these attempts to manipulate the budget: "I'm damn tired of Air Force sales programs. In 1946 they argued that if we can have seventy groups, we'll guarantee security for ever and ever and ever." Now, under the guise of national security, the air force came up with a "trick figure of 141. They sell it. Then you have to abide by it or you're treasonous."
Eisenhower was so angry partly because the air force was creating aircraft wings that were, in effect, paper tigers. Although it was building up to 141 wings, many of these units had aircraft but lacked trained personnel to fly or service them and facilities to maintain and operate them. The air force was overzealous in setting aircraft production rates and determining the lead time for procurement. Although the number of wings or the date when they would be operational did not change, the aircraft, associated equipment, and trained personnel were often far behind schedule. Eisenhower was willing to accept fewer wings as long as they were fully operational. His approach, in other words, did not involve a reduction of real strength; however, this did not satisfy air force supporters in the Senate.
Senator Symington, the air force's spokesman on Capitol Hill, took up the fight for more money by attacking the administration. Charging that the cutbacks would leave the United States open to a Soviet strategic attack, he pushed for reinstatement of funds to the air force budget and for enlarging the air fleet to 141 air groups, an idea that Eisenhower totally rejected. As a general he had played the budget game, and he knew that the military overstated its needs, seeking thereby to build what Samuel Huntington called "political castles" to ensure its continued existence. Many people in the House and Senate had no background in such matters and tended to trust the military. Congressional leaders who knew better often accepted demands, especially when doing so was politically expedient or beneficial to their constituents. Supporters of the air force in both houses tried unsuccessfully to increase its share of defense spending during debates on the budget for FY 1954.
Debates over the budget provoked even more heat in the years that followed. The budget for FY 1955 (prepared in 1954)—the first test of the New Look—generated fierce opposition from Symington and other air force supporters, who eventually lost in a vote of 50–38. Later in 1955 General Ridgway (among others) aired his misgivings before Congress, arguing that cuts to the army jeopardized national security. Unfortunately for him the budget passed without major increases. Within days the air force released information concerning a possible Soviet increase in bomber strength. On May 17, 1955, Symington again took up the fight for defense spending. Suggesting that the nation might already have lost air superiority, he stressed that the Soviet threat was expanding in quality as well as in quantity. Pointing to Soviet bombers and hinting at the possible development of ICBMs, he attacked the administration's stance. Even when funds for bomber production increased slightly in the budget, that did not satisfy the senator. By June he was pushing again, this time to increase the number of air force personnel.
Attacks by Symington and other air force proponents continued in 1956. Between April and July most of the testimony at hearings on air power before the Senate Armed Services Committee came from air force personnel asking for more funds. Gen. Curtis LeMay, commander of Strategic Air Command and a longtime hawk and advocate of strategic bombing, played up the Soviet threat. Predicting that by 1958–60 the United States would lag behind the Soviet Union in bomber forces by as much as 2:1, he warned that a surprise attack could eliminate the United States if the air force did not receive an additional $23.8 billion in the budget by FY 1958. A key forum for the "bomber gap" debate, these hearings (and those that followed the launch of _Sputnik_ in 1957) clearly illustrate the problem facing the administration. During the budget hearings for FY 1957, LeMay used his professional knowledge of "additional unspecific information" about Soviet capabilities in an effort to add $3.8 billion to the air force budget. The danger that he presented was clear to the president: without some means of control, defense spending could rapidly increase without real justification.
Constant pressure by the military and the tendency by many members of Congress to follow blindly posed a grave threat to national security, as Eisenhower saw it. Attempts to bring military leaders, especially the JCS, on board had failed. The chief executive was constantly dealing with officers willing to go behind his back to gain what they wanted. He went out of his way to convince the JCS and the military in general to accept the overall goals of the New Look. During a candid conversation in December 1954 with the joint chiefs and Secretary Wilson, he outlined the "big picture," stressing economic and fiscal policy. First, he based the defense budget on his study of security matters, and in his judgment it was sound. Second, as commander in chief he expected the loyal support of his subordinates for his decisions. If anyone had to complain, it should be to him in private, not to the media. Yet this appeal was to no avail. As congressional attacks mounted over FYs 1955 and 1956, so too did Eisenhower's frustration. He expressed some of this to Swede Hazlett. In 1955 he attacked the narrow military mind-set: "So what I need to make the Chiefs realize is that they are men of sufficient stature, training and intelligence to think of this balance—the balance between minimum requirements in the costly implements of war and the health of our economy."
In an effort to curb Symington's calls for increased defense spending, Eisenhower took the unusual step of arranging three special briefings for the senator by Allen Dulles, head of the CIA. Symington left their first meeting, on July 21, 1958, skeptical that the Soviets were not ahead in ICBM development. He based his opinion on information from Tom Lamphier, a former colleague and the assistant to the president of Convair, a major contractor for the missile program. Lamphier claimed to receive his insights from unnamed officials in the intelligence community who disagreed with the official CIA position. While most of the evaluation of Lamphier's intelligence remains classified, Dulles apparently believed that he obtained his information through his work at Convair.
Dulles and the accepted opinion of intelligence circles could not put off the senator. On August 29 Symington wrote to the president detailing his assessment of the missile program, using information from Lamphier. He was emphatic that American efforts were inadequate. The intelligence community had underrated both the scale and the capabilities of Soviet missile development. The CIA had predicted roughly five hundred Soviet ICBMs operational by 1960–61; the United States would have only half that number by 1962. Symington questioned the CIA figures on Soviet missile tests, citing U.S. intelligence to show that American efforts were insufficient. He was confident that the United States was not doing enough for defense, and it obviously disturbed him that no one would agree with him.
Again confiding in Hazlett the president outlined the threat that the relationship between Congress and the military posed. His personal knowledge and experience allowed him to counter many of their efforts, but he worried, "Someday there is going to be a man sitting in my present chair who has not been raised in the military services and who will have little understanding of where slashes in their estimates can be made with little or no danger. . . . If that should happen while we still have the tension that now exists in the world, I shudder to think what could happen to this country."
## Three Challenges, Three Responses
Historians have tended to explain the advent of satellite reconnaissance in terms of fear of a surprise Soviet attack and the military's need for information about Soviet strategic capabilities. Eisenhower was fully aware that such an event was a possibility that the United States had to prevent. However, this was not the driving force for the creation of the first spy satellites within his administration. Other concerns helped to motivate him to experiment with a radical new system for collecting intelligence.
The three-pronged threat—from the misunderstanding of the true nature of the cold war, the economic threat facing the nation, and pressure from the military—served as a major impetus for Eisenhower's acceptance of space-based reconnaissance. These were the concerns that were present from the beginning of his administration and dominated almost every decision and policy related to national security that he made while he was in the White House.
Eisenhower responded to these problems by adopting a nuclear deterrent, trying to stabilize defense spending, and addressing interservice rivalry. Massive retaliation, expansion of the nuclear arsenal, and incorporation of atomic weapons down to the tactical level provided (in theory) a strong deterrent to Soviet aggression. Soviet fear of a nuclear response was the big stick that the president expected to preserve American safety throughout the cold war. Nuclear weapons, ironically enough, allowed the United States to wage the cold war indefinitely by providing a "cheap" counter to conventional forces. For them to be effective, however, the United States had to keep abreast of Soviet military capabilities to ensure that they maintained sufficient defenses.
The difficulties that Eisenhower encountered in curbing military demands demonstrated the importance of accurate intelligence. The president needed an extremely reliable means of gathering intelligence that would provide the information necessary to keep the American deterrent effective for several generations. Such a system was essential to determine sufficient force levels and to assess future military and economic needs during the cold war. Equally important, it could provide future presidents with information to counter military demands for ever-escalating defense spending. A satellite that could constantly monitor the Soviet Union would be the ideal solution.
#
# Eisenhower and Satellite Reconnaissance
# Three Projects (1954–58)
Intelligence applications warrant an immediate program leading to very small artificial satellites in orbits around the earth.
—Report of the Technological Capabilities Panel, February 1955
The need for information on the Soviet Union to manage the cold war is one thing, but this need does not necessarily lead to the development of spy satellites. In 1954 the idea of space-based reconnaissance was at best something from a science fiction novel. For it to become reality, politically the Eisenhower administration had to first be aware of the idea of spy satellites and it had to take political steps to make it happen. Between 1954 and 1958 three major developments proved seminal to fostering and protecting overhead reconnaissance.
First, in July 1954, at the president's request, Eisenhower's Scientific Advisory Committee set up the Technical Capabilities Panel (TCP), also known as the Killian Commission (July 1954–February 1955). In November 1954 it reported that the state of U.S. intelligence was distressing, and on its advice the president made a visionary commitment to pursue the U-2 high-altitude aircraft and space reconnaissance so as to obtain better information about the Soviet Union.
The resulting TCP report directly shaped U.S. intelligence and space policy from that point forward, and its importance cannot be overstated. In terms of cold war policy, the report (sometimes referred to as the Killian Report) has been described as the most influential study to come out of the Eisenhower administration. Former CIA official Richard Bissell argued that this report played a major role in initiating the development and deployment of a series of reconnaissance systems that drastically expanded the scope of the whole U.S. intelligence collection process. More notably the TCP report led to the merging of American space, defense, and intelligence policy. As a result the United States began to pursue specific objectives meant to guarantee continued access to intelligence gathered from space for decades to come. This intelligence became a major counterweight against Eisenhower's worst fears for the future of the nation and ultimately changed the course of the cold war.
Second, and following up on the TCP report, the United States began planning to provide legal protection for both the U-2 and spy satellites. In 1954–55 Eisenhower attempted to finesse the legal issues of overflight for intelligence gathering. To protect the U-2 program he enunciated his Open Skies proposal at an international meeting in Geneva in July 1955. To help establish the legal precedent that delineated the line where airspace ended and outer space began, a major issue in the final TCP report of February 1955, the United States announced its support for a small scientific satellite as part of the International Geophysical Year (IGY, 1957–58). He hoped to use the project to help legitimize satellite reconnaissance. This was formalized in NSC paper 5520 (May 1955), which enunciated a national satellite policy and committed the United States to launching an IGY satellite. This policy was augmented in the wake of _Sputnik_ in June 1958 in the form of NSC 5814. A more comprehensive policy, it helped shape U.S. space activities for the next decade.
Third, the United States began Project VANGUARD, a scientific satellite in support of the IGY. Started in July 1955 as a completely civilian program, it was meant to establish the legal precedent in a way that would not antagonize the Soviet Union. Although it failed to beat _Sputnik_ into space, it displayed the commitment of the president to solving the legal issues involved in space-based reconnaissance.
Together these efforts—the Technical Capabilities Panel, the IGY satellite, and VANGUARD—and the formal U.S. space policy chart a cohesive, long-term endeavor to enhance and safeguard U.S. intelligence gathering. They show the extent of the president's gamble in applying innovative methods to solve the problems that he faced. Yet in 1958 the very satellite that he was trying to safeguard was still in the drafting stage.
## The Technical Capabilities Panel (July 1954–February 1955)
As president Eisenhower had ultimate responsibility for national security at a time when the nation lacked strategic intelligence on its major rival: the Soviet Union. In light of apparent increases in Soviet long-range bombing capabilities, there was cause for concern, and the matter aggravated the president's problems in managing national security. Already facing calls by the armed forces and some members of Congress to increase defense spending, particularly on the air force, Eisenhower realized that without adequate information he could not justify the expenditures or counter demands to increase them. Although he doubted that the Soviet Union would choose to attack the United States directly, should a mistake occur the lack of information could hasten a global disaster.
To clarify the issue and provide a long-term answer that would help guarantee security for generations to come, President Eisenhower turned in early 1954 to a group of noted scientists who sat on the Science Advisory Committee under the Office of Defense Management(ODM-SAC). The ODM-SAC was already well aware that the administration worried about a surprise attack. As early as March 15, 1953, it had warned of the country's vulnerability to such a strike. In the following year Trevor Gardner, assistant to the secretary of the air force for research and development, encouraged it to look more actively at defense policy; he also invoked his knowledge of RAND studies that warned of how vulnerable U.S. retaliatory forces were to maximum effect. He pushed the Science Advisory Committee to play a more active role and alerted them to the president's concerns about American vulnerability prior to their meeting with him on March 27, 1954.
When ODM-SAC scientists met with the president he revealed his concerns about gathering intelligence and the risk of surprise attack. Facing conflicting perspectives from the military and the CIA, he sought to outmaneuver them and asked the group to draft a "root and branch," detailed study of how to lessen that risk. Chair Lee DuBridge asked James Killian, president of MIT, to convene a subcommittee to examine the feasibility of such a study and to make recommendations.
## The Killian Commission (July 1954–February 1955)
Killian's subcommittee quickly convened to consider the matter and in less than a month reported back to the ODM-SAC. It recommended creation of a technical task force to study methods of countering surprise attacks, look at the technology underlying American weapons and intelligence, and pass on its findings to the White House. DuBridge recommended to Dr. Arthur Flemming (director of the Office of Defense Management) that he recruit such a group and that the president give it official authority and support. Three panels would examine three critical subjects: continental defense, striking power, and intelligence. The president approved the proposal on July 26, 1954, in effect giving these civilian scientists complete access to national military and intelligence secrets. Many military experts did not like the idea of allowing scientists to look at such sensitive information. The decision was particularly courageous in light of the paranoia about communist sympathizers and spies that emerged from the congressional hearings under Senator Joseph McCarthy (R-WI).
DuBridge recommended Killian to chair the steering committee of what became the Killian Commission, with Dr. James Fisk to act as associate director. Knowing Killian from his own years as president of Columbia University (1948–52), Eisenhower readily accepted him as chair. Killian embraced the job wholeheartedly and quickly formed a steering committee, assigning personnel to various tasks. Some forty-one scientists, engineers, and military and communications experts joined sixteen military and civilian members of the government. The steering committee accepted the three-panel approach, to which it added several supporting studies in communications and technical personnel to round out the report.
Overall the TCP report addressed five key goals. The first and foremost was development of American capacity to gain more intelligence about a potential enemy's intentions and capabilities—principally to receive warning of any planned attack. The second was increasing retaliatory power through the use of new technology, which might strengthen the deterrent force and decrease the risk of an attack. The third was to strengthen defensive capabilities in order to blunt any form of assault. Fourth was assuring secure and reliable communications. Finally came the need to understand the effect of technology on the military's personnel requirements.
## Land's Intelligence Panel
All three panels—on continental defense, striking power, and intelligence—proved vitally important, but the third had the most far-reaching impact on the cold war. It was not only instrumental in the decision to develop satellite reconnaissance but it was the major factor in spurring development of the U-2 aircraft. It also stimulated formulation of major policies relating to space and to the Open Skies proposal of 1955. It is this panel and its findings that I now examine in greater detail.
Killian chose Edwin Land to head the intelligence panel. A genius and an innovator, he was an unorthodox thinker and a solver of impossible problems. Creator of the Polaroid Corporation, with hundreds of patents to his name, he also had significant influence in Washington's intelligence circles. As a major player in the photography industry, he was a natural choice to advise the administration on aerial reconnaissance. In the words of his biographer, his approach to innovation was consistent: "Defining a need and the shortest path to a practical answer, he did not think there was a law of nature forbidding what you wanted or specified." Land's six-member panel was relatively small for the task, reflecting his own style of research and his belief that a committee must be small enough to "fit into a taxicab" in order to avoid conflicts of will and personality. His panel was a veritable who's who of experts. It included James Baker (associate director) from the Harvard Optical Laboratory, the leading lens designer for aerial photography; Joseph W. Kennedy, a renowned chemist from Washington University in St Louis; Allan Latham Jr., a longtime friend of Land's and on the staff at Polaroid; Edward Purcell from Harvard, a Nobel laureate in nuclear physics; and John W. Tukey of Princeton and Bell Laboratories. This group, responding to Land's demand that the United States have the best intelligence system possible, received frequent briefings from various intelligence officials.
The intelligence panel, in fact the entire Killian Commission found the state of U.S. intelligence appalling despite its best attempts to locate people in the CIA and the military who supposedly knew about gathering and processing such information. Years later, during an interview, Land recalled, "We would go in and interview generals and admirals in charge of intelligence and come away worried. Here we were, five or six young men, asking questions that these high-ranking officers couldn't answer." The only individual who impressed the panel was the former naval officer Arthur Lundahl, who came over to the CIA in 1952–53 to head their photographic-interpretation group, which at the time relied on air force photos from both periphery and limited penetration flights of the Soviet Union. He was a strong and vocal advocate for gathering intelligence via photographing the Soviet Union.
The commission's final report, which it submitted on February 14, 1955, sharply criticized the quality of intelligence. Echoing the president's concern about the matter, it noted, "We must face up to the reality of the actual situation: our estimates of the capabilities of the Soviets have, at their center, only a very limited core of hard facts, and we know even less about their intentions." In one of the report's most forceful statements, Land wrote, "We must find ways to increase the number of hard facts upon which our intelligence estimates are based, to provide a better strategic warning, to minimize surprise in the kind of attack, and to reduce the danger of gross overestimation or gross underestimation of the threat. To this end, we recommend adoption of a vigorous program for the extensive use, in many intelligence procedures, of the most advanced knowledge in science and technology." Land considered intelligence absolutely essential, thinking about it and its role in the policy process "in its most constructive and benign sense, as a search for the knowledge to reach sound national policies." Like the president, he believed it was vital for limiting or eliminating surprise attacks and for reducing the tendency to overestimate or underestimate a possible threat. Irrefutable facts would prevent the military or civilians in government from overreacting and allow instead for cool and rational decision making. Thus solid intelligence would allow the United States to "better cope with those fantasies about our weaknesses and the enemy's superiority that occur occasionally among the military or the politicians." Clearly Land was thinking along the same lines as Eisenhower.
The TCP report stressed that the U.S. government should be the best informed in the world. This would not only protect the nation but also help to resolve internal differences concerning national security. More significant, "if intelligence can uncover a new military threat, we may take steps to meet it. If intelligence can reveal an opponent's specific weakness, we may prepare to exploit it. With good intelligence we can avoid wasting our resources by arming for the wrong danger at the wrong time. Beyond this, in the broadest sense, intelligence underlies our estimate of the enemy and thus helps to guide our political strategy." The president liked this argument. His concerns about curbing the military's spending and his criticism of its demands and lobbying found support in the TCP report.
## Recommendations of the Land Panel
Land and his intelligence panel made two sets of recommendations about increasing the quantity and quality of hard data for security. The first set appeared in the report's section on intelligence. These were official and "open" recommendations, emerging from the intelligence panel's insight and experience. The second did not form part of the report. The panel deemed this material too sensitive for the full National Security Council and discussed it only with the president in November 1954; they handed the president a single copy of these findings for his eyes only in February 1955. Together these two sets changed the direction of U.S. intelligence collection.
The Land panel's official, open recommendations were more general and aimed at improving intelligence gathering without specifically defining methods or equipment. The most important for our purposes involved calls for increased use of science to penetrate Soviet security and prevent hoaxes that could mislead the United States. One urged development of safeguards to ensure that National Intelligence Estimates did not accidentally release their sources of information.
Land's intelligence panel was especially worried about electronic intelligence gathering. Members of the panel felt that technical and administrative problems were severely hampering that effort. They wanted a speedy solution via "a combination of technical knowledge and adequate authority at a high level." At a later stage both of these concerns had long-term repercussions for satellite reconnaissance. Within the administration stringent security and compartmentalized information on a "need-to-know" basis emerged in response.
Only two of the panel's recommendations dealt directly with satellite reconnaissance in the context of "freedom of space." Neither recommendation called for the immediate start of a "crash" program. Rather they indicated the satellites' potential for reconnaissance and the need for a framework to protect them.
The first of these two recommendations responded to legal issues relating to freedom of space. At that time there was no legal definition for where airspace ended and "space" began. Now that launching a satellite into orbit was technically feasible, the question of who controlled space became a central issue. Land's intelligence panel saw a major opportunity for the United States to establish the legal precedent. The TCP report argued that it was necessary to reassess the principles and practices of international law in this area. The departments of Defense, Justice, and the Treasury examined the issues of satellites and international law in the wake of the panel report. They concluded that jurisdiction in space was really not a concern since a satellite in space orbited above the atmosphere and thus would not violate the Civil Aviation Convention of 1944 or international law. By July 1955 the NSC agreed with the panel on the importance of a satellite to test the principle of freedom of space.
The second recommendation also related to freedom of space in the context of overflights. Looking to the long-term goal of space-based reconnaissance, the panel pushed for a small satellite as a stepping stone to future operations. Significantly "intelligence applications warrant an immediate program leading to very small artificial satellites in orbits around the earth. Construction of large surveillance satellites must wait upon adequate solutions to some extraordinary technical problems in the information gathering and reporting system and its power supply, and should wait upon development of the intercontinental ballistic missile rocket propulsion system. The ultimate objective of research and development on the large satellite should be continuous surveillance that is both extensive and selective and that can give fine-scale detail sufficient for the identification of objects (airplanes, trains, buildings) on the ground." Establishing a legal precedent in space, the small satellite would help to protect future reconnaissance satellites. Thus overflight would be legal in space, preventing the Soviet Union or other nations from attempting to stop such activity. The call for a satellite was ahead of its time, as the relevant technology had not yet matured. Certainly satellite reconnaissance fit with the TCP report's call for use of the latest technology in gathering intelligence.
No other open recommendations of the intelligence panel emphasized specific solutions beyond the use of science and technology to create a U.S. advantage. The emphasis on satellites takes on greater significance in light of a secret meeting of Eisenhower, Killian, and Land in November 1954.
## Toward the U-2 and Satellite Reconnaissance
Arthur Lundahl, the only CIA official who grasped the concept of increasing data collection, reinforced the idea of overhead reconnaissance. As the head of the CIA photographic interpretation group, he played an important role in the interpretation of many photographs, including the product of the U-2 aircraft when it became available. In August 1954 Trevor Gardner of the air force and Philip Strong of the CIA discovered plans for a revolutionary high-altitude reconnaissance plane, designed by Kelly Johnson, that eventually became the U-2. The air force had shown no interest in the aircraft. In mid-August Strong passed his knowledge of the plane on to Land, who expressed great surprise that the air force had rejected what appeared to be one of the most promising intelligence platforms available. He took the information to James Baker, who had been trying to figure out what type of aircraft would be necessary for overhead reconnaissance. He found the U-2 design ideal for the task. Thus began a long-term relationship between Edwin Land and the U-2.
Although it would become an extraordinary aircraft, the U-2 could not be a permanent solution for all intelligence needs. To do its job the U-2 had to violate international law by penetrating another nation's airspace, which could have severe political repercussions. A more permanent, and politically more feasible, answer was necessary. The TCP report's identification of the key problems relating to satellites—the information-gathering/forwarding system, power supply, and booster rocket—shows that Land and his panel were informed about satellites and their technical problems.
We do not know exactly how much the panel knew and, more important, how much it told Eisenhower. By November 1954 the entire panel found the intelligence situation so depressing that its members needed to speak with the president. Later that month they met with him secretly in the Oval Office, probably on the 24th. No official record of this secret conclave exists (Eisenhower often had such gatherings in the White House); most of the information about it comes from memoirs. According to secondary sources, the major participants, along with Land and Killian, were Secretary of State John Foster Dulles, CIA Director Allen Dulles, Secretary of Defense Charles Wilson, Air Force Secretary Harold Talbott, and air force generals Nathan Twining and Donald Putt. The president's appointment schedule for this date lists an off-the-record meeting at 8:15 that morning but does not show Killian and Land as attending. A memorandum by Andrew Goodpaster about the U-2 decisions of that day indicates that this was the gathering.
This secret meeting has raised a great deal of speculation over the years. While what exactly was discussed at the meeting will likely never be known, it is clear from the remaining evidence that Land and Killian presented two highly compartmentalized pieces of information to the president. This material became part of a secret annex to the TCP report. It was hand-delivered to the president in February 1955 after the formal submission of the TCP report. The annex itself has since disappeared. According to R. Cargill Hall, official historian for the National Reconnaissance Office, the annex did exist but has since either been lost or destroyed—probably the latter.
One part of the annex dealt with the future Polaris ballistic missile system, the other with overhead reconnaissance and a radical new aircraft design that Land and Killian felt was an essential source of intelligence. This aircraft, the CL-282 (later the U-2), seemed to offer a short-term solution to the intelligence problem, helping with defense planning, assessing the Soviet bomber threat, and, in the long run, warning of an attack. Eisenhower authorized its development and gave the CIA responsibility for its operation. The overall cost was probably going to be in the range of $35 million, with both the CIA and the air force providing the funds. Eisenhower realized that since its use would violate international law, its _other_ costs would be far higher should it become public knowledge. The secretary of state also anticipated difficulties but argued, "We could survive through them."
Years later Killian indicated that the participants discussed satellites, but as a long-term goal for intelligence. Unfortunately, no memorandum of the meeting dealing with satellites has survived. Eisenhower's penchant for security was very effective in this case. Killian's memoirs describe the meeting and the TCP report, which came out on February 14, 1955. He praises Eisenhower for his willingness to listen to radical proposals and act on them. He also mentions an important point: the most significant and sensitive recommendations on overhead reconnaissance appeared not in the report but in the annex. Worrying that information on the U-2 might leak out, the president ordered its omission, except for the special appendix that was for his eyes only. Killian also states that the report left out a specific recommendation on satellites. Both he and Land accepted that omission. Like Assistant Secretary of Defense for Research and Development Donald Quarles, they saw satellites as a long-term answer, not an immediate possibility.
Eisenhower's decision on November 24, 1954, to go ahead with the U-2 and eventually with satellite reconnaissance was a unique and difficult one. Recognizing that U-2 overflights would violate international law, he showed by his actions his commitment to closing the "intelligence gap" that obscured American knowledge of the Soviet Union. The equally challenging decision about an expensive satellite program, even though its legality and its technical feasibility were still questionable, was clearly visionary. The official General Operational Requirement (March 1955) for this satellite program followed the TCP report by a month and indicated the speed at which the president committed himself. These were not the decisions of a hands-off chief executive.
## Eisenhower's "Open Skies" Proposal (March–August 1955)
Realizing that the long-term viability of such systems would necessitate government support, the administration took steps to provide as much legal protection for the U-2 and satellites as possible. To legitimize and protect the U-2 Eisenhower attempted to link photographic overflight with arms-control verification. The closed nature of Soviet society was the major impediment to any form of arms control. There was no practical method short of on-site inspection to verify compliance with agreements, and the Soviets rejected this completely. Grasping the opportunity Eisenhower proposed the use of politically sanctioned aircraft overflights as a means of circumventing Soviet hostility to inspection. This proposal would provide legal protection for the U-2 program then under development and allow both superpowers to halt the escalating arms race.
The "Open Skies" initiative represented Eisenhower's attempt to legalize overflight—an innovative effort to regain the initiative in the cold war and move toward arms control. The concept emerged from a study group that met in Quantico, Virginia. With Nelson Rockefeller (special assistant to the president) as chair, the group consisted of Deputy Secretary of Defense Robert B. Anderson, Adm. Arthur Radford, Harold Stassen (the president's special assistant on disarmament), and eight other people. It convened in June 1955 to evaluate the possible role of aerial reconnaissance in achieving substantial disarmament; its members were unaware of plans for the U-2 or for a satellite system. The core of the proposal was a U.S.-Soviet exchange of data on military capabilities, organization, and the location of units. Mutual overflight and aerial reconnaissance would then verify and monitor this information. In other words, this exchange would help to establish a base-line understanding of both countries' order of battle. Overhead reconnaissance could then monitor any arms agreements to ensure compliance. This arrangement would reduce the risk of surprise attack, ease tensions, and increase trust. To expedite this plan both countries would make facilities available for landing and refueling planes, and both sides would agree not to interfere with each other's aircraft directly or to conceal activities below them.
Eisenhower could justify the proposal in two ways. First, it was a grand and dramatic gesture that promised big propaganda dividends. At the height of the cold war he could show that he was seeking some means of impressing on the Soviet Union his resolve to find a way of ending the arms race. His scheme was unlike previous proposals for arms verification: it demonstrated American willingness to be innovative in this area. It also captured people's imagination by seemingly extending an olive branch to establish peace. Second, it shrewdly attempted to establish legal precedent and legitimacy for overhead reconnaissance; if it accomplished that, then the U-2 and its successors would not be in danger, and in the long run U.S. security would greatly increase.
Whether or not the Soviet Union accepted the Open Skies proposal was of little concern to Eisenhower. Even before making the proposal, he had committed his administration to developing both the U-2 and satellite reconnaissance. The proposal was a clever attempt to gain Soviet acceptance of American overhead reconnaissance. It gave the United States the moral high ground, while providing a rationale for something that Eisenhower was going to do anyway. For him Soviet acceptance was not a requisite. The United States had nothing to lose and much to gain both militarily and politically from the proposal. The Soviets, however, had everything to lose and little to gain if overhead reconnaissance became standard practice. The Kremlin relied on a closed society to conceal its weakness and ensure the illusion of strength. Therefore most people in the Eisenhower administration expected Soviet rejection almost from the start.
Over John Foster Dulles's strong opposition, Eisenhower presented the Open Skies proposal on July 21, 1955, at the Geneva Summit. In the middle of his speech on disarmament in which he asserted, "No sound and reliable agreement can be made unless it is completely covered by an inspection and reporting system adequate to support every portion of the agreement," he removed his glasses, turned to the Soviet delegation, and spoke:
Gentlemen, since I have been working on this memorandum to present to this Conference, I have been searching my heart and mind for something that I could say here that could convince everyone of the great sincerity of the United States in approaching this problem of disarmament.
I should address myself for a moment principally to the Delegates from the Soviet Union. . . . I propose, therefore, that we take a practical step, that we begin an arrangement, very quickly, as between ourselves—immediately. These steps would include:
To give each other a complete blueprint of our military establishments, from beginning to end, from one end of our countries to the other; lay out the establishments and provide the blueprints to each other. Next to provide within our countries facilities for aerial photography to the other country . . . ample facilities for aerial reconnaissance, where you can make all the pictures you choose and take them to your own country to study, you to provide exactly the same facilities for us . . . and by this step to convince the world that we are providing as between ourselves against the possibility of great surprise attack, thus lessening danger and relaxing tension.
Although politely agreeing that Open Skies had merit, Soviet leaders never seriously considered it. In one sense, however, the proposal was successful: it paved the way for future acceptance of overhead reconnaissance by establishing the link between overflight and verification of arms control. Eventually this evolved into the term "national technical means of verification" in Article 5 of the SALT treaty of 1972.
## The IGY Satellite (February–May 1955)
Having failed to provide legal safeguards for U-2 aerial surveillance by the summer of 1955, Eisenhower turned to protecting satellite reconnaissance. In the wake of the TCP report he had immediately asked all the relevant branches of government to review and comment on the areas of the document relating to their spheres of interest; he had asked the State Department to comment on international law issues. Before these studies could bear fruit, however, other events accelerated satellite development.
The TCP report emphasized the need to determine legally where airspace ended and outer space began. Scientific and psychological issues aside, it was the potential legal ramifications of satellite reconnaissance that most concerned Land and his panel. The mere idea of orbiting a satellite had already sparked the interest of other people within the administration. Indeed the call for a scientific satellite antedated the TCP report. In the fall of 1954 the worldwide scientific community called for a satellite to support scientific research. As a result in October 1954 the Comité Special Année Géophysique Internationale (CSAGI) 1957–58 (the Special Committee for the International Geophysical Year) invited all participating nations to launch a satellite during the IGY. The U.S. National Committee for the IGY concurred and on March 14, 1955, recommended that the United States begin such a program.
Concern over the future of "international" space led Deputy Secretary Donald Quarles to lead the drive to send up an American satellite to mark the IGY. Quarles was an unusual supporter of the IGY satellite concept. Not a believer in untried technology, he doubted that a reconnaissance satellite would be available in the near future. It was not until 1958, when the American IGY satellite got off the ground, that he and others finally realized that a spy satellite was possible. Long before that, however, Quarles was aware of satellites' potential value for intelligence gathering and of air force initiatives in this regard and the resulting need to establish legal precedent. In early 1955 he quietly pushed the military to develop a scientific satellite, which in the long term would assist in establishing the legal precedent for freedom of space. He did this at two levels. Internally, within the military, he began investigating possible avenues for fulfilling the TCP report and IGY calls for a satellite by initiating a study of military satellite options. Externally he supported the American IGY committee's calls for a satellite program, initially through correspondence and later by pushing space policy at the NSC level.
## Quarles's Quest for Military Support
Quarles's internal quest for a satellite program began on April 1, 1955, with an inquiry into the value of such a program. He asked the Department of Defense's Research and Development Coordinating Committee on General Sciences (CGS) to review the military's research and development plans for satellites and submit a recommendation on action. Robert W. Cairns, chair of the CGS, replied on May 4. Limiting the scope of its review to research and development on scientific satellites, the CGS found many benefits in the development of both a small, inert, trackable satellite and a satellite with scientific instruments and recommended establishment of a satellite program within the Department of Defense. The main concern was one of scale.
A small, inert satellite (weighing between five and ten pounds) would probably supply a great deal of information about the upper-atmosphere conditions through which satellites, and missile warheads, would have to travel. Ground-based observation of the vehicle would obtain the data. A satellite with scientific instruments would yield a great deal more information, including a range of details about orbital space that was currently unknown. It could monitor conditions such as temperature, available solar energy, solar radio noise, meteor collision, and cloud patterns. Valuable insights would emerge from tracking small satellites, from information relating to gravitational variation and ion content in the atmosphere, and from better understanding of how satellite orbits would shift because of the earth's oblateness and rotation. All these elements would of course have direct implications for any broader space program.
The CGS noted four possible military options for a satellite program. The first, which the navy initially proposed on March 23, 1955, was for a triservice satellite project called ORBITER. The navy envisioned using the REDSTONE missile as a primary booster to deploy a five- to ten-pound satellite with an elliptical orbit ranging from two hundred to eight hundred miles. The outlook was quite discouraging; probably only about one in three attempts would succeed. To maximize the chances the navy argued for four attempts. By adding additional stages, the rocket could increase the system's payload and reliability. Estimates for the cost of ORBITER ran at about $5.5 million plus logistical expenses (and an additional $3 million for instrumentation), and launch would occur probably in autumn 1957.
Second, on April 15, 1955, the navy suggested a satellite configuration that had a VIKING missile as a first stage. Using fewer motors than ORBITER, this prototype would probably have a more reliable propulsion system and hence slightly better performance. The navy confidently predicted that such a missile could lift a ten-pound satellite into a 334-mile-high orbit, and a larger satellite (roughly forty pounds) was possible, if a lower orbit at around 200 miles was acceptable. Overall cost for this program (which the navy saw as a backup and second-generation system to ORBITER) would be roughly $7.5 million.
Third, the air force proposed a satellite closely related to the WS-117L satellite concept. It would use an ATLAS ICBM as a primary booster and an AEROBEE-HI rocket as a second stage. Because of ATLAS's higher thrust potential, the air force predicted that its design could lift a satellite payload of one hundred to two hundred pounds into orbit. Expecting to be able to orbit the satellite by the end of 1958, the air force estimated the program's cost at $50 million to $100 million. The air force's final proposal involved the CONVAIR test missile (then part of the ICBM program) as a booster to lift a satellite weighing about two thousand pounds into orbit. However, the incorporation of a satellite role for this system seemed to the air force neither necessary nor desirable because the satellite would slow ICBM development.
While acknowledging that the Department of Defense probably possessed the technical capability to launch a scientific satellite during the IGY period of July 1957 to December 1958, the CGS's review concluded that the many complex technical problems meant that there was no guarantee that the satellite would be ready in time. As well, any program had to be part of continuing research and development, which would produce a series of improved satellites. Unfortunately the CGS did not recommend any one program. Noting the prestige and value of satellites as a basic research tool, it argued that Defense should pursue development of all three (ORBITER, VIKING, and ATLAS-AEROBEE) to make success more likely and provide a range of capability.
## Quarles's Quest for Nonmilitary Support
Even as the CGS was preparing this report, Quarles was mustering nonmilitary support for the IGY program. Aware of the TCP report's call for a satellite to support freedom of space, Quarles went to the U.S. National Committee for the IGY in February and March 1955 to gain its backing for a satellite program. He prompted Alan Waterman, director of the National Science Foundation, to contact both the State Department and the CIA concerning the U.S. IGY committee's request for such an effort. State responded favorably in discussions on March 22 and in a memorandum of April 27. Robert Murphy, a deputy undersecretary at State, thought the program in the national interest. However, he felt that State would be unable to judge whether a satellite was strategically feasible and so deferred to the military on this issue. More important, he argued that "arrangements among the pertinent government agencies should go forward, with the understanding that the U.S. representatives of CSAGI will report on our intentions in the IGY at the appropriate time."
CIA Director Allen Dulles and his deputy, Richard Bissell, were also responsive to the concept of the IGY satellite. Both thought expediency was a necessity and suggested two approaches. The proposal could go forward from the Department of Defense or it could be directly presented to the NSC. To accelerate the process Dulles agreed to take up the matter with the NSC's Operations Coordinating Board, but Waterman got the impression from both State and the CIA that it was up to either him or Quarles to initiate formal action. Quarles, however, decided to act on his inside knowledge. Quietly he pushed the United States to develop a scientific satellite that would work toward establishing legal precedent. He expected the U.S. IGY committee to formally request development of a scientific satellite, which it did on May 18, 1955.
## NSC 5520 (May 1955)
Keen to see a national space policy to support satellite reconnaissance, two days later, on May 20, 1955, Quarles submitted a draft statement on a policy to the NSC—the first articulation of a national space policy. It related not to the two military programs but to the value of a satellite launched under IGY auspices. Quarles's policy paper was similar to the TCP report's recommendation about establishing the legal basis for freedom of space.
In the "Draft Statement of Policy on U.S. Scientific Satellite Program" (NSC 5520), Quarles confirmed the TCP's findings that the United States had the technology to produce a satellite of five to ten pounds, and, more important, it would be able to orbit one in 1957–58. The draft also stated that since the TCP report had appeared the Soviet Union had apparently taken the initiative. On April 15, 1955, Quarles announced that the Astronomic Council of the Soviet Academy of Sciences had established a permanent high-level "interdepartmental commission for interplanetary communications." There were also significant indications that the Soviet Union might be working on a satellite of its own. Quarles went on in NSC 5520 to identify some of the advantages of a satellite, which included the gathering of scientific data, psychological benefits, and increased prestige from being first in space. Among the most valuable benefits was the opportunity to test freedom of space. Preliminary studies by the executive branch uncovered no legal obstacles under international law to orbiting such a satellite. Therefore the IGY could establish the legal precedent of overflight by a satellite. Based on these factors NSC 5520 called for government support of the IGY satellite. Arguing that a satellite was not a militarily offensive weapon, the paper suggested that the benefits of orbiting a satellite would greatly outweigh the $20 million that the program might cost. NSC 5520 expected the military to take the lead in launching the satellites, as it had the most experience with rockets. Emphasizing the peaceful nature of American space efforts, in conjunction with the IGY commitment, it claimed that those factors would ease the possible backlash over a satellite launch.
Quarles was not alone in comprehending the value of the scientific satellite. The CIA also backed the notion in its discussion of the TCP report. Asserting that "successful launching of the first satellite will undoubtedly be an event comparable to the first successful release of nuclear energy," the CIA argued that the first nation into space would gain incalculable prestige. More significant, the satellite would be a stepping stone toward a larger satellite capable of providing early warning and reconnaissance of the Soviet Union. The potential psychological impact of being first in space and the military intelligence advantages of a satellite impressed Nelson Rockefeller, who firmly backed the idea. Aware that Soviet propaganda would criticize a U.S. satellite, he emphasized the value of IGY auspices for the effort.
NSC 5520 committed the United States to launch a civilian satellite during the IGY period. Eisenhower hoped that a civilian program would resolve the legal issue and minimize the risk of an international dispute. The IGY's blessing gave it additional legitimacy and made strong Soviet opposition to overflight less likely. Despite the psychological and prestige effects of a successful launch, at least one member of the NSC grasped its more subtle implications. During the discussion of NSC 5520, Allen Dulles observed "that it was very important to make this attempt"—the IGY satellite was crucial for the future of satellite reconnaissance.
## VANGUARD (July 1955–June 1958)
Although the United States could have waited for one of the international scientific meetings in August 1955 to announce its IGY program, the administration was increasingly worried that the Soviet Union could tell the world about its own satellite program at any time. So it seized the initiative, announcing its satellite in a press statement on July 29, 1955. Careful wording stressed its scientific benefit for everyone concerned. The press release stated, "This program will for the first time in history enable scientists throughout the world to make sustained observations in the regions beyond the earth's atmosphere." Press Secretary Jim Hagerty pointed out that the president was pleased that American scientists would help colleagues from all nations to benefit from this project. The announcement was also the first step in securing legal protection for satellite reconnaissance.
The scientific satellite program, VANGUARD, proved more of a challenge than anyone expected after its inauguration on September 23, 1955. Due to Eisenhower's desire to separate the ICBM program from civilian space activities, the scientific program was only loosely placed under naval management. Initial financial projections were overly modest, and a small workforce that had to do everything from scratch quickly produced cost overruns. Politically the decision to create a civilian program was sound and stemmed from the emphasis on the endeavor's scientific and nonmilitary nature; a launch on a military missile would have sent the wrong message. Also at risk was the secrecy of American missile technology when the scientific data became public. Furthermore the ICBM program could not afford a delay for a satellite shot. Accordingly the president kept the two programs separate.
Air force historian Lee Bowen argues that a lack of monetary support by the secretary of defense intensified financial problems. Money for the program arrived too sporadically to support rapid development. Following the launch of _Sputnik_ , difficulties with finances and the army's claim that it could have launched a satellite far quicker led to attacks on the program and the president. Eventually the army's Jupiter C experimental booster helped to orbit the first American satellite, EXPLORER, on January 31, 1958.
Concerns over possible conflict between the ICBM efforts and VANGUARD were troubling. Less than a year after the public announcement of VANGUARD, the NSC learned about this problem. In early May 1956, during an NSC meeting, Secretary of Defense Wilson recommended that to prevent conflict between the two programs VANGUARD should be given lower priority that the ICBM efforts. This seems to support Bowen's assessment of Wilson's support for VANGUARD. The financial problems were already evident, and the meeting revealed that costs were rapidly escalating. The National Science Foundation had asked for an additional $28 million for more satellites, and Secretary of the Treasury Humphrey warned the NSC that the projected costs for the VANGUARD program were approaching $60 million to $90 million, a far cry from the original estimates. The president then reminded Humphrey that the higher figure was for the proposed doubling of the number of satellites to launch. This would be reviewed as needed, but Eisenhower did not expect the price to exceed $60 million.
The hemorrhage of funds in the VANGUARD program did not stop at a mere $60 million. By the end of January 1957 the cost had risen from the original $20 million estimate to $83 million, while the experts expected the launch of a satellite only in October 1957. The National Science Foundation sought an additional $30 million to increase the number of satellites, raising total expenditures to $113 million, something both Eisenhower and the Department of Defense were unwilling to countenance. This pattern repeated itself in May 1957, when cost overruns indicated that the anticipated total would reach $110 million. Expensive technical problems had slowed development of a reliable booster. VANGUARD was not just attempting to establish a legal precedent; it was also trying to achieve a major technological breakthrough.
In the meantime the legal questions that NSC 5520 and VANGUARD were to solve were proving a major stumbling point for satellite reconnaissance. In the wake of the TCP report the State Department had begun to reassess the legal ramifications of satellite overflight, believing that no state could claim territorial sovereignty at such altitudes. It expected that space would be open to all countries so long as those seeking to use it could surmount the technical problems.
Eisenhower hoped that the scientific satellite would define space as international, just as the earth's oceans are free domain outside the territorial limits of nations. This "great common" in space would allow open access and prevent interference. To achieve this he emphasized VANGUARD's scientific and peaceful purpose. Once the Soviets accepted the principle of overflight, reconnaissance would receive at least tacit legal sanction, which would help to prevent countermeasures against satellites. For both Eisenhower and Quarles the Soviet launch of _Sputnik_ in October 1957 solved the problem to some extent. Quarles suggested that the Soviet Union had in fact "done us a good turn, unintentionally, in establishing the concept of freedom of international space." It was not until the mid-1960s, however, that the United Nations was able to develop the rudiments of a legal regime for space. Initially satellite reconnaissance existed in quasi-legal limbo. The Soviet Union stopped protesting about U.S. space espionage once its space capabilities matched those of its great rival.
## NSC 5814 (August 18, 1958)
After _Sputnik_ U.S. space policy adapted quickly. In June 1958 the NSC reviewed a new policy on outer space in NSC 5814, which offered a more comprehensive approach than NSC 5520. NSC 5814 systematically assessed the value of American security in space, reconsidering the issue of the legal boundaries of outer space. For example, a theoretical upper limit of air space might be the highest altitude at which an aircraft could maintain enough lift for flight and/or the lowest orbit a satellite could maintain without encountering atmospheric drag.
Officially NSC 5814 emphasized the American desire for peaceful use of outer space, but the document shows that the United States considered some military uses of space essential for national security. The most important of these involved reconnaissance satellites to gather military and other intelligence on the Soviet Union and to detect possible missile launches against the United States. The political challenge was to create a legal position that allowed certain satellite activities, such as reconnaissance, without creating excessive political repercussions. Reconnaissance satellites seemed a possible means of implementing the Open Skies proposal that the Soviets had rejected in 1955.
The TCP report was a pivotal moment, when Eisenhower solicited advice from sources outside the national security bureaucracy. Their expertise could both invigorate defense efforts and challenge the conventional wisdom, especially that of the military. Land's intelligence panel, which the Science Advisory Committee of the Office of Defense Management created ostensibly to advise the president on issues relating to possible surprise attacks, in fact had a far greater impact. It dramatically clarified the military and civilian need for intelligence and drew the chief executive into the process. Advocating both the U-2 and satellite reconnaissance as sources of intelligence, the panel, with Eisenhower's full support, changed the course of the cold war.
The TCP report linked Eisenhower's principal challenges: ensuring national security through intelligence and strategic warning, defending economic security, and blunting the military's opposition to sound fiscal planning. Good intelligence would, in theory, allow the government to balance defense spending by silencing the shrill calls of the military and its supporters for larger budgets. More significant, overhead reconnaissance might monitor Soviet military activity, verify arms-control agreements, and in the long term strongly counterbalance undue military influence over the political leadership.
We can see how important these goals were to Eisenhower by the effort he was willing to put into achieving them. He realized that overhead reconnaissance as an intelligence source would work only within a legally benign international system. The U-2 aircraft had to violate international law in order to do its job, but reconnaissance satellites had no legal status because of the ambiguity of space law. Eisenhower used the power of his office to protect the U-2 program as much as possible. He probably knew that the Soviets would never accept Open Skies. He made it as a legitimate offer, however, because it had the potential to protect aerial reconnaissance. He may also have theorized that it would put the United States on the moral high ground by linking photographic reconnaissance with the peaceful goal of verifying arms control.
With satellite reconnaissance Eisenhower likewise tried to protect a future intelligence asset by securing legal protections. U.S. acceptance of the IGY satellite proposal and NSC 5520, even though it would cost millions of dollars (something Secretary Humphrey repeatedly railed against), shows how highly Eisenhower valued the potential for this form of intelligence collection. The IGY satellite was an attempt to use a scientific satellite to achieve a military/intelligence goal; it would establish the legal precedent that could then somewhat justify spy satellites.
VANGUARD thus has to be seen as more than the American contribution to the IGY efforts. It was also part of Eisenhower's attempts to create legal protection of space-based reconnaissance. By gaining worldwide acceptance of satellite overflight, no nation could legitimately protest military satellites. VANGUARD, in essence, was running interference for WS-117L. The fact that the Soviets orbited the first satellite only made it easier for Eisenhower. Once they had established the legal principle of freedom of space, they would be unable to backtrack by protesting an American satellite. From 1958 on, U.S. space policy shifted to block military uses of space while supporting peaceful ones. Of course it defined reconnaissance satellites, which provided stability and protection from surprise attack, as "peaceful."
Individually and collectively the Technical Capabilities Panel, the IGY satellite, and VANGUARD were seminal to Eisenhower's strategy for national security in the cold war. They charted a cohesive, long-term effort to safeguard the future of U.S. intelligence gathering. They show the extent of the president's gamble, applying innovative methods to solve the problems that he faced. Yet in 1958 the very satellite that he was trying to safeguard had yet to get off the ground.
# Part 2
# WS-117L
#
# Origins
# RAND and Satellite Reconnaissance (1945–54)
Despite years of research on the subject, the space issue never reached the upper levels of the Truman White House. There was no Truman space policy, and space issues remained largely the realm of a small group of engineers and analysts.
—Dwayne Day
In an era dominated by the doctrine of massive retaliation, cities were the main targets and bombers were the main threat. For such a military outlook, reconnaissance from space represented a useful but scarcely essential capability.
—Robert Perry
By 1954 a variety of forces were converging within the Eisenhower administration to produce a drive for satellite reconnaissance. Concurrent with the growing awareness of the value of intelligence and rising political will to take action, a small group of civilian consultants and military officers was looking for solutions. These experts were seeking to conceptualize a new reconnaissance system—a process that began in 1946—under the auspices of the RAND Corporation.
RAND played a threefold role. First and foremost, after it became clear in 1946 that satellites were technically feasible, RAND examined the elements necessary for a launch, such as stabilization and power supplies and possible political and psychological repercussions. Second, by 1950 RAND had helped the air force (which Truman created in 1947) through its early jurisdictional battles to emerge as the dominant agency for satellite development. Third, its studies helped to define a military use for satellites.
This chapter first explains the context and content of RAND's report of 1946 on the feasibility of satellites and shows how the air force employed the results to achieve dominance in the satellite turf wars of 1945–50. It then examines RAND's work on satellite reconnaissance that led to its groundbreaking 1951 study and looks at the aftermath of that report. Finally it considers RAND's FEEDBACK project of 1953–54, which dealt in detail with technical issues concerning a reconnaissance satellite. As I discuss in chapters 5 and 6, a number of key players in RAND's preparation of its crucial reports of 1951 and 1954 later played major roles in WS-117L.
## RAND and the Emergence of the Air Force (1945–1950)
By the end of 1945 the U.S. Army had already launched a concerted effort to dominate missile projects, using captured German scientists and missiles, creating the White Sands Missile Range in New Mexico, and funding several rocket-related programs. On October 3, 1945, the Navy Department expanded its Bureau of Aeronautics (a research-focused bureau) to include the Committee for Evaluating the Possibility of Space Rocketry. It did so at the prompting of Cmdr. Harvey Hall, special scientific assistant to the head of the Radar Section in the Bureau of Aeronautics. The most important of its many specified goals was conducting feasibility studies for employing an artificial satellite to relay naval communications. Within a month the committee agreed that it was technically possible to launch a satellite into orbit, and a contract went to California Institute of Technology's Jet Propulsion Laboratory (JPL) for it to undertake further study. In 1945, however, the engineering and preliminary work alone looked likely to cost in the range of $5 million to $8 million. In the face of financial cutbacks and demobilization, there were no funds for a project that lacked direct military application.
In an effort to preserve and expand on the existing work, the navy turned to the U.S. Army Air Forces (USAAF) to try to pool resources for a program. On March 7, 1946, it suggested the formation of a satellite committee to the members of the USAAF on the joint-service Aeronautical Board. The government had formed the board during the First World War to review new developments in aeronautics. Consisting of high-ranking members from the army and naval air arms, this board met monthly and acted as a bridge between the two services on air matters. In 1946 the Joint Research and Development Board took over some roles from the Aeronautical Board, which was finally disbanded in 1948. The new body's members agreed in March 1946 that the potential of satellites justified further discussion. The army representatives to the Aeronautical Board's Research and Development Committee agreed to investigate the "extent of Army interest by discussions with [Major] General [C. E.] LeMay [director of research and development]."
According to R. Cargill Hall, this consensus did not last long. Whereas most accounts portray the USAAF as delaying a decision on a joint program until after it could consult LeMay, pushing back the meeting from April 9 to May 14, 1946, Hall maintains that the USAAF rejected the joint effort prior to the April 9 meeting because it believed that missiles were in its bailiwick. Interservice rivalry quickly ended any hopes of such a collaborative effort.
At issue was the question of roles and missions, problems that plagued the military for years. The army saw the navy's proposal as a possible threat to its own rocket research, Project HERMES, which was to develop and test missile technologies using the German V-2, with General Electric holding the contract. The army saw development of rockets as an extension of the artillery's role on the battlefield, so it regarded the navy's proposal as poaching on what it considered an army mission. To forestall naval encroachment, the office of the USAAF's commanding general took the position that its branch had to demonstrate its competence and interest in space research to prevent an interservice conference from outflanking it.
Immediately after the decision to defer discussion of the navy's idea, LeMay called for a comprehensive study to establish the USAAF's competence in space matters. A tough-minded officer whose combat experience was in long-range strategic bombing, LeMay realized that simply turning down the navy's joint project would not end that service's interest or efforts. The navy had already contracted several feasibility studies with JPL and several private companies. To prevent further naval encroachment on army air force turf, LeMay realized that he needed scientific studies to demonstrate the air force's interest and dominance in the field. He turned to Douglas Aircraft's Research and Development project, or Project RAND, for a scientific report on the feasibility of orbiting a satellite. Douglas had set up Project RAND in March 1946 to continue the close cooperation between the military and the scientific community that had begun during the war. Realizing that science and technology had transformed warfare, Gen. H. A. P. Arnold and several key individuals from the aerospace industry tried to keep the relationship alive, and RAND was the result.
LeMay needed the report from RAND in time for the meeting of the Aeronautical Board's Research and Development Committee on May 14, 1946. RAND put in a rush effort, delivering its preliminary report on May 2 and a revised copy on May 12. Although not directly focused on reconnaissance, RAND's first report, "Preliminary Design for an Experimental Earth Circling Spaceship" (1946), proved seminal. First, it established that it was technically feasible to orbit a satellite. Second, it showed that any satellite would have significant repercussions. The RAND scientists expected that such a vehicle would become one of the most potent scientific instruments of the twentieth century. Third and more significant, an American device would impress the world with U.S. technology and have a psychological impact on other nations.
The document provided an in-depth analysis of the possibility of building and launching a satellite. Discussing various configurations for the rocket booster, it theorized about the possibility of sending up a payload of roughly five hundred pounds to an altitude of about three hundred miles. Such a satellite could stay in orbit for about ten days to gather scientific data. The report envisioned a multistage rocket (with either two or four stages, depending on the fuel) and analyzed the conditions in which the satellite would operate, the problems of attitude control, the threat from meteors, orbital trajectories, and the possibility of recovery. It determined moreover that it was theoretically possible to control the forces involved sufficiently to place a human being into orbit. The overall cost to orbit the first satellite would be perhaps $150 million over five years.
This pivotal work linked the scientific and military importance of both rockets and satellites. The only difference between a satellite booster and an ICBM was that a satellite required slightly more energy for orbit. Thus developments for one would aid the progress of the other. Predicting (correctly, as it turned out) that the missile would become the primary delivery vehicle for nuclear weapons, the document suggested several roles for a satellite, the most important being reconnaissance. The value of the satellite was very clear: "It should also be remarked that the satellite offers an observation aircraft which can not be brought down by an enemy who has not mastered similar techniques. In fact, simple computation from the radar equation shows that such a satellite is virtually undetectable from the ground by means of present-day radar. Perhaps the two most important classes of observation which can be made from such a satellite are the spotting of the points of impact of bombs launched by us, and the observation of weather conditions over enemy territory." Support missions, such as attack assessment and communications, were also feasible for satellites. As early as 1946 scientists had proposed observing the earth from space, and the report's demonstration that a satellite was theoretically and technically feasible set the stage for RAND to look carefully at questions of utility.
LeMay's attempt to use the RAND report to demonstrate the army air force's interest and competence in space was partially successful. It showed that the USAAF was looking at the problems and potential of satellites and effectively blocked the navy's proposal for a joint effort. It did not, however, prove the army air force's dominance in space matters; rather it muddied the waters. During its May meeting the Aeronautical Board's Research and Development Committee did not assign responsibility for developing a satellite to either contender. Instead it forwarded a summary to the Aeronautical Board, which, with half its members from the army and half from the navy, deferred judgment until higher authorities made a decision.
The issue of institutional control of satellites remained unresolved, and in January 1947 Rear Adm. Leslie Stevens (assistant chief, research and development, in the Navy Bureau of Aeronautics) went over the heads of the Aeronautical Board's committee directly to the Joint Research and Development Board (JRDB). He requested the formation of an ad hoc committee to coordinate all phases of an Earth satellite program and assign the control over such efforts to one of the services. In essence, as one historian of the pre-NRO satellite program, Robert Perry, maintains, the navy hoped to circumvent the army air force's dominance in the satellite field. Unfortunately the JRDB was a coordinating body with no authority to decide policy or act. To make matters worse, by trying to circumvent the Aeronautical Board and its committee, the navy only raised the ire of its competitors. According to Hall, the JRDB took note of Stevens's ideas and remanded them back to the Aeronautical Board for review before a final decision.
Events soon overtook the process. Congress created the National Security Act, which President Truman signed into law on July 26, 1947, and which reorganized the players in the early satellite drama. One of the act's major elements was the reorganization of the military in the light of new technology. It separated out the old army air forces as the core of the new independent air force. With the emergence of the independent air force in 1947, the two-way war over satellites became a slugging match among the three services. To be sure, the satellite issue was only a small part of the interservice rivalry. The three branches were fighting not just for control of the latest technology but for their share of everything from defense spending to personnel, industrial support, and popular opinion. For the air force this rivalry took on greater importance. Seeing itself as the first line of defense because of its nuclear-strike role, the air force thought of itself as the dominant American striking arm, relegating the other services to secondary roles. Yet this put the air force in a difficult position. Needing to keep its bomber and fighter forces strong enough to act as a deterrent, it also had to counter other services' perceived attempts to undercut its roles. Being the youngest service, it was extremely sensitive about its independence. Thus for the air force interservice rivalry for cash, resources, and popular opinion quickly became a fight for its survival as an independent military arm.
The Research and Development Board (RDB) absorbed the JRDB on September 30, 1947, and became part of the new Department of Defense. The RDB was to assist the secretary of defense in coordinating the armed forces' research and development activities and answered to him; Vannevar Bush was its chair. No longer a coordinating organization, the RDB could now create policy and exercise some direct control over the making of decisions. Aware of the security implications of satellites and the need for an effective policy, the RDB directed its Committee on Guided Missiles to coordinate satellite policy, which removed the roadblock at the Aeronautical Board. The committee assigned its Technical Evaluation Group to examine the issues and assign responsibility. The RDB killed the navy's plans to orbit a satellite because their proposed program lacked immediate military utility. In 1946–47 cuts in the defense budget severely curtailed or canceled many programs. A highly speculative and risky program that did not fill a pressing military need did not stand much chance. The only activity that the RDB sanctioned was continued study of satellite utility through RAND.
The former army air forces had been more successful. Although the first RAND study did not create much enthusiasm for a satellite program, it did demonstrate the program's feasibility. Unable to obtain control over satellite efforts, the army air forces were able to gain the Aeronautical Board's approval to continue examining space projects (and reaffirm it later with the JRDB). It ordered a second RAND study, which over the winter of 1946–47 produced a series of documents on various aspects of satellite flight that defined the problems and benefits of a project. These papers triggered no immediate action but laid the groundwork for future planning. Following the creation of the air force on September 18, 1947, they received further consideration. On September 25 the air force ordered its Air Material Command to evaluate all the RAND studies from the perspective of technical and operational feasibility.
The Air Material Command's response was conservative, and its report certified technical feasibility but cautioned that the complexity and cost might be untenable. The paper questioned the air force's ability to maintain funding for such large-scale research and development. Noting that the practicality of this kind of system was in question, it urged a study to determine air force requirements, functions, and scheduling that would permit development at a later date. In the meantime guided missiles had priority. Satellites remained on the back burner, although RAND continued doing relatively cheap and useful feasibility studies.
Although the air force did not pursue a satellite program immediately, it decided to assert its supremacy in space research. Lt. Gen. H. A. Craig, deputy chief of staff for material, concluded that, despite current financial limitations, progress in research on guided missiles meant that a full-scale satellite program would become feasible in time. On January 12, 1948, Craig urged Gen. Hoyt S. Vandenberg (vice chief of staff) to assert the air force's responsibility for space, thus assuring its dominance. Vandenberg agreed and three days later signed a formal policy document—the first clear declaration by any service regarding control of a space program.
Vandenberg's "Statement of Policy for a Satellite Vehicle" affirmed the air force's claim on space research and development policy: "The USAF, as the service dealing primarily with air weapons, especially strategic, has logical responsibility for the satellite." Indicating that it was making rapid progress with guided missiles, he linked satellites with the missile sciences: "The problem will be continually studied with a view to keeping an optimum design abreast of the art, to determine the military worth of the vehicle—considering its utility and probable cost—to insure development in critical components, if indicated, and to recommend the initiation of the development phase of the project at the proper time." This statement put the other services on notice that the air force would study the matter of orbiting a satellite to assess its military value and costs.
The air force's director of research and development authorized its Wright Field agency near Dayton, Ohio, to put Vandenberg's ideas into effect. It fell to RAND to monitor the state of technology and inform the air force of technical developments. In February 1948 Brig. Gen. Alden R. Crawford (Wright Field) instructed RAND to develop components and techniques for an eventual launch, thereby maintaining the effort at a study level only. RAND in turn subcontracted important research to aerospace companies such as North American and RCA and to institutions of higher learning such as Boston University. It helped determine the feasibility and effectiveness of complex satellites, and by 1950 research was focusing on two future aspects of the space program. These illustrated changes within the RAND community that had not yet reached the military or the U.S. administration. RAND had begun to examine satellites' psychological/political implications and, more important, their military utility.
## Satellite Reconnaissance and the RAND Report (1950–53)
The shift in emphasis from feasibility to utility helped RAND convince the air force's Directorate of Intelligence to undertake further research into satellites. Within RAND satellites had strong support from many people, including Merton Davies and Amrom Katz, who became advocates, monitors, and even guiding voices of "expert knowledge when it came to satellite reconnaissance from 1954 to 1960." These enthusiasts transcended their role as objective providers of knowledge to become partisan advocates.
The first step was to determine the psychological and political ramifications of orbiting a scientific or a military satellite. In a study that appeared in early October 1950, Paul Kecskameti directly addressed these issues. While examining them from the perspective of the United States being first into space, he focused on the impact this would have on the Soviet Union and world opinion. In many regards his report was prophetic.
He believed that a satellite, whether scientific or military, would have significant political ramifications. The worldwide reaction depended on how the United States was viewed. The technological achievement would inevitably increase perceptions of American strength and technical and scientific superiority. Rival or hostile states would perceive a satellite as a threat to their interests. The Soviet Union in particular would see it as a challenge and a demonstration of its own weakness. Its propaganda would emphasize the American threat. Friendly nations would see the success as an affirmation of American scientific, technical, and military prowess and thus find it reassuring.
Kecskameti's interpretation is entirely logical and represents a clear understanding of the likely situation. To counter these problems he advised the United States to neither conceal the launch nor play up its military value. To maximize its political and psychological value, the U.S. government had to keep the public abreast of its development, while emphasizing its peaceful and scientific roles. This approach would prevent the illusion of a nefarious military purpose. The alternative approach of total secrecy and then a sudden release of information would have a greater psychological impact. Unfortunately this was likely to push even neutral countries toward hostility. In the long run, no matter what approach the United States followed, hostile states would still react negatively to an American satellite. Although advance publicity would blunt some of the dramatic effect, it would be better to diminish the effect rather than risk a strong backlash. In either case the Soviet Union, which depended on secrecy to conceal its military weakness, would react harshly and equate an American satellite with reconnaissance and perceive it as a direct threat.
Finally Kecskameti raised concern about the legality of overflight. There was no legal precedent for a satellite flying over a sovereign nation. Without resolution of these issues, such a use of space could lead to serious international legal repercussions. Kecskameti was accurate with his prediction that a satellite was "bound to be a spectacular event, causing a worldwide sensation." The post- _Sputnik_ reaction certainly lived up to his expectations, complete with "estimates of aggressive intent behind the development and use of the instrument." His only incorrect prediction was that the United States would orbit a satellite first.
Kecskameti's report shows that RAND clearly anticipated the benefits and risks of satellite reconnaissance. It also set the stage for later actions by the Eisenhower administration to establish the legal framework for both satellites and overhead reconnaissance, but it would not lead to a satellite program. Before that could begin the air force had to demonstrate a compelling military need—only that would win funding. Beginning with their initial study, RAND scientists had looked at a variety of roles for a satellite, among which one crucial mission stood out: reconnaissance. But before a satellite for that purpose could emerge, RAND had to prove its technical feasibility in gathering useful intelligence and work out a viable design. By 1954 it had produced both. This achievement coincided with the Eisenhower administration's desire for better surveillance of the Soviet Union and its conclusion that a reconnaissance satellite might be the right vehicle.
The idea of overhead reconnaissance was not new in 1950; it was as old as aviation itself. By 1945 aerial photography allowed for extremely high-resolution color photographs. Researchers, including Dr. James Baker, had developed camera lenses with focal lengths as large as 240 inches, which automatically compensated for alterations in air pressure as an aircraft's altitude changed. By war's end photographic equipment had become very precise, and the interpretation of photographic images had become much more technically sophisticated. Army air force doctrine, however, considered overhead strategic reconnaissance only in terms of specific functions in wartime: for identifying and locating targets vital for aerial attack, for their defenses, and for assessing bomb damage. The concept of preconflict reconnaissance to gather general intelligence simply did not exist within the U.S. military.
The advent of atomic weapons had forced the United States to reassess intelligence and strategic reconnaissance. Breaking away from the narrow military view, many began to equate "pre-D-day," or peacetime, reconnaissance with an ability to warn of surprise attack. In the wake of Pearl Harbor and the dawning of the nuclear era, peacetime reconnaissance became more important. The first clear call for a new type of reconnaissance came from Gen. Henry H. "Hap" Arnold (USAAF) in November 1945. Like Eisenhower, he understood that atomic weapons made the nation's wartime experience obsolete. While Pearl Harbor had been costly, the United States was able to recover and rebuild its military to wage a global war; but an atomic Pearl Harbor would be quite different, threatening the American heartland itself. The general warned Secretary of War Robert Patterson that if U.S. leaders were to prevent atomic attack they needed "continuous knowledge of potential enemies." This included information not just on troop deployments but on economic, industrial, political, and scientific developments. Arnold was unwilling or unable to say how to collect such data; the man who could—and did—was a junior officer, air force lieutenant colonel Richard S. Leghorn.
Leghorn was a graduate of the Massachusetts Institute of Technology with a bachelor's degree in physics. At MIT he became friends with James R. Killian Jr. Leghorn's career encompassed military and civilian occupations that helped to shape his views on reconnaissance and science. Following graduation in 1939, he worked for Eastman Kodak before joining the army reserves as a lieutenant in the Ordnance Corps. In 1940, following a meeting with Maj. George W. Goddard, commander of the army air corps' Aeronautical Photographic Laboratory at Wright Field, Leghorn transferred to the corps and began work in aerial photography. As a member of Goddard's team, he searched for methods to glean more data from photographs. He also met several major figures in reconnaissance circles who came to play a major role in satellite reconnaissance, including Amrom Katz (a civilian physicist and future RAND scientist), Lt. Walter J. Levison (a physicist), and astronomer James G. Baker, who designed camera lenses both during and after the war.
Leghorn reunited with his Dayton colleagues to provide aerial reconnaissance during the CROSSROADS nuclear testing at Bikini Atoll in October 1945. The military needed trained crews, and Leghorn became deputy to Col. Paul T. Cullen, commander of the photographic unit. During the journey to the test site Leghorn had access to the _United States Strategic Bombing Survey (Europe)_ , which assessed the effectiveness of bombing operations in Europe. The report noted that a lack of strategic reconnaissance before the war and during the initial stages of the bombing had hampered targeting and wasted effort. Leghorn studied the conclusions, specifically the call to improve coordination among intelligence services in collecting and evaluating information. The document also encouraged involvement of civilian scientists in intelligence gathering and concluded that "the combination of the atomic bomb with remote-control projectiles of ocean-spanning range stands as a possibility which is awesome and frightful to contemplate."
The Bikini testing in July 1946 impressed on Leghorn the case for prehostility reconnaissance. For the first time he saw the result of an atomic blast. Although there is no evidence that he knew about General Arnold's comments on the importance of intelligence, he shrewdly reached the same conclusions. Reliable peacetime intelligence would decrease the threat of a nuclear Pearl Harbor. Leghorn quickly realized that regular overhead strategic observation would solve the intelligence dilemma. He also decided that a specialized aircraft would reduce the chance of detection. He repeatedly shared his views on overhead reconnaissance with his colleagues from Wright Field.
Among the early converts was the physicist Duncan Macdonald, head of Boston University's Optical Research Laboratory (BUORL). Macdonald thought highly of Leghorn's ideas and invited him to speak at BUORL's dedication on December 13, 1946. Leghorn's comments were the first public discussion of peacetime overhead reconnaissance. He argued that atomic weapons made peacetime aerial reconnaissance imperative. Nuclear war's destructiveness made surviving a surprise assault and mounting a successful counterattack unlikely. Only information from aerial reconnaissance could prevent such a surprise.
With the scars of Pearl Harbor still fresh in the American psyche, the consequences of a lack of vigilance were clear. Aerial reconnaissance of the sea approaches to Hawaii could have provided some warning. Now the constant threat of total destruction from a single attack created a pressing need for intelligence. The Pearl Harbor effect was dramatic in the military: it led to calls for larger budgets, more striking power, and a stronger defensive stance. However, this outlook prepared the United States for a possible confrontation but provided no warning. Leghorn called for something different: the ability to warn of an attack before it happened.
Leghorn envisioned a world in which most intelligence came from overhead reconnaissance in daylight and, if the technology allowed, at night as well. He believed that this was the only way to penetrate a totalitarian regime keen to preserve its secrets. He admitted that this activity was illegal under international law but insisted that it was essential for national security. Thus a method to minimize the chance of detection was necessary. He envisioned an aircraft that flew at extreme altitudes, _not_ satellites. At this point in time spy satellites were still a speculative idea.
Following the nuclear testing Leghorn returned to Eastman Kodak, but he did not stay out of military intelligence circles for long. The army air forces started overflights of the Soviet Union as early as 1946, when they employed aircraft—mostly RB-29s (modified Boeing B-29 Superfortresses)—patrolling the Soviet periphery to collect intelligence. Carrying cameras of limited focal length (thirty-six inches or less), they flew within a few miles of Soviet territory taking oblique photographs. The army air forces were especially curious about the areas around the Kola Peninsula (especially the port of Murmansk), the area north of Leningrad, and the Chukotskiy Peninsula (across the Bering Strait from Alaska). They saw these regions as potential staging points for any attack on the United States. Since the planes had to stay outside Soviet territory and had only cameras with short focal length, the pictures were of limited value. Hence these flights concentrated on gathering electronic intelligence. Despite gradual improvements to the cameras, oblique photography never provided the depth of coverage that the United States required.
In the wake of the first Soviet atomic bomb test in 1949, U.S. restrictions on the penetration of Soviet airspace were quickly pushed aside. As the cold war chilled East-West relations, the air force (especially the Strategic Air Command, the nuclear striking force) argued for penetrating Soviet air space to gather intelligence. With evidence of Soviet nuclear testing in hand, General LeMay (then commander of SAC) recommended to President Truman that the United States begin overflights of the Soviet Union to collect evidence of possible preparations for a surprise attack. Still using converted bombers, these missions inflamed emotions in the Soviet Union and were very risky in the face of its air defenses. Penetrating its airspace posed two problems. First, the planes' limited range meant that there were vast areas that they could not observe. Second, they were extremely vulnerable to enemy fire, so each flight increased the chances of an international incident. From 1952 to 1955 seven incidents did occur, in which thirty crewmen were either killed or missing in action.
These problems forced the air force to find another method of obtaining intelligence. By 1955 it had tried a variety of schemes, the most innovative being Project GOPHER, which unsuccessfully released hot-air balloons carrying cameras across the Soviet Union. But GOPHER and the development of the U-2 aircraft during 1955–56 were only stopgap solutions.
By the time Kecskameti's 1950 report appeared, RAND was already on the way to solving this problem: using satellites for reconnaissance. This work took the original RAND studies on orbiting a satellite and married them to the concept of overhead reconnaissance that Leghorn advanced in 1946. Culminating in two separate studies, the idea of overhead reconnaissance evolved by 1954 into a clearly defined and articulated call for a satellite to provide vital intelligence for the next fifty years. By November 1950 RAND's thinking had advanced to that point, and it recommended to air force headquarters that it extend research into some aspects of a proposed reconnaissance system.
Following a meeting with RAND, Col. Bernard Schriever requested that the air force's Directorate of Intelligence provide specific intelligence requirements for a reconnaissance satellite. The directorate's reply on March 17, 1951, identified high-quality images as the primary requirement. The proposed satellite would need picture resolution sharp enough that experts could identify harbors, oil storage areas, large residential and industrial targets, and airfields and collect weather data. The images had to provide enough clarity to allow for correction of maps and charts and be able to cover the entire Soviet Union in a period of weeks, provide continuous daytime coverage, and record the information in more or less permanent format. Representatives from the directorate visited RAND on March 2, 1951. The scientists simulated a satellite photograph by taking a photo of Los Angeles and relaying it via television to Mount Wilson, California, and back before photographing the monitor screen. Photo interpreters examined the photo and agreed that it would satisfy the minimum requirements. Maj. Gen. C. P. Cabell, the author of the memo from the Directorate of Intelligence, recommended starting the reconnaissance satellite project "with a view toward the present urgent need for such a reconnaissance system, rather than a future need." 44 The demonstration that a satellite could provide the necessary type of intelligence was only the beginning.
RAND's study, "Utility of a Satellite Vehicle for Reconnaissance" (April 1951), examined five different areas of importance relating to satellite reconnaissance. Starting with a discussion of orbits and the ground area covered, the report went on to cover the reconnaissance process, control of the satellite, power supplies, and reliability of a robotic system. It analyzed comprehensively the capabilities of satellites for reconnaissance, paying particular attention to the use of television, communications, and the problem of the electrical power supply. These were all limiting factors of the satellite.
The report found that the optimum orbital altitude for a spy satellite was about 350 miles, with an inclination of roughly 53 degrees from the equator. This meant that the satellite would orbit the earth fifteen times per day. Due to the earth's rotation, the satellite's orbital path over the main areas of interest would shift by about eight hundred miles with each orbit. Thus every eighty-seven or so days the satellite would return to its original flight path, having covered the entire area of interest. Unfortunately, because of the earth's rotation, daylight coverage was possible only during alternating thirty-five-day periods. Thus it would take two satellites to give adequate coverage.
This altitude seemed to be the best balance between the forces that limited the satellite. Higher orbits promised the vehicle longer life and more time to convey information but needed larger booster rockets to reach orbit; either image quality would be poorer because of the greater distance, or a larger payload would be necessary to allow for bigger camera components. Lower orbits meant smaller boosters and higher image resolution but shorter life because of friction in the denser atmosphere. Friction slows a satellite, causing it to ride lower in the atmosphere, and generates heat. Eventually the satellite will reenter the atmosphere and burn up. An altitude of roughly three hundred miles seemed likely to make for a satellite life span of two years.
The RAND scientists felt that the ICBM program provided the obvious answer as to how to launch the satellite. An ICBM was a suitable primary booster if high accuracy was possible in attitude control. This was essential to ensure that the camera system faced the earth properly at all times. The scientists at RAND discussed several options, including the use of gyros, flywheel stabilization, and attitude sensing, but presented no definitive solution.
Power consumption was the main limiting factor for the system. The satellite needed enough power to run equipment for an extended period of time. This included not only the camera and attitude control but also the radio, ground communications, storage of ground commands, and timers. J. E. Lipp and the other RAND scientists predicted that the satellite would need about five hundred watts of power for up to a year, while staying within strict weight requirements. With a 1,000-pound payload, only 250 pounds could go to produce electricity. Conventional means of producing power were ruled out because of the huge amount of fuel they would need. Therefore the RAND report proposed a nuclear reactor employing radioactive isotopes to produce heat to generate electricity.
The study team now tried to figure out how to retrieve data based on the predicted orbit pattern. It quickly dismissed the idea of recovering the intelligence through the use of a reentry capsule. Just as with the ICBM program, any attempt at physically recovering anything from space was seen as impossible at that time. The extreme heat of reentry was expected to destroy the capsule. While the USAF was working on solving this problem for the ICBM program, the technology had not yet been found to allow a successful reentry. Until the technology caught up to the idea, alternative methods of information recovery were needed. Two alternative methods for gathering photographic intelligence were examined in detail: photostatic facsimile transmission and the adaption of standard television technology.
Photostatic facsimile transmission was a reliable technology then in use. It used standard camera film to record images. These images were then electronically scanned, converting them into electrical impulses in a method similar to the standard wire photo of the time. A radio link could then relay this information back to Earth. While tried and tested already on Earth, the main limit of the system was that it required a lot of film. A month's operation was expected to consume about three-fourths of a ton of film. Weight wise, this was cost prohibitive unless reusable film became available.
The alternative option was to adapt current television technology. While only allowing daylight observation, the scientists felt that it would work for pioneering reconnaissance efforts. Basing their predictions on current technology, RAND scientists predicted photographic resolution of roughly two hundred feet. Though suitable for weather data, such low resolution was not good enough for surface surveillance. It would detect major airfields, highways, railways, and large factories but not much more. Productive intelligence needed fifty-foot resolution. The study also predicted that assessing bomb damage would necessitate resolution of about ten feet. Increasing resolution by increasing lens magnification meant either decreasing the area covered per photo (so total coverage would take longer) or technically improving the camera system. The scientists were confident that resolution of forty feet would eventually be feasible.
The study's final issue was reliability. A satellite in space would have to operate independently and perform perfectly for a year or more in a vacuum and at extreme temperatures. In addition it had to survive the excessive vibration of the launch, motion changes, and gravitational forces. None of these expectations seemed unrealizable. The scientists believed that technology "on the shelf" could sustain at least thirty-five days of operations. Through research and development, this might be extended to a year.
In conclusion the 1951 RAND study assessed available technology and related it to aerial reconnaissance. It also helped to prove that a military satellite was feasible and made it clear that any problems with television reconnaissance—the most likely system—would relate to engineering. RAND saw the basic science as already extant.
The RAND scientists disseminated and promoted their work at various levels of the military. After release of the report, military representatives and members of BUORL received briefings at Wright Field. In the audience was Amrom Katz, chief physicist at the Air Force Reconnaissance Lab and one of the greatest skeptics about satellite reconnaissance. The briefing by Lipp and his colleagues generated a mixed response. Some listeners found the findings encouraging, except for the expected poor resolution of satellite images. The air force photo interpreters thought the resolution inadequate for their purposes, which seriously challenged the RAND team. Katz and three of the BUORL participants used the resolution problem to discount the whole proposal.
Katz, Walter Levison (assistant director, BUORL), Duncan Macdonald (director, BUORL), and Col. Richard W. Philbrick (air force liaison officer to BUORL) found a test to refute RAND's proposals. The test was conducted at Wright Field in November 1951. It used an 8-mm camera and coarse film to take reconnaissance photos from an altitude of thirty thousand feet to mimic the satellite's performance. Katz and his colleagues were stunned by the results. Expecting to see nothing of value, they were able to clearly identify roads and bridges in the Dayton area, the airfield itself, and several landmarks. Katz became an instant convert, joining RAND in 1954 and pushing for satellite development.
During the spring of 1951 three developments occurred at roughly the same time as RAND's report. The first was the activation of the air force's Air Research and Development Command (ARDC) in April 1951. Working with the deputy chief of staff for development, the ARDC was to improve research and development within the air force. Since war's end that service had concentrated on building up its combat arms, but this effort had failed to provide for future needs. The ARDC began to play a greater role as the satellite program moved toward development.
Second, the USAF recalled Colonel Leghorn to active duty in April 1951 because of the Korean War. Initially stationed at Wright Field, he was reassigned to the weapon-system procurement office as chief of reconnaissance systems. There he surveyed the requirements for reconnaissance and assessed possible appropriate systems. Initially he worked in coordination with Colonel Schriever, assistant for Air Force Development Planning (AFDAP) in Washington. Schriever was preparing Development Planning Objectives (DPOs) for the service's roles in areas such as tactical air operations. In August 1951 Leghorn was transferred to Washington under Schriever's command to help prepare a new DPO for intelligence and reconnaissance and to act as the air force's liaison officer with RAND.
Leghorn was in a unique position in this assignment. He worked with some of the most influential people in Washington: Edwin Land, Carl Overhage (chief of Eastman Kodak's color laboratory), James Baker, Edward Purcell, and Burton Klein from the RAND Corporation. These men shared their views on intelligence gathering. Curtis Peebles maintains that Leghorn's liaison duties also put him in contact with Merton Davies and the RAND satellite proposals. Thus Leghorn was at the focal point, both within and outside the Pentagon, for much innovative thinking.
Leghorn further refined his views on intelligence in 1951–52 to help the "counter force" strategy for nuclear weapons that he had begun to advocate. Such a strategy calls for targeting of a potential enemy's nuclear arsenal with your own. In the event of war your nuclear strike would then attempt to disarm your enemy. Leghorn realized that intelligence was the key here and that the military would need prior intelligence of Soviet strategic assets. As Leghorn put it, "Our qualitative intelligence and reconnaissance capabilities constitute the primary problems, and without extraordinary action, these might delay adoption at operational planning levels of strategies with emphasis on counter force operations."
Overhead reconnaissance was the only practical method of gathering this intelligence. Leghorn emphasized use of an unmanned aircraft with a credible cover story, such as pursuit of scientific or weather data. Recognizing that manned aircraft would have to suffice, he thought them vulnerable and useful only over areas with weak air defenses. How much he knew about RAND's satellite proposals is in question. According to Davies, Leghorn's contacts at RAND and conferences on intelligence gathering exposed him to the idea of satellites. Davies claims that he had convinced Leghorn about reconnaissance satellites and their inclusion within the framework of air force needs; however, Leghorn considered these long-term goals.
A third development in 1951 coincided with release of the RAND report, emergence of the ARDC, and Leghorn's return. An agreement between MIT and the air force to study defense issues led to the creation of the Beacon Hill Study Group under the auspices of Project Lincoln. Schriever's AFDAP branch had called for such a body to consider U.S. intelligence problems and to suggest better ways of gathering and processing intelligence. These civilian and military experts worked from July 1951 until completion of their report the following June.
The Beacon Hill experts included Baker, Killian, Land, Leghorn, and Overhage. We can see Leghorn's influence in one of the report's crucial recommendations: "We have now reached a period in history when our peacetime knowledge of the capabilities, activities, and dispositions of a potentially hostile nation is such as to demand that we supplement it with the maximum amount of information obtainable through aerial reconnaissance. To avoid political involvement, such aerial reconnaissance must be conducted either from vehicles flying in friendly airspace, or—a decision on this point permitting—from vehicles whose performance is such that they can operate in Soviet airspace with greatly reduced chances of detection or interception." The report recommended improvements in sensors and development of aircraft and other means of flying over the Soviet Union. Borrowing from Leghorn's ideas (as they appeared in his DPO), its writers emphasized vehicles capable of peacetime reconnaissance with a minimal risk of detection and preferred unmanned flights for penetration of another nation's airspace. They outlined several options for reconnaissance, including high-altitude balloons (this became Project GOPHER in the mid-1950s), higher-altitude aircraft, and either the Snark or the Navaho air-breathing missile. The group considered aircraft overflight politically risky except for lightly defended areas. Whatever means the military selected, it needed a credible cover story if diplomatic complications occurred.
The Beacon Hill Group did not ignore the RAND satellite proposals. According to Davies and Peebles, Leghorn eventually became a convert to satellite reconnaissance. But the Beacon Hill Group received no direct RAND input and invited no RAND personnel to speak, let alone to participate on the steering committee. Several RAND personnel did attend meetings and had contact with members outside these sessions, but they were there as guests.
The Beacon Hill report (June 1952) assumed that satellites were possible in the long term. Believing that making satellite reconnaissance a reality in a timely fashion would be very expensive, the group concluded that it would be better to apply such funds to developing other, more promising technologies. Both Land and Leghorn were strong supporters of balloon and aircraft reconnaissance and advocated these options instead.
Leghorn's role in the development of overhead reconnaissance was played out in two final acts. First, in November 1952 he briefed the Air Force Air Council, the CIA, and the NSC on reconnaissance requirements and progress to date on efforts to meet these needs. Second, he submitted his DPO to Schriever in January 1953, after which he returned to Eastman Kodak before eventually starting the Itek Corporation, which produced payloads for the future CORONA program. In this DPO he argued that high-altitude reconnaissance was the best means of gaining intelligence from the Soviet interior. He forcefully supported development of high-altitude balloons and specialty aircraft for overhead reconnaissance, with balloons providing area coverage and planes penetrating areas of interest that photo interpreters deemed worthy of closer observation. Satellites would play a role in the _distant_ future. Until then the endeavor required a specially built aircraft capable of operating at over seventy thousand feet. In the wake of Leghorn's DPO, the air force began looking at designs of high-altitude aircraft for strategic reconnaissance, and the winner was the U-2.
Within a few months of the Beacon Hill report's completion in June 1952, Americans elected Eisenhower as president. For such a cost-conscious, intelligence-aware chief executive, the report became more than just an air force document. As a foundational work its conclusions were picked up in the even more important TCP report of 1955. The Beacon Hill paper reinforced the value of intelligence and of overhead reconnaissance (the very foundation of the RAND study of April 1951). More important, its emphasis on an unmanned vehicle with a small likelihood of detection and interception seemed to fit satellite reconnaissance. The problem lay not in convincing experts of feasibility but in doing things quickly.
Despite the best intentions to expedite satellite development, inertia within the air force meant progress was slow and tedious. Even as the Beacon Hill experts were considering satellites, some air force officers saw development as premature. They suggested instead a further feasibility study. Thus on December 19, 1951, the air force authorized RAND to make recommendations on development of reconnaissance satellites. This slow movement frustrated the RAND experts. The old notion that reconnaissance was a wartime operation was hard to kill. Ironically the one agency that should have been advocating peacetime photographic reconnaissance, the CIA, showed no interest. It still focused on human intelligence, which emphasized penetration agents and clandestine collection. Its very small photographic-interpretation unit had no real authority.
Despite air force and CIA resistance, the ideas from RAND, Leghorn, and the Beacon Hill group were slowly filtering up through the air force chain of command. In December 1952 Lt. Gen. Thomas D. White (deputy chief of staff for operations) promoted the idea of satellite reconnaissance. Citing the Soviet hydrogen bomb, he argued that Washington desperately needed to "obtain reconnaissance and surveillance data leading to knowledge of Soviet capabilities and intentions before the beginning of hostilities." White mentioned a high-flying aircraft but thought a satellite reconnaissance vehicle the best solution. Invoking RAND's predictions for resolution and satellite specifications, as well as political and psychological values, he recommended that the air force immediately issue a requirement for a satellite.
## Project FEEDBACK and Fine Details (June 1953–March 1954)
Despite frustration RAND returned to the drawing board for yet another study, called Project FEEDBACK. This effort was aimed not only at justifying a satellite system but at specifically defining the components required. To support this effort, in 1952 RAND signed contracts with a number of companies to examine particular aspects of a reconnaissance satellite. These contracts included several with RCA to look at adapting television cameras, radiation-recording devices, and other equipment, and North American Aviation, which studied orbital sensing and control systems. The air force also supported these studies. In July 1953 the Communication and Navigation Laboratory at the Wright Air Development Center in Dayton contracted with North American Aviation for a study of pre-orbital guidance systems. The air force even arranged for the Atomic Energy Commission to begin work on small reactors suitable for a satellite. The government funded most of these projects under its existing contract with RAND through a special supplement for fiscal year 1953 specifically for the satellite research.
By this time the air force was ready to take a more active role in satellite reconnaissance. In May 1953 it ordered the ARDC to take over the FEEDBACK program by June 1 and to investigate the feasibility of starting to develop auxiliary nuclear power plants for satellites. This led to a series of meetings between the ARDC and RAND. The ARDC found RAND's efforts to date impressive and quickly grasped the importance of satellite reconnaissance. In September, with ARDC support, RAND pushed the air force to authorize contracts for system design within a year and then begin full-scale development.
In the waning months of 1953 pressure for satellite development was building. Seeking to pull together elements of the endeavor for better management, the ARDC gave the program its first official designation. Identified as Project 409-40, "Satellite Component Study," the program was also given an unofficial project number (WS-117L) for later system development. The Wright Air Development Command took responsibility for the program and began work to demonstrate the feasibility of major satellite elements—specifically television components, attitude and guidance control, and auxiliary power-plant subsystems. In January 1954 the project received the unclassified title "Advanced Reconnaissance System" and an engineering-project designation, MX-2226. Lacking only official authorization WS-117L was ready for development.
In March 1954 the long-awaited, two-volume FEEDBACK study finally appeared. It was released just as the Defense Department was reporting that ICBMs were technically feasible and at about the same time as the Killian Commission was working on the TCP report. Unfortunately, very little detail on the report has found its way into the historiography because it has only recently been declassified in its entirety. Portions became available in the 1960s for Robert Perry's work "Origins of the USAF Space Program," but they indicated only some of the RAND study's conclusions. Most early historians refer to the FEEDBACK report but give only vague indications about a reconnaissance satellite using television technology. More recent work by Jeffrey Richelson, R. Cargill Hall, and William E. Burrows provides more details about image resolution and the satellite's longer-term implications. They maintain that the report's description of a satellite capable of continuous Soviet surveillance spurred the creation of a system for satellite reconnaissance. Unfortunately, these sources do not convey either the report's scope or its impact on the decision to develop satellite reconnaissance. To better understand the report and its significance to the air force, it needs to be examined at some length.
The FEEDBACK report, like RAND's April 1951 studies on satellite reconnaissance, represented a refinement of not only the engineering data but also the fiscal and political aspects of such a system. It was the culmination of work begun in 1946, in the sense that many of the same scientists who had been advocating satellites then played a role in FEEDBACK. As such it represented the collective wisdom of some of the top minds in the field. In the report RAND formally recommended that the air force undertake the development of a satellite vehicle at the earliest possible opportunity. It felt that this decision ought to be made at a high policymaking level and that both the decision and the program itself should be covered in a blanket of secrecy. Predicting that the process would take about seven years at a cost approaching $165 million, the experts cautioned that this estimate could double or triple depending on a variety of circumstances.
To help explain the satellite system that FEEDBACK proposed, specific aspects of the program need to be examined. The report recommended a two-stage launching rocket to put the vehicle into orbit. With an overall length of about eighty-one feet and a total takeoff weight of about 178,000 pounds, the rocket would use two main boost engines and two gimbaled motors to generate a takeoff thrust of 285,000 pounds. The gimballing of the smaller engines was part of the rocket's control system. Small motors linked these engines and the control systems, allowing the rocket to adjust the exhaust and thus change direction. Approximately 80 percent of each stage went for storage of the fuel and oxidizer (in this case, gasoline and oxygen). The second stage was the satellite itself, which consisted of the reconnaissance payload, weighing 1,500 pounds (about one-third of the second stage's total dry weight), related items (recorder-playback systems and so on), guidance and attitude control, transmitters and receivers, and the power plant. The camera, antennae, and horizon scanners would be mounted at the rear of the satellite, above the motor for the second stage.
The only prerequisite for a launch site would be proximity to the latitudes from which the military wanted intelligence. Thus for surveying the Soviet Union a location near White Sands, New Mexico, or Patrick Air Force Base on the California coast would be acceptable. The actual ascent would follow a pattern that has become almost the norm: after a vertical launch, the two stages would be fired in sequence to achieve sufficient altitude. At that point, using an inertial guidance system, the rocket would tip over to the correct angle for the desired orbital path before coasting into a generally circular orbit. The second stage would ignite again briefly for a final application of power to create a stable orbit. The inertial guidance system acted as a point of reference to measure satellite motion to maintain course and attitude. To stabilize the platform a system of gyros, accelerometers, analog computers, and servo controls measured and compensated for any pitch, yaw, or roll. A horizon scanner would keep the camera aiming at the earth. The operational life was likely to be at least one year.
The power source for the satellite was to be a water-moderated nuclear reactor designed to heat mercury into a gas that turned a turbine to generate electricity. The reactor itself was a sphere two feet in diameter that would produce about eighty kilowatts of heat energy. Radiators along the satellite's hull would dissipate excess heat and cool the mercury. The RAND experts chose a nuclear reactor due to the need for a long-term power source. Chemical power sources and the radio-isotope-heated power plant from the 1951 report seemed inadequate, and using conventional fuels and oxidizers was impractical due to the massive amounts needed. Solar energy was also rejected due to the primitive state of the technology at the time. The only limit on nuclear reactors, however, was the need to radiate excess heat outside the spacecraft. Because of the satellite's size the experts concluded that such a device would generate about one kilowatt of electricity; more power would require diversion of more of the satellite's weight to the reactor. The overall weight of a plant producing a single kilowatt of electricity would be roughly 450 pounds. For an additional fifty pounds of satellite weight, the reactor could produce two kilowatts. The plans called for a power source of one to two kilowatts.
The satellite that FEEDBACK described was to be part of an integrated system for obtaining photographic intelligence. The report also dealt with testing, ground control, data handling, and other elements. Ground stations would download imagery, aid in tracking and locating the satellite in orbit, relay command instructions to the vehicle, and monitor satellite systems. They would relay intelligence data to a central intelligence center, which would interpret photos, compile them into mosaics, and store them. A two-way radio link would handle communications. The onboard television would relay the image to a receiver at the ground station, whence a transmitter would instruct the satellite. All communications between ground stations and satellites would use microwave-transmission frequencies, which require line of sight for communication. Scientists anticipated that this would reduce jamming and decrease the risk of interception by the Soviet Union. Ideally the satellite would be able to store information before transmitting it to ground stations within the continental United States. Otherwise the ground receiving stations would have to be located closer to the Soviet Union, where they would be under greater security threat.
The satellite's success of course depended on its ability to produce photos with enough resolution for interpretation. RAND understood that initial imagery would at best be the bare minimum acceptable. Conventional aerial photography was using imagery with a map scale down to 1:80,000, the minimum useful for interpretation; resolution of satellite photos was to be only about 1:500,000. Lines of communication and distribution and transportation systems (including highways, railroads, and pipeline and power-line right of ways) would be visible; however, as they appear as long lines they would be mutually indistinguishable, whereas harbors, docks, ships, and related structures would be readily visible. Equally important, repeated observation could also indicate the level of activity in these areas. Photographs with a 1:500,000 scale can reveal airfields, larger military installations, and urban and industrial areas with some degree of definition.
Thus early on experts expected satellites to provide only what the USAF director of intelligence described in 1951 as "pioneer-level" imagery and mapping capabilities, which were adequate and useful for planning bombing operations. Barring weather problems, the system could probably survey the entire Soviet Union in three or so days. Identifying the roles and functions of buildings and other structures would be difficult.
Both RAND and RCA proposed a camera system employing a standard Image Orthicon television camera of the period, attached to somewhat better equipment, which would project the ground image onto two sequentially operating cameras. Because the satellite would operate beyond the line of sight of ground stations, magnetic tape would store images until the satellite was close enough to relay them to the ground station. The state of the art at the time of the report was a video magnetic-tape recorder with a bandwidth of about 1.5 megacycles (Mc). The designers thought that they could increase its speed to 8 Mc while shrinking it to roughly two hundred pounds—a simple engineering problem. The study proposed development of two cameras: a pioneering one, operating at 1:500,000, and one with higher resolution, at 1:125,000, to provide more detailed information on ground targets and wider surveillance coverage.
Employing a scale to indicate an image's quality is not as effective as defining resolution in terms of the size of an object that the camera will see. The 1:500,000 pioneering system would provide ground resolution of perhaps seventy feet in diameter under ideal conditions (if the object was alone in a uniform background), but ideal conditions were extremely unlikely. Therefore RAND and RCA experts expected a two hundred–foot ground resolution with the original system. The camera itself would not move throughout the process. A scanning drum would rotate so that it could record sequentially a strip of ground under the satellite. Each strip would be approximately 374 statute miles wide, with each frame covering roughly 77 square miles. The camera would also compensate for the satellite's forward momentum to minimize blurring of the image. The camera with a scale of 1:125,000 would use similar techniques but with increased resolution.
The satellite system that Project FEEDBACK's two-volume report proposed presented its potential users a groundbreaking concept. Overall it was not exactly what the air force had hoped for, but it was technologically feasible. This document accomplished what the air force had asked RAND to do in 1951. The primary difference between the two reports rests in the depth of detail and documentation. FEEDBACK contained far more technical detail than the 1951 report. Volume 1 dealt very precisely with comparisons between camera systems and justifications for choosing one, details about metals, effects of a vacuum on gasses within the satellite, and so on. It also placed more emphasis on the reconnaissance mission of the satellite. It included samples from a series of test shots that simulated space photos. Volume 2 served in effect as a technical annex. The demonstration of photographic products reflects the strength of the FEEDBACK document, which did not simply recommend satellite reconnaissance but _sold_ it. The system could obtain photographic coverage of previously unseen Soviet regions clearly enough to make the enterprise worthwhile.
The overall impact of RAND's efforts from 1946 to 1954 was immeasurable. From 1946 on, RAND scientists were convinced that launching a satellite was technically feasible and would benefit the United States politically, psychologically, and militarily. Every study from 1945 to 1954 strengthened their faith. Colonel Leghorn's work and his views about prehostility reconnaissance meshed well with emerging views on intelligence. By 1954 the technology was finally catching up to RAND's ideas.
A number of people who worked within RAND on both the 1951 report and FEEDBACK went on to key roles in WS-117L. Other people with whom the RAND group came into contact also had a major impact. Edwin Land, James Killian, Herbert York, and others all became prominent figures in intelligence gathering and within the Eisenhower administration. Just as important, a few air force officers realized not only the feasibility of satellites but their utility. Even before FEEDBACK forward-thinking officers had pushed for satellite reconnaissance. Most writers seem not to have known about RAND's jump-starting satellite development. To quote Bruno W. Augenstein, "The impetus given to satellite work by RAND studies of this era seems mostly forgotten now; but it is doubtful if the program could have obtained a running start without it."
Despite all it achieved RAND could not create the satellite program nor ensure its completion, but it germinated the seed of the satellite effort. The need for intelligence and a president eager to control defense spending allowed the seed to grow. All these forces came together in 1953–54 to produce the first satellite program. The RAND studies were not _the_ decisive factor, but they established the necessary specifications for satellites to fly and created a constituency of influential supporters who could sell the feasibility and utility of satellites. The administration and a handful of the air force accepted the need for them. The success of the reconnaissance satellite would depend on research and development by many interested and passionate parties who believed wholeheartedly in the concept.
#
# WS-117L
# Two Stages (1954–57)
By the end of the Eisenhower administration, the foundations of each of the major military space programs had been laid.
—William J. Durch
The year 1954 saw the convergence of all the forces I have discussed so far. First, in March 1954 the concept of a satellite for reconnaissance reached fruition in the form of RAND's FEEDBACK satellite study, which defined the shape of a satellite reconnaissance system and became the foundation for development of the WS-117L satellite. Second, as RAND was finishing its report, the desire for military intelligence was leading to creation of the Technological Capabilities Panel, its two reports, and the U-2. Third, President Eisenhower began working to protect satellite reconnaissance by securing legal protection for it through his Open Skies proposal and the VANGUARD satellite for the International Geophysical Year. By the end of 1955 he had also accepted the WS-117L program. It fell to the air force to develop satellite reconnaissance.
The WS-117L effort laid the foundation for every military satellite after 1953. Many historians mention it but provide only fragmentary and very incomplete details. For example, the program received considerable criticism after the success of _Sputnik_ in October 1957 because of its slow progress and lack of results. But the reasons for these delays have eluded most authors, who cannot isolate and explain the underlying problems.
There are few available sources about development of WS-117L, especially from 1954 to 1957, partly because of the slow declassification of relevant material. Daily logs, correspondence, and even plans and reports are still inaccessible. The only window that currently exists into these years of development comes from two sources: the writings of Robert Perry and James Coolbaugh. Perry, an air force historian, wrote several works, including the multivolume official history of the satellite reconnaissance program. Despite many blacked-out sources, his history outlines the official documentation that has yet to surface. For insight into daily workings, one may turn to the memoirs of Capt. James S. Coolbaugh, the WS-117L's project officer from December 1953 to March 1957, at the Wright Air Development Center's (WADC) New Development Project Office within the Bombardment Missiles Branch. Coolbaugh joined the WADC in September 1952 when he began to work with the Bombardment Missile Branch under Maj. Sidney Greene, chief of the New Development Project Office. The idea of space-based reconnaissance was not unknown to Coolbaugh. He had knowledge of the RAND efforts on satellite reconnaissance from Maj. Quenten Riepe, the WADC's liaison with RAND, who had kept him up to date on developments. Coolbaugh's memoirs, put together at the request of R. Cargill Hall, are unique as they detail his daily activities as project officer from 1953 to 1957.
The danger of relying on only two sources, one of which is a personal memoir, is that the picture presented of the program will be skewed and incomplete; thus wherever possible, I use alternative sources to verify both Perry's and Coolbaugh's accounts. So although it cannot be considered a definitive account of the early development of WS-117L, this chapter conveys a fascinating glimpse into the development of American satellite reconnaissance prior to _Sputnik_.
As I explained in chapter 4, the Air Research and Development Command's support of the RAND proposals led the air force to merge satellite efforts into a single project. Once work began in January 1954 the system received the unclassified title Advanced Reconnaissance System. Perry maintains that on December 3, 1953, the ARDC ordered the WADC to start work on demonstrating the feasibility of the major satellite components. The WADC assigned the program to the Bombardment Missiles Branch in its Systems Management Organization. Perry's description of how the WADC was assigned responsibility for the program seems accurate and consistent. However, he does not explain clearly three additional designations: project number 1115; the cover name "Pied Piper," which the press and several authors picked up and took to mean many things; and WS-117L.
Coolbaugh, in contrast, explains all three designations. Project number 1115 and Pied Piper refer to the program's process for bidding on contracts. His wording is very clear: "The evaluation of the three proposals completed the Project 1115, PIED PIPER portion of the satellite program." As I discussed earlier, the project received the name WS-117L unofficially at the end of 1953, but it became official only in 1955, according to Coolbaugh. As 1954 began, then, before RAND's FEEDBACK report appeared, the air force's work on satellites had changed from a semiofficial planning project to a proposed system complete with project number.
Satellite reconnaissance developed in two stages between 1954 and 1957. The first began in December 1953 with Coolbaugh's appointment and ran until roughly October 1955. This period included initial planning for the contract process as well as research on specific components in the WADC's laboratories. In 1955, however, the program transferred from Wright Field to the Western Development Division (WDD). This decision was logical since the satellite's primary booster was likely to be the ATLAS ICBM, which the WDD was developing under Gen. Bernard Schriever. The group at WDD supervised the selection of contracts, further work on components, and contract development. The two stages overlapped, as many personnel who worked at the WADC transferred with the program to the WDD; much of the effort carried over from one period and facility to the other.
## First Stage: Wright Air Development Center (December 1953–October 1955)
In the twenty-two-month initial stage, research and development started from scratch at the Wright Air Development Center. Coolbaugh first obtained a description of RAND's proposal from Major Riepe (the RAND-WADC liaison officer), who was the ideal man to brief him. Intimately aware of the satellite concept, Riepe had turned down Coolbaugh's position as project officer and elected to remain the liaison officer. Riepe's orientation and a RAND briefing on FEEDBACK in mid-January 1954 brought Coolbaugh quickly up to date on the concept for a reconnaissance satellite. Unfortunately the air force had not budgeted for the WS-117L in 1954, so there was no money for research and development and no guarantee about how much funding would be available for the next fiscal year. The lack of funding was the reason Riepe had rejected the position of project officer.
Coolbaugh found that everyone involved in the project at RAND was enthusiastic about the FEEDBACK proposal. He returned to the WADC with helpful information and strong indications of areas requiring immediate action. After discussing the issues with his superior, Major Greene, he put the WADC's laboratories to work on developing components for every major system on the satellite, including a rocket motor for the second stage, an onboard power supply, recording of data via videotape, reliability of the electronics, and atmospheric properties. Coolbaugh was also faced with financial problems and the need to find a system contractor. He makes it clear in his memoirs that he spent a lot of time keeping up with all the work done by the various labs at WADC on the WS-117L project. This led to a very close working relationship with both the on- and off-base labs involved in the program.
Coolbaugh was fortunate that some work had already been started under RAND contract with outside companies. So in some areas he had only to convince the appropriate WADC lab to continue working with the original contractor. This was certainly the case with respect to the horizon scanner for the satellite. One of the most pressing challenges to be overcome was how to stabilize the satellite so that the camera was always pointed at the earth. The solution was the horizon scanner, which, in tandem with control systems, would align the cameras properly. North American Aviation had already started to work on this device. In order to maximize this advantage Coolbaugh approached the WADC's own Communication and Navigation Laboratory to convince it to cooperate with the company to develop and test it. Eager to participate in space research, the lab agreed to contribute expertise and to fund it itself. Because a working horizon scanner was so critical to the success of the WS-117L satellite, Coolbaugh felt that a backup program was needed. So he established a backup program using the Armament Laboratory at the WADC and the Instrumentation Laboratory at MIT, using money left over from other projects. By working on two different designs for the horizon scanner, Coolbaugh felt that he was maximizing his chances for success.
Next Coolbaugh turned to the satellite's need for power. This was a dual problem as the satellite needed both a second-stage engine to establish its orbit and an energy source to perform its tasks. Coolbaugh was fortunate when it came to the rocket engine for the satellite. During discussions with the air force he learned that the Bell Aircraft Company was already developing a rocket motor for the B-58 HUSTLER strategic bomber. With most of the cost for development paid for under its contract for the B-58, all he had to see to was retrofitting the motor to the satellite and related costs. Providing power for the satellite was not as simple. The FEEDBACK report called for a nuclear power plant, but no one had ever conceptualized such a small reactor. The best that Lt. Col. Edward Hall of the Power Plant Laboratory could do was introduce Coolbaugh to people doing research on nuclear-powered aircraft engines. Although they could not solve his problem immediately, they agreed to review RAND's findings and make suggestions.
Related to the power supply were various problems with the satellites' electronics, the most pressing of which was how to record the satellite imagery. This was a vital element since without recorded imagery, the satellite was useless as an intelligence platform. Because such electronics did not form part of the lab structure at the WADC, and RCA had undertaken the original study for RAND, Coolbaugh met with Jim Huckaby, project manager of RCA's video-recording development team. Many technical problems had to be worked out. The prototype system at that time ran standard studio-quality audiotape at very high speeds (360 inches per second) past two recording heads. A seventeen-inch reel of tape could hold only about four minutes of video information. Since the satellite was to operate in an environment almost devoid of gravity, the reels' rapid spinning would create destabilizing forces in the satellite. The stabilization system would be hard-pressed to compensate for even the seventeen-inch wheel of tape. Anything larger, and the vehicle could easily become uncontrollable. Despite these concerns, Coolbaugh agreed to view a demonstration of the system.
The recording problem was only one of the electronic issues. From the outset electronic reliability was crucial. The system would have no human maintenance once it was in orbit, and it was expected operate for up to a year (if not longer) in an incredibly harsh environment after experiencing a great deal of vibration during launch. Turning to the RCA Electronic Components Laboratory, Coolbaugh probed for the best method to ensure reliability, but there were no conclusive answers. The best that the facility could do was recommend that Coolbaugh meet with members of RCA's David Sarnoff Laboratory to discuss the matter and solicit the views of J. M. West, vice president of Bell Laboratories. Reliability, moreover, was also a function of quality control during production, and no one knew what level of control would ensure the satellite's reliability.
Coolbaugh met frequently with his superiors to discuss progress and plans, but they most often talked about funding. On March 15, 1954, he met with Col. John Kay at ARDC headquarters. After outlining his plans for the satellite program, Coolbaugh briefed Kay on lab work and the problems that arose due to lack of funds. Aware of the financial problem, Kay promised his best efforts to release money from the budget for FY 1955, but he did not anticipate substantial amounts until FY 1956, so he urged the use of laboratory funds as much as possible. Coolbaugh also inquired about contracting both the Rome Air Development Center and the Air Force Cambridge Research Center to conduct research and development in communication and atmosphere work, respectively.
To tap into laboratory funds, Coolbaugh exploited a loophole that his friend Pete Murray had pointed out to him in the spring of 1953. The fiscal year started on July 1; every year the labs reviewed their budgets in late March and early April to be sure they had applied all their allotted funds. If they had excess funds on July 1, their budget would fall by that amount for the next fiscal year. This situation created an incentive for the labs to help Coolbaugh in late spring and use up their excess funds. To speed up this process Coolbaugh and Capt. Buford B. Biggs, who worked in the procurement section of the WADC, developed a system that accelerated the processing of the paperwork. Normally a work order could take days to weeks to be processed. Under the Coolbaugh-Biggs system, it could be processed in an hour. The result was the most effective use possible of scarce resources.
With the laboratories at the WADC working on technical issues, Coolbaugh turned to writing the development plan for the satellite system. Aware that he could expect little money for FY 1955, he assumed that lab resources could pay for technical efforts in the first year. Creative funding could be stretched only so far, however. Once he required a contractor and began actual development, the financial situation would be critical. As his planning progressed, Coolbaugh came to realize that the selection of a contractor was going to create a major problem. There were no potential candidates with any experience in this field. Normally the air force chose a prime contractor, who then selected subcontractors to develop components, with the air force approving them after a review process. Companies generally took on subcontractors with whom they had previous experience or who had submitted the lowest bid. But since no single firm could develop a satellite, the regular development process wouldn't work. After much deliberation Coolbaugh turned to the resourceful Captain Biggs.
Biggs advised that they try a new approach. First, he argued that they should look not for a single contractor but for a team approach. Due to the unique technical hurdles involved, only a strong team of contractors had a chance of success, and the contract should be awarded accordingly. Second, Biggs suggested a change in how the contract bidding would be done. Following teams' submission of their initial bids, the best two or three would receive a study contract to prepare definitive proposals, which would reflect a better developed understanding of both the task involved and the strengths of the contractor-subcontractor relationship. Both Major Greene and the ARDC headquarters accepted these unorthodox suggestions.
Adopting Biggs's proposal, Coolbaugh set up a rough schedule for tendering bids. Taking into account the time needed to both produce the bids and study them, he figured that the request for proposals would go out in May or June 1955, with the naming of three finalists by October or November. The ultimate selection would then take place by the beginning of FY 1956 (July 1). To cover the costs of the proposals, ARDC headquarters agreed to grant $500,000 each for the three finalists. It assured Coolbaugh that it would include $1.5 million for the proposals and $1 million for ongoing development of subsystems in the budget for FY 1956. Thus the first program funding would become available only in FY 1956.
The program underwent a few changes in mid-1954. In July it established a weapon system project office—a major advance. By this time Major Riepe had reconsidered his earlier decision concerning management and took overall charge of the program office, with Coolbaugh as technical director. The two men split the responsibilities, with Riepe handling the financial wrangling and political work and Coolbaugh concentrating on the technical issues. Ten days later the program transferred, on paper at least, to ARDC headquarters, but the office and technical activities remained at the WADC. On November 27, 1954, the air force issued System Requirement No. 5, thus officially starting to develop a satellite.
Despite all this progress, serious technical hurdles remained. This became evident at the end of July 1954, when Coolbaugh witnessed a test of the new video recorder system at RCA. The device was the size of a room, far outside the design parameters, and it could not handle the tape's speed; within a minute of starting, tape spilled off the reels onto the floor at the rate of three hundred inches per second. Coolbaugh wondered whether "we had better start looking around to see who else was working on video recorders." A second test failure convinced him to search for an alternative.
Huckaby observed that there were two other companies working on video recorders, both in California: Bing Crosby Enterprises and Ampex. In February 1955 Huckaby accompanied Coolbaugh to California on a fact-finding tour. Bing Crosby Enterprises had nothing to offer, but the talks with Ampex were productive. To founder A. M. Poniatoff and Chief Engineer Charles Ginsburg, Coolbaugh laid out his requirements: the system had to be as small as possible, use little power, and be able to record a signal of 4.5 Mc and, hopefully another signal at 6 Mc. When told about the unsuccessful tests at RCA, Poniatoff immediately recognized the problem: RCA was moving the tape so rapidly over the recording heads that the system could not handle the speed. Ampex had the opposite approach, spinning the recording heads rapidly to "paint" the information across the tape. With the smaller and lighter heads spinning rapidly, the tape moved slowly and was under control. After watching a demonstration of the machinery, Coolbaugh signed a contract with Poniatoff to provide progress reports on the Ampex system to keep Coolbaugh's office up to date, with RAND footing the bill.
Coolbaugh now pursued two methods of powering the satellite: nuclear and solar. The first was the nuclear power plant that RAND had called for. Scientists working on the nuclear-powered aircraft project were sure that a small reactor capable of producing one to five kilowatts was feasible. They put Coolbaugh in contact with Atomic International, a division of North American Aviation. This firm had been designing small reactors for domestic use and gave him a plan for such a system. Coolbaugh also began investigating solar power in September 1954. He contacted the Electronic Components Laboratory and convinced it to start experiments with photovoltaic crystals. The man in charge of crystals for the lab, Don Reynolds, persuaded Coolbaugh to support his research with cadmium sulfide crystals, which promised to produce more electrical power per square inch than conventional crystals. By the end of 1954 the first large cadmium sulfide crystals were produced. In the long term such crystals in solar arrays would solve many electrical problems.
The solar array was practical only if the satellite could store power for periods when it was out of direct sunlight. Batteries were the obvious choice. In early 1954 the best that the industry could produce was a battery that could store ten watt hours for every pound of battery. This was far too inefficient for a satellite. To get enough power storage, smaller and more efficient batteries were needed, but no research in this field was being done. So Coolbaugh contacted various manufacturers; in less than two years this produced a tenfold improvement in battery efficiency. By 1958 the WS-117L program had combined solar cell technology and improvements in battery capacity, which has became the norm for satellite power.
Early in 1955 the satellite program gained access to new facilities for research and development. The Rome Air Development Center in New York State joined up to support work on the satellite's communication elements. This included all the ground-to-space microwave systems for relaying data and command instructions between the satellite and ground stations. The air force's Cambridge Research Center, located at Hanscom Air Force Base outside of Bedford, Massachusetts, also became a partner. Already active in space research as part of VANGUARD, the Cambridge Center examined the atmospheric conditions in which a satellite would travel and operate. This effort helped to determine the main threats to the satellite and thus how robust it had to be and also assisted the researchers in defining vehicle characteristics and identifying any special requirements.
While components formed an ongoing focus of Coolbaugh's program, the bureaucratic system brought its own burdens. Caught within the USAF bureaucracy, Coolbaugh found that a great deal of effort was devoted to briefing his superior officers. This was necessary to keep them current on progress and, since money was a major problem in the first years of the program, to encourage financial support. Occasionally these briefings did produce good results. Coolbaugh's greatest convert to satellite reconnaissance was Gen. Donald Putt. Described as a "space cadet," a term of endearment for the enthusiastic supporters of space programs, Putt was WS-117L's highest-ranking supporter. Behind the scenes he exerted a great deal of influence on its behalf. Unfortunately General Putt was the exception, not the rule.
In March 1955, one month after the TCP report appeared, Coolbaugh gave a crucial program briefing to a group that included many members of the top air force brass—most notably, Assistant Secretary of Defense Donald Quarles. Displaying models of the reactor and the inertial guidance system, he told his audience about the program, its goals, and its current status. Although Putt and two or three other people showed marked interest, Quarles was not enthusiastic. Except for a few sharp questions his lack of enthusiasm was manifest to everyone. According to Hall, Quarles did not oppose the program but saw it as a long-term effort, and so he was unwilling to give it too much support.
At roughly the same time, Coolbaugh and Riepe learned that less funding than they had expected was available. The FY 1956 budget for the program had dropped from $2.5 million to only $1.5 million, just enough to cover the design studies. Whether this shortfall resulted from Quarles's lack of interest, a desire to cut corners, or Coolbaugh's success in using excess funds from the labs, there was money only for contract bidding.
On March 15, 1955, the air force formally issued General Operational Requirement No. 80, calling for the development of a strategic reconnaissance satellite system and providing technical requirements for it. This General Operational Requirement represented top-level approval for the program and laid out for the first time its full and formal requirements. It defined a satellite that could survey the world's entire surface, determine a potential enemy's ability to wage war, supply information for national intelligence, and warn of an attack on the United States. The images had to be clear enough to allow interpreters to identify airfields, cities, factories, and other strategically important structures. There were also requirements for gathering intelligence (through photographs, signals intelligence, and infrared and other sensor systems) and for ground stations to monitor the satellite and process the data.
Following the document's release, a staff team began preparing for contract bidding, and by the spring of 1955 this group had expanded. Riepe was still director of the satellite project; Coolbaugh remained technical director and specialized in power supply and propulsion. First Lt. John C. Herther took control of guidance and stabilization, and Capt. William O. Troetschel was placed in charge of communications and command and control. This foursome brought the program together during the initial development.
In May these senior officials held a Request-for-Proposal meeting with prospective contractors at Wright Field. They invited representatives from all of the top aircraft and electronics companies and briefed attendees on the proposed program and technical work to date. Riepe also explained the contract-bidding process and the crucial role of the chosen team of contractors. No single firm could develop the program, so the final choice would depend heavily on the quality of the team of contractor and subcontractors.
The favored combination was Douglas Aircraft and Bell Telephone Laboratories because of their outstanding reputations and joint experience on military contracts, but they declined to bid. Quarles was a former vice president at Bell, and when J. M. West from the company asked him about the program Quarles strongly opposed the firm's bidding on the contract. In his opinion the air force would not be undertaking a serious satellite effort for at least a decade, so the program would do little more than keep the idea alive until then.
The fact that the senior Department of Defense official in charge of research and development was not enthusiastic explains a great deal about the program's slow evolution, despite the staff's best efforts. Without the support of highly placed officials, the program was doomed to progress slowly. Although no documentary evidence has been found to show that Quarles specifically blocked funding during the first year of the program, the circumstantial case is clear. His attitude toward the program before the launch of _Sputnik_ in 1957 can be described as indifferent at best.
## Second Stage: Western Development Division (1955–57)
The summer and fall of 1955 was a busy time for the WS-117L team. Besides the many meetings with labs and consultants, the group had to deal with the transfer of WS-117L to the WDD and evaluation of the first-stage contract bids. News of the transfer arrived as the team prepared to evaluate the bids. As mentioned, the WDD had already taken over the ATLAS missile program. Since the ATLAS was likely to be WS-117L's first-stage booster, the transfer would facilitate consultation on matters relating to both programs. The head of the WDD, Maj. Gen. Bernard Schriever, had advocated the move in an attempt to prevent likely competition for resources.
Accordingly staff members from the WDD attended the deliberations on contractors' bids. Its senior representative, Capt. Robert Truax (USN), received a warm reception. Called the "first space cadet" because of his long-standing interest in space, his personality and similar outlook impressed his counterparts. A small group of scientists accompanied him from Ramo-Woolridge, a private company that provided scientific and technical support to the WDD, much as the various laboratories did at the WADC. The Ramo-Woolridge contingent, under Dr. Robert Cornog, came across as arrogant and condescending. Overemphasizing the "superior" support that they would provide, they alienated every air force officer present. Their questions during the contract presentations were also insulting. Despite the tension between them and the WS-117L group, and the pending transfer of the program to the WDD, the contract proposals went forward. The evaluation team consisted of representatives of all the laboratories supporting the program at the WADC, as they were the most familiar with the various components. The three most successful bidders—Lockheed–CBS Laboratories, Martin Aircraft–IBM, and RCA–North American Aviation—received contracts for follow-on studies.
On October 10, 1955, WS-117L was formally transferred to the WDD, effective early in 1956. Lt. Gen. Thomas Power, commander of ARDC, authorized the move, overriding the objections of virtually all the general officers of the WADC and his own staff. The need for a close link between the WS-117L and the ATLAS programs weighed heavily in his decision. The staff at the WADC was not happy about Ramo-Woolridge's taking over its program; its members worried, quite legitimately, that progress would slow once WS-117L no longer had air force supervision and Ramo-Woolridge tried to expand its expertise.
The WS-117L team suspected that it would be difficult for Ramo-Woolridge and the WDD to support both a satellite and an ICBM effort and that the pairing of efforts would in fact dilute the resources available. General Schriever agreed, and Ramo-Woolridge did not enter the program. WADC personnel working with the WS-117L and the air force labs continued under Schriever's direction at the WDD.
The move to WDD ended the program's first phase. Technical studies and development continued unabated, taking on greater importance as a contractor came on board in 1956. Budget and support remained problematic. By moving west the program was closer to fruition. With the backing of the WDD and Schriever's ICBM team, the WS-117L took on high priority. From October 1955 to October 1957 it moved slowly toward completion of a prototype system.
The order for the transfer of the WS-117L program to the WDD took effect in February 1956. The transition was fast and seamless largely because most of the WADC's staff, including key members such as Coolbaugh, Herther, and Troetschel, went along. Overall control of the program was vested in Col. Otto J. Glasser, who was responsible for both WS-117L and the ATLAS program. Daily control rested with the WS-117L's new office head, Captain Truax, who arranged support from higher levels of authority. The net result of this division of labor was that Truax, through daily control of the office, oversaw most things directly related to the satellite program itself.
The first major task for the new WDD office was to complete the plan for developing the program. This involved settling myriad issues and conducting many briefings to win support for the program's budget. The plan was to explain the system's goals, its expected completion date, technical details and specifications for equipment, and how the components would work together. Part of the plan explained the tasks that remained before the system's first flight and its subsequent operational deployment; it also authorized the steps to reach these goals. As such it was a crucial document, especially with regard to obtaining funding.
Colonel Glasser insisted on completion of all basic planning by April 1, 1956, to allow for the filing of the development plan before the next budget year started. This demand put a great deal of pressure on the WDD team to fit the program's elements into a coherent description. The result was a plan that called for a first orbit by May 1959, with complete operational capability by the third quarter of 1963. The document described the satellite and supporting equipment. Exclusive of facility costs, research and development would require roughly $114.7 million. General Schriever received the plan on April 2, 1956, approved it, and passed it to the office of General Power, who gave it his blessing about three weeks later.
While writing the development plan, the WDD was also busy with contract issues. The three in-depth proposals for the satellite program were due in March 1956. The program brought in many original staff members from the WADC to help it make informed judgments. Of the three presentations, Lockheed's seemed the most intriguing. Its engineers believed that adding a second stage to the ATLAS could allow the payload to increase by 10,000 to 15,000 pounds. According to Curtis Peebles, this proposal called for two separate satellites: a pioneer version, weighing no more than about 3,500 pounds, and an advanced version of roughly 7,800 pounds. Peebles's descriptions are somewhat sketchy. With the development of the advanced satellite, operational reconnaissance would be possible. The WDD could not give Lockheed the contract immediately, however, because funding would only follow the approval of a development plan.
Air Force Headquarters issued this formal approval on July 24, 1956, and a development directive was issued on August 3. But financial problems continued. The plan called for $39.7 million for fiscal year 1957, but the program received only a maximum of $3 million, less than 10 percent of what had been requested. The ARDC invoked "severe limitations" on the budget for FY 1957 as justification, but even it admitted to underfunding the satellite program. The WDD did, however, receive enough money to award Lockheed the contract to develop the first American reconnaissance satellite. The delay in assigning the contract—until October 29, 1956—reflected the difficult financial situation.
Lockheed, the prime contractor for WS-117L, now had the daunting task of bringing a complicated system together under Jack Carter, head of the project, and his chief scientist, Louie Ridenour, who had presented Lockheed's proposal to the WDD staff. Most of the early problems at Lockheed came not from the satellite itself but from the usual start-up challenges: inexperienced managers, miscommunication, and problems with suppliers. The early delivery of HUSTLER engines to Lockheed best illustrates this problem. Bell Aircraft hoped to have all engines delivered before Lockheed had made final modifications so it could bill its customer for later changes. Careful management by Carter and the Lockheed team, as well as the WDD, ironed out the problems, and the engines arrived on time.
A small army of subcontractors supported this enterprise behind the scenes. Some, such as Ampex and RCA, had helped develop the initial components. The new ones included some of the biggest laboratories in the United States. For example, CBS Laboratories under Dr. Peter Goldmark took on development of the read-out system (part of the photographic subsystem), while Dr. Charles Draper's Instrumentation Laboratory at MIT worked on the satellite's guidance and stabilization equipment. These subcontractors played major roles in the program, although much of their work was behind the scenes.
Funding continued to limit progress, however, and the WDD found it impossible to circumvent these restrictions. The stumbling block remained Air Force Secretary Donald Quarles, who controlled the rate of development—providing only a trickle of funds and forbidding construction of components past the design stage without his approval. The program could not prepare mockups of the satellite or its components, let alone actual experimental vehicles. Truax remembered being in Quarles's office when the secretary told him, "I don't mean to throw too much cold water on the program, but I don't want any tin-bending yet." 45
Coolbaugh's insightful memoirs end in early March 1957. He moved temporarily to the THOR IRBM program, and his only contact with WS-117L as a THOR representative occurred in early 1958. Although he did play a role in CORONA and the DISCOVERER program, he did not return to WS-117L until January 1, 1960. Unfortunately Perry's "History of Satellite Reconnaissance" is a poor substitute; it tells us little about daily activities or overall progress before October 1957.
It is possible, however, to piece together fragments of the remaining WS-117L story before _Sputnik_. We know, for example, that financial problems continued. General Schriever believed that he had figured out the air force's resistance during his attempts to find backing for the program in 1957. He blamed both Quarles and the administration's emphasis on "space for peace" over national security.
First, of all people to oppose the program, Quarles seemed an unlikely suspect. He had pushed for the IGY scientific satellite and produced the first draft statement on Outer Space Policy (NSC 5520). He saw satellite reconnaissance as beneficial and important in the long run but favored low-risk technologies and mistrusted projects that had not demonstrated complete reliability. VANGUARD would prove the technology and provide legal precedent, so he was willing to wait. Thus he was not against WS-117L, or even reconnaissance satellites, just hesitant to devote considerable resources at that time. This inclination was reinforced by his strong support of the administration's desire to cut defense costs. An untried program that promised results years in the future was a marginal priority; further study was a relatively inexpensive alternative that ensured some progress. His lack of enthusiastic support, both in meetings and in funding battles during the first three years, grew out of these views. As Eisenhower's representative and the man charged with research and development, his ambivalence is emblematic of the fact that the WS-117L program did not enjoy the support of the top-level military officers and civilian leadership of the air force.
Second, the administration's stance of "space for peace" also helped to generate program restraint. By April 1957 Schriever was blaming this policy for the inertia in funding. The core of the problem rested with the idea of separating the military and civilian space programs and the legal question of satellite overflight. VANGUARD was created to establish the principle of freedom of space in a manner least likely to antagonize the Soviet Union. There were other reasons for separating the programs—most notably Eisenhower's desire to prevent interference between the military and civilian programs. Hence the administration decided to keep VANGUARD exclusively civilian.
To this end both military proposals for scientific satellites and military attempts to accelerate the VANGUARD program proved pointless. Both the army and the air force claimed that they could orbit a satellite sooner than the VANGUARD effort and more economically, but the administration consistently blocked them. The army was pushing its own REDSTONE-JUPITER missile combination to launch a scientific satellite. The air force also entered the effort, pushing a variation of the WS-117L program that used nonreconnaissance components to launch a scientific satellite. Unsuccessful attempts to promote this endeavor detracted from the WS-117L effort in general. The irony here is that while the administration publicly urged "space for peace," it never clearly defined _peaceful_ , aiming to keep the Soviets pliable vis-à-vis freedom of space.
To circumvent the funding problem Schriever developed a plan that may have inspired development of the CORONA program. Understanding that the United States had to advance the program for intelligence purposes, in April 1957 he ordered "Fritz" Oder to devise a policy to boost the status of the air force's satellite program. In the meantime he maintained pressure on the air force to supply financial support through regular channels. This effort failed, however, when he had to accept a reduction following the first review of the development plan in April 1957. Sure that the administration was intent on sacrificing the program, he reevaluated its schedule, making cuts wherever possible. By July he had placed severe spending ceilings on the Lockheed contract, which further delayed the program. It became clear that WS-117L was in serious peril.
Oder's financial solution went to the heart of "space for peace." Since financial difficulties arose from the administration's reluctance to pay for an expensive endeavor that might endanger U.S.-Soviet relations should it become public knowledge, the solution was to make WS-117L vanish. Oder's scheme—with the code name "Second Story"—was a clever piece of sleight of hand that rested on three key principles: satellite reconnaissance had to be covert, the CIA had to take an active part in it, and the effort required a massive infusion of money.
The plan was simple: the air force should publicly cancel WS-117L, but the CIA would covertly reactivate it. To provide cover for the program the air force would establish a major scientific satellite program as a follow-up to VANGUARD, which would explain the WDD's satellite efforts. The CIA in turn would keep the effort secret and, using the WDD for technical purposes, provide the satellites for an active reconnaissance program. If the plan succeeded, it would never compromise "space for peace," and the administration could support it. With the plan in hand, Schriever secretly approached key members of the administration in June 1957 to gain their support. Among those he briefed were General Putt, James Killian, Edwin Land, and Richard Bissell from the CIA, who went on to run the CORONA program.
Despite Schriever's attempts to increase funding to $10 million, the administration refused to bend, though it extended permission to procure items that had long lead times. Schriever felt he had to approve tentatively a schedule that would put Second Story into practice. This required that General Putt "request" that the air force develop a scientific satellite program to replace VANGUARD, backing it up with a proposal for a Ballistic Missile Division for such an effort. This division would have to be in place by the start of September 1957. To facilitate the plan, and to ensure it would interest the White House, Maj. Gen. Andrew Goodpaster (the president's military aide) and several other members of the White House staff were briefed in August. In light of later developments it is likely, but not certain, that the president received a briefing.
By late September Schriever's program had bogged down. The need to coordinate efforts with numerous officials in both the scientific and the military programs while maintaining secrecy proved too difficult. Even as the Stewart Committee, which had recommended VANGUARD over other military proposals, was undergoing reactivation to help plan for a follow-up program, Schriever faced signs of trouble. The secrecy on which he depended was on the verge of evaporating. A consultant with the Department of Defense, while working on a memorandum calling for a national policy for space and totally unaware of Schriever's plans to save WS-117L, stumbled on a 1956 proposal to use the WS-117L program to develop a scientific satellite. This was the original submission of January 16, 1956, proposed as an option for the IGY satellite use of all of WS-117L's nonreconnaissance elements.
The consultant raised several questions over the feasibility of using a military satellite in a scientific role, and his rejection of the idea endangered Second Story. From Schriever's perspective, if the scientific program came under attack, that would compromise the cover story. With the consultant totally unaware of Second Story and its secrecy, Schriever devised a three-stage plan to move the project forward. First, the Ballistic Missile Division had to develop a detailed proposal for a scientific satellite that he could present to the Department of Defense. It would have to demonstrate not only scientific value but air force unity on the matter. Second, Schriever needed a public-relations policy to manage information about the project's scientific and covert intelligence sides. Third, he had to reassure the Stewart Committee that the scientific program would work and be beneficial.
Schriever's gamble to increase funding for the WS-117L soon proved futile, for within a month the Soviet Union had orbited the first satellite, demonstrating technical feasibility. The administration and the air force, under harsh criticism for the failure of American efforts, immediately increased funding for space research and development. In effect the launch of _Sputnik_ on October 4, 1957, solved the air force's budget problems by removing all the justifications for slowing efforts. With the Soviet success providing strong motivation, the WS-117L program was to leap forward dramatically in October 1957.
Due to the slow declassification process, the story of WS-117L is incomplete in some key areas. The largest element missing from the record is a description of the satellite project on the eve of _Sputnik_. Still we can reconstruct what the program was aiming for in 1957. The satellite development plan at this stage differed little from the original concept, which used a television system that stored images on magnetic tape for relay to Earth. Yet it was clear by August 1957 that this configuration would be technically unfeasible. Predicting resolution of such low quality as to negate the program's value, RCA convinced Lockheed and the WS-117L team to reject this approach. The television system gave way to a camera with more conventional film, which would receive processing on board the satellite before scanning to allow transmission. Although the resolution would not equal that available from looking at the photo directly, it was better than the television would have produced. As early as 1956, however, the RAND corporation had discussed a camera system that would return the film to Earth, but the WS-117L team had rejected it, owing to technical problems with recovery.
The plan had been for the program to progress in stages. The original Lockheed proposal of 1956 called for two distinct phases: a test phase, or pioneering system, to validate components and hone the process of intelligence gathering, and a second phase to transition to operational capability. Several historians argue that the program actually involved three phases. The initial test phase, using a THOR-AGENA launch configuration, would begin in November 1958. This system would give way in June 1959 to phase 2, in which an ATLAS-AGENA system would test higher payload launches and longer-term orbits. Finally a third phase was to start in March 1960 and consist of three satellite systems: a pioneer photographic-reconnaissance system with a six-inch focal-length camera lens, an advanced photographic system using a thirty-six-inch lens, and a long-term surveillance system; clearly the emphasis was on the photographic systems. However, these satellites would contain not only photographic-reconnaissance systems but also electronic intelligence and later infrared sensors. These dates contradict the 1956 development plan, which anticipated an initial test series beginning in 1959 and achievement of operational capability in mid-1963. The difference in dates is striking, given that funding was almost nonexistent. One can conclude only that the phases I describe derive either from speculation and are totally incorrect or from accelerated post- _Sputnik_ plans—probably the latter. There are indications that these dates may reflect planning after October 4, 1957. For example, Oder's account refers to camera designations that clearly relate to the program after _Sputnik_.
Confusion over dates is the smallest problem facing historians. Some accounts refer to systems that were clearly not part of the original satellite concept. Robert Divine's account describes the WS-117L system as having three satellites: DISCOVERER, a system for film recovery; SAMOS, to relay images; and MIDAS, an infrared early-warning satellite. MIDAS and SAMOS were definitely descendants of the WS-117L system, but Divine's descriptions seem to assume post- _Sputnik_ program data because these names emerged only after October 1957. DISCOVERER actually may have been the initial test system for WS-117L or part of Schriever's Second Story project. Either way it definitely was part of the program _after Sputnik_.
The satellite program, as it evolved in the years leading up to _Sputnik_ , was a coherent one. It arose out of RAND studies and tried to remain true to its original design concept. The program was extremely ambitious and called for a level of technology not yet available in a reliable form. Consequently, as technical problems appeared, they necessitated new technology and/or adaptation of existing technology. In the meantime, however, the program ran into serious external problems, especially the absence of financial and political support. These factors directly encumbered development. Unlike the technical issues, these difficulties were beyond the control of people working on the program; only other external factors could change them. That didn't happen until October 1957, but when it did it would have profound impact on WS-117L.
#
# Satellite Photography, Film Return, and the Birth of CORONA (1957–58)
The successful launching of a satellite instrument is bound to be a spectacular event, causing worldwide sensation.
—Paul Kecskameti
The Soviet launch of _Sputnik_ on October 4, 1957, had far-reaching consequences. No other single event had exerted so much domestic pressure on the Eisenhower administration. In the wake of the beeping satellite, the political backlash and the whole "missile gap" controversy shook the foundations of American confidence and security.
## A President under Pressure (October 1957)
To recount the entire debate over _Sputnik_ 's implications or to explore the various positions on the issue is not necessary for this study since it is amply covered by historians elsewhere. The controversy's basic elements, however, can help us grasp _Sputnik_ 's impact on the WS-117L program. The missile gap antagonized relations between the White House, Congress (particularly the Democrats), and the military. Just as they had done with the "bomber gap," the military and its legislative supporters urged increased defense funding in response to _Sputnik_. Exploiting the press and fear of Soviet technological "superiority," the military (especially the air force) and several elected officials tried to use the opportunity for maximum political benefit. One of the most vocal critics of the administration was again Senator Stuart Symington, who used information that the air force leaked to him. Charging (correctly, as the previous chapter showed) that Eisenhower's economy measures had caused the satellite program to fall behind, he argued that the United States was also lagging in ICBM development. Accusations from Symington and others, such as Senate Majority Leader Lyndon Johnson (D-TX), placed the administration on the defensive. The press, sensing a "big story," picked up the rhetoric and intensified the attacks.
At the heart of these scathing critiques lay the argument that inadequate defense spending had squandered American technological superiority and thus left the country vulnerable to the Soviet Union. Republican policy in general and the president in particular were to blame for the Soviet space coup. As with the debate over the bomber gap, the military went before Congress to request more money. Beginning in November 1957 Senator Johnson gave the military an open venue for its attacks through his inquiry into satellite and missile programs. During these hearings defense spending received a thorough airing. Both Gen. Nathan Twining, chairman of the Joint Chiefs of Staff, and air force chief of staff Gen. Thomas White argued that the United States was not behind in ICBM development but had to increase spending for bombers and missile development over the long term to prevent a Soviet lead. Many pundits in the media supported their claims, predicting a major discrepancy in ICBMs. The _New York Times_ journalist Joseph Alsop predicted that by 1963 the Soviets would have 2,000 ICBMs—a far cry from the 130 that experts expected the United States to have by 1962. The press helped to stoke a damaging set of accusations against Eisenhower, whom it characterized as a "do-nothing" president whose lack of resolve led to this crisis. Congressional leaders, invoking dire warnings of the Soviets' being far ahead, sought more money for expanded programs.
The press and Congress's virulent attacks shocked Eisenhower, who had no idea that the nation's perception of itself was so fragile. Many Americans could not grasp the idea that another country could surpass the United States in science and technology. _Sputnik_ forced many of them to question their assumptions. The administration's response to _Sputnik_ was not very astute. Eisenhower thought the Soviet threat to be rather insignificant. Having the advantage of U-2 imagery during the bomber and missile controversy, Eisenhower knew there was no gap in favor of the Soviet Union. The forays by the U-2 revealed the locations of launching facilities and kept the Americans generally abreast of Soviet missile work. Unfortunately the president felt that he could not reveal this publicly and endanger the U-2 as an intelligence source. To reveal it to the public would humiliate and provoke the Soviet Union to take action. Thus the president refrained from refuting the doomsayers in the Senate.
But this did not stop him from looking into the matter further. On October 8, 1957, shortly after the launch of _Sputnik_ , he requested clarification of the status of U.S. missile and space efforts (both civilian and military). He asked both Gen. Robert Cutler, his special assistant for national security affairs, and Deputy Secretary of Defense Donald Quarles to look at three issues for him. First, he wanted information about studies of the development of guided missiles since 1953. Second, he hoped for an account of work on Earth satellites during that period: progress reports, recommendations, and program priorities. Third, always worrying about fiscal issues, he insisted on receiving "the chronology regarding the costs of the program." In essence he wanted to know why the United States had not yet launched a satellite.
Having activated the government machinery to answer his questions, the chief executive turned his attention to calming the public. He and key members of his cabinet sought to create the impression of "business as usual." Saying that the satellite did not raise his apprehensions by "one iota," Eisenhower maintained that _Sputnik_ did not endanger U.S. security and that there was no crisis. At the prompting of members of the Science Advisory Committee of the Office of Defense Management, he took the opportunity to increase public awareness about the need for science and education for American youth rather than increase the defense budget. Through speeches and meetings with leading scientists, he stressed that his administration valued technological developments and that the best minds available were advising him on these matters.
He also took several steps to blunt the attacks from the military and Democrats. First, in October 1957 he named James Killian, who had shaped the TCP report, to the new post of special assistant for science and technology and to lead the President's Science Advisory Committee. Second, in early February 1958, he asked that committee to recommend the outlines and organization of a space program. This move led eventually to the creation of the National Aeronautics and Space Administration, which incorporated existing elements, such as the old National Advisory Committee on Aeronautics, into a coherent civilian space program.
Third, Eisenhower refused to accelerate the VANGUARD program in the wake of _Sputnik_. He had been blocking military attempts to do so prior to the launch of _Sputnik_ , as it would have meant using military technology to implement the program. That approach ran counter to his goal of using civilian satellites to establish the legal precedent of the freedom of space. It also seemed unnecessary since the Americans were never in competition with the Soviets to be first in space. The scientific mission remained in place. Since VANGUARD had every prospect of success, Eisenhower saw no reason to advance the effort through either schedule changes or use of military boosters, as long as progress remained satisfactory. Fourth, he asked for briefings on the progress of the military satellite program in response to his questions of October 8, 1957.
## Photography and Film Return (March 1956–October 1957)
The historical record of WS-117L becomes increasingly unclear after _Sputnik_ , partly because of the flurry of activity that followed it and partly due to the scarcity of information about developments. The gaps in our understanding are ironic and unfortunate, for the most pivotal moment of the whole story occurred between October 1957 and March 1958, a period of fundamental reorganization. On October 6, 1957 WS-117L had consisted of a single satellite, but in November 1958 the Department of Defense revealed that it consisted of three satellites: DISCOVERER, SENTRY, and MIDAS. DISCOVERER was a research-and-development satellite for testing system components and for research in various areas, including biomedical studies. In reality this was the cover for another reconnaissance satellite, CORONA, which used a system that returned film to Earth for processing. The film readout system, the core of the WS-117L concept, was dubbed SENTRY, an innovation discussed in detail in chapter 7. Finally, the infrared detection system designed to warn of a Soviet missile attack, originally part of WS-117L, was given its own satellite, called MIDAS.
The historians Jeffery Richelson and Robert Divine refer to these three programs as subparts of the entire effort and as being in place prior to _Sputnik_. However, they do not explain how or when a single satellite suddenly became three divergent systems. Most scholars have argued that the film-return system was part of WS-117L before _Sputnik_. They portray it as part of the overall program, a continuation of the WS-117L concept; its separate development and operation became essential after it seemed apparent that WS-117L could not meet its schedule. These theories, however, are derived from information before declassification. No historian has provided a specific account of when the program added film return or of the impact this had.
Before exploring _Sputnik_ 's effects on U.S. satellite reconnaissance, we should place the origins and development of the CORONA concept in its proper context. It was the idea of a second satellite program, revolving around film return, that emerged in early 1956 and was ready for serious consideration on the eve of _Sputnik_. CORONA joined WS-117L in _Sputnik_ 's wake, and its association with the earlier effort was brief. With this development in mind we can see _Sputnik_ 's impact for what it was: a catalyst to American satellite reconnaissance.
RAND was not idle following release of the FEEDBACK report in 1954. Rather it expanded its staff and kept current about the latest developments in what it still considered its program. It was not until early in 1956, however, that its team became vocal again about satellite reconnaissance. Aware of problems with WS-117L's film system and of new developments in the field, the group reconsidered a satellite that returned its film to Earth for processing, even though it had rejected the prospect a few years earlier.
Richard C. Raymond, a physicist and expert in information theory and a member of RAND's Electronics Division, looked at how much information was theoretically available from intelligence photographs and negatives. When comparing WS-117L's film-readout system then in development (similar to television technology of the period) and the amount of information available from direct examination of photos and negatives, he came to a startling conclusion. Assuming that both types of images came from the same altitude, the standard film would produce roughly two orders of magnitude more data. To make matters worse, the radio bandwidth for transmitting data was very narrow, so relays to Earth would be very slow. Television imagery, then, would be far inferior to photography. Admittedly no one had expected that the television system could compete with normal photography, but no one had realized just how much information was at stake.
In light of this revelation and of advances in reentry technology that came from ICBM research and development, RAND's president, Frank Collbohm, with the help of a RAND researcher named Brownlee W. Haydon, prepared a formal recommendation to the air force's air staff in March 1956. "Photographic Reconnaissance Satellites" called for development of a recoverable satellite system. RAND withdrew the recommendation, however, within a few weeks for reasons that remain unclear. According to Merton Davies, one of the firm's experts, the materials relating to the decision were destroyed after the withdrawal, so the reasons may never be known.. Raymond, however, recalled that the air force was concerned about the time lag between the taking of the photo and the information becoming available. WS-117L offered the theoretical prospect of quick access to the data, almost in real time. A film return or drop-film system, as RAND proposed, would mean a built-in delay between intelligence gathering and film recovery. Furthermore no one knew yet if recovery was feasible. Until an in-depth study comparing WS-117L and the drop-film proposal could be completed, the air force had no reason to accept this proposal. Hence later that same year (1956) it contracted with Lockheed to develop the WS-117L system.
Nevertheless RAND continued to explore the technology of returning film to Earth. By June 1956 enough progress had been made that it could issue a research memorandum on the physical recovery of payloads. J. H. Huntzicker and H. A. Lieske's "Physical Recovery of Satellite Payloads—A Preliminary Investigation" clearly demonstrated that a film-returning satellite system had strong merit. First and foremost, it could collect more information than a relay satellite. Photos offered greater resolution (and therefore more information), and captured data about the environment the satellite worked in would increase scientific knowledge and provide an understanding of, among other things, the conditions affecting film, the camera, and orbits. Second, film return was likely to provide intelligence sooner. The complicated nature of the WS-117L system, and its projected 1960s operational date, meant that it would be years before it was ready to work. The simple film-recovery satellite could be utilized almost immediately for intelligence gathering. Third, the recovery technology for film was seen as a major step toward manned flight in space.
The successful recovery of a capsule from orbit depended on three elements. First, the satellite trajectory or orbital path would need modification so that the satellite would reenter the atmosphere at a predetermined point and have time for successful landing and recovery. Second, it was essential to protect the payload from heat to ensure the integrity of intelligence materials. Third, a means of locating and retrieving the payload was crucial.
Huntzicker and Lieske found that current technology could meet all three conditions. Orbital adjustments were the easiest goal to achieve. Since all orbits decay over time, recovery necessitated premature orbit decay, which was a function mainly of atmospheric drag slowing the satellite, forcing it to orbit lower in the atmosphere. The use of a rocket to decelerate the satellite (or part of it) would suffice. In addition it was possible to adjust the direction of travel and thus to identify a general area of recovery.
Protecting the payload from the heat of reentry presented the greatest challenge. ICBM research and development had already begun to work through this problem; warheads, like the satellite payload, had to reenter the atmosphere and survive to reach their targets. The two programs' similarities meant that one could borrow technology from the other. Heating the surface of the reentry vehicle was a function of the satellite's flight path, speed, size, and surface material and area. To minimize the heat of reentry, only a part of the satellite should return to Earth. Furthermore layers of fiberglass insulation and metal could protect the payload and diffuse the heat that builds up as the satellite descends into denser layers of the atmosphere. Since the most intense heating would probably occur in the first thirty seconds of reentry, the RAND group recommended separation of the outer heat shield only after this point. This shield would have built up a great deal of heat, and its ejection would prevent that energy from radiating to the inner layers of protection.
Detecting and recovering the payload seemed straightforward in comparison. A recovery system would probably weigh roughly 227.6 pounds, including the film payload (about 50 pounds), the reentry rocket, insulation, beacon, power supply, and parachute. Since most of the weight—the heat shield and reentry rocket—would jettison shortly after reentry, recovery would be a great deal simpler. While a parachute covered final descent, increasing the time in the air, radio beacons would indicate location. RAND expected easy payload detection and recovery. Being able to predict roughly the area of recovery made the odds of success very good. The report did not define the method of recovery; rather it anticipated use of conventional ships and aircraft.
RAND's work on returning film did not stop there. For RAND the inherent simplicity and value of recovered film meant giving the recovery system high priority, but the air force and the WS-117L program offices were not very supportive. At this time WS-117L was not yet under way, contract selection had just started, no funding had arrived, Assistant Secretary Quarles was offering no support, and he and other senior officials in the air force had still not accepted development of the FEEDBACK satellite. A satellite that returned photos to Earth via a capsule seemed even more radical than WS-117L. Until recovery of a payload proved feasible and a detailed comparison between FEEDBACK and film return was available, the military rejected the idea.
RAND, however, continued to support the idea of a film-return satellite. Two of its main thinkers on satellites had picked up the idea and kept the work alive. Merton E. Davies, a longtime "space cadet," and Amrom H. Katz, a convert to satellite reconnaissance who had years of experience in photo interpretation, quickly grasped the merit of Huntzicker and Lieske's study. They continued work on the drop-film camera system throughout 1956 and early 1957, and slowly an advanced satellite took shape.
Two external groups helped them greatly. First, the Boston University Optical Research Lab provided the basic idea for a new, higher resolution camera. On February 19, 1957, several of the Boston Lab staff, including Walter Levison, Dr. Duncan Macdonald, and James Baker, briefed Davies and Katz on their ideas for creating a camera capable of providing 120-degree coverage. Second, their concept gained reinforcement with information they obtained in early March from Fred Willcox, vice president of Fairchild Camera and Instrument Corporation. Willcox had been designing a panoramic camera to provide wide-area coverage for aircraft reconnaissance. By spinning the camera he increased considerably the area that it could photograph. Davies and Katz adapted both these concepts—higher resolution and wider coverage—into a new system. The "Hyac" (for high acuity) camera promised a theoretical resolution of one hundred lines per millimeter of film (roughly ten times the resolution of a Second World War system).
By early 1957 the duo had merged this high-resolution panoramic camera with a long-focal-length lens to generate a new high-altitude system. Since the main problem of photography from space was the difficulty of obtaining high-resolution shots that covered large areas, the combination seemed to solve the principal problem that Katz found in the WS-117L system. In early 1957 Davies and Katz believed that they had developed the best means of accelerating the WS-117L program by merging the Hyac camera, the film-return concept, and the THOR IRBM as a first-stage booster.
To accomplish this goal, however, the WS-117L program had to overcome air force resistance, and do it with enough momentum to sustain development after the initial impetus had dissipated. _Sputnik_ created the essential catalyst to release the bureaucratic brakes and affected the American effort in four ways. First, _Sputnik_ demonstrated feasibility; it forced skeptics within Defense and the administration—Quarles, for example—to recognize the possibilities of satellite reconnaissance. Second, _Sputnik_ had legal repercussions. The American VANGUARD scientific satellite program had aimed to establish the legal principle of freedom of space. With _Sputnik_ 's success the entire issue became moot since now the Soviet Union could hardly complain about a U.S. satellite flying over its territory. Ironically, by being first in space the Soviets had done the WS-117L program a sizable favor. Third, the Soviet triumph drove home the need for intelligence. Clearly the Soviet Union was close to developing a feasible ICBM design. The possibility of a deployed ICBM force increased the risk of surprise attack and illustrated the unequivocal necessity to track Soviet activities closely for possible signs of deployment. Such tracking would be easiest during construction of launch sites, when the large amount of activity would be observable.
Fourth, _Sputnik_ heated the political temperature in the United States. A firestorm quickly obscured the entire effort on reconnaissance satellites. While partisan attacks and congressional inquiries dominated the landscape, the military services again looked to turn the situation to their benefit. Having warned of the increased Soviet threat, the services (particularly the air force) saw _Sputnik_ as vindication. Taking advantage of the situation, all three services sought carte blanche for large-scale missile and satellite programs, along with a bigger share of the defense budget. Congressional testimony and the press became vehicles toward these ends, and a plethora of proposals swamped and distorted the political process. Beneath this layer of infighting, the WS-117L program was doing its best to ride out the deeper currents that _Sputnik_ created.
Many sought to accelerate the program, but these attempts ran into trouble. Concern about WS-117L first emerged in a report by the President's Board of Consultants on Foreign Intelligence Activities. This board was created by Executive Order 10656 on February 6, 1956, to monitor foreign intelligence activities, review them, and give recommendations to the president on how to improve intelligence. The board's report on October 24, 1957, displayed serious doubts about the American ability to gain intelligence on the Soviet Union. The board insisted on a thorough review of the advanced reconnaissance systems then under development. It received briefings on two such systems—the WS-117L satellite and the Project Oxcart/SR 71 aircraft, a supersonic successor to the U-2 but designed to be more invisible to radar—and determined that the programs were not receiving adequate support. Neither seemed likely to provide intelligence soon, and Killian worried particularly about the satellite program, pleading for clear decisions on its priorities.
According to secondary sources, the board's report of October 1957 concluded that the WS-117L program was far from accomplishing its goals. The board, which included Killian and Edwin Land, discovered several reasons for its tardiness. The original development timetable was far too optimistic and proved unrealizable. Air force mismanagement had resulted in a lack of focus, and shortage of funds only made the situation worse. Finally, the program was at the edge of current technology. The board found its camera/radio-relay system to be far below the necessary level for practical intelligence gathering. The camera, with its short focal length, offered only one-hundred-foot ground resolution. Integrating it with the narrow-bandwidth radio transmission would substantially slow return of the imagery to Earth. Although the timetable called for active testing by late 1959, 1960 seemed the earliest realistic date. The board also learned that the air force, already mismanaging the program, was publicizing it in an effort to win more money for its research—clearly an attempt to exploit the situation that _Sputnik_ created.
Air force efforts frustrated Killian and Land and their colleagues on the panel. In order to speed acquisition of vital intelligence about Soviet missile development, the group called for a simpler accelerated WS-117L program that could be operational as soon as possible. In conjunction with this recommendation, it asked for an interim intelligence system. Borrowing RAND's concept of a film-return satellite, this second satellite system seemed less complicated and therefore more practical for rapid development than WS-117L. Killian and Land argued that the CIA should develop the system with support from the air force, in much the same way as the two bodies had developed the U-2; this method would avoid the inefficiency and bungling of the air force effort. Thus as the initial shock of _Sputnik_ began to dissipate, a small group of enthusiasts was already proposing a second system, to return film to Earth in a capsule. This system would not replace WS-117L but merely accelerate collection of intelligence on the Soviet Union.
In late 1957, as the President's Board of Consultants on Foreign Intelligence Activities reported on its findings and questions concerning an interim program were being discussed, WS-117L supporters sought ways to overcome its funding problems. Initially they thought that the response to _Sputnik_ would overcome resistance to satellite reconnaissance and free up funding, but surprisingly there was still opposition at high levels of the air force and the Department of Defense and skepticism from the White House itself. This grew out of concerns not directly related to the program but rather with respect to the very idea of an "open" reconnaissance effort. Although classified, the WS-117L project was not "black" in terms of its secrecy, as there had been newspapers articles about the program. This raised concerns of a Soviet reaction to reconnaissance satellites.
Eisenhower's reaction intensified the reluctance to accelerate the program. Resenting inferences that the administration had ignored space and missile development, he would not approve any form of acceleration if it appeared to be a "crash effort." He worried lest the press see such action as an admission of culpability. The strongest opposition at Defense came, not surprisingly, from Deputy Secretary Quarles. Still opposing rapid development, he blocked the acceleration plan on October 16, 1957. To circumvent his decision, General Putt secured permission from Air Force Secretary J. H. Douglas to present his case for a speed-up directly to the new secretary of defense, Neil McElroy, who took over on October 9, 1957. This briefing, on October 29, finally overcame Quarles's opposition. On November 1 Secretary McElroy authorized the program, which was to proceed "at the maximum rate consistent with good management."
The WS-117L program did receive a much-needed financial boost in the wake of _Sputnik_. The pre- _Sputnik_ budget of the air force's Ballistic Missile Division had stood at $991 million, virtually all of it for ICBM development, which was on a tight schedule. For fiscal year 1958, however, the WS-117L's budget increased from $13.6 million to $65.8 million—not enough to overcome years of neglect but a step in the right direction.
Attempts to solve the funding problems were bolstered by the National Security Council when it issued NSC Action 1846, "Priorities for Ballistic Missiles and Satellite Programs," on January 22, 1958. This document helped the situation by opening the way for placing satellite reconnaissance at a higher level of importance. Now the secretary of defense could assign high priority to satellite programs that had "key political, scientific, psychological or military import." On June 20, NSC 5814, "U.S. Policy on Outer Space," placed the SENTRY/SAMOS and DISCOVERER programs on the priority list. By giving satellite reconnaissance such a high designation, the president made it clear that WS-117L was vital to national interests, without creating the appearance of a crash program. This stance helped to eliminate problems relating to funding and access to resources and personnel.
Neither RAND nor the air force had been idle. In the months between the initial suggestion of a film-returning satellite system in March 1956 and the launch of _Sputnik_ , RAND kept the idea alive. By November 1957 Katz and Davies were ready to submit a more comprehensive report on its feasibility and a formal recommendation for its development, which they did on November 12. The document was the product of a massive joint effort within RAND and represented one of its most significant research memoranda.
It attacked the problems in gathering intelligence much more systematically than earlier studies had. Looking at satellite reconnaissance in terms of intelligence requirements and the current state of technology, the report urged a series of progressively more complicated systems. Calling for better acquisition in three major types of intelligence (warning of attacks, estimation of enemy capabilities, and targeting information), especially vis-à-vis Soviet ICBMs, Davies and Katz described aerial reconnaissance in terms of four distinct levels of information gathered. The difference between these levels was based on the size of the area a satellite's cameras covered, ranging from thousands of square miles to tens of square miles.
The first level, A, was a system for searching a large area and covering vast expanses quickly. Poor resolution would prevent detection of small objects, but A was useful for monitoring activities and providing data that would permit precise coverage by more advanced systems. In short it was the foundation level of intelligence gathering. Level B would allow identification of many major installations, aircraft on runways, and minor lines of communication (smaller roads and shorter railway and canal lines). Level C would facilitate photography of specific objects on airfields and in industrial zones, and level D was to provide even finer photos. With a scale that was very small, this information would satisfy most of the technical intelligence needs. The trade-off, of course, was that as resolution increased, coverage decreased, so it would take more time to cover the entire Soviet Union.
Davies and Katz proposed a reconnaissance system that could move from level A to level D in a continuum of technical improvements. Unlike the WS-117L satellite, it used a spin-stabilized satellite that rotated the camera shutter across the earth's surface, exposing the film with each revolution. The initial satellites would probably employ an Aerecon camera with a focal length of twelve inches to test the system and to begin photographic reconnaissance. Since the entire camera rotated with the satellite, stability was less of a concern. The camera, sweeping across the line of flight, would photograph large areas. The film had to move through the camera fast enough to compensate for the satellite's speed so as to prevent ground blurring. The shutter speed (1/4,000 second) meant that substantial changes in altitude, speed, and so on would not degrade the image. Once it proved itself reliable, the more advanced Baker twenty-four-inch camera would come into use. Recovery of the film would require the de-orbiting of the entire satellite. Katz and Davies projected that the resolution for such a system would be about sixty feet, with a single shot covering about 18,000 square miles. Thus a reel holding five hundred feet of film could photograph about 4 million square miles, or roughly half of the Soviet land mass. All major targets, including airfields, lines of communications, and urban and industrial areas, would be visible.
The basic configuration of the system that Katz and Davies proposed was both simple and practical, and it harnessed existing technology, greatly reducing development time. This prototype was to serve as the foundation for more complicated satellite systems. Second- or third-generation satellites would probably have camera lenses of longer focal length (twenty-four and thirty-six inches) for higher resolution and greater film capacity. This first system, however, needed little time for development; a satellite could be in orbit within a year of the date of the contract. Davies and Katz hoped that this system would augment and support the WS-117L program by supplying intelligence in the short term while work proceeded on the follow-on system.
Davies and Katz described clearly both what the intelligence the system could provide immediately and its growth potential. The initial system would provide level A, or area, surveillance with resolution comparable to air force reconnaissance aircraft. Through repeated coverage, it could monitor changes in any area that indicated construction or development and indicate patterns of activity. In terms of growth potential, improvements to camera and lens systems would enhance resolution, allowing for far superior photos and thus better data. The initial lens of twelve-inch focal length would probably give way to a thirty-six-inch system within eighteen months. Three years after the initial contract improvements might lead to ten-foot resolution.
With such a study behind it, RAND's recommendation for development of a film-returning satellite was compelling. Because of the expected intelligence dividend from such a system, RAND suggested reprogramming of WS-117L to emphasize tasks requiring space-to-ground communication capability—specifically level A coverage. Because of its lower resolution and slower relay capability WS-117L would be ideal for area surveillance, identifying targets for the superior resolution of the second and third generations.
The WS-117L staff and the Ballistic Missile Division paid attention to Davies and Katz's studies for RAND. In November 1957, when the Ballistic Missile Division asked it to consider modifications to speed up the program, Lockheed's Missile and Space Division proposed two alternatives. The first included use of intermediate range ballistic missiles (IRBMs) as a booster for testing and early flights. The second incorporated a film-returning satellite along the lines of RAND's proposal. The combination would mean even an earlier satellite launch.
Lockheed included these ideas when it briefed the air force and General Schriever on program acceleration late in 1957. Schriever was impressed enough to request a formal proposal for hastening the program. Missile and Space Division's proposal of January 6, 1958, drew heavily on both the RAND work and some of the original concepts that Lockheed had developed for the WS-117L contract bid in 1956 but had shelved because of funding limits.
Lockheed still had to meet air force requirements for visual reconnaissance as well as for electronic intelligence and infrared systems. The plan called for a series of development phases. The first phase would test a series of satellites to verify the orbital capabilities of the booster/AGENA system. The plan slated the inaugural flight in this series for October 1958, the final one for February 1960. The THOR IRBM—available in larger numbers than the Atlas ICBM and thus able to sustain more flights—was to see service in some of these tests. While gauging orbital capability, this phase would also test the visual, electronic intelligence, and infrared systems. The second phase, Program II, was to achieve pioneer visual reconnaissance; the intention was to provide resolution of at least one hundred feet and accuracy of up to one mile in mapping a location. The goal was locating airfields, major cities, and industrial centers, with the first flight likely in March 1960.
The proposed RAND system received a development position as part of Program II, namely Program II-A. It was to be a 7,200-pound second stage launched from the top of a THOR IRBM. Using a RAND-style method of returning film, II-A would supply imagery equal to the early stages of WS-117L. The launch schedule mentioned six satellites between January and July 1959.
Program IV of the WS-117L system was the more advanced photographic configuration, with a total resolution approaching twenty feet and location accuracy on the order of one-half mile. Finally, Program VII featured the infrared system. Initially part of the overall WS-117L program, it had become the basis of its own network of satellites to transmit early warning of an attack. Programs III, VI, and VIII remain classified but probably made up the electronic intelligence satellite system, called ferret satellites in the vernacular of intelligence operations. Programs VI and VIII seem to have included better visual and more advanced ferret capabilities for continuous surveillance.
The most crucial questions about technical feasibility concerned the new requirements of Program II-A, notably the first-stage IRBM booster and the recovery of film from space. The booster was straightforward. The THOR, a smaller missile than the ATLAS, could not support as large a payload. Its limited lift capability of about one hundred pounds to an orbit altitude of three hundred miles was its main liability. However, when combined with the AGENA second stage, then in development for the WS-117L system, it could carry a projected payload of two hundred to four hundred pounds. The THOR had thus become a viable booster at least for testing and returning film. Therefore Lockheed's Missile and Space Division rejected the three-stage booster configuration that RAND put forward in memorandum RM-2012. A simpler two-stage version seemed adequate for a useful interim reconnaissance mission until the more advanced WS-117L system became available.
The recoverable film payload was another matter. The proposed recovery system would require a method of activating the process on command over Alaska for recovery in the Pacific north of Hawaii. A heat shield using the ablation technology for ICBM reentry would protect the film capsule. Impact down range would occur probably within a circular error probability of about thirty miles, allowing air and naval units to close in and retrieve the capsule fairly quickly. This system differed from the RAND proposal just in its recovery of only the nose cone with film inside. The rest of the satellite (electronics, fuel cells, camera, and so on) remained in orbit, thus reducing the necessary amount of heat shielding to roughly sixty pounds.
Except for a shift in the schedule for the first flights, the accelerated Lockheed plan did not change the original WS-117L system to any appreciable degree. It relied on the THOR IRBM and RAND's film-returning satellite. Although smaller and less powerful than the ATLAS, the THOR was available in larger numbers and more reliable. This meant a much faster pace of launchings and tests and thus quicker detection and elimination of flaws in the components. The acquisition of intelligence could begin that much sooner. The interim addition of the RAND film-return system almost in its entirety did not accelerate WS-117L, but it did speed up acquisition of intelligence. The WS-117L system remained as it was; experts simply grafted the Lockheed plan onto it to make intelligence gathering faster and to provide a testing platform. The addition of Program II-A is significant, however, for understanding the division of the WS-117L in early 1958. It was a short-term supplement that by February 1958 had separated for independent development as CORONA.
## Film Return, Project II-A, and the CIA's CORONA (November 1957–February 1958)
Even as Lockheed was drawing up plans for expediting the WS-117L, forces beyond its control were eclipsing its efforts. The RAND design proposal had circulated within the air force and among senior scientific advisors in the White House in November 1957. The ideas behind RAND's drop-film system had appeared in the October 1957 evaluation of the WS-117L program by the President's Board of Consultants, and the most vocal proponents of a dramatic change in the program, Killian and Land, strongly endorsed the idea.
Eisenhower took heed of their advice and on October 28, 1957, had his executive secretary advise Secretary of Defense McElroy and CIA Director Allen Dulles that he wanted a joint briefing on the status of advanced reconnaissance systems. Because the issue was so sensitive, McElroy (speaking on behalf of Dulles and himself) proposed oral briefings with no written records. Always security conscious, the president agreed. What is clear is that during this blackout period, running from December 5, 1957, until roughly February 28, 1958, the chief executive decided that the program needed major changes.
The alteration occurred in absolute secrecy. On February 6, 1958, Killian and Land met with Dulles, McElroy, and Quarles to discuss a new proposal for a film-return satellite. The following day they met again to brief Eisenhower. They proposed removal of those portions of the WS-117L program that promised the most rapid success—namely the film-return satellite (Lockheed's Program II-A)—for independent development. Unlike the WS-117L program, this was not a "pie-in-the-sky" scheme; Land made it very clear that it was a "specific small project for bona fide intelligence purposes."
To be effective the whole program had to be as covert as possible, hiding "under the cloak of other activities" to reduce the political ramifications of a reconnaissance satellite. There was, however, a technical and strategic reason for this as well. The resolution of the initial camera system was likely to be inferior to that of the U-2 (resolution of fifty to one hundred feet, as opposed to approximately six inches for the U-2 Baker camera). Careful precautions were essential to guard against Soviet discovery of the spy satellite; the Soviets could easily deceive and confuse American photo interpreters with dummies and camouflage. Fortunately the capsule radiated no signals, so the greatest threat to its covert nature lay in espionage or leaks during development and operation. Land and Killian called for joint air force–CIA production, with the air force dominant—an idea that Eisenhower rejected. He favored putting the CIA in complete and exclusive control of all the intelligence phases, with as few people as possible knowing about it.
Eisenhower's distress over satellite management lay behind his insistence on CIA control. The military, particularly the air force, appeared to concentrate more on lobbying for funds than on producing the reconnaissance satellite it was already developing. All three services had advocated their own brand of space program in the wake of _Sputnik_ , with the army's virtually identical to the WS-117L system. However, it was the air force, which was actually developing satellite reconnaissance, that sought to translate the WS-117L program into political clout. Congressional hearings discussed the program, and details about it soon appeared in the media. A mere ten days after _Sputnik_ orbited, _Aviation Week_ revealed the existence of a reconnaissance satellite program under the name PIED PIPER (WS-117L's moniker during contract bidding). Giving considerable detail about the program, the article noted the participation of both RAND and Lockheed, the satellite's expected configurations, and such diverse tidbits as company nicknames for the project.
Eisenhower acted decisively. In February 1958 he ordered the separation out of WS-117L's Program II-A for independent development. He stressed two priorities: rapid development for operation by spring 1959 and covert work to avoid antagonizing the Soviet Union, conceal the satellite's capabilities, preserve its intelligence value, and prevent military hindrance. Recalling the CIA's effective role in developing the U-2, Eisenhower ordered that agency to take the lead in development and to work closely with select elements of the air force. This decision reflected his view that the CIA should control national intelligence, as the National Security Act intended it to. If the air force had developed the program, then it would have had considerable control over strategic intelligence. Because it tended to leak information and exaggerate Soviet capabilities, Eisenhower would not trust it to dominate collection and evaluation of intelligence. By giving the CIA control, as with the U-2 program, he expressed confidence in its ability to produce results quickly, quietly, and objectively. He believed that he was initiating an interim program, but it outlasted WS-117L by many years.
The decision to accelerate satellite reconnaissance by removing Program II-A from the WS-117L effort and giving it to the CIA under the name CORONA was understandable. With few details available about Soviet missile deployments, and under a great deal of pressure from all sides to increase research and development on ICBMs and space programs, Eisenhower desperately needed a system that could begin obtaining intelligence as soon as possible. At the same time, he suspected that the old WS-117L program could not do the job. The film-returning satellite offered the irresistible promise of a speedy solution.
The program that Eisenhower gave the CIA had the most lasting impact of any on U.S. intelligence. Officially canceled on February 28, 1958, Program II-A was formally restarted on April 21, 1958, when Andrew Goodpaster gave the CIA the go-ahead. For the next twenty-eight months the CORONA program underwent an intensive and often painful development effort, which included many spectacular failures before it produced success on August 19, 1960, when it managed to orbit the first successful reconnaissance satellite. The satellite provided the first images from space and served as the backbone of U.S. space reconnaissance for well over a decade. People responsible for its development included Richard Bissell from the CIA (who had also run the U-2 program) and his deputy, Brig. Gen. Osmond Ritland (General Schriever's vice chief of staff). Having collaborated together on the U-2, Bissell and Ritland already had a good working relationship. The CIA and the air force both made major contributions to CORONA. The CIA brought experience in developing and maintaining clandestine reconnaissance systems, along with money that no one could trace back to any specific program. The air force had missile expertise and experience in research and development, along with the thousands of person hours that it had already put into the WS-117L program.
All of this would serve as the foundation for many of CORONA's basic concepts. The air force was responsible for all aspects relating to operation of the satellite system, including development, launch, physical control, communications, and recovery of the space vehicle. Besides providing security, the CIA supplied camera systems and looked after creation of special film, developing of photos, and processing of data from them. The two sides complemented each other well largely because of Bissell's approach. Believing in a small team that facilitated fast decision making and individual responsibility, he and his staff combined their talents to maximum benefit. Bissell answered directly to Allen Dulles—a marked improvement over air force management. There was no elaborate, ponderous chain of command to contend with when the program needed swift changes.
The CIA employed clever sleight of hand to make the requisite parts of the WS-117L system vanish. Bissell and his staff used a variant of Second Story, the cover program for a "scientific satellite" that Schriever proposed in 1957. On February 28, 1958, the WS-117L system formally canceled Program II-A. It so informed Lockheed and made arrangements to cover costs to date. This came as a severe shock to everyone outside the loop, and the outcry from some people, such as Katz, lent credence to the cancellation order. By complaining loudly and bitterly and pushing for a rethinking, Katz helped to convince many people that the decision was real and final. The very short list of people with "need-to-know" clearance did not include him and Davies.
Since some explanation was needed for the large number of THOR-based satellite shots, the CIA decided to hide CORONA in plain sight. The WS-117L plan added a new program called DISCOVERER. The agency described it as a scientific satellite program to support future manned space efforts by providing environmental and test data as well as validating the recovery concept and other aspects of space flight. It was also to help test some WS-117L components, specifically the AGENA second stage.
Thus the public DISCOVERER program gave cover for the clandestine satellite system, as did a handful of satellites that went into orbit for biomedical test shots—to generate scientific data for public consumption to hide the gathering of intelligence. This was not CORONA's sole protection, however. The fact that the public associated WS-117L with a spy satellite helped to deflect attention from CORONA. Continuing the development of WS-117L as a follow-up system (although no public announcement took place) drew the attention of interested parties. With the visible military program eclipsing it, DISCOVERER/CORONA could develop in the shadows of the program that spawned it.
As a system CORONA was far simpler than WS-117L. It achieved many firsts for the CIA and the air force: the first reconnaissance satellite to photograph the earth, the first recovery of an artificial object to have orbited the earth, and the first airborne recovery of a satellite component. In its simplest form the program consisted of a two-stage launcher using the THOR IRBM as a first stage and the AGENA booster as the second. The system was to establish a near-polar orbit, allowing it to photograph all key areas of the Soviet Union. Operating for twenty-four to forty-eight hours, it employed a timer so that it would photograph only areas within the Soviet Union, thus maximizing use of the film on board. After the film's exposure it spooled up in the nosecone of the AGENA rocket. At the appointed time, on command from a ground station, the satellite disconnected the nose cone and fired a rocket to slow it down for reentry. Once in the atmosphere and through the initial heating stage, the capsule deployed a parachute, and the armed forces recovered the entire package in the air or once it had reached the surface, somewhere in the Pacific north of Hawaii.
CORONA was clearly a separate program from WS-117L. Film return was not part of the WS-117L system except from the end of 1957 to early 1958, when the program added it to speed up acquisition of intelligence. It was not an integral part of WS-117L so much as a graft onto the program. CORONA was only superficially similar to the WS-117L satellite concept. With a different style of management, a different operational plan, and a decidedly different camera system, it had few similarities to WS-117L, except in a handful of components, particularly with respect to the AGENA booster. Conceptually, however, the two programs had one central connection: without the WS-117L development program CORONA would have been beyond reach in 1958–59. It was only the initial, groundbreaking discoveries of the RAND studies and air force work that made orbiting a system like CORONA possible. Though a major success in its own right, CORONA owed a great deal to the WS-117L program that preceded it.
The CORONA program transformed U.S. intelligence gathering and the nation's understanding of the cold war. But its later history will not be discussed in any greater detail here. The reasons for this are threefold. First, CORONA was separate from the rest of WS-117L and so branches off from its story. Second, CORONA is already the subject of much high-quality work by some of the best historians in the field. Interested people may consult any of the published sources that are cited here. Third, from the start this book has sought to redress the historical imbalance between WS-117L and CORONA. For our purposes the CORONA story effectively ends here.
With an understanding of how CORONA came into existence, its shape, and its scope, we can now return to WS-117L and its development from the launch of _Sputnik_ to its end in the 1960s. The program underwent significant changes and suffered dramatic setbacks in the wake of _Sputnik_. In the shadow of CORONA, this part of the story of satellite reconnaissance has attracted little attention and found even less understanding.
#
# SENTRY/SAMOS, MIDAS, and the Dissolution of WS-117L (1958–60)
As far as the satellite [ _Sputnik_ ] itself is concerned . . . that does not raise my apprehension one iota. I see nothing at this moment, at this stage of development, that is significant in that development as far as security is concerned.
—Dwight D. Eisenhower
Members of the Samos organization, engaged in an enterprise tenfold larger and more costly than _Corona_ , and convinced that the highly sophisticated E-6 would shortly displace the theoretically less capable _Corona_ system, tended to be a bit more superior about the older program.
—Robert Perry
With the origins of CORONA firmly established, the development of the WS-117L satellite program in the wake of _Sputnik_ can now be clearly examined. Eisenhower and the USAF continued work on WS-117L from October 1957 through the end of his second term in office, albeit with some significant changes and restructuring. The story of WS-117L post- _Sputnik_ is about the attempts to straighten out management issues as it is about the technology. A new management structure, complete with the creation of a new management body in the form of the Advanced Research Projects Agency (ARPA), was tried but failed. By the end of Ike's second term in January 1961, the satellite that I have traced so carefully was all but gone. In addition to the technical problems of satellites, serious management problems continued to plague the program, and in the end, with its division into separate satellites, for all intents and purposes WS-117L ceased to exist.
## ARPA and Its Three Systems (January–May 1958)
As we saw in chapter in chapter 6, on February 7, 1958, to help facilitate both the WS-117L program and the secret CORONA effort, Eisenhower took significant action to straighten out problems in research and development within the military. Displeased with how the USAF had handled the program, he ordered establishment of the Advanced Research Projects Agency within the Department of Defense to manage all research and development for military space projects. Prior to ARPA the services had created their own projects from scratch, which led to duplication and wasted effort. With Roy Johnson (from General Electric) as head and physicist Herbert York as chief scientist, ARPA was to coordinate research and development in the military space program and eliminate waste, duplication, and unrealistic projects. The armed forces continued to propose program requirements, but ARPA evaluated them and assigned those appropriate for development to the applicable service. It ignored proposals with interplanetary goals and instead emphasized near-earth orbit systems.
The military services did not like the creation of ARPA. They worried that Johnson would operate it as an independent command, ignoring their wishes and arbitrarily imposing its will. On March 27, 1958, their worst fears were confirmed when the service secretaries learned that ARPA would bypass them. Instead of being under control of the military chiefs, ARPA answered directly to the secretary of defense and had direct access to the three agencies—the Air Research and Development Command, the Army Ballistic Missile Agency, and the Naval Ordnance Test Station—active in military space research and development. In theory direct access to these bodies gave ARPA a great deal of control over them; in reality this was far from the truth. Since ARPA had no facilities or personnel to conduct research and development, it had to depend on the agencies to do the work. ARPA received funding only to test the feasibility of systems and components, which in the case of WS-117L meant roughly $186 million in fiscal year 1959. The armed forces had to pay for systems' development, production, and deployment. ARPA emphasized streamlining and the elimination of duplication and mismanagement during research and development.
ARPA established control of WS-117L relatively quickly. Secretary of Defense McElroy authorized acceleration of the program on February 24, 1958, with the proviso that it take place under ARPA's control. Four days later Johnson canceled WS-117L's film-returning portion, Program II-A, supposedly because it duplicated the ATLAS-WS-117L effort. In fact, as I explained earlier, this directive was part of the cover plan for CORONA and was the creation of Richard Bissell and his CIA staff. The same order that canceled CORONA initiated the DISCOVERER program. It authorized the use of the THOR booster as part of the engineering and system tests for WS-117L and for biomedical experiments, thereby ensuring a plausible cover for CORONA. For CORONA, ARPA thus did little more than fund the overt portion of a covert program.
In May 1958 the air force transferred the WS-117L program to ARPA management—a temporary formality, as ARPA Order No. 9-58 soon returned it to the air force's Ballistic Missile Division for technical development. The division had to submit a detailed development and financial plan to ARPA, which was not impressed with the progress to date or the expected form of the WS-117L satellite. Suffering from neglect and inadequate funding, the program also had become extremely complicated. It had augmented the visual reconnaissance mission with a plethora of other functions, including a subsystem to monitor the electromagnetic spectrum, an infrared alert mechanism to warn of Soviet missile attack, and a capsule-recovery system for biomedical experiments, to say nothing of other subsystems for satellite flight, including power supplies and guidance.
The WS-117L program also called for thirty-two satellite test vehicles, with the last eight probably to carry at least a rudimentary reconnaissance capability over the Soviet Union by March 1960. Initial test flights were to begin in 1959 and involve about nineteen THOR-boosted vehicles, with second-phase testing using ATLAS as the primary booster in the first half of 1960. These test vehicles formed part of the WS-117L's budget for fiscal year 1959 of $186 million. Because of the reconnaissance capabilities of the last ATLAS shots in 1960, James Killian recommended that the Department of Defense seek presidential approval prior to these launches.
The complex combination of systems that composed WS-117L was a major problem. By jamming in such a wide variety of sensors and systems, the engineers had created two distinct problems. First, each component had to be as compact as possible so that everything could fit into a small package, which complicated the technical design challenges. Second, the array of equipment meant greater complexity and more opportunities for problems. It also represented a psychological barrier to the program's progress and success.
In effect the air force had decided to forgo a functional but less effective satellite in the short term for a superior system that would be unavailable for many years. Any one of the systems—photo reconnaissance, infrared warning, or electronic intelligence gathering—might have emerged far earlier if it had been the program's sole effort. The air force's desire for perfection exacerbated the many problems and complexities. The "all or nothing" mentality meant that it reached none of its goals in a timely manner. ARPA, as well as the White House's Office of the Special Assistant for Science and Technology, worried about such issues. As a member of the President's Science Advisory Committee and through his contact with Herbert York and Edwin Land, Killian requested that York and ARPA review the WS-117L program in July 1958. The classified results are not yet available, but ARPA certainly looked at the complexity and development problems.
Eventually ARPA decided that one large satellite could not fulfill all of the WS-117L system's missions. The more complex elements were holding back progress on those with higher priority or less challenging technical problems. ARPA considered dividing the effort into separate programs that could advance at their own pace, which would lead to faster system development. At the same time, people active in CORONA worried about public knowledge of the WS-117L system. The air force was using a press campaign to link itself with observation satellites and thus secure funding. Information leaked to trade journals and glossy pamphlets that described DISCOVERER in some detail raised warning flags among those responsible for the security of CORONA.
Bissell and Colonel Oder worried that the identification of WS-117L with space-borne reconnaissance would lead the public to conclude—correctly—that DISCOVERER had links with it. The reason for canceling the drop-film camera system and restarting it under the name DISCOVERER had been to prevent just this linkage. Fearing that a backlash would lead to cancellation of CORONA, Bissell and others pushed for some means of distinguishing it from WS-117L, resulting in a compromise. According to Curtis Peebles, on October 20, 1958, Johnson ordered a change of program names. Afraid that the designation "weapon system" (WS) had aggressive implications, he ordered the air force to stop using it.
To improve management, in September 1958 ARPA also ordered the division of WS-117L into three programs. DISCOVERER, still the cover for CORONA (formerly the WS-117L's Program II-A and now under CIA control), officially remained a test bed for experiments on generic problems relating to space operations, along with biomedical support missions. The film-readout system, the core of WS-117L, it renamed SENTRY. This moniker lasted until June 1959, when Eisenhower questioned having code names with military implications and had the label changed to SAMOS. The infrared system, the other key element of WS-117L, separated out to form its own satellite, MIDAS (Missile Defense Alarm System).
ARPA's association with WS-117L and its descendant systems did not end there. Its supervisory role often placed it in the middle between the air force and the administration over funding for programs. To prevent rampant overspending to achieve a satellite at "any cost," ARPA's leadership often had to fight with the Bureau of the Budget and the Department of Defense for the conflicting goal of sufficient funding. For example, in the budget for fiscal year 1959, citing the need to complete the program as soon as possible, the air force sought to increase its budget from $152 million to $220 million. ARPA acted as a counterweight, arguing the budget down to $185 million and then, because of its position between the program development offices and the administration, fought to gain that money for the program.
The splitting of WS-117L into three systems was a pivotal move in the history of U.S. reconnaissance satellites. By allowing the various elements to progress at their own pace, the United States could develop a wide range of future satellite options because there was no single design. The most technically challenging programs no longer prevented the country from gaining the benefit of systems that could be available earlier. It also permitted DISCOVERER/CORONA to progress with less attention and interference from outside, and, in the long term, it would be easier and faster to modify any one program.
## SENTRY/SAMOS: Readout and Film Return
The SENTRY system remained basically the same as that in the WS-117L development plan of January 1958. Fortunately ARPA required regular status reports, which give us glimpses into the shape of the program in 1958–59. By the end of January 1959 SENTRY consisted of several elements. With an operational goal of providing a timely and "continuous (visual, electronic or other) coverage of the USSR and satellite nations for surveillance purposes," the program envisioned two intelligence packages for the satellite: film relay and electronic intelligence.
SENTRY was to supply equipment capable of mapping, locating targets and defenses, gathering information on military and industrial strength, and monitoring electronic signals. With an intended operational life of one year or longer, it needed a high degree of reliability. The visual reconnaissance program would take photographs and store them on film. The satellite would then convert these images into an electronic signal to relay to a ground station. Initially the satellite was to use standard photograph technology with a special system to develop film on board. After developing photos the system would scan the images and relay them to Earth. Later versions envisioned electrostatic sensors and a high-resolution television system working in conjunction with a magnetic tape-recording system. At the start the resolution would probably be twenty feet but would eventually reach five feet or less. Thus the original television system that RAND's FEEDBACK report proposed in 1954–55 remained a long-term goal.
SENTRY's photographic subsystem consisted of six different cameras, E-1 through E-6, which would provide sequential improvements in performance. E-1 through E-3 used readout systems, and E-4 through E-6 film return. The E-1 camera was the pioneer system and would use very fine-grained film and a lens with a focal length of six inches to produce images with a hundred-foot ground resolution. Operating over the target area for only five minutes, it would return images to three U.S. ground stations, delivering about 10 percent of the intelligence to a central analysis station within the first hour after transmission, and the rest over the next eight hours. To be effective several satellites would provide virtually constant coverage. The E-2 system followed the same pattern but had an extended operational life and used a thirty-six-inch camera lens, reducing the resolution to probably twenty feet. The E-3 was an electrostatic system that relied on television cameras and videotape to record the images for relay.
SENTRY's three remaining camera configurations consisted of film-return systems, despite formal cancellation of that part of the program and its transfer to CORONA. The E-4 camera was a mapping system that duplicated the army mapping satellite ARGON. When ARGON received approval on July 21, 1959, it replaced the redundant E-4 and was grafted onto CORONA. The E-5 involved a panoramic camera with a large recovery capsule. According to Dwayne Day, it had a focal length of about sixty-six inches and a likely resolution of two feet. In reality it never achieved better than a six-foot resolution and never flew as part of the SENTRY/SAMOS program. Going into mothballs following SAMOS's cancellation, the E-5 cameras later became part of the CORONA program. With the code name LANYARD and designation KH-6 (as part of the Key Hole satellite system), it flew only a handful of missions in 1963 before a second cancellation. The E-6 camera, an upgrade of E-5, was a late addition in 1960 as part of a revamping of SAMOS. It was a recovery system with a focal length of almost thirty-six inches and a ground resolution of roughly eight feet and would probably have had full-area surveillance capability with better-quality images.
The January 30, 1959, SENTRY development plan also included some details concerning the Signals Intelligence system that was to form part of SENTRY. The ferret component was likely to progress from the most basic package with only limited capability to a sophisticated surveillance system and to collect data from electronic radiation sources in the Soviet Union. The system was to provide more accurate measurements and better location techniques while monitoring signals in the range of 30,000 to 40,000 Mc/second. This would allow the United States to establish an indication of the imminence of possible hostility as well as an electronic order of battle and data useful for technical intelligence purposes. The ferret system would store data on the direction to the signal source, its frequency, and its power, using a filter and indexing system, and then relay the information to the ground via a signal link similar to the one for the photographic system. To date very little declassified information relating to this area has become available, but even this small cache is very revealing.
No other major differences are evident in the rest of the satellite configuration. Still anticipating that a nuclear power supply would give way to a solar power/battery to power the satellite, the development plan also detailed some of the other systems. Along with the airframe, propulsion, guidance, and control, it discussed ground-space communication and the complex data-processing systems. Apart from outlining equipment for data retrieval and calibration, the document called for systems to maintain accurate positional data to ensure targeting, a method for initial interpretation of data to find indicators of problems or changes, and a mechanism for saving information in a manner conducive for use by standard equipment and operating agencies.
Lockheed's Missile and Space Division remained the program's primary contractor. Responsible for management and systems development, it provided the central direction and monitored the many subcontractors to ensure work of an acceptable standard. Subcontractors for the SENTRY system were varied. The rocket motor, for example, was originally a Bell Aircraft engine, and the company maintained a subcontract for any modification to it. Eastman Kodak was responsible for the design and fabrication of the visual reconnaissance equipment and for conducting simulations to facilitate system development. Airborne Instruments Laboratory developed the entire ferret package.
The SENTRY program envisioned a lengthy testing process. The first launch of the film-readout system was likely in April 1960, with following tests in June, August, and October through December 1960. The recovery system would probably not begin flights until January 1961, and eight flights would complete the testing cycle. There would also be wind tunnel, environmental, and component tests. The program was expensive; for fiscal year 1959 costs totaled $108,969,000, most of it from ARPA. Lockheed received the largest amount: $66,700,000. Among the increasing costs were amounts for ATLAS missiles (up from $600,000 to $14.2 million). Planning for fiscal year 1960 indicated that expenses might reach $170,500,000. ARPA was again providing the lion's share ($160 million), and the air force the rest ($10.5 million). As before, Lockheed received the most funding ($104 million), and acquisition of ATLAS missiles amounted to the next highest cost, at $34.4 million.
Despite this massive investment, the SENTRY/SAMOS system was not very successful. Although it was to be CORONA's successor, severe problems prevented it from reaching its goals. It proved incapable of achieving the high resolution necessary for productive reconnaissance. Carrying about 4,500 feet of 70-mm film, the satellite had a far lower film capacity than CORONA due to the extra space needed to develop and store the film for scanning. Once developed, the film was to go to a series of storage reels and loops to await scanning and relay to Earth. The readout system was far too slow to be practical. Scanning the film required focusing a beam of light on it through a flat window. Concentrating on only a small portion of the film at a time, the light had to move back and forth across its width to scan the entire image. Behind the film a lens/receiver system collected about 75 percent of the light that came through, multiplied it, and transformed it into an electronic signal.
The transmission process was tedious. For an image at 200-mile altitude, resolving objects of fourteen feet in size, the scale in the photo amounted to 1:400,000. Thus each frame of 70-mm film covered a ground area of 270 square miles. Because of the limited bandwidth (6 Mc/second) available to transmit the imagery to Earth, the satellite would take three hours to relay five minutes of exposures, far slower than the designers had expected. Making matters worse, the scanning system was reading only white, black, and one shade of gray. The more shades of gray for which the system scanned, the longer it took to process the film. Since each ground station could accept only about sixty-two frames (roughly 16,740 square miles of target area) each day because of the short contact with the satellite, the readout system seemed far from acceptable. By comparison, one early CORONA satellite could scan about 1.5 million square miles in a day with higher resolution because the image did not degrade during transmission.
The image-readout system that was the hallmark of SAMOS had a very short operational life. The first system, the E-1 (one-hundred-foot resolution), consisted of three satellites, of which the program launched only two. The first attempt failed because of a launch accident on October 11, 1960. The second (SAMOS 2) made orbit on January 31, 1961, but operated for only one month. The third never orbited. The follow-on system, consisting of the improved E-2 camera and an F-2 ferret system, was no more successful. The program built only two of these satellites, with resolution of roughly twenty feet; there followed one unsuccessful launch and the abandonment of the second satellite.
Because of the poor image quality from the SAMOS 2 satellite and a damning study by the Directorate of Defense Research and Engineering, SAMOS was canceled in 1961. The directorate's study was extremely hard on SAMOS because of weaknesses in its readout system and the concept of giving operational control of both the satellite and interpretation of intelligence to the air force. Its author, Dr. B. H. Billings, echoing the views of Herbert York and Undersecretary of the Air Force Joseph Charyk, argued for a national intelligence capability. How much influence the Billings report had on Eisenhower's decisions of 1960–61 to create the National Reconnaissance Program is not clear. What is certain, however, is that a real-time satellite had to wait until the technology could catch up to the concept.
In the meantime the film-return versions of SAMOS were not much more successful. The E-5 system commenced operation in 1961. Using solar panels and a more sophisticated camera system than CORONA, it was likely to produce higher-resolution images of smaller areas. The first E-5 launch in November 1961 failed to reach orbit, and a second attempt in December orbited but with no film recovery. The third E-5 system, which went up in March 1962, also failed to provide any imagery. The program was canceled and its remaining cameras put in mothballs. The more advanced E-6 system began to fly in April 1962. Like its predecessors, it experienced a series of setbacks before finally recovering film in November 1962. The imagery was no better than the CORONA system, and the program canceled the last remnants of the SAMOS system.
## MIDAS: The Road to Early Warning
The MIDAS satellite was more successful than SAMOS. The November 5, 1958, ARPA Order No. 38-59 separated out MIDAS and gave it to the air force's Air Research and Development Command, with the goal of providing an operational satellite system by April 30, 1960. This order mandated several tasks, including production of two completely functional infrared satellites, continued studies of infrared tracking with the goal of creating more advanced tracking capabilities, and sustaining an experimentation program to provide physical data on infrared-spectrum phenomena to improve the engineering of the MIDAS system. To facilitate these efforts an initial $750,000 became available for the period November 1, 1958 to January 31, 1959, to cover immediate costs. For fiscal year 1959 ARPA had budgeted $22.8 million for the program; this went up for the next fiscal year to $46.9 million. This was a far cry from its treatment of SENTRY/SAMOS. The division of responsibility between the ARDC and ARPA was very straightforward. While the ARDC would supply technical direction for the program, ARPA provided policy and technical guidance. Thus the ARDC had to conform to ARPA's policy direction. As with the SAMOS system, regular monthly reporting would inform ARPA of developments and needs within the program.
Many officials in the Eisenhower administration certainly recognized the system's potential worth. The report by the Ad Hoc Technical Advisory Board to ARPA evaluated the MIDAS proposal in February 1959. Strongly supporting it, partially because of its value in defending the nation and its technical simplicity, the report argued for more research and advocated pursuit of the program. The Early Warning Panel, part of the President's Science Advisory Committee, also kept abreast of MIDAS and recognized its long-term value to national defense. In its March 1959 review of possible methods for early warning of a missile attack on the United States, the panel portrayed MIDAS as the most promising of the numerous options available. Although more costly than the U-2 and the Ballistic Missile Early Warning System, MIDAS was likely to have a greater range than any other proposed system because of the altitude at which it operated. The review was realistic, however, in assessing the remaining questions. Expressing concern over the effects of cloud cover and atmospheric absorption, the ability to discriminate between missiles and high-altitude jet aircraft, system reliability, and so on, the panel nevertheless felt that MIDAS deserved development. With operational costs ranging between $200 million and $600 million, it recommended only initial-phase testing until answers became available for many of the outstanding scientific research questions.
The MIDAS program was as ambitious as the SENTRY/SAMOS program. By December 1959 Lockheed's Missile and Space Division expected that a prototype operational satellite would be ready in March 1962. The plan called for six satellites to be in orbit by the end of June 1962 and for this number to increase until deployment of a fully operational system (eight satellites) by December 1962. The system concept called for a web of MIDAS satellites to cover Soviet territory continuously; it would detect any launched missile at an altitude of thirty-five thousand feet. The information would go to ground stations in Britain or the North Pacific. From there it would travel to a MIDAS operational center for assessment and relay to North American Aerospace Defense Command and the White House. The satellite would identify the approximate location of a missile launch, thus allowing the United States to locate Soviet ICBM fields, the approximate direction the ICBM was taking, and the number launched. The MIDAS network of eight satellites in two groups of four, orbiting at an altitude of two thousand miles, with each group moving in a different orbital plane, permitted maximum coverage with the fewest possible satellites.
MIDAS used the same basic AGENA booster as the SENTRY and DISCOVERER systems but carried a different payload. The seven-unit ground system for handling data was also more complex. Three tracking and acquisition stations, all within the continental United States, would acquire precise orbital information, adjust orbits, relay commands, and monitor equipment. They had links to the Technical Operations Control Center, which would monitor the three tracking stations, provide orbital data to people processing information, and schedule and control changes in satellite operations. The final three units all related to monitoring of missile launches. Two readout stations, one in the United Kingdom and one in Alaska, would monitor the satellites. They would receive warnings of Soviet missile launches and relay them to the MIDAS Operations Center, which would combine them with orbit data and accurately assess the warnings. The MIDAS Operations Center would then alert the president and the military, while also directing and coordinating the entire operational system.
## A Plague of Problems and the President's Panel (1957–60)
Progress on satellite reconnaissance was problematic from 1957 to 1960, with two types of problems hampering it. The first problem was technical and stemmed from the need to invent the technology and the science simultaneously. All three satellites (SAMOS, MIDAS, and CORONA) presented daunting challenges that called for technology that was years ahead of its time. We can see the problems clearly in CORONA, a far simpler system than WS-117L. From the first failed launch ( _Discoverer 0_ ) on January 21, 1959, through to the launch attempt of _Discoverer XII_ on June 29, 1960, problems plagued the CORONA program. Only the second launch, on April 13, 1959, could be considered a partial success. Once the satellite achieved orbit, ground control mistakenly activated the recovery sequence too early, and the capsule landed on the island of Spitzbergen, north of mainland Norway, where the Soviets recovered it, although they never acknowledged having done so. Since the satellite contained only instruments and telemetry equipment, the loss of the capsule did not endanger the program. The next two launches, _Discoverer XIII_ and _XIV_ (August 10 and 18, 1960, respectively), were successful; _XIV_ returned the first photographic reconnaissance of the Soviet Union from space on August 19, 1960. System unreliability and technical problems continued for years. MIDAS and SAMOS both suffered from mechanical failures as well.
The second major problem facing the programs had to do still with management and funding. ARPA's control of the military space program had not improved management but created more confusion and instability, further slowing progress. Brigadier General Ritland and Lieutenant General Schriever, both intimately connected with the spy satellite programs, strongly criticized the management. ARPA's lack of leadership and direction often resulted in "inadequate or untimely decision, or the lack of a decision at all." Financial difficulties made the whole situation even worse; the program received money month by month, which made long-term planning and development more difficult. In September 1959 Schriever complained bitterly to air force chief of staff Gen. Thomas D. White about this hurdle to both SAMOS and MIDAS. Because of constantly changing support levels, he wrote, "we are forced into a day-to-day type planning for conduct of the program." Consequently "establishment of a logical plan and execution of that plan in terms of procurement, scheduling, production, test and operation of SAMOS has been rendered essentially impossible."
ARPA's management had proved so inept that Secretary of Defense Neil McElroy finally acted. On September 18, 1959, he ordered the removal of SENTRY/SAMOS from ARPA's control and returned it to the air force, pending acceptable development and operational plans. As soon as it learned of this decision, the Strategic Air Command indicated that it expected to gain operational control of the program upon its completion, and its leadership immediately began to prepare plans to accelerate SAMOS's development. Pushing for a broad-front approach to research and development, before the satellite's designers had finished their work SAC was generating plans for ground-support structures and training of staff. Without any conclusive evidence that the system would work, the air force had planned how to spend its extra money on the program. In short, it was "putting the cart before the horse."
Management for the program did not return to the air staff directly. SAC's concept for program development and operation ran afoul of the Directorate of Defense Research and Engineering, which it had set up in 1958, with Dr. Herbert York in charge. The directorate was above ARPA in the chain of command and was responsible for authorizing military and Department of Defense research projects. With ARPA removed from program management, the Directorate of Defense Research and Engineering assumed responsibility for development. York's views on the matter led to strong clashes with SAC and the air force. He and his staff refocused the program away from image relay—the foundation of the WS-117L program since its inception in 1954—and advocated film recovery instead. Supporters of the real-time potential of film readout, especially SAC, and those favoring use of as many approaches as possible to provide intelligence as early as possible (the Ballistic Missile Division and the ARDC) fought this change. Realizing that the air force had to accept the restructuring in order to regain control of the program, the ARDC finally bowed to the directorate's pressure. The secretary of defense officially transferred SAMOS from ARPA back to the air force on November 17, 1959. The ARDC again furnished development plans and requested funds for the programs on February 18, 1960. For SAMOS the air force requested $160 million for FY 1960 and $199.9 million for FY 1961 in order to maintain at least minimum levels of research and development. To achieve the same for MIDAS, it asked for $31.1 million and $40 million for FYs 1960 and 1961, respectively, above and beyond money remaining from research and development in this area under SAMOS.
The unilateral decision to change the SAMOS program to a film-return system shows that the directorate's leadership style was far more divisive than necessary. The order to restructure the program met with acquiescence, but the Ballistic Missile Division still fought to maintain the readout system. Arguing that abandoning work on it would delay a reconnaissance system until mid-1963 unless there was a rapid increase in funding, the Ballistic Missile Division stubbornly maintained SAMOS's readout capability as part of the program. Such resistance led to longer delays in program development. The infighting over funding and priorities between the readout systems (E-1 to E-3) and the film-return systems (E-4 to E-6) prevented decisive leadership in satellite development.
SAC's attempt to usurp responsibility for SAMOS before it completed research and development caused greater confusion and diluted efforts to create a single, unified program. York and his directorate saw SAC's efforts to speed up SAMOS as unrealistic and harmful in the long run. In December 1959 he openly attacked these attempts, claiming that they "inevitably interfere with the research and development program." He found SAMOS "confused and slowed down" by the concentration on an operational system in advance of establishing capability. Favoring film return himself, York discouraged development of ground-support systems until there was certainty about the satellite's actual form. The Ballistic Missile Division's approach of relying on a readout satellite system and its development simultaneously with all the ground support elements did not change in the face of this opposition. It resolutely included these concepts in every presentation on its development plan.
The loss of a U-2 over the Soviet Union in May 1960 reinforced the Eisenhower administration's belief in the importance of satellite reconnaissance. In January 1960 the president had asked for a presentation on the status of the reconnaissance satellite effort where the problems with SAMOS seemed self-evident. On February 5, at Land's request, York briefed Eisenhower on the progress of DISCOVERER, SAMOS, and MIDAS. Dr. George B. Kistiakowsky (the president's science advisor) and Land questioned the air force's emphasis on the SAMOS readout system. Accepting its long-term value, Land worried that it was not yet technically feasible and was rather an ultimate objective. He felt that recoverable systems should receive priority because they were likely to be available sooner and generate immediate intelligence products.
A May 1960 meeting between Eisenhower, Kistiakowsky, National Security Advisor Gordon Gray, and General Goodpaster, the president's staff secretary, emphasized the management failures of the SAMOS program and its inability to meet intelligence needs. Kistiakowsky argued that its bad management and poor guidance had led to a program and a satellite that was beyond the practical feasibility of existing technology. Eisenhower questioned how such a program could spin out of control and decided that he wanted sound advice on the satellite effort.
## The President's Panel (May–August 1960)
Facing problems that seemed to be delaying SAMOS and MIDAS indefinitely and a lack of progress in the DISCOVERER/CORONA program, the president took direct action to fix the problem again. The presentation on satellite reconnaissance became a review of the overall program. In May 1960 Eisenhower directed Kistiakowsky and Secretary of Defense Thomas S. Gates Jr. to form a panel to assess the SAMOS program. Consisting of Joseph V. Charyk, Herbert F. York, two staff members of the President's Science Advisory Committee, and Edwin Land's Intelligence Panel, this group was to examine SAMOS's likely intelligence requirements and determine their validity. The study would also look at the technical feasibility of the planned satellite systems "in relation to requirements, development schedules and technical direction of the program, together with the effectiveness of control over the scope and characteristics of the operational systems." 44 The panel would also recommend ways to improve organization and management.
This review played an important role in the future of satellite reconnaissance in the United States. Eisenhower received the results of the study in an off-the-record meeting on August 25, 1960, and it confirmed previous findings that the SAMOS program was facing major problems. Citing the U.S. Intelligence Board's statement "Intelligence Requirements for Satellite Reconnaissance" of July 5, 1960, the report concluded that the technology under development could not meet the intelligence community's requirements.
The board had listed three desiderata. First, it wanted sufficient resolution in images to permit recognition of objects no greater than twenty feet on a side. This meant that the satellites needed a photographic resolution of about five to eight feet. Second, the satellite had to be able to traverse the Soviet Union fully about once every month. This would provide a general search capability, while higher-resolution systems would follow up and do repeat coverage of key locations. The follow-up system, probably with the ability to recognize five-foot objects, was to supply descriptive information on items from earlier images. Third, the board called for a system with better than five-foot resolution, able to reveal the technical characteristics of objects that it photographed. The president's panel found these requirements reasonable in light of the technology then available. Prior to the Intelligence Board's statement, the only guideline for SAMOS development had been the pronouncements of a 1958 ad hoc committee on satellite reconnaissance. However, these requirements were constantly being modified, making effective management very difficult.
SAMOS was a concept far exceeding existing technology. Two different readout camera systems were under development, but their resolution was likely to be very poor. Capable of resolving objects measuring only 250 and 50 feet on a side, respectively, these cameras were far below the Intelligence Board's stipulations, and the slow relay of images to Earth compounded the problem. Studies predicted that the higher-resolution camera would take about five hundred days to cover the Soviet Union and convey the information to Earth, assuming good weather. To make matters worse, SAMOS was not likely to be ready before 1962. Thus at every level the SAMOS program fell short of intelligence requirements. It had long-term promise, but the air force was ignoring more promising technology. The president's panel therefore recommended converting SAMOS to a long-term research program until the technology could catch up to the concept.
The panel felt that film recovery was the only workable solution. Believing that it held great promise to deliver both very high-resolution imagery and faster coverage of the Soviet Union earlier than SAMOS, it urged reorientation of the program toward this technology. SAMOS was already developing a film-return system (E-5 and E-6). Defying orders (when CORONA began), the air force had refused to delete them from their development plans. The panel's report noted that SAMOS was neglecting film return in favor of the readout satellite, which could not supply appropriate intelligence. It argued that with effort the recovery system could very quickly produce useful results. Its emphasis on film recovery is interesting, as there is no indication that its members knew about CORONA. They did know about DISCOVERER's problems with payload recovery, even indicating that those difficulties might be soluble. However, in light of their recommendations for SAMOS, it is unlikely that they were aware that DISCOVERER was a film-recovery reconnaissance program.
The main problem was getting the air force to reorient and maintain its focus on such a system, considering its ineffective management to date. The air force had divided control between the ARDC and the Air Material Command, which caused many of the difficulties. To rectify this situation the panel's report provided three solutions, all involving centralized management. The first was the concentration of responsibility and authority within the ARDC, eliminating internal division. The second was creation of a new air force command, which would manage all research and development but would not procure systems, thus eliminating the Air Material Command from the command chain. The third option would have the president remove SAMOS from the air force entirely, placing it within another agency. The report placed greater confidence on the second or third options. Regardless of choice, however, centralization was necessary for early operational capability.
According to Perry, one of the air force's chronic problems was its incessant pursuit of publicity. The panel found that the air force had failed to maximize its efforts on the satellite program because it used the project to gain funding, thereby creating widespread organizational problems. It milked the satellite program for its publicity value but actually stalled work on it to keep research and development funds flowing. For example, on September 23, 1959, the Department of Defense's Office of Public Information announced the transfer of MIDAS and SAMOS to the air force from ARPA. It even described SAMOS as a reconnaissance satellite and MIDAS as an early-warning system against ballistic missiles. The air force's drive to gain public support for its role in space overshadowed the secrecy of satellite reconnaissance. Though it was never a "black" program, release of such information could antagonize the Soviets, thereby endangering SAMOS. Ironically such leaks did help to further mask CORONA by drawing Soviet attention away from the small "scientific" elements of DISCOVERER. Moreover the air force was not complying with the secretary's explicit instructions on SAMOS. According to Col. W. G. King, project office chief for SAMOS, the air force was "deliberately obstructionist."
The program's setbacks and its pursuit of technology decades ahead of the times meant that SAMOS could not fly until at least 1963. The panel's three recommendations concerning management reflected its desire to see a simplified command structure. Eisenhower readily accepted its reasoning and on October 31, 1960, removed all the commands that were causing confusion. The new line of responsibility ran from the general officer in operational control of the whole program up to the secretary of the air force, through a new secretariat-level bureau—the Office of Missile and Satellite Systems, under Air Force Undersecretary Joseph Charyk—that handled administration and liaison. This change removed the program from direct military control, thus rendering it more accountable to the administration and making satellite reconnaissance part of a national intelligence program rather than an air force effort. This shift in turn served the needs of national security, not the desires or demands of a particular service. As for the technology, Eisenhower made it perfectly clear that the film readout system—cornerstone of satellite reconnaissance since 1954—was to receive very low priority but remain a research project under very tight fiscal control. He decided to emphasize film recovery, which promised better resolution and faster development; more advanced systems would emerge as follow-on satellites.
On the same day (August 25, 1960) that the review panel briefed Eisenhower, just before that meeting, the president received a copy of the roll of film from _Discoverer XIV_ , containing the first reconnaissance photos from space. The panel's report of that day led to the creation of the Office of Missile and Satellite Systems, forerunner of the National Reconnaissance Office, which the secretary of the air force ran and whose existence remained a secret until the end of the cold war. This organization took over control of operations and development of reconnaissance satellites and, while not assessing intelligence, provided the materials to the consumers—mainly the CIA and the military—for assessment.
The post- _Sputnik_ period saw major changes to the satellite reconnaissance effort. The original system, WS-117L, experienced two major modifications. First, in late November 1957 it took on a film-return system, which later became a separate unit, CORONA, in February 1958. The need for intelligence informed the decision to run the film-return elements as a distinct satellite effort. Since WS-117L was having problems, the desire to expedite this portion of the program was logical and beneficial in the long term. The resulting CORONA program ran for years and supplied a wealth of information. It was only the first reorganization for the program.
Second, the introduction of ARPA into the management system to eliminate duplication and waste in military research was not as beneficial as some had hoped. It generated a backlash within the air force, which saw ARPA as a threat to its independent programs of research and development. ARPA acted in effect as a middle man between the military and the administration. Because it had its own goals and desires, its interference in development processes and its inability to manage research projects effectively meant that it just added an additional layer of red tape.
Ironically removal of SENTRY/SAMOS from ARPA only created more problems for satellite reconnaissance. The air force, free from ARPA's control, saw return of the program as vindication and a sign to move full-speed ahead on development and deployment. Jumping ahead to system deployment and integration into the various commands, the air force forgot that it had to develop the satellite first. Instead of concentrating on making an effective system, it diverted its energies into creation of accouterments, such as crew training and the creation of the infrastructure to support it. The result was a "cart before the horse" approach that created more confusion than benefit.
Division of the WS-117L system by the end of 1959 into three parts—DISCOVERER, SENTRY (later SAMOS), and MIDAS—effectively ended the original program. Despite the sound logic underlying the decision, the results were not particularly effective. DISCOVERER (really CORONA), which the CIA ran, had a simpler development process and went on to dramatic success. SENTRY, which the air force's development system swallowed whole, became so complicated that its development program lost cohesion. Instead of putting its effort and limited resources into a single satellite reconnaissance effort, the air force worked on six systems plus a huge logistical and support program for them. Thus its efforts to produce a satellite were problematic at best. Seeking more funding, the air force failed to keep its collective and corporate eye on the ball. MIDAS, which has received little discussion in the literature, fell almost to the wayside. Designed to warn of attack, it made little progress until the 1960s. One of the few accounts that discuss the results of MIDAS indicates that there were some serious problems with the program. For example, despite the frequencies that it used, it picked up the sun's heat reflecting from clouds, creating false alarms. Work continued until it evolved into the Defense Support Program's early-warning satellites, which are now in geostationary orbit of roughly 22,300 miles, using twelve-foot-long infrared telescopes to monitor the world for missile launches. It is unfortunate that a great deal of this story is not yet available.
In the end the most telling information concerning the development of satellite reconnaissance remains two simple facts. First, the program that began in 1954 languished because of mismanagement. Focusing too closely on other goals and padding the budget, the air force failed to produce the satellite that the United States required to monitor the Soviets. Seeking a "perfect" satellite, capable of a wide variety of missions, the air force closed its eyes to the mission of acquiring photographic intelligence on the Soviet Union as soon as possible. Aiming too high, the program failed to live up to its goals.
Second, the WS-117L satellite (later SENTRY and then SAMOS) was a concept years ahead of the technology. The slow relay of images, problems with resolution, and the challenges of orbiting a satellite doomed it, and by 1962 it was in mothballs. Although some of its cameras (E-6s) came out of storage for CORONA, it was, in the end, a bust. Ironically CORONA, the "short-term effort" to provide reconnaissance until SAMOS was operational, went on to great success. It was able to do this because it harnessed technology already available in a far simpler way. In addition its management concentrated on accomplishing its aim—orbiting a satellite as soon as possible—and its descendants were still in operation in 1972. The long-term implications of these programs, however, deserve serious attention.
# Epilogue
# WS-117L in Perspective
By the time Eisenhower left the White House, on January 20, 1961, his administration had laid the foundation for U.S. satellite reconnaissance and the space effort for the next forty years. Although not successful in and of itself, the WS-117L program played a pivotal role, and its descendants still orbit the earth today. These satellites have supplied valuable intelligence, allowing the U.S. government to chart a more informed course through the turbulence of the cold war to the present day.
How should we assess the WS-117L program? Was it a success or a failure? The design that was its hallmark from conception—the readout satellite—had only one unit orbiting before cancellation. Does this make the program a failure? The answer is no. The WS-117L system had a major impact on U.S. national security, the course of the cold war, and the future of space exploration.
Ironically it was the Soviets who pushed the right button— _Sputnik_ —to accelerate a program to spy on them. In one instant the Soviets demonstrated that they had rocket capability superior to that of the Americans, that satellites were feasible, and that the Soviets needed monitoring by the United States. The Soviet propaganda effort helped to create the "missile gap" controversy. By demonstrating what appeared to be superior scientific technology, the Soviets shook American confidence. The United States had seen itself as scientifically superior and now was suddenly second best. _Sputnik_ galvanized Americans into action and legitimized the notion of the free use of space by establishing the legal precedent that paved the way for U.S. satellite reconnaissance. Unintentionally the Soviet Union had done the United States a big favor; the end result was acceleration of the WS-117L program.
As the first U.S. satellite program WS-117L broke many theoretical barriers. Its multistage booster configuration and corresponding work relating to orbital mechanics, communications, and alien environments set the stage for manned space flights. Equally groundbreaking were its research on possible satellite payloads, its requirements for weather reconnaissance, and its efforts on communication and navigation, which propelled RAND beyond direct scientific uses and created applications that are now the norm.
WS-117L's work in space and reconnaissance technology was exceedingly ambitious. Not only was it planning to orbit a satellite—before _Sputnik_ the realm of science fiction and dreamers—it also expected to deliver high-resolution photography. The technical challenges were enormous, as the idea of a real-time camera system was years ahead of the technology. The complex system had to perform perfectly in a totally alien environment. Mechanical reliability was therefore deficient, as the necessary technology was still in its infancy.
Despite all the seemingly insurmountable problems WS-117L laid the foundation of every other satellite reconnaissance system in the U.S. arsenal to date. In many respects the APOLLO lunar missions were descendants of RAND's efforts. CORONA, though not part of the initial plans for the program, was the most visible sign of success. From August 1960 to May 1972 CORONA satellites were in orbit during every major foreign policy crisis, feeding vital data to the U.S. government. That CORONA could orbit successfully so soon after _Sputnik_ is testimony to the WS-117L efforts.
Similar to CORONA the MIDAS satellite profoundly influenced military space activities. The goal of the MIDAS program was to produce an early-warning satellite. While not completely successful, MIDAS spawned generations of such satellites, with the Defense Support Program satellite currently orbiting and guarding against missile attacks on the United States. The SAMOS readout system also has descendants orbiting the earth.
The story of the WS-117L satellite also reveals a great deal about Eisenhower's character and administrative style. His goal as president was to preserve the nation and leave it stronger than when he took office. For the good of the nation Eisenhower embraced the U-2 and the spy satellite, and it was his vision that initiated the program. The WS-117L program was of fundamental significance, for it had the potential to help stabilize the cold war.
The original concept of WS-117L was thus achieved, albeit decades after its inception. CORONA was the product of the failure of the air force to develop WS-117L. Only by understanding WS-117L can we put CORONA into the proper context. An offshoot of the original program, it stood on the shoulders of its big brother to succeed.
The world has greatly changed since _Sputnik_ went into orbit, making electronic noises. The satellite forever altered the tranquil U.S. skies. As Americans look upward at the stars at night, they can feel secure in the knowledge that reconnaissance satellites are guarding national security and keeping them safe. The saga of satellite reconnaissance and space exploration continues to evolve, and the next chapter promises to be even better.
# Appendix
# Historiography of Eisenhower and Space Reconnaissance
While most readers probably do not require an in-depth analysis of the historiography relating to space and satellite reconnaissance, some grasp of the problems in the literature can help explain why we know so little about WS-117L. The overall lack of information, and the fragmentary state of what does exist, directly affect our understanding of the birth of U.S. space-based reconnaissance. The history of the subject is difficult to research, in part because it bridges so many historical fields; it does not fit snugly into one area of American history. Rather it involves presidential history and the history of the cold war, U.S. intelligence gathering, and U.S. space science and technology. It is also hard to dig into because of the often confusing differences of opinion among authors.
The core of the problems for researchers rests within three main areas. First, the original spy satellite program falls between the cracks. While most people would expect that the CIA had a major role in its inception, the truth is quite the opposite. Up until the Soviet success with _Sputnik_ , the development of space-based reconnaissance rested squarely with the U.S. Air Force. It was only in February 1958 that the CIA began to take a role, with the creation of the CIA-run CORONA satellite program, an offshoot of WS-117L. The fact that its satellite effort was so successful has also overshadowed the earlier WS-117L program, which failed to produce a working satellite. Second, increased levels of security and secrecy effectively blanketed satellite reconnaissance, preventing the "loss" of information for national security reasons. Third, those few intrepid authors working in the field prior to 1995 had to rely more on speculation and rumor than hard fact. The result was a confusing account of spy satellites' development and characteristics.
The best place to begin an examination of the historical literature on American satellite reconnaissance is with Dwight D. Eisenhower. President during WS-117L's inception and development, he was the most important figure in determining early cold war foreign policy and strategic posture. How have historians treated his presidency? As both general and president Eisenhower led the United States through some crucial transitions: the Second World War, decolonization of European empires, the rise of communism, and the emergence of the national-security state.
The historiography on his administration boasts two separate and clearly defined schools of thought. The "orthodox" approach emerged during Eisenhower's years in the White House and held sway into the 1970s. This school viewed the president in a decidedly unfavorable light, portraying him as an aging hero—inarticulate, unintelligent, bland, and lacking the motivation and political skills to have a major impact on Washington and the world at large. It depicted him as little more than a puppet to John Foster Dulles, his secretary of state, or to various corporate executives and other forces within his administration. These historians admit that the public loved Eisenhower yet argue that he contributed only a sense of security and stability. As Charles C. Alexander argues in _Holding the Line: The Eisenhower Era_ (1975), his two terms were uneventful, dull, and boring. The vibrancy of the Kennedy administration (1961–63) reinforced this interpretation. Measured against his young and active successor, he appeared to fall well short of the ideal chief executive. Historians came to see him as merely a custodian, presiding blandly over a transitional phase between two stronger and more significant periods.
By the early 1980s this interpretation was the subject of serious criticism by a revisionist school. From its humble start in the 1960s, the revisionist challenge began to dominate perceptions by the late 1970s. Since then this school of thought has only consolidated its gains and extended its influence.
In contrast to the orthodox school, revisionists have had access to the monumental document collection at the Eisenhower Presidential Library in Abilene, Kansas. Members of Eisenhower's White House staff were avid diarists and note takers. By the 1970s the declassification process had made large portions of the documentary record accessible to scholars. This included the minutes of all of Eisenhower's National Security Council meetings, his personal diaries, and the Ann Whitman file, which contains a record of Oval Office business and related materials. The records of many of the president's aides augment this hoard. Together they provide the clearest picture of the daily workings of the administration of any American president. With this abundance of documentary evidence, revisionist historians have produced better accounts of the administration and portray the chief executive as the exact opposite of the orthodox interpretation: decisive, intelligent, and perceptive.
There are many examples of revisionist thought, but several works stand out as especially significant. Arthur Larson, _Eisenhower: The President Nobody Knew_ (1968), Herbert S. Parmet, _Eisenhower and the American Crusades_ (1972), Peter Lyon, _Eisenhower: Portrait of the Hero_ (1974), and Douglas Kinnard, _President Eisenhower and Strategy Management: A Study in Defense Politics_ (1977) are all admirable early revisionist studies. These authors reject the old notion of a "do-nothing" president; instead they see him as an active and skilled man with a strong leadership style, who sacrificed political opportunities for self-aggrandizement to preserve the dignity of the office. His strong abilities in foreign and strategic policy particularly impressed them.
The idea of an active leader in control of his administration found masterful development by Fred Greenstein in his watershed essay, "Eisenhower as an Activist President: A Look at New Evidence" (1979–80). Greenstein depicts a "hidden-handed" president who played a major role in his administration, putting his "personal stamp on public policy" through application of "a carefully thought-out conception of leadership to the conduct of his presidency." The difference between him and other presidents rested with the fact that he was not overt about his control. His style of governance, Greenstein argues, involved cultivating the appearance of being above politics and decision making.
Eisenhower's leadership methods reflected his views on command, encouraging debate among his subordinates, gathering of information, and consideration of different viewpoints before he made a final decision. Once he decided on a course of action, he delegated responsibilities to individuals he trusted and respected. Preferring informal influence to open confrontation, he was willing to work via intermediaries to achieve his ends. Dulles and Sherman Adams, the White House chief of staff, are perfect examples. They served as intermediaries between the president and the public (or parts of it). This allowed Eisenhower to control events while seeming to remain aloof from them. It also deflected blame for unpopular decisions. Finally, he emphasized results over publicity. The presidency was not a political game to him but rather an office responsible to the American people. He sought to represent and serve them, with a dignity that transcended the sort of egotism that so often accompanies great power.
Greenstein is not alone in identifying Eisenhower as a savvy and skilled leader. Many historians who examine his policies find him highly competent. The best examples of the revisionist school, however, remain the works of Stephen Ambrose and Craig Allen. Ambrose has become synonymous with Eisenhower studies. In books such as _Ike's Spies_ (1981) and his monumental two-volume biography, _Eisenhower: Soldier, General of the Army, President-Elect, 1890–1952_ (1983) and _Eisenhower: The President_ (1983–84), Ambrose depicts a chief executive in control of his administration and ardently dedicated to peace. He convincingly elaborates and supports the view of a supremely competent keeper of the peace. Ambrose portrays Eisenhower as thoughtful, decisive, and pragmatic, while experimenting with unconventional methods—such as the CIA, the U-2, and covert operations—in order to avoid war while protecting the nation's interests. He details Eisenhower's deep concern about the impact of continuous heavy defense spending on the United States. Fearing both for civil liberties and economic stability, Eisenhower sought to decrease world tensions in order to preserve American domestic institutions. Ambrose's biography epitomizes the revisionist school in its meticulous attention to detail concerning Eisenhower's part in the creation of policy and the daily affairs of the White House.
Craig Allen's _Eisenhower and the Mass Media: Peace, Prosperity and Prime-Time_ TV (1993) suggests something of the diversity among revisionists. Examining Eisenhower's use of the media to convey his message to the American people, Allen portrays an astute political manipulator who managed the press with consummate skill in order to garner public support. Political communication was vital for Eisenhower, and he learned to use the media effectively to get his message out. This interpretation confirms the revisionist view of a chief executive acutely aware of events around him and extraordinarily adept at channeling the flow of events in ways that advanced his own agenda.
The great diversity of revisionist methods and materials has resulted in a much better understanding of a dynamic leader, but the complexity of the story seems to preclude interpretive closure. Nonetheless the revisionist perspective has gained wide acceptance. Eisenhower the active president certainly sets up the historical context for any study of the programs and policies that emerged during his tenure.
The main limitation remains Eisenhower's strong tendency to maintain secrecy at all costs. Even when use of such secrets could have been politically advantageous, he refused to reveal information. He carefully edited his own memoirs to hide programs that he felt should remain secret. Thus WS-117L is glaringly absent from the historical record. While it is clear that he hid the original satellite reconnaissance program from public scrutiny, his overriding concerns and his struggles certainly give us an understanding of the political context in which it appeared and developed. He worried deeply about the welfare of his nation and its long-term security. To understand WS-117L we must explore other avenues of inquiry.
It is somewhat ironic that the one agency most people associate with spy satellites and intelligence gathering, the CIA, had so small a role in the original satellite program. It joined only in the wake of _Sputnik_ , and most histories of the CIA at best glance over the WS-117L program. Many authors do not even recognize that it existed. For example, Ray S. Cline's pioneering _Secrets, Spies and Scholars: Blueprint of the Essential_ CIA (1976) reveals nothing about the program. Other treatments, such as Victor Marchetti and John D. Marks, _The_ CIA _and the Cult of Intelligence_ (1974), provide only a cursory discussion of satellite reconnaissance. These texts offer no detailed analysis and little sense of the program's size or importance. In fact they imply that the CORONA program was the original American effort in satellite reconnaissance. WS-117L and the air force work prior to 1958 have practically vanished from the historical record.
In recent years works on the CIA have provided more information on satellite reconnaissance within the framework of intelligence needs. While giving some indication of the field's origins, these studies do not provide details concerning the program from 1946 to 1958. Rather they focus on the operational program, CORONA, and the CIA's role in the use of intelligence. The unavailability of declassified information, however, has meant that the accounts in these books are sketchy at best. John Ranelagh's _The Agency: The Rise and Decline of the_ CIA (1987) is typical. It discusses the WS-117L program and the decision to give control of satellite systems to the CIA. Unfortunately it presents the entire WS-117L system as a CIA program from the start. It considers neither the program's development before 1958 nor its details or differences from other satellite efforts.
More recent works on the CIA and the U.S. intelligence community also fall short. This is certainly the case with Christopher Andrew's _For the President's Eyes Only_ (1995) and Rhodri Jeffreys-Jones and Christopher Andrew's _Eternal Vigilance? 50 Years of the_ CIA (1997). _For the President's Eyes Only_ provides a general history of the CIA, tracing American intelligence gathering since the presidency of George Washington. While appreciating Eisenhower's pivotal role in developing photographic intelligence, Andrew does not discuss satellite reconnaissance in any meaningful way. He focuses instead on covert operations and other aspects of the CIA's role in the cold war. Nor does _Eternal Vigilance_ provide any significant information about Eisenhower's satellite efforts. Concentrating on the post-1960 period, this collection refers to the development of satellite reconnaissance only in passing and as it relates to the career of Richard Bissell. Although it notes the importance of satellites as a means of gathering intelligence, it offers no history of their development. Where did CORONA come from? The book offers no clues. As with earlier works, moreover, there is no mention of pre-1958 developments. Jeffrey T. Richelson's _The U.S. Intelligence Community_ (1999) is equally disappointing on the history of satellite reconnaissance. It considers the current state of satellite technology but provides only a two-page synopsis of developments between 1946 and 1958. These histories of the agency make no mention of WS-117L prior to 1958.
The scarcity of information on WS-117L is attributable partly to intense secrecy in relation to intelligence matters and the cold war. Authorities kept top-secret military satellite projects away from public view out of concern for the Soviet response to satellite surveillance of their territory. To flaunt their development seemed an unnecessary provocation, especially once the program started to produce intelligence. If the capabilities of American photographic satellites become common knowledge, the Soviets would be able to counter their effectiveness. Concealing the program was critical to preserving its usefulness. Thus, starting with the CORONA program in 1958, public discussion of the subject was virtually nonexistent, and succeeding presidents have blanketed these programs with ever greater levels of secrecy. Unfortunately the air force leaked or made publicly available some information before CORONA and the security blackout, but the Kennedy administration and its successors were very good at limiting subsequent knowledge of satellites. Thus any account emerged more from hint and rumor than from hard fact.
Works on satellite reconnaissance before declassification began to appear in the 1970s and continued sporadically until 1990. Using rumors, leaked information, and memoirs, authors tended to write impressionistic books that were confusing and lacking in both detail and supporting documentation. A mixture of popular historical works on the space program and more specialized studies, this literature does provide at least two common reference points: recognition that there was a military satellite program aiming to provide photographic intelligence on the Soviet Union and that it began during the Eisenhower administration. Most accounts mention specific elements of the "program"—the role of RAND, for example, and some specific project names: SENTRY, MIDAS, and DISCOVERER—but there is no consensus on details of programs. Ironically these projects had no connection with WS-117L until after _Sputnik_ , when the space program underwent fundamental reorientation; the pre- _Sputnik_ efforts effectively remained secret. Such accounts only add to the confusion about the first satellite program. Perhaps the most influential of the early pre-declassification offerings are Philip J. Klass, _Secret Sentries in Space_ (1971), and Anthony Kenden, "U.S. Reconnaissance Satellite Programmes" (1978). These studies introduce the problems and ask the questions that would preoccupy a subsequent generation of researchers. Following in their wake, a number of authors, including Curtis Peebles, Robert Divine, and William Burrows, have expanded on the story to the extent that the information would allow.
Of all the people writing prior to declassification, the most important is Walter A. McDougall. His monumental work, _. . . The Heavens and the Earth: A Political History of the Space Age_ (1985), was the first successful account of both the military and the civilian space programs within the political context of the post- _Sputnik_ space race. While admitting the difficulties that security restrictions imposed, McDougall ambitiously attempts to synthesize the divergent materials available. His discussions of cold war nuclear strategic planning and U.S. policy concerning spending on defense and research before Eisenhower provides a solid historical foundation for an understanding of the space race and thus of the reconnaissance satellite program. The greatest portion of his study, however, focuses on the politics of space exploration, research, and the civilian space effort. The satellite reconnaissance program is not central to McDougall's account.
The difficulties in penetrating secrecy all began to change in 1995. On February 22 President Clinton issued Executive Order 12951, releasing satellite imagery associated with CORONA, ARGON, and LANYARD. This executive order, coupled with the Freedom of Information Act (which allows for the review and declassification of documents on request), has begun to yield a more complete picture of CORONA, which the CIA began in 1958. Kevin Ruffner's CORONA _: America's First Satellite Program_ (1995) combines documents and historical commentary from the CIA's history staff and the Center for the Study of Intelligence. It consists almost entirely of declassified materials for the May 1995 conference "Piercing the Curtain: CORONA and the Revolution in Intelligence" and provides a wealth of technical information about the CORONA program. Further, it includes material from the Committee on Overhead Reconnaissance, correspondence among many of the key players in the development of satellite technology, important memoranda relating to various aspects of the program, and intelligence reports based on satellite reconnaissance. Kenneth E. Greer's long-classified article, "CORONA," which had appeared only in the CIA's restricted journal, _Studies in Intelligence_ (1973), makes this work particularly valuable. Unfortunately, while the information on CORONA is illuminating at many levels and is helping to provide a clear account of that satellite effort, WS-117L still remains a virtual mystery.
The confusing picture of the WS-117L satellite program began to change only with requests under the Freedom of Information Act and the presidential declassification order of 1995. The act permitted declassification of some important materials, including most of the official histories on satellite reconnaissance that Robert Perry compiled in the early 1960s, the complete Technological Capabilities Panel (Killian) Report (1955), and several of the most highly classified RAND studies on satellites, such as Project FEEDBACK. This slow process has begun to uncover new aspects of the program and supporting contextual information. Yet the available material is still far from complete. Recent trends in security classification also indicate that it will be harder and harder to gain access to this material. Although what is open to access has yielded many answers, it has also provoked a host of new questions.
Authors working since declassification obviously have had a marked advantage. Declassified documents released in conjunction with articles relating to the American space effort have greatly increased our knowledge of the civilian program. The two-volume compilation by John M. Logsdon, _Exploring the Unknown: Selected Documents in the History of the U.S. Civil Space Program_ (1995, 1996), is an example of the impact of released documents on the historiography. Articles such as R. Cargill Hall, "Origins of U.S. Space Policy: Eisenhower, Open Skies, and Freedom of Space," and Dwayne Day, "Invitation to Struggle: The History of Civilian-Military Relationships in Space," make these volumes very useful. For example, Day traces the struggle between the air force and the army over control of satellites, which caused both services to lose focus on space matters until _Sputnik_ forced change. Using numerous documents, including RAND studies and various National Security Council papers, such as NSC 5520, "Draft Statement of Policy on U.S. Scientific Satellite Program" (May 20, 1955), these works mark a major step forward. They place the history of space and satellites within a wider context that helps explain the reconnaissance program.
The _Exploring the Unknown_ anthology and Ruffner's compilation on the CORONA project are the most significant additions to the early history of the satellite program to date and confirm the importance of the RAND studies. They do not, however, go into great detail concerning WS-117L or the reasons for its failure. Like the work on satellite reconnaissance before declassification, they reveal a few pieces of the puzzle but do not provide an overall picture of what the program entailed or of the forces affecting its development and failure. Their greatest contribution remains the documents they make available to the reader.
Some recent authors have expanded our horizons. Three books—Curtis Peebles, _The Corona Project: America's First Spy Satellites_ (1997); Dwayne A. Day, John M. Logsdon, and Brian Latell, _Eye in the Sky: The Story of the Corona Spy Satellites_ (1998); and Philip Taubman, _Secret Empire: Eisenhower, the_ CIA _, and the Hidden Story of America's Space Espionage_ (2003)—all examine satellite reconnaissance during the Eisenhower years. They present the most coherent account of the development of satellite reconnaissance to date. Taking into account the role of RAND and various government-sponsored studies on intelligence, they concentrate on CORONA, the operational CIA satellite system, not on the WS-117L program.
Discussing the period 1945–57 only briefly, these volumes concentrate on the operational CORONA program. For both Peebles and Day the twin problems of lack of funds and technical issues slowed WS-117L development. RAND's calls for a recoverable satellite system, _Sputnik_ 's success, and the need for program acceleration forced the development of CORONA. By 1961 CORONA's success and WS-117L's problems resulted in cancellation of the latter. Taubman's _Secret Empire_ summarizes much of the historiography relating to WS-117L and CORONA. Focusing on Eisenhower and attempts to gain intelligence on the Soviet Union from 1946 to 1957, the writer concentrates on the U-2 and CORONA programs and mentions WS-117L in passing, mainly as background for the founding of CORONA in 1957–58. _Secret Empire_ nonetheless places these programs within the Eisenhower administration.
As is clearly evident, recent treatments of the subject fail to illuminate the air force's WS-117L program. They offer some surface information, such as Lockheed's role as primary contractor, but nothing of the story's particulars—personalities, problems, internal politics. Because many scholars emphasize the successful CORONA satellite effort WS-117L remains an enigma.
The relative obscurity of WS-117L, even in the historiography that one would expect to take heed of this program, creates a number of problems. Most important, while we know a great deal about Eisenhower's concerns for the safety of the nation, we know very little about how satellite reconnaissance factored into his efforts to address these issues. His need for intelligence appears in the literature of both the CIA and satellite technology, but the development of a spy satellite system seems to exist in a vacuum. Why did Eisenhower support a satellite reconnaissance program before _Sputnik_? Was it solely because he wanted photographs of the Soviet Union (as the CORONA historians would lead us to believe), or did broader national interests drive his decision? The literature does not make the link between overhead space reconnaissance and Eisenhower's underlying concerns about economic health as the foundation of national security.
The pre-CORONA foundations of satellite reconnaissance—namely the WS-117L program—receives almost no attention. WS-117L is largely uncharted territory. Several historians look at efforts by RAND and the Defense Department to spur development of satellite intelligence, but this effort is no more than a starting point. Thus this book goes beyond that frame of reference into terra incognita. It uses the WS-117L program as a vehicle to show that Eisenhower was a visionary and activist president who saw spy satellites as more than a means to solve a difficult problem in intelligence collection. He thought of them as a way to generate knowledge that would facilitate his management of the cold war. This volume fills the void that surrounds the WS-117L program. It establishes the parameters of the air force's satellite effort, thus filling a large gap in the literature on the cold war, Eisenhower, and satellite reconnaissance.
# Notes
In an effort to provide both proper citation and a consistent flow of information for the reader, this book uses the Kate Turabian style for citation. Unfortunately, for some document collections—specifically the materials at the Eisenhower Library—there is no universally accepted method of citation. Due to the vast quantity of materials that I use from that source, and the length of full citations for some of the collections, I provide the following list of short forms. Unless I indicate otherwise, all special collections documents are from the Dwight D. Eisenhower Presidential Library, Abilene, Kansas.
AWF, Admin. Series: Dwight D. Eisenhower, Papers as President of the United States, 1953–61 (Ann Whitman File), Administration Series
AWF, A. W. Diary Series: Dwight D. Eisenhower, Papers as President of the United States, 1953–61 (Ann Whitman File), Ann Whitman Diary Series
AWF, DDE Diary Series: Dwight D. Eisenhower, Papers as President of the United States, 1953–61 (Ann Whitman File), DDE [Eisenhower] Diary Series
AWF, NSC Series: Dwight D. Eisenhower, Papers as President of the United States, 1953–61 (Ann Whitman File), NSC Series
DDE Personal Diary: Dwight D. Eisenhower, Diaries, 1935–38, 1942, 1948–53, 1966, 1968, 1969
Hagerty Diary: James C. Hagerty, Press Secretary to the President, Papers, 1953–61, Diary Entries
Harlow Records: Bryce N. Harlow, Records, 1953–61
Hazlett Papers: Edward E. "Swede" Hazlett, Papers, 1941–65
PSAC Records: U.S. President's Science Advisory Committee, Records, 1957–61
Quarles Papers: Donald A. Quarles, Papers, 1952–59
RG 59, GRDS: RG 59, General Records of the Department of State, National Archives and Records Administration II, College Park, Maryland
RG 340, ROSAF: RG 340, Records of the Office of the Secretary of the Air Force, General Correspondence—Secret, Confidential, National Archives and Records Administration II, College Park, Maryland
SPI, GWU: Space Policy Institute, George Washington University, Washington DC
SSB/GCSWS: Spy Satellites Box (copies; originals in the National Security Archive), Gregg Centre for the Study of War and Society, University of New Brunswick, Canada
WHO, NSCSP, Exec. Sect. Subject File Series: White House Office, National Security Council Staff, Papers, 1948–61, Executive Secretary's Subject File Series
WHO, OSANSA, NSC Series, Briefing Notes Subseries: White House Office, Office of the Special Assistant for National Security Affairs, Records, 1952–61, NSC Series, Briefing Notes Subseries
WHO, OSANSA, NSC Series, Policy Papers Subseries: White House Office, Office of the Special Assistant for National Security Affairs, Records, 1952–61, NSC Series, Policy Papers Subseries
WHO, OSANSA, NSC Series, Status of Projects Subseries: White House Office, Office of the Special Assistant for National Security Affairs, Records, 1952–61, NSC Series, Status of Projects Subseries
WHO, OSANSA, NSC Series, Subject Subseries: White House Office, Office of the Special Assistant for National Security Affairs, Records, 1952–61, NSC Series, Subject Subseries
WHO, OSANSA, Spec. Assist. Series, Chronological Subseries: White House Office, Office of the Special Assistant for National Security Affairs, Records, 1952–61, Special Assistant Series, Chronological Subseries
WHO, OSAST: White House Office, Office of the Special Assistant for Science and Technology, Records, 1957–61
WHO, OSAST (Killian/Kistiakowsky): White House Office, Office of the Special Assistant for Science and Technology (James R. Killian and George B. Kistiakowsky), Records, 1957–61
WHO, OSS, Minnich Series: White House Office, Office of the Staff Secretary, Records, 1952–61, L. Arthur Minnich Series
WHO, OSS, SUB, ALPHA: White House Office, Office of the Staff Secretary, Records, 1952–61, Subject Series, Alphabetical Subseries
WHO, OSS, SUB, DoD Subseries: White House Office, Office of the Staff Secretary, Records, 1952–61, Subject Series, Department of Defense Subseries
###### 1. On the Cold War
"President Eisenhower Press Conference on the U-2 Incident and Summit Conference," in Schlesinger, _Dynamics_ , 2:625.
Louis Ridenour, "There Is No Defense," in Freedman, _Evolution_ , 33.
1. The literature on the debate over the reasons for the Pearl Harbor failure is massive. Some of the best works on the matter are Wohlstetter, _Pearl Harbor_ ; Prange, _At Dawn We Slept_ ; Clausen and Lee, _Pearl Harbor_. For a discussion of the impact it had on the United States, see Day et al., _Eye in the Sky_ , 2–3; Kaplan, _Wizards_ , 9–10, 16–39; Bissell et al., _Reflections_ , 92.
2. The various investigations are Roberts Commission (December 22, 1941–January 23, 1942); the Hart Inquiry (February 22, 1944–June 15, 1944); the Army Pearl Harbor Board (July 20, 1944–October 1944); the Navy Court of Inquiry (July 24, 1944–October 19, 1944); Hewitt Investigation (May 15, 1945–July 11, 1945); the Clausen Investigation (November 23, 1944–September 12, 1945); the Clarke Investigation (September 14–16, 1944, July 13–August 4, 1945); Joint Congressional Committee Investigation (November 15, 1945–July 15, 1946). For the most thorough historian's assessment of the Pearl Harbor attack, see Prange, _At Dawn We Slept_ , 818–21, 841–42.
3. Kaplan, _Wizards_ , 33.
4. Freedman, _Evolution_ , 25–30; York, _Arms_ , 5–8.
5. DDE Personal Diary, January 27, 1949, box 1, file [December 13, 1948–March 5, 1951], 1; Aronsen, "Seeing Red."
6. Freedman, _Evolution_ , 33–35.
7. Starting in 1945 Brodie wrote some of the most important works in the field and is considered by many to be the key thinker in this area. See Brodie, _The Atomic Bomb and American Security_ ; Brodie, _Absolute Weapon_ ; Brodie, _Strategy in the Missile Age_ ; Brodie, _War and Politics_ ; Brodie and Brodie, _From Crossbow to H-Bomb_.
8. Kaplan, _Wizards_ , 24–30. See also Brodie, _War and Politics_ ; Brodie, _Absolute Weapon_.
9. On the OSS and its dismantling, see the documents in U.S. Department of State, _Foreign Relations of the United States_ [hereafter FRUS] _1945–1950_ , 1–2, 15–19, 20–23, 44–46, 89–94, 108–11; Andrew, _For the President's Eyes Only_ , 156–61.
10. Presidential Directive on Coordination of Foreign Intelligence Activities, FRUS _: Emergence_ , 178–79.
11. For a discussion of the CIG and the creation of the CIA, see the documents in FRUS _: Emergence_ , 19–21, 166–69, 316–17, 329–31, 333–34, 364–65, 518–19, 525–33, 586; Andrew, _For the President's Eyes Only_ , 168–71; Gaddis, _United States_ , 306–7; Leffler, _Preponderance_ , 150, 175.
12. "National Intelligence Authority Directive No. 2," February 8, 1946. See also "Central Intelligence Group Administrative Order No. 3," April 19, 1946; "National Intelligence Authority Directive No. 5," July 8, 1946, all in FRUS _: Emergence_ , 332, 343–44, 391–92.
13. "Memorandum from the Chief of the Interdepartmental Coordinating and Planning Staff, Central Intelligence Group (Edgar) to the Assistant Director for Reports and Estimates (Huddle)," January 13, 1947; "Memorandum From the Chief of the Intelligence Staff, Central Intelligence Group (Montague to the Assistant Director for Reports and Estimates (Huddle)," January 29, 1947, all in FRUS _: Emergence_ , 458–87.
14. "Memorandum by the Director of Central Intelligence (Souers), and Enclosure," April 29, 1946; "Memorandum From the Director of Central Intelligence (Souers) to the National Intelligence Authority," June 7, 1946; "National Intelligence Authority Directive No. 7," January 2, 1947, all in FRUS _: Emergence_ , 345–47, 358–63, 478–79. Leffler, _Preponderance_ , 12. Freedman, _U.S. Intelligence_ , 1.
15. Central Intelligence Agency, "Threats to the Security of the United States," ORE 60-48, September 28, 1948, pp. 1–11; "CIA Research Reports: The USSR 1946–1976," Harriet Irving Library (hereafter HIL), University of New Brunswick, microfilm, reel 1, frame 0387.
16. Central Intelligence Agency, "Soviet Capabilities for the Development and Production of Certain Types of Weapons and Equipment"; "CIA Research Reports: The USSR 1946–1976," ORE 3/1, October 31, 1946, HIL, microfilm, reel 1, frame 0017.
17. Shulsky, _Silent Warfare_ , 75–78.
18. "Soviet Capabilities," frame 0017.
19. Aronsen, "Seeing Red," 112, 115. "Soviet Capabilities," frame 0017–0018.
20. "Threats to the Security of the United States," pp. 1–11.
21. Ziegler and Jacobson, _Spying_ , 16, 23–24.
22. Ziegler and Jacobson, _Spying_ , 21–33.
23. Freedman, _Evolution_ , 25–29; Kaplan, _Wizards_ , 33–39; Day et al., _Eye in the Sky_ , 2–4; York, _Arms_ , 6–8.
24. Rear Adm. R. H. Hillenkoetter, director of Central Intelligence, memorandum for the president, "Estimate of the Status of the Russian Atomic Energy Project," July 6, 1948, CIA Research Reports: USSR 1946–1976, HIL, microfilm, reel 1, frame 0250.
25. "Threats to the Security of the United States," pp. 1–11.
26. Kaplan, _Wizards_ , 39. For further information, see Burrows, _This New Ocean_ , 135–36; Ziegler and Jacobson, _Spying_ , 203.
27. "Estimate of the Effects of the Soviet Possession of the Atomic Bomb upon the Security of the United States and upon the Probabilities of Direct Soviet Military Action," ORE 91-49, April 6, 1950, CIA Research Reports, The Soviet Union 1946–1976, HIL, microfilm, reel II, frame 0059, pp. 3–5.
28. "The Effect of the Soviet Possession of Atomic Bombs on the Security of the United States," ORE 32–50, June 9, 1950, CIA Research Reports, The Soviet Union 1946–1976, HIL, microfilm, reel II, frame 0120, p. 2.
29. "The Effect of the Soviet Possession," pp. 2–3.
30. Koch, _Selected Estimates_ , 165–73, 189–92.
31. "Basic National Security Policy, NSC 5440, Section A: Estimate of the Situation," December 14, 1954, WHO, OSANSA, NSC Series, Policy Papers Subseries, box 14, file NSC 5440/1—Basic National Security Policy; "NSC 5501" (undated), WHO, OSANSA, NSC Series, Policy Papers Subseries, box 2, file NSC 112/1—Disarmament (3); Koch, _Selected Estimates_ , 165–73, 189–92.
32. Leffler, _Preponderance_ , 9–10, 81–89, 96, 246–51, 331–33, 361, 364–69; LaFeber, _America, Russia, and the Cold War_ , 85–94, 101–7.
33. "Memorandum for the Record," October 3, 1956, WHO, OSS, SUB, ALPHA, box 14, file Intelligence Matters (2); "148th Meeting of NSC," June 4, 1953, AWF, NSC Series, box 4, file 148th Meeting of NSC, June 4, 1953. For an excellent discussion of Soviet espionage in the United States, see Sibley, "Soviet Industrial Espionage."
34. "Discussion at the 148th Meeting of the National Security Council," June 4, 1953, AWF, NSC Series, box 4, file 148th Meeting of NSC, June 4, 1953; Aid, "The National Security Agency," 29–31; Albats, KGB, 37–50, 55–58; R. Cargill Hall, "Origins of U.S. Space Policy: Eisenhower, Open Skies, and Freedom of Space," in Logsdon et al., _Exploring the Unknown_ , 1:215–19.
35. Aid, "The National Security Agency," 29–31; Aid and Wiebes, "Introduction."
36. Growing out of the Sky Hook balloon research program of the 1940s, projects like Moby Dick, GOPHER, and GENETRIX (WS-119L) are all similar in nature. They all called for the use of balloons to carry cameras and other equipment either for scientific research of the upper atmosphere or in support of intelligence operations against the Soviet Union. The culmination of these efforts was the 1956 GENETRIX program, a clearly defined photographic reconnaissance effort. "Memorandum of Conference with the President," December 28, 1955, WHO, OSS, SUB, ALPHA, box 14, file Intelligence Matters (1); Gaddis, "Learning to Live with Transparency: The Emergence of a Reconnaissance Satellite Regime," in _The Long Peace_ , 197. For a well-documented account, see Peebles, _Moby Dick Project_ , 1–4, 28–99.
37. For a detailed discussion of the importance of the NSA and SIGINT, see Aid, "The National Security Agency"; Aid and Wiebes, "Introduction"; Bamford, _Body of Secrets_.
38. Richelson, _American Espionage_ , 42–48; Grose, _Gentleman Spy_ , 314–22.
39. Bernstein, "The Challenges and Dangers," 78–81; May, _American Cold War Strategy_.
40. Kaplan, _Wizards_ , 56–58; Schwiebert, _A History of the U.S. Air Force Ballistic Missiles_ , 42–44; Gorn, _Prophesy Fulfilled_.
41. "Index of ICBM Development," AWF, Admin. Series, box 17, file Guided Missiles 1958; York, _Arms_ , 8–10; Kaplan, _Wizards_ , 74–80, 85–110. See also Schilling, "The H-Bomb Decision"; Rhodes, _Dark Sun_ , 250–53, 298, 394, 397–98, 400, 418–19.
42. Ambrose, _Eisenhower: The President_ , 257.
43. "The President's Appointments, 27 March 1954," AWF, A. W. Diary Series, box 1, file ACW Diary, March 1954 (1). See also Peebles, _Corona Project_ , 18–19; Hall, "The Eisenhower Administration," 62–63.
44. Immerman, "Confessions," 331; Day et al., _Eye in the Sky_ , 29; Cook, _Declassified Eisenhower_ , 164.
45. "Report to the National Security Council by the Special Evaluation Subcommittee of the National Security Council," NSC 140/1, May 18, 1953, in FRUS _1952–1954: National Security Affairs_ , vol. 2, pt. 1, 332–34; ANSC 140/1, Summary Evaluation of the Net Capabilities of the USSR to Inflict Direct Injury on the United States up to July 1, 1955, May 18, 1953, WHO, OSANSA, NSC Series, Policy Papers Subseries, box 3, file NSC 140/1—Special Evaluation Subcommittee of the NSC.
46. "U.S. Policy on Control of Armaments (NSC 112)," AWF, NSC Series, box 6, file 236th Meeting of NSC, February 10, 1955.
47. U.S. "Atoms for Peace" proposal, address by President Eisenhower to the General Assembly, December 8, 1953, in U.S. Department of State, _Documents on Disarmament_ , document 92: 396.
48. Ambrose, _Ike's Spies_ , xi; Peebles, _Shadow Flights_ , 3–4; Hall, "The Eisenhower Administration," 61.
49. For a discussion of the origins of the NIE process, see Freedman, _U.S. Intelligence_ , 30–31; Steury, _Sherman Kent and the Board of National Estimates_ , vii, x–xvi.
50. Steury, _Intentions and Capabilities_ , viii; Freedman, _U.S. Intelligence_ , 31–32.
51. Letter, Brig. Gen. W. M. Burgess, deputy chief of staff/intelligence, to Brig. Gen. W. M. Garland, Air Technical Intelligence Center, May 26, 1953, SSB/GCSWS; "Briefing on Significant World Developments Affecting U.S. Security," February 17, 1954, AWF, NSC Series, box 5, file 185th Meeting of NSC, February 17, 1954.
52. Memorandum from HQ Air Defense Command to director of intelligence, Headquarters USAF, July 17, 1953, SSB/GCSWS.
53. By the time of its release the SNIE was already dated. In June 1953 the CIA increased the number of TU-4 bombers on hand to 1,600 based on a projected higher production rate of thirty-five planes per month. See "Memorandum for: Executive Secretary, NSC, Subject: CIA Comments on NSC 140/1," June 1, 1953, WHO, OSANSA, NSC Series, Policy Papers Subseries, box 3, file NSC 140/1—Special Evaluation Subcommittee of the NSC.
54. "Soviet Capabilities for Attack on the United States through Mid 1955," July 31, 1953, CIA Research Reports: The Soviet Union 1946–1976, HIL, microfilm, reel 2, frame 0649, pp. 1–3; "Appendix A: Elements of the World Situation and Outlook, the Soviet Threat through Mid-1959, NSC 5422/1," July 26, 1954, WHO, OSANSA, NSC Series, Policy Papers Subseries, box 11, file NSC 5422/2—Guidelines under NSC 162/2 for FY 1956 (2); "Memorandum of Discussion at the 157th Meeting of the National Security Council," July 30, 1953, AWF, NSC Series, box 4, file 157th Meeting of NSC, July 30, 1953.
55. "Soviet Capabilities for Attack on the United States through Mid 1955," July 31, 1953, microfilm, reel 2, frame 0649, pp. 4–5.
56. Ambrose, _Ike's Spies_ , 253; Herken, _Cardinal Choices_ , 87.
57. "Report to the President of the United States by the Chief of Staff, USAF, Subject: Visit of the U.S. Air Delegation to the USSR, 23 June–1 July 1956," AFW, Admin. Series, box 1, file Air Force, Department of (1).
58. Estimates of Soviet aircraft production rates could not be verified by any means available. These numbers were highly subjective and easily manipulated. A May 1953 comparison of U.S. and Soviet production of aircraft shows marginal differences in overall production whether examining the current predicted rate or an all-out production effort. See "Memorandum for Secretary Wilson," May 26, 1953, AWF, Admin. Series, box 1, file Aircraft Power.
59. "Discussion at the 172nd Meeting of the National Security Council," November 23, 1953, AWF, NSC Series, box 5, file 172nd Meeting of NSC, November 23, 1953.
60. Aronsen, "Seeing Red," 113–14, 118–19, 121–23.
61. "Memorandum for the Record," October 3, 1956, WHO, OSS, SUB, ALPHA, box 14, file Intelligence Matters (2); "The Reminiscences of Andrew J. Goodpaster," April 10, 1982, 18–20, Dwight D. Eisenhower Presidential Library.
62. Kaplan, _Wizards_ , 85–110; Richelson, _America's Secret Eyes in Space_ , 9–10.
63. "Significant World Developments Affecting U.S. Security," April 29, 1954, AWF, NSC Series, box 5, file 194th Meeting of NSC, April 29, 1954; "Briefing on Significant World Developments Affecting U.S. Security," May 13, 1954, AWF, NSC Series, box 5, file 197th Meeting of NSC, May 13, 1954; "Significant World Developments Affecting U.S. Security," April 28, 1955, AWF, NSC Series, box 6, file 246th Meeting of NSC, April 28, 1955; NIE 11-4-54, "Soviet Capabilities & Probable Courses of Action through Mid 1959," September 14, 1954, in Steury, _Estimates_ , 1–37.
64. NIE 11-5-54, "Soviet Capabilities and Main Lines of Policy through Mid-1959," in Koch, _Selected Estimates_ , 201–11. See also memorandum by the acting special assistant to the secretary of state for intelligence (Howe) to the acting secretary of state, subject: SNIE 11-2-54: Soviet Capabilities for Attack on the U.S. through 1957, March 1, 1954, in FRUS _: National Security Affairs_ , vol. 2, pt. 1, 634–737.
65. For example, the White House received telegrams and phone calls pushing for rapid increases in spending on air power. See "Telegram from Congressman Samuel W. Yorty (Republican, California) to DDE," October 4, 1953, AWF, A. W. Diary Series, box 3, file ACW Diary, October 1954 (5).
66. The figures for these bomber numbers come from the Natural Resources Defense Council, which has published a great deal of information regarding nuclear force size since 1984. Two of the most important working papers dealing with nuclear data are "U.S.-USSR/Russian Strategic Offensive Nuclear Forces 1945–1996" (January 1997) and "U.S. Inventories of Nuclear Weapons and Weapon-Usable Fissile Material" (revised September 26, 1995), Natural Resources Defense Council, "Archive of Nuclear Data," <http://www.nrdc.org/nuclear/nudb/datab7.asp#foot1>.
67. NIE 11-3-55 (May 1955) and NIE 11-56 (March 1956) predicted forty Bison bombers by January 1, 1956, and eighty by July 1, 1956. In NIE 11-4-56 (August 1956) this number was dropped to thirty-five. The decrease was due to slower Bison production. Observations of an aircraft plant in Moscow and new data on Soviet long range aviation bases supported this. See "Memorandum of Discussion at the 280th Meeting of the National Security Council," March 22, 1956, AWF, NSC Series, box 7, file 280th Meeting of NSC, March 22, 1956; "Memorandum for Brig. General Goodpaster from DCI Dulles," March 1, 1957, WHO, OSS, SUB, ALPHA, box 14, file Intelligence Matters (3); "Memorandum for the Record," February 12, 1959, WHO, OSS, SUB, ALPHA, box 15, file Intelligence Matters (8); Freedman, _U.S. Intelligence_ , 66–67; Peebles, _Shadow Flights_ , 136–37, 149.
68. "Soviet Gross Capabilities for Attack on the U.S. and Key Overseas Installations and Forces through Mid-1959 (NIE 11-56)," in Steury, _Intentions and Capabilities_ , 5–6, 9–19; "Memorandum of Discussion at the 409th Meeting of the National Security Council," June 4, 1959, AWF, NSC Series, box 11, file 409th Meeting of NSC, June 4, 1959; "Memorandum for the President, Subject: Supplemental Appropriations for FY '57," March 29, 1956, AWF, Admin. Series, box 9, file Budget 1957 (2); diary entry for April 4, 1956, and "Memorandum for the Record," April 4, 1956, AWF, A. W. Diary Series, box 8, file Apr. '56 Diary-ACW (2); diary entry for April 26 and 27, 1956, AWF, A. W. Diary Series, box 8, file Apr. '56 Diary-ACW (1); "Address by General Thomas D. White, Vice Chief of Staff, USAF to the General Electric Dinner," February 9, 1956, WHO, OSS, SUB, ALPHA, box 6, file Military Program (Missiles) [January 1956–September 1957] (2).
69. SNIE 11-6-57, "Soviet Gross Capabilities for Attacks on the Continental U.S. in Mid-1960," was produced in October 1956 (in Steury, _Intentions and Capabilities_ , 39–46, 47–56). See also Koch, _Selected Estimates_ , 215–21; Freedman, _U.S. Intelligence_ , 21–22; Herken, _Cardinal Choices_ , 87.
70. "Index of ICBM Development," AWF, Admin. Series, box 17, file Guided Missiles 1958; Schwiebert, _History_ , 42–46; Neufeld, _Development_ , 44–48.
71. "Summary Presentation on History of Development of U.S. Long-Range Guided Missiles by Herbert F. York, Director of Defense Research and Engineering," May 5, 1960, WHO, NSCSP, Exec. Sect. Subject File Series, box 1, file Miscellaneous (File #2) (7); Kistiakowsky, _A Scientist_ , 95–96; Levine, _Missile_ , 29–34; Killian, _Sputniks_ , 11–13; Schwiebert, _History_ , 70–78.
72. Diary entry, April 4, 1956, AWF, A. W. Diary Series, box 8, file Apr. '56 Diary-ACW (2); diary entry, April 27, 1956, AWF, A. W. Diary Series, box 8, file Apr. '56 Diary-ACW (1); "Meet the Press, Transcript for Sunday 5 February 1956," WHO, OSS, SUB, DoD Subseries, box 6, file Military program (Missiles) [January 1956–September 1957] (2); "Address by General Thomas D. White, Vice Chief of Staff, USAF at the General Electric Dinner," February 9, 1956, WHO, OSS, SUB, DoD Subseries, box 6, file Military program (Missiles) [January 1956–September 1957] (2).
73. "Memorandum of Discussion at the Special Meeting of the National Security Council," March 31, 1953, AWF, NSC Series, box 4, file Special Meeting of NSC, March 31, 1953; "Memorandum of Discussion at the Special Meeting of the National Security Council," March 31, 1953, in FRUS _: National Security Affairs_ , vol. 2, pt. 1, 268.
74. "Continental Defense, NSC 5408," February 11, 1954, WHO, OSANSA, NSC Series, Policy Papers Subseries, box 9, file NSC 5408 Continental Defense (1).
75. Gaddis, "Transparency," 197; Hall, "Origins of U.S. Space Policy," 218–19; Burrows, _This New Ocean_ , 155; Taubman, _Secret Empire_ , 24–25.
76. These concerns were not new. On June 4, 1953, President Eisenhower raised his concerns over not being able to penetrate the Soviet Union to gain any useful intelligence. See "Discussion at the 148th Meeting of the National Security Council," June 4, 1953, AWF, NSC Series, box 4, file 148th Meeting of NSC, June 4, 1953.
77. McElheny, _Insisting_ , 278–79; Ambrose, _Ike's Spies_ , 267–68; Burrows, _This New Ocean_ , 152–59. Peebles, _Corona Project_ , 18–20; Killian, _Sputniks_ , 11–13.
78. Ferris, "Coming in from the Cold War," 90.
###### 2. Eisenhower and Defense
"Farewell Address by President Eisenhower on the 'Military Industrial Complex,'" in LaFeber, _Dynamics_ , 645–46.
1. Metz, "Eisenhower," 51.
2. Metz, "Eisenhower," 51.
3. Presidential News Conference, May 14, 1953, in Branyan and Larsen, _Eisenhower Administration_ , 1:39; Taubman, _Secret Empire_ , xii.
4. "Informal Condensation of NSC 20/4, 68/2, 135/3, and 141 (for discussion purposes at NSC meetings)," February 6, 1953, WHO, OSANSA, NSC Series, Subject Subseries, box 8, file President's Meeting with Civilian Consultants, March 31, 1953 [re: Review of Basic National Security Policy] (1).
5. May, _American Cold War Strategy_ , 25.
6. May, _American Cold War Strategy_ , 23–80. See also Leffler, _Preponderance_ , 355–60; Bernstein, "The Challenges and Dangers," 79–80.
7. These views continued in NSC 135/3 (September 25, 1952) with no major changes. "Informal Condensation of NSC 20/4, 68/2, 135/3, and 141 (for discussion purposes at NSC meetings)"; May, _American Cold War Strategy_ , 23–80; Leffler, _Preponderance_ , 355–60; "The Reminiscences of Andrew J. Goodpaster," April 10, 1982, 1–3.
8. May, _American Cold War Strategy_ , vii; Robert J. Donovan, "Truman's Perspective," in Heller, _Economics_ , 18–19; John F. Snyder, "The Treasury and Economic Policy," in Heller, _Economics_ , 29–30.
9. Ambrose, _Eisenhower: The President_ , 88; Donovan, _Eisenhower_ , 51; Bernstein, "The Challenges and Dangers," 80–81.
10. DDE Personal Diary, January 22, 1952, box 1, file [January 1–February 28, 1952], 3. See also "Letter to Charles E. Wilson from President Eisenhower, Jan. 5, 1955," AWF, DDE Diary Series, box 9, file DDE Diary, January 1955 (2).
11. "Statement of Policy, General," April 29, 1953, WHO, OSANSA, NSC Series, Policy Papers, box 4, file NSC 149–2—Basic National Security Policies & Programs in Relation to Their Costs (1); "The Reminiscences of Andrew J. Goodpaster," April 10, 1982, 1–3; "Press Release of an Exchange of Correspondence between the President and Secretary of Defense, 5 January 1955," WHO, OSS, SUB, DoD Subseries, box 6, file Military Planning, 1954–1955 (1).
12. DDE Personal Diary, January 22, 1952, box 1, file [January 1, 1950–February 28, 1952], 2; DDE Personal Diary, June 4, 1949, box 1, file [December 13, 1948–March 5, 1951], 1; Darrell B. Montgomery, "New Evidence of the Evolution of a Postwar Air-Atomic Strategy," in Levantrosser, _Harry S. Truman_ , 158–59.
13. DDE Personal Diary, December 17, 1948, January 8, 1949, June 4, 1949, box 1, file [December 13, 1948–March 5, 1951], 1; January 22 1952, box 1, file [January 1, 1950–February 28, 1952], 2; "Notes on Legislative Leadership Meeting, 30 April 1953," AWF, DDE Diary Series, box 4, file Staff Notes January–December 1953.
14. "Notes on Legislative Leadership Meeting, 30 April 1953"; Ambrose, _Eisenhower: The President_ , 88–89.
15. Diary entry, December 13, 1954, Hagerty Diary, box 1A, file December 1954; letter to Charles E. Wilson from President Eisenhower, January 5, 1955, AWF, DDE Diary Series, box 9, file DDE Diary, January 1955 (2).
16. Letter to Edward Hazlett from General Eisenhower, Hazlett Papers, box 1, file 1946 July 1.
17. Letter to John Foster Dulles and Neil McElroy from Robert Cutler, April 7, 1958, WHO, OSANSA, NSC Series, Briefing Notes Subseries, box 14, file Nuclear Policy (1958).
18. "Memorandum for Admiral Arthur W. Radford, USN, Chairman, Joint Chiefs of Staff, 17 February 1956," WHO, OSS, SUB, DoD, box 4, file Joint Chiefs of Staff (2) [January–April 1956]; "The Reminiscences of Andrew J. Goodpaster," OH-378, 104–5.
19. Diary entry, June 19, 1954, Hagerty Diary, box 1, file June 1954. See also "The Next Ten Years," WHO, OSANSA, NSC Series, Policy Papers Subseries, box 2, file NSC 112/1—Disarmament (3).
20. Diary entry, February 1, 1955, Hagerty Diary, box 1A, file February 1955.
21. "Memorandum: Discussion at the 272nd Meeting of the National Security Council, January 12, 1956," AWF, NSC Series, box 7, file 272nd Meeting of NSC, January 12, 1956.
22. "Memorandum: Discussion at the 272nd Meeting of the National Security Council, January 12, 1956." See also memorandum of conference with the president, May 24, 1956, AWF, DDE Diary Series, box 15, file May '56 Goodpaster.
23. The worst-case scenario called for an attack without any warning until U.S. radar detected the bombers. The "better"-case scenario allowed the United States to have approximately one month's warning, though no information on the date of such an attack. See "Memorandum: Discussion at the 201st Meeting of the National Security Council, June 10, 1954," AWF, NSC Series, box 5, file 201st Meeting of NSC, June 10, 1954; diary entry, January 23, 1956, AWF, DDE Diary Series, box 9, file Diary—Copies of DDE Personal [1955–56] (2); "Report to the NSC by the Special Evaluation Subcommittee of the NSC," May 18, 1953, in FRUS _1952–1954: National Security Affairs_ , vol. 2, pt. 1, 329–35.
24. "Report by the Net Evaluation Subcommittee," December 20, 1956, AWF, NSC Series, box 8, file 306th Meeting of NSC, December 20, 1956; "Report by the Net Evaluation Subcommittee," November 12, 1957, AWF, NSC Series, box 9, file 344th Meeting of NSC, November 12, 1957; "Memorandum of Conference with the President, 4 November 1957," WHO, OSS, SUB, ALPHA, box 23, file Science Advisory Committee (3).
25. Letter from Eisenhower to Richard L. Simon, April 4, 1956, AWF, DDE Diary Series, box 14, file April 1956 Miscellaneous (5). See also "Legislative Leadership Meeting, Supplementary Notes, 24 June 1958," AWF, DDE Diary Series, box 33, file June 1958—Staff Notes (2); "Press Release of an Exchange of Correspondence between the President and Secretary of Defense, 5 January 1955," WHO, OSS, SUB, DoD, box 6, file Military Planning, 1954–1955 (1).
26. "Guidelines under NSC 162/2 for FY 1956," July 26, 1954, WHO, OSANSA, NSC Series, Policy Papers Subseries, box 11, file NSC 5422/2—Guidelines under 162/2 for FY 1956 (2); memorandum for the secretary of state from President Eisenhower, September 8, 1953, AWF, DDE Diary Series, box 3, file DDE Diary—August–September 53 (2); memorandum: "Subject: The Meaning of Paragraph 39b, NSC 162/2, as Understood by the Department of Defense," December 1, 1953, WHO, NSCSP, Exec. Sect. Subject File Series, box 5, file #19 Policy re Use (of nuclear weapons) (file #1) (1).
27. "Letter to General Alfred M. Gruenther, Chief of Staff, SHAPE, from President Eisenhower, 4 May 1953," AWF, DDE Diary Series, box 3, file DDE Diary, December 1952–July 1953 (3); "Bipartisan Legislative Meeting, 5 January 1954," AWF, DDE Diary Series, box 4, file Staff Notes, January–December 1954.
28. Memorandum to the NSC by Executive Secretary Lay, "Review of Basic NSC Policies," February 6, 1953, in FRUS _: National Security Affairs_ , vol. 2, pt. 1, 223–24; Freedman, _Evolution_ , 81–82; "Notes by the Assistant Staff Secretary to the President (Minnich) on the Legislative Leadership Meeting," December 14, 1954, in FRUS _: National Security Affairs_ , vol. 2, pt. 1, 826; _Congress and the Nation_ , 274–75; "Basic National Security Policy," January 7, 1955, WHO, OSANSA, NSC Series, Policy Papers Subseries, box 14, file NSC 5501—Basic National Security Policy.
29. "Annex A," WHO, NSCSP, Exec. Sect. Subject File Series, box 11, file General Papers (Colonel Bonesteel); "The Reminiscences of Andrew J. Goodpaster," April 10, 1982, 13; "The Reminiscences of Andrew J. Goodpaster," April 25, 1967, 12–14, Columbia Center for Oral History.
30. Memorandum for the record by the special assistant to the president for national security affairs (Cutler), "SOLARIUM Project," May 9, 1953, in FRUS _: National Security Affairs_ , vol. 2, pt. 1, 323–26; memorandum for the National Security Council, "Project Solarium," July 22, 1953, WHO, OSANSA, NSC Series, Subject Subseries, box 9, file Project Solarium, Report to the NSC by Task Force "A"; "Project Solarium: Summary of Basic Concepts of Task Forces," July 30, 1953, WHO, OSANSA, NSC Series, Subject Subseries, box 10, file Project Solarium [1953] (1).
31. Memorandum for the National Security Council, "Project Solarium," July 22, 1953.
32. "Annex A"; memorandum for the National Security Council, "Project Solarium," July 22, 1953; "Task Force B," July 16, 1953, WHO, OSANSA, NSC Series, Subject Subseries, box 9, file Project Solarium, Report to the NSC by Task Force "A" [1953] (2); "The Reminiscences of Andrew J. Goodpaster," April 10, 1982, 13.
33. "Annex A."
34. "Task Force C," July 16, 1953, WHO, OSANSA, NSC Series, Subject Subseries, box 9, file Project Solarium, Report to the NSC by Task Force "A" [1953] (2); memorandum for the National Security Council, "Project Solarium," July 22, 1953; "The Reminiscences of Andrew J. Goodpaster," April 10, 1982, 13.
35. "Project Solarium," July 16, 1953, AWF, NSC Series, box 4, file Minutes of 155th Meeting of NSC, July 16, 1953; "The Reminiscences of Andrew J. Goodpaster," April 10, 1982, 14–15.
36. "The Reminiscences of Andrew J. Goodpaster," April 10, 1982, 14–15.
37. Gaddis, _Strategies_ , 133.
38. "Meeting with Secretary Dulles, 7/20/54," AWF, A. W. Diary Series, box 2, file ACW Diary, July 1954 (3); Freedman, _Evolution_ , 76–78; Kinnard, _Secretary of Defense_ , 51–52; Bernstein, "The Challenges and Dangers," 82–83.
39. Diary entry, December 9 and December 13, 1954, Hagerty Diary, box 1A, file December 1954; "Memorandum of Conference with the President, 6 November 1957," WHO, OSS, SUB, ALPHA, box 23, file Science Advisory Committee (3); "Memorandum for the National Security Council, Subject: Basic National Security Policy, 31 March 1960," WHO, NSCSP, Exec. Sect. Subject File Series, box 13, file NSC 5906/1—Basic National Security Policy [1959]; memorandum of conference with the president, May 24, 1956, AWF, DDE Diary Series, box 15, file May 1956; "The Reminiscences of Andrew J. Goodpaster," April 10, 1982, 14–15; Freedman, _Evolution_ , 76–78; Kinnard, _Secretary of Defense_ , 51–52; Bernstein, "The Challenges and Dangers," 82–83.
40. All amounts, except percentages, are in U.S. dollars; figures from U.S. Bureau of the Census, _Historical Statistics of the United States: Colonial Times to 1970_ , 1975, pt. 1, 224, 230, 233; pt. 2, 1116, <https://fraser.stlouisfed.org/title/?id=237>, accessed July 15, 2015; Divine, _Since 1945_ , 7–30; Craufurd A. Goodwin, "The Economic Problems Facing Truman," in Heller, _Economics_ , xv–xvii, 1–2, 13–20; Gosnell, _Truman's Crises_ , 257–390.
41. All amounts, except percentages, are in U.S. dollars; figures from _Historical Statistics of the U.S., Colonial Times to 1970_ , pt. 1, 224, 230, 233; pt. 2, 1116; letter to Eisenhower from Secretary Humphrey, July 29, 1953, AWF, DDE Diary Series, box 3, file DDE Diary December 1952–July 1953 (1); notes on legislative leadership meeting, April 30, 1953, AWF, DDE Diary Series, box 4, file Stuff Notes January–December 1953; Divine, _Since 1945_ , 7–30; Goodwin, "The Economic Problems Facing Truman," xv–xvii, 1–2, 13–20; Gosnell, _Truman's Crises_ , 257–390.
42. Harris, _Economics_ , xv–xxiii, 3–8; Heller, _Economics_ , 6–7; letter to Clifford Roberts from Eisenhower, March 27, 1958, AWF, DDE Diary Series, box 31, file DDE Dictation, March 1958.
43. Speech before the American Legion Convention, New York City, August 25, 1952, in Eisenhower Library, "Campaign Statements of Dwight D. Eisenhower: A Reference Guide," 111.
44. Campaign speeches in Philadelphia, September 4, 1952, Jackson, Michigan, October 1, 1952, New Brunswick, New Jersey, October 17, 1952, in Eisenhower Library, "Campaign Statements of Dwight D. Eisenhower: A Reference Guide," 111.
45. Letter to Eisenhower from G. M. Humphrey, July 29, 1953, AWF, DDE Diary Series, box 3, file DDE Diary December 1952–July 1953 (1); letter to Brig. Gen. Benjamin F. Caffey (Ret.) from Eisenhower, July 27, 1953, AWF, DDE Diary Series, box 3, file DDE Diary December 1952–July 1953 (1).
46. "Memorandum of Discussion at the 138th Meeting of the National Security Council," March 25, 1953, in FRUS _: National Security Affairs_ , vol. 2, pt. 1, 262–63; letter to Brig. Gen. Benjamin F. Caffey (Ret.) from DDE, July 27, 1953; letter to Swede Hazlett, July 22, 1957, AWF, DDE Diary Series, box 25, file July 1957—DDE Dictation; Ambrose, _Ike's Spies_ , 275–76.
47. "Review of the 1954 Budget," August 27, 1953, AWF, Admin. Series, box 12, file Dodge, Joseph M. 1952–1953 (1).
48. Discussion at the 165th meeting of the National Security Council, October 7, 1953, AWF, NSC Series, box 4, file 165th Meeting of NSC, October 7, 1953; "Condensed Statement of Proposed Policies and Programs (Draft), 31 March 1953," AWF, NSC Series, box 4, file Documents Pertaining to Special NSC Meeting, March 31, 1953.
49. Memorandum of conference with the president, April 2, 1956, WHO, OSS, SUB, DoD Subseries, box 4, file Joint Chiefs of Staff (2) [January–April 1956]; memorandum of conference with the president, November 6, 1957, WHO, OSS, SUB, ALPHA, box 23, file Science Advisory Committee (3); undated statement by Eisenhower, AWF, A. W. Diary Series, box 8, file January '57 Diary-ACW.
50. "Review of Basic National Security Policy, NSC 162," October 7, 1953, AWF, NSC Series, box 4, file 165th Meeting of NSC, October 7, 1953; diary entry December 14, 1954, Hagerty Diary, box 1A, file December 1954; bipartisan legislative meeting, December 14, 1954, AWF, DDE Diary Series, box 4, file Staff Notes, January–December 1954.
51. DDE Personal Diary, January 8, 1949, box 1, file [DDE Diary, December 13, 1948–March 5, 1951], 1; Divine, _Since 1945_ , 7–8, 28–30; Montgomery, "New Evidence," 158–60.
52. Eisenhower left Korea on December 5, 1952, flying to Guam, where he boarded the cruiser USS _Helena_. From there he sailed to Wake, where members of his future cabinet met the ship. Dulles, Humphrey, McKay, General Clay, and Dodge joined up with the president-elect to discuss a variety of matters. See Eisenhower, _White House Years_ , 1:96; Ambrose, _Eisenhower: The President_ , 30–31.
53. "Capsule Statement of National Security Policy," AWF, NSC Series, box 4, file Documents Pertaining to Special NSC Meeting, March 31, 1953; letter to Clifford Roberts from Eisenhower, March 27, 1958, AWF, DDE Diary Series, box 31, file DDE Dictation, March 1958; Ambrose, _Eisenhower: The President_ , 32–34; Cook, _Declassified Eisenhower_ , 300; Freedman, _Evolution_ , 156.
54. "State of the Union Address," February 2, 1953, in _Public Papers_ , 1:17.
55. "Draft Statement of Policy Proposed by the National Security Council," February 11, 1954, in FRUS _: National Security Affairs_ , vol. 2, pt. 1, 611. For similar statements, see "Restatement of Basic National Security Policy, General Considerations [DRAFT STATEMENT]," June 1, 1953, WHO, OSANSA, NSC Series, Policy Papers Subseries, box 5, file NSC 153/1—Basic National Security Policy.
56. Notes on meeting of April 4, 1956, AWF, A. W. Diary Series, box 8, file April '56 Diary-ACW (2).
57. _Congress and the Nation_ , 274. See also memorandum of conference with the president, September 30, 1958, AWF, DDE Diary Series, box 36, file Staff Notes—September 1958.
58. Kistiakowsky, _A Scientist_ , xviii–xix.
59. Radio address to the American people on National Security and its costs, May 19, 1953, in _Public Papers_ , 1:306–16; "Condensed Statement of Proposed Policies and Programs (Draft Statement)," April 2, 1953, AWF, NSC Series, box 4, file Documents Pertaining to Special NSC Meeting, March 31, 1953; DDE Personal Diary, January 22, 1955, box 1, file [January 1, 1950–February 28, 1952], 3.
60. _Congress and the Nation_ , 275–76.
61. "Memorandum of Conference with the President," December 22, 1954, AWF, A. W. Diary Series, box 3, file ACW Diary, December 1954 (2); DDE Personal Diary, January 22, 1952, box 1, file [January 1, 1950–February 28, 1952], 2; diary entry, December 13–14, 1954, Hagerty Diary, box 1A, file December 1954; president's news conference, April 30, 1953, in _Public Papers_ , 1:245–46.
62. "Notes on Legislative Leadership Meeting, 30 April 1953," AWF, DDE Diary Series, box 4, file Staff Notes January–December 1953.
63. Ambrose, _Eisenhower: The President_ , 88–89; Gaddis, _Strategies_ , 139–40. See also diary entry, December 13, 1954, Hagerty Diary, box 1A, file December 1954.
64. Memorandum of conference with the president, December 20, 1956, WHO, OSS, SUB, DoD Subseries, box 2, file Budget, Military (4) [May–December 1956]; discussion at the 293rd and 294th meetings of the National Security Council, August 16 and 17, 1956, AWF, NSC Series, box 8, file 293rd and 294th Meetings of NSC, August 16 and 17, 1956; Gaddis, _Strategies_ , 133.
65. Eisenhower, as quoted in Gaddis, _Strategies_ , 133–34. Andrew Goodpaster maintains that this speech clearly showed the president's commitment to both disarmament and the control of defense spending. Author's phone interview with Goodpaster, March 7, 2003.
66. Gaddis, _Strategies_ , 133–34; Ambrose, _Eisenhower: The President_ , 224; Eisenhower, _White House Years_ , 1:144–45; Divine, _Eisenhower and the Cold War_ , 108.
67. "Notes on Legislative Leadership Meeting," April 30, 1953; diary entry, June 1, 1953, DDE Personal Diary, box 1, file December 1952–8/19/53 (2); discussion of the 138th meeting of the National Security Council, March 25, 1953, AWF, NSC Series, box 4, file 138th Meeting of NSC, March 25, 1953.
68. "Memorandum for the Director of the Bureau of the Budget from DDE," December 1, 1953, AWF, Admin. Series, box 12, file Dodge, Joseph M. 1955 Budget (2); "Review of National Security Programs, Memorandum for the Executive Secretary, National Security Council," March 24, 1953, WHO, OSANSA, NSC Series, Subject Subseries, box 8, file President's Meeting with Civilian Consultants, March 31, 1953 [re: Review of Basic National Security Policy] (8).
69. Joseph Alsop and Stewart Alsop, "Defense 'New Look' and New Weapons," _Washington Post_ , February 22, 1954, AWF, NSC Series, box 5, file 187th Meeting of NSC, March 4, 1954; "The Reminiscences of General Nathan F. Twining," 142–44; Presidential News Conference, March 17, 1954, in Branyan and Larsen, _Eisenhower Administration_ , 1:40–41.
70. Discussion at the 160th meeting of the National Security Council, August 27, 1953, AWF, NSC Series, box 4, file 160th Meeting of NSC, August 27, 1953.
71. Discussion at the 166th meeting of the National Security Council, October 13, 1953, AWF, NSC Series, box 4, file 166th Meeting of NSC, October 13, 1953; Alsop and Alsop, "Defense 'New Look'"; draft memorandum, "Subject: Policy Regarding Use of Nuclear Weapons," December 31, 1953, signed by Eisenhower January 2, 1954; memorandum "Subject: The Meaning of Paragraph 39b, NSC 162/2, as Understood by the Department of Defense," December 1, 1953; memorandum for the record from Robert Cutler, December 2, 1953, WHO, NSCSP, Exec. Sect. Subject File Series, box 5, file #19 Policy re Use (of Nuclear Weapon) (File # 1) (1).
72. _Congress and the Nation_ , 275; diary entry, December 13, 1954, Hagerty Diary, box 1A, file December 1954; "Principal Budgetary Elements of New Obligational Authority and Expenditures," May 27, 1953, AWF, Admin. Series, box 12, file Dodge, Joseph M. 1952–1953 (1); Eisenhower, _White House Years_ , 1:452.
73. "Budget Message of the President," January 12, 1954, AWF, Admin. Series, box 12, file Dodge, Joseph M. 1954–56 (5); _Congress and the Nation_ , 280; Ambrose, _Eisenhower: The President_ , 223; Eisenhower, _White House Years_ , 1:452.
74. _Congress and the Nation_ , 275, 279–80, 285–92, 294–307; Department of Defense Military New Obligational Authority, undated, AWF, Admin. Series, box 8, file Brundage, Percival 1955–57 (2).
75. Letter to Secretary of Defense Charles E. Wilson from Eisenhower, January 5, 1955, WHO, OSS, SUB, DoD Subseries, box 5, file Man Power and Personnel (2) [January 1955–August 1957]; Ambrose, _Eisenhower: The President_ , 144, 171–72; Ambrose, _Rise to Globalism_ , 135–36.
76. "President's Interview with John Taber, Congressman from New York," October 21, 1953, AWF, A. W. Diary Series, box 1, file ACW Diary, August–September–October 1953 (1).
77. Exchange of correspondence between Secretary Wilson and Eisenhower, January 5, 1955, WHO, OSS, SUB, DoD Subseries, box 6, file Military Planning, 1954–1955 (1); _Congress and the Nation_ , 274–75, 279–80.
78. Letter to George Humphrey from Eisenhower, November 22, 1957, AWF, DDE Diary Series, box 28, file November '57 DDE Diary.
79. Memorandum of conference with the president, December 20, 1956, WHO, OSS, SUB, DoD Subseries, box 2, file Budget, Military (4) [May–December 1952]; memorandum of conference with the president, April 2, 1956, WHO, OSS, SUB, DoD Subseries, box 4, file Joint Chiefs of Staff (2) [January–April 1956]; diary entry, February 1, 1955, Hagerty Diary, box 1A, file February 1955.
80. DDE Personal Diary, December 17, 1948, January 27, 1949, box 1, file 1948, 1; letter to Swede Hazlett from Eisenhower, April 27, 1949, Hazlett Papers, box 1, file 1949 April 27.
81. DDE Personal Diary, January 7, 1949, box 13, file 1948 (Washington).
82. DDE Personal Diary, January 7–8, 1949, February 2, 1949, box 13, file 1948 (Washington).
83. DDE Personal Diary, February 2 and 4, 1949, box 13, file 1948 (Washington); letter to Swede Hazlett, February 24, 1950, box 1, file 1950 February 24; letter to Swede Hazlett, November 14, 1951, box 1, file 1951 November 14.
84. DDE Personal Diary, January 22, 1952, box 1, file [January 1, 1950–February 28, 1952], 3; "Discussion at the 163rd Meeting of the National Security Council," September 24, 1953, AWF, NSC Series, box 4, file 163rd Meeting of NSC, September 24, 1953.
85. An excellent example was the open journal discussions of new aircraft capabilities, which only aggravated competition between the services. See "Memorandum of Conference with the President," April 18, 1956, AWF, DDE Diary Series, box 15, file Apr. '56 Goodpaster; "Memorandum of Conference with the President," October 7, 1958, AWF, DDE Diary Series, box 36, file Staff Notes—October 1958; discussion at the 141st meeting of the National Security Council, May 6, 1953, AWF, NSC Series, box 4, file 143rd Meeting of NSC, May 6, 1953.
86. "Memorandum for the Record," November 6, 1957, AWF, DDE Diary Series, box 28, file November '57, Staff Notes.
87. "Memorandum for the Record," November 6, 1957, WHO, OSS, SUB, DoD Subseries, box 1, file Department of Defense, Vol. 2 (3) [November–December 1957]; "Memorandum for the Record," November 6, 1957, AWF; phone conversation with Gen. Andrew Goodpaster, March 7, 2003.
88. "Discussion on Continental Defense," November 23, 1953, AWF, NSC Series, box 5, file 172nd Meeting of NSC, November 23, 1953; "Memorandum of Conference with the President," December 6, 1954, AWF, A. W. Diary Series, box 3, file ACW Diary, 1954 (5).
89. Diary entry, October 26, 1953, AWF, A. W. Diary Series, box 1, file ACW Diary, August–September–October 1953 (1); "Discussion at the 293rd and 294th Meetings of the National Security Council," August 16 and 17, 1956, AWF, NSC Series, box 8, file 293rd and 294th Meeting of NSC, August 16 and 17, 1956; "The Reminiscences of General Nathan F. Twining," 113–17; "Meeting at the Pentagon," January 25, 1958, WHO, OSS, SUB, DoD Subseries, box 1, file Department of Defense, Vol. 2 (4) [January 1958].
90. Diary entry, December 9, 1954, Hagerty Diary, box 1A, file December 1954; "Discussion at the 138th Meeting of the National Security Council," March 25, 1953, AWF, NSC Series, box 4, file 138th Meeting of NSC, March 25, 1953.
91. Diary entry, February 1, 1955, "Legislative Leaders Meeting," Hagerty Diary, box 1A, file February 1955; memorandum of conference with the president, April 5, 1956, AWF, DDE Diary Series, box 15, file Apr. '56 Goodpaster; diary entry, December 9, 1954, Hagerty Diary, box 1A, file December 1954; discussion at the 176th meeting of the National Security Council, December 16, 1953, AWF, NSC Series, box 5, file 176th Meeting of NSC, December 16, 1953; discussion at the 227th meeting of the National Security Council, December 3, 1954, AWF, NSC Series, box 6, file 227th Meeting of NSC, December 3, 1954.
92. "The Reminiscences of John S. D. Eisenhower," 14–38; "The Reminiscences of General Nathan F. Twining," 113–17, 144–45.
93. Ambrose, _Eisenhower: Soldier and President_ , 321–22; _Congress and the Nation_ , 279–80.
94. Notes on legislative leadership meeting, April 30, 1953, AWF, DDE Diary Series, box 4, file Staff Notes January–December 1953.
95. Ambrose, _Eisenhower: The President_ , 88.
96. "Cardboard Wings vs Real Strength," June 1953, Harlow Records, box 4, file Air Force Appropriations [1953 and 1957–58] (4); "Salient Facts about Air Force Expenditures," June 5, 1953, Harlow Records, box 4, file Air Force Appropriations [1953 and 1957–58] (5); "Air Force Strength," January 5, 1954, WHO, OSS, Minnich Series, box 1, file Miscellaneous—A (2) [July 1953–February 1954]; "Special Staff Note on Military Aircraft Production," July 27, 1957, AWF, DDE Diary Series, box 25, file July 1957 Staff Memos (1).
97. Ambrose, _Eisenhower: The President_ , 88–91; Ambrose, _Eisenhower: Soldier and President_ , 321–22; Hall, "The Eisenhower Administration," 60–61; Divine, _Sputnik Challenge_ , 20–21; "The Reminiscences of Andrew J. Goodpaster," April 10, 1982, 18; Huntington, _Changing Patterns_ , 14.
98. The budget for fiscal year 1954 was not the first of the New Look; most of the groundwork and long-term planning that went into it was a carryover from the Truman administration.
99. _Congress and the Nation_ , 279–89; memorandum for executive secretary, NSC, from Allen Dulles, DCI, June 1, 1953, WHO, OSANSA, NSC Series, Policy Papers Subseries, box 3, file NSC 140/1—Special Evaluation Subcommittee of the NSC.
100. Diary entry, April 27, 1956, AWF, A. W. Diary Series, box 8, file April '56 Diary-ACW (1); _Congress and the Nation_ , 290–92; Freedman, _Evolution_ , 29; "Pre–Press Conference Briefing," February 8, 1956, AWF, DDE Diary Series, box 13, file February '56 Miscellaneous (5).
101. Ambrose, _Eisenhower: The President_ , 223–24.
102. Eisenhower et al., _Ike's Letters_ , 109; diary entry, February 1, 1955, Hagerty Diary, box 1A, file February 1955.
103. Ambrose, _Eisenhower: The President_ , 225.
104. Letter to the president from Senator Symington, August 29, 1958, AWF, Admin. Series, box 36, file Symington, Senator Stuart.
105. "Memorandum for the Record, Subject: Discussion of Soviet and U.S. Long Range Ballistic Missile Programs," August 18, 1958, WHO, OSS, SUB, ALPHA, box 24, file Symington Letter.
106. Letter to the president from Senator Symington, August 29, 1958; "Memorandum of Conversation, Subject: DCI Briefing of Senator Stuart Symington on _Soviet Ballistic Missile Programs and Capabilities_ ," December 16, 1958, WHO, OSS, SUB, ALPHA, box 24, file Symington Letter.
107. Letter from Eisenhower to Everett Hazlett, August 20, 1956, quoted in Ambrose, _Eisenhower: The President_ , 225.
108. "Memorandum for the National Security Council, Subject: Basic National Security Policy, 31 March 1960," WHO, NSCSP, Exec. Sect. Subject File Series, box 13, file NSC 5906/1—Basic National Security Policy [1959]; Freedman, _Evolution_ , 76–78; Kinnard, _Secretary of Defense_ , 51–52.
109. Bissell et al., _Reflections_ , 124–25.
###### 3. Satellite Reconnaissance
"Comments on the Report to the President by the Technological Capabilities Panel of the Science Advisory Committee," WHO, OSANSA, NSC Series, Policy Papers Subseries, box 16, folder NSC 5522 Technological Capabilities Panel, S23.
1. Burrows, _This New Ocean_ , 157–58.
2. Bissell et al., _Reflections_ , 92.
3. Davies and Harris, RAND _'s Role_ , 55, RAND Corporation; Beschloss, _May-Day_ , 73–74. Project RAND was an army air forces–sponsored think tank that used a multidisciplinary approach to examine the long-term issues and questions that the air force itself could not. Davies and Harris, RAND _'s Role_ , 3–6; Kaplan, _Wizards_ , 1–5, 51–63.
4. The numerous RAND studies on American vulnerability and the need to improve warning include "The Cost of Decreasing Vulnerability of Air Bases by Dispersal," R-235, June 1, 1952; "The Military Value of Advanced Warning of Hostilities and Its Implications for Intelligence Indicators," July 1953; "Vulnerability of U.S. Strategic Air Power to a Surprise Attack in 1956," SM-15, April 15, 1953. See Davies and Harris, RAND _'s Role_ , 48–51; Kaplan, _Wizards_ , 85–110, 116–27; Taubman, _Secret Empire_ , 13–14.
5. Killian, _Sputniks_ , 68–69; Welzenbach, "Din Land," 22; "The Reminiscences of James R. Killian," pt. 1, p. 13, pt. 8, pp. 224–25, 228–34.
6. Killian, _Sputniks_ , 69–70; Welzenbach, "Din Land," 22–23; Peebles, _Corona Project_ , 18; letter to Gen. Curtis E. LeMay from James R. Killian Jr., September 2, 1954, SPI, GWU, file Technological Capabilities; Oral History OH-216, pt. 1, pp. 13–14.
7. Welzenbach, "Din Land," 22–23; Burrows, _This New Ocean_ , 156–58; Peebles, _Corona Project_ , 18–19; "The Reminiscences of James R. Killian," pt. 1, pp. 13–14; Taubman, _Secret Empire_ , 89.
8. Killian, _Sputniks_ , 71.
9. McElheny, _Insisting_ , 3; Taubman, _Secret Empire_ , 91.
10. Welzenbach, "Din Land," 23; R. Cargill Hall, "Strategic Reconnaissance in the Cold War," _Prologue_ 28, no. 2 (1996): 118; Peebles, _Shadow Flights_ , 78; Peebles, _Corona Project_ , 19–20; McElheny, _Insisting_ , 293–95; Taubman, _Secret Empire_ , 96–97.
11. Interview with Edwin Land, 1984, in Welzenbach, "Din Land," 23; McElheny, _Insisting_ , 294.
12. Peebles, _Shadow Flights_ , 79; McElheny, _Insisting_ , 293–94; Welzenbach, "Din Land," 23; Taubman, _Secret Empire_ , 98–99.
13. "Part V, Intelligence: Our First Defense against Surprise," WHO, OSS, SUB, ALPHA, box 16, file Killian Report—Technological Capabilities Panel (2).
14. Comments on the report to the president by the Technological Capabilities Panel of the Science Advisory Committee, WHO, OSANSA, NSC Series, Policy Papers Subseries, box 16, file NSC 5522 Technological Capabilities Panel, p. S21.
15. TCP report, as quoted in Killian, _Sputniks_ , 79–80.
16. TCP report, as quoted in Killian, _Sputniks_ , 80.
17. Comments on the report to the president by the Technological Capabilities Panel, S21–S22.
18. Comments on the report to the president by the Technological Capabilities Panel, S21–S22.
19. Comments on the report to the president by the Technological Capabilities Panel," S5; "Annex A: Department of State Reaction to Technological Capabilities Panel Recommendations Nos. 7 and 9," June 3, 1955, WHO, OSANSA, NSC Series, Briefing Notes Subseries, box 17, file [Technological Capabilities Panel of the Science Advisory Committee] (3) [1954–56].
20. Comments on the report to the president by the Technological Capabilities Panel," S23.
21. Many historians have described the development of the U-2. For a short summary of Land and his role, see McElheny, _Insisting_ , 294–300. For more details, see Beschloss, _May-Day_ , or the first two chapters of Brugioni, _Eyeball to Eyeball_ ; Welzenbach, "Din Land," 23–28; Taubman, _Secret Empire_ , 99–102, 113–89.
22. Taubman, _Secret Empire_ , 102–5; Beschloss, _May-Day_ , 366, 372, 388, 411–12; phone interview with R. Cargill Hall, official historian, National Reconnaissance Office, October 17, 2002; "The Reminiscences of James R. Killian," pt. 2, pp. 35–36; "The Reminiscences of Andrew J. Goodpaster," April 10, 1982, 39–44.
23. Killian, _Sputniks_ , 82–83; Hall, "The Eisenhower Administration," 62; Welzenbach, "Din Land," 28; Bissell, "Origins," 15–16; Peebles, _Corona Project_ , 19; Taubman, _Secret Empire_ , 107.
24. "The Reminiscences of Andrew J. Goodpaster," April 25, 1967, pt. 1, pp. 14–16, 34–36; "The Reminiscences of Andrew Goodpaster, Ann Whitman, Raymond Saulnier, Elmer Staats, Arthur Burns, and Gordon Gray," 15–17; President's appointments, November 24, 1954, AWF, A. W. Diary Series, box 3, file ACW Diary, November 1954 (1); memorandum of conference with the president, November 24, 1954, AWF, A. W. Diary Series, box 3, file ACW Diary, November 1954 (1).
25. Phone interview with R. Cargill Hall, October 17, 2002.
26. Phone interview with R. Cargill Hall, October 17, 2002; "The Reminiscences of James R. Killian," pt. 1, pp. 20–21, pt. 2, pp. 34–36; "The Reminiscences of Dr. Richard M. Bissell," June 5, 1967, 38–41, Columbia Center for Oral History.
27. Memorandum of conference with the president, November 24, 1954; "The Reminiscences of Andrew J. Goodpaster," June 26, 1975, 79–83, Dwight D. Eisenhower Presidential Library.
28. "The Reminiscences of James R. Killian," pt. 1, pp. 19–22; phone interview with R. Cargill Hall, October 17, 2002.
29. Phone interview with R. Cargill Hall, October 17, 2002; "The Reminiscences of James R. Killian," pt. 1, pp. 21–22, pt. 2, pp. 34–37.
30. Gaddis, _The Long Peace_ , 198; Alexander, _Holding the Line_ , 96–97; Rostow, _Open Skies_ , 10, 26–48; Davies and Harris, RAND _'s Role_ , 62–63.
31. Gaddis, _The Long Peace_ , 198; Alexander, _Holding the Line_ , 96–97; Rostow, _Open Skies_ , 10, 26–48; Davies and Harris, RAND _'s Role_ , 62–63.
32. Rostow, _Open Skies_ , 29–30; Karas, _New High Ground_ , 97–98; Stassen and Houts, _Eisenhower_ , 321–30; Beschloss, _May-Day_ , 98–104.
33. Gaddis, _The Long Peace_ , 198–99; Beschloss, _May-Day_ , 99–100.
34. Gaddis, _The Long Peace_ , 199; Albertson, _Eisenhower_ , 83–84.
35. The Open Skies proposal remained a fixed element of the U.S. disarmament program following its proposal. "Brief Statement of the Joint Chiefs of Staff Relative to the Problem of Disarmament," January 25, 1956, WHO, OSANSA, NSC Series, Policy Papers Subseries, box 2, file Memorandum for the Secretary of Defense; Stassen and Houts, _Eisenhower_ , 334–36.
36. Statement by President Eisenhower at the Geneva Conference of Heads of Government, "Aerial Inspection and Exchange of Military Blueprints," July 21, 1955, in U.S. Department of State, _Documents on Disarmament_ , 486, 487–88. See also Stassen and Houts, _Eisenhower_ , 339.
37. "Discussion at the 256th Meeting of the National Security Council," July 28, 1955, AWF, NSC Series, box 7, file 256th Meeting of NSC, July 28, 1955; "Discussion at the 257th Meeting of the National Security Council," August 4, 1955, AWF, NSC Series, box 7, file 257th Meeting of NSC, August 4, 1955; Gaddis, _The Long Peace_ , 200; Stassen and Houts, _Eisenhower_ , 340–41.
38. Article 12 of the Anti-Ballistic Missile Treaty (1972) and Article 15 of the SALT II treaty also used the term. See Herman, _Intelligence Power_ , 158–60.
39. Robert R. Bowie, Policy Planning Staff, Department of State, "Memorandum for Mr. Phleger," March 28, 1955, SPI, GWU, file TCP and Overflight.
40. "Tab C: Recommendations under Which Primary Responsibility Was Assigned to Other Agencies, Subject to Coordination with the Central Intelligence Agency," December 22, 1954, WHO, OSANSA, NSC Series, Briefing Notes Subseries, box 17, file [Technological Capabilities Panel of the Science Advisory Committee] (3) [1954–56].
41. "Statement by the President: Summary of Important Facts in the Development by the United States of an Earth Satellite," October 9, 1957, WHO, OSS, SUB, ALPHA, box 23, file Satellites [October 1957–February 1960] (1); Burrows, _This New Ocean_ , 169–72.
42. Phone interview with R. Cargill Hall, October 17, 2002; "Tab E: Memorandum RE U.S. Participation in International Geophysical Year: Artificial Satellite Project," June 3, 1955, and "Letter to Mr. Meeker from Mrs. Fleming, Subject: U.S. Participation in International Geophysical Year: Artificial Satellite Project," April 15, 1955, WHO, OSANSA, NSC Series, Briefing Notes Subseries, box 17, file [Technological Capabilities Panel of the Science Advisory Committee] (3) [1954–56].
43. "Memorandum for the Assistant Secretary of Defense (Research and Development), Subject: Scientific Satellite Program for the Department of Defense," May 4, 1955, WHO, OSANSA, NSC Series, Briefing Notes Subseries, box 7, file Earth Satellites (3) [1955–58].
44. "Memorandum for the Assistant Secretary of Defense (Research and Development), Subject: Scientific Satellite Program for the Department of Defense," May 4, 1955.
45. "Memorandum for the Assistant Secretary of Defense (Research and Development), Subject: Scientific Satellite Program for the Department of Defense," May 4, 1955.
46. "Memorandum for the Assistant Secretary of Defense (Research and Development), Subject: Scientific Satellite Program for the Department of Defense," May 4, 1955.
47. "Memorandum for the Assistant Secretary of Defense (Research and Development), Subject: Scientific Satellite Program for the Department of Defense," May 4, 1955.
48. "Memorandum for the Assistant Secretary of Defense (Research and Development), Subject: Scientific Satellite Program for the Department of Defense," May 4, 1955.
49. "Memorandum for Mr. Robert Murphy, Deputy Under Secretary of State, from Alan T. Waterman, Director, National Science Foundation," March 18, 1955, WHO, OSANSA, NSC Series, Briefing Notes Subseries, box 7, file Earth Satellites (3) [1955–1958].
50. "Memorandum for Dr. Alan T. Waterman, Director, National Science Foundation from Mr. Robert Murphy, Deputy Under Secretary of State," April 27, 1955, WHO, OSANSA, NSC Series, Briefing Notes Subseries, box 7, file Earth Satellites (3) [1955–1958].
51. Letter to Donald A. Quarles, assistant secretary of defense (research and development) from Alan T. Waterman, May 13, 1955, WHO, OSANSA, NSC Series, Briefing Notes Subseries, box 7, file Earth Satellites (3) [1955–1958]; "Memorandum for Mr. Robert Murphy, Deputy Under Secretary of State from Alan T. Waterman, Director, National Science Foundation," March 18, 1955, SPI, GWU, file "Bissell and Right of Overflight"; Peebles, _Corona Project_ , 22–23; Burrows, _This New Ocean_ , 173–74.
52. NSC 5520, "Draft Statement of Policy on U.S. Scientific Satellite Program," May 20, 1955, in Logsdon et al., _Exploring the Unknown_ , 1:308; "Memorandum for General Cutler, Subject: U.S. Earth Satellite Program (NSC 5520)," January 18, 1957, WHO, OSANSA, NSC Series, Briefing Notes Subseries, box 7, file Earth Satellites (3) [1955–58]; phone interview with R. Cargill Hall, October 17, 2002.
53. NSC 5520, in Logsdon et al., _Exploring the Unknown_ , 1:309; "NSC 5520," SPI, GWU, file NSC 5520, 1.
54. "Tab C: Recommendations," December 22, 1954; phone interview with R. Cargill Hall, October 17, 2002.
55. "Memorandum for General Cutler, Subject: U.S. Earth Satellite Program (NSC 5520)," January 18, 1957, WHO, OSANSA, NSC Series, Briefing Notes Subseries, box 7, file Earth Satellites (3) [1955–58]; NSC 5520, in Logsdon et al., _Exploring the Unknown_ , 1:309–24; "Draft Statement of Policy on U.S. Scientific Satellite Program," in FRUS _1955–1957: United Nations_ , 724–30; "NSC 5520," SPI, GWU, file NSC: 5520, 4–5.
56. "Recommendation on Which CIA Has Secondary Responsibility for Reporting to the NSC, Subject to Coordination with Other Agencies," May 12, 1953 [RECOMMENDATION for TCP], WHO, OSANSA, NSC Series, Briefing Note Subseries, box 7, file Earth Satellites (2) [1955–1958], 2, 3; "Discussion at the 250th Meeting of the National Security Council," May 26, 1955, AWF, NSC Series, box 6, file 250th Meeting of NSC, May 26, 1955.
57. "Memorandum for Mr. James S. Lay Jr. from Nelson A. Rockefeller," May 17, 1955, AWF, Admin. Series, box 31, file Rockefeller, Nelson 1952–55 (5).
58. "Memorandum of the Discussion at the 250th Meeting of the National Security Council, Thursday," May 26, 1955, AWF, NSC Series, box 6, file 250th Meeting of NSC, May 26, 1955; "Draft Report on NSC 5520, U.S. Scientific Satellite Program," November 9, 1956, WHO, OSANSA, NSC Series, Briefing Notes Subseries, box 7, file Earth Satellites (2) [1955–58]; Burrows, _This New Ocean_ , 167; Levine, _Missile_ , 53.
59. Editorial note, FRUS, 11:734.
60. "Memorandum to the President, Subject: Public Information Program with Respect to Implementation of NSC #5520," July 27, 1955, and "Memorandum for General Cutler," January 18, 1957, WHO, OSANSA, NSC Series, Briefing Notes Subseries, box 7, file Earth Satellites (3) [1955–58].
61. "Satellite Programs in the Department of Defense," October 25, 1957, Harlow Records, box 1, file DoD Report to Senate Preparedness Investigating Subcommittee, Missiles (October 1957) (1).
62. "Memorandum for the President, Subject: Project VANGUARD," April 30, 1957, and "Memorandum for: The Secretary of Defense, The Director, Bureau of the Budget, The Director, National Science Foundation, Subject: U.S. Scientific Satellite Program," May 14, 1957, WHO, OSANSA, NSC Series, Briefing Notes Subseries, box 7, file Earth Satellites (2) [1955–58].
63. "Memorandum of Discussion at the 339th Meeting of the National Security Council," October 11, 1957, AWF, NSC Series, box 9, file 339th Meeting of NSC, October 10, 1957.
64. Bowen, _Threshold of Space_ , 10–12; "Memorandum for the Deputy Secretary of Defense from C. C. Furnas," July 10, 1956, "Memorandum for Deputy Secretary of Defense from E. V. Murphree, Special Assistant for Guided Missiles," Subject: Use of the JUPITER Re-entry Test Vehicle as a Satellite, July 5, 1956, and "Memorandum for the Assistant Secretary of Defense (R&D) from Homer J. Stewart, Chairman, Advisory Group on Special Capabilities," Subject: VANGUARD and REDSTONE, June 22, 1956, SPI, GWU, file 1957 Jupiter Satellite Launch.
65. "Discussion at the 283rd Meeting of the National Security Council," May 3, 1956, AWF, NSC Series, box 7, file 283rd Meeting of NSC, May 3, 1955; NSC Action No. 1545, as referred to in "Draft Report on NSC 5520," 2, 4–6.
66. "Memorandum of Discussion at the 310th Meeting of the National Security Council," January 24, 1957, and "Memorandum of Discussion at the 322d Meeting of the National Security Council," May 10, 1957, in FRUS, 11:743–53; "U.S. Scientific Satellite Program: Cost Estimates," May 9, 1957, WHO, OSS, SUB, DoD Subseries, box 6, file Missiles and Satellites, Vol. 1 (1) [January 1956–May 1957].
67. "Recommendations in the Report to the President by the Technological Capabilities Panel of the Science Advisory Committee, ODM," October 2, 1956, RG 59, GRDS, box 87, NSC 5522 Memoranda; "Memorandum of Discussion at the 339th Meeting of the National Security Council," October 11, 1957, AWF, NSC Series, box 9, file 339th Meeting of NSC, October 10, 1957.
68. John L. Gaddis attributes this quote to Eisenhower. It was really said by Donald Quarles, quoted in Gaddis, _The Long Peace_ , 199. See also Hall, "The Eisenhower Administration," 64; memorandum of discussion at the 339th meeting of the National Security Council, October 11, 1957, AWF, NSC Series, box 9, file 339th Meeting of NSC, October 10, 1957.
69. For a more detailed discussion of space law, its evolution, and its current state, see Benko et al., _Space Law_ ; Fawcett, _International Law_ ; Jasentuliyama, _International Space Law_ ; Gal, _Space Law_ ; Morenoff, _World Peace_.
70. NSC 5814, June 20, 1958, RG 59, GRDS, box 96, file NSC 5814 (Memorandum), 3–4.
71. Letter to Dr. L. V. Berkner, president, Associated Universities, Inc., from O. G. Villard Jr., National Academy of Sciences, National Research Council, Space Science Board, January 22, 1959, WHO, OSAST (Killian/Kistiakowsky), box 15, file Space [January–June 1959] (6); "Memorandum to J. R. Killian Jr. from David Z. Beckler, Subject: Legal Aspects of Outer Space," April 24, 1959, WHO, OSAST (Killian/ Kistiakowsky), box 15, file Space [January–June 1959] (6); NSC 5814, RG 59, GRDS, 7–8.
72. Taubman, _Secret Empire_ , 193.
###### 4. Origins
Dwayne A. Day, "Invitation to Struggle: The History of Civilian-Military Relations in Space," in Logsdon et al., _Exploring the Unknown_ , 2:238; portions of this quote paraphrased from Bulkeley, _Sputnik Crisis_ , 83.
Robert L. Perry, "A History of Satellite Reconnaissance, Vol. 5: Management of the National Reconnaissance Program, 1960–1965," unpublished official history, National Reconnaissance Office, 1969, 2, declassified November 26, 1997.
1. McDougall, _. . . the Heavens and the Earth_ , 89; Davies and Harris, RAND _'s Role_ , 3–6; Kaplan, _Wizards_ , 1–5, 51–63; Burrows, _This New Ocean_ , 129.
2. Department of the Navy, Naval Historical Center, "United States Naval Aviation 1910–1995: Part 3. The Twenties 1920–1939," 50, <http://www.history.navy.mil/research/histories/naval-aviation-history/united-states-naval-aviation-1910-1995/part-3-the-twenties-1920-1920.html>; Ludwig, _Opening Space Research_ ; Gruntman, _Blazing the Trail_ , 187.
3. Hall, "Earth Satellites: A First Look by the United States Navy," in Hall, _Essays_ , 2:253–54; Burrows, _This New Ocean_ , 124–25; Robert L. Perry, "Origins of the USAF Space Program, 1945–1956," 1961, SPI, GWU, file Origins of the USAF Space Program, 8–9; Hall, "Earth Satellites," 254–59.
4. Hall, "Earth Satellites," 259, 273–74 (note 41).
5. As quoted in Perry, "Origins of the USAF Space Program," 9.
6. Hall maintains that LeMay rejected the joint program in mid-March 1946 and reconfirmed this stance on April 8, 1946, before the RAND study was commissioned. See Hall, "Earth Satellites," 258–60; Perry, "Origins of the USAF Space Program," 9; Augenstein, "Evolution," 3.
7. Burrows, _This New Ocean_ , 111–12; author's correspondence with Jim Eckles, Public Affairs Office, White Sands Missile Range, February 27, 2002.
8. Burrows, _This New Ocean_ , 125–26; McDougall, _. . . the Heavens and the Earth_ , 85–87; Perry, "Origins of the USAF Space Program," 10.
9. McDougall, _. . . the Heavens and the Earth_ , 89; Davies and Harris, RAND _'s Role_ , 3–6; Kaplan, _Wizards_ , 1–5, 51–63; Burrows, _This New Ocean_ , 129.
10. Burrows, _This New Ocean_ , 125–29; Peebles, _Corona Project_ , 5–9; Perry, "Origins of the USAF Space Program," 10–11.
11. Day, "Invitation to Struggle," 236; Peebles, _Corona Project_ , 5–6; Perry, "Origins of the USAF Space Program," 11–15.
12. Logsdon et al., _Exploring the Unknown_ , vol. 1: document II-2, 236–41.
13. Logsdon et al., _Exploring the Unknown_ , vol. 1: document II-2, 242; Perry, "Origins of the USAF Space Program," 11–15.
14. Logsdon et al., _Exploring the Unknown_ , vol. 1: document II-2, 242.
15. Perry, "Origins of the USAF Space Program," 16; Hall, "Earth Satellites," 258–59.
16. The JRDB was created in June 1946, under the War Department. With Vannevar Bush as chair, it expanded and formalized many of the Aeronautical Board's functions. Its primary responsibility was to prepare an integrated program of research and development to allow the military air arms to evaluate their own programs. It eliminated duplication of effort and had an influence on weapon design. It gave way in 1947 to the Research and Development Board, which emerged in the reorganization of the armed forces in 1947. See Hall, "Earth Satellites," 273–74 (note 41).
17. Perry argues that bureaucratic issues prevented a decision. See Perry, "Origins of the USAF Space Program," 16–17; Hall, "Earth Satellites," 263.
18. Burrows, _This New Ocean_ , 126–27.
19. Perry, "Origins of the USAF Space Program," 16–17; Hall, "Strategic Reconnaissance," 107; Hall, "Earth Satellites," 264.
20. Perry, "Origins of the USAF Space Program," 18–20. See also R. Cargill Hall, "Origins of U.S. Space Policy: Eisenhower, Open Skies, and Freedom of Space," in Logsdon et al., _Exploring the Unknown_ , 1:214–15; Hall, "Earth Satellites," 263 (notes 57, 275), 264.
21. Perry, "Origins of the USAF Space Program," 20–21; Day, "Invitation to Struggle," 236; Peebles, _Corona Project_ , 5–6; Richelson, _America's Secret Eyes in Space_ , 3–4.
22. Perry, "Origins of the USAF Space Program," 21–22; Day, "Invitation to Struggle," 236; Augenstein, "Evolution," 4.
23. Perry, "Origins of the USAF Space Program," 22–23; Day, "Invitation to Struggle," 236–37; memorandum for the vice chief of staff, from Lt. Gen. H. A. Craig, air force deputy chief of staff for materiel, "Subject: Earth Satellite Vehicle," January 12, 1948, SPI, GWU, file Air Force in Charge of Space; Augenstein, "Evolution," 4–5.
24. "Statement of Policy for a Satellite Vehicle," Gen. Hoyt S. Vandenberg, vice chief of staff, USAF, January 15, 1948, SPI, GWU, file Air Force in Charge of Space.
25. Perry, "Origins of the USAF Space Program," 21–24; "Statement of Policy"; letter to commanding general, Air Material Command, from Maj. Gen. L. C. Cragie, "Subject: Satellite Vehicle," January 16, 1948, SPI, GWU, file Air Force in Charge of Space; Day, "Invitation to Struggle," 236–37.
26. Merton E. Davies was a mathematician and engineer who in 1947 joined RAND, where he worked on some of its most important studies on space. Amrom H. Katz was a photo interpreter and expert on shutters and lenses who brought fifteen years of experience from the Wright-Patterson AFB Aerial Reconnaissance lab to RAND in 1954. Davies and Harris, RAND _'s Role_ , vii; Richelson, _America's Secret in Space_ , 14–15; Peebles, _Corona Project_ , 28.
27. Kecskameti used the earlier RAND research as the technical foundation for a satellite. Predicting a device orbiting at a 350-mile altitude with an orbital period of ninety minutes, he sought an oblique orbit (an orbital path that remains fixed between two defined latitudes) for reconnaissance and outlined requirements for a television reconnaissance system. Paul Kecskameti, "The Satellite Rocket Vehicle: Political and Psychological Problems," RAND Research Memorandum RM-567, October 4, SSB/GCSWS, 1–4.
28. Kecskameti, "The Satellite Rocket Vehicle," 5–8.
29. Kecskameti, "The Satellite Rocket Vehicle," 8–11, 13–14.
30. Kecskameti, "The Satellite Rocket Vehicle," 15–17, 8, 12.
31. The focal length is the distance from the center of a lens to the point where the light rays that make up a visual image come into focus. Thus for a lens with a focal length of 240 inches, the image that the camera sees is focused on a point 240 inches behind the lens. The greater the length, the closer objects appear and the greater the clarity. See Giancoli, _Physics_ , 533–34.
32. Burrows, _Deep Black_ , 26–51; R. Cargill Hall, "Postwar Strategic Reconnaissance and the Genesis of CORONA," in Day et al., _Eye in the Sky_ , 86.
33. Hall, "Strategic Reconnaissance," 107; Peebles, _Shadow Flights_ , 4.
34. Hall, "Postwar Strategic Reconnaissance," 87–88; Hall, "Strategic Reconnaissance," 108.
35. Hall, "Strategic Reconnaissance," 109; Peebles, _Shadow Flights_ , 4–7; Hall, "Postwar Strategic Reconnaissance," 88–90; Taubman, _Secret Empire_ , 35–38; _United States Strategic Bombing Survey_ , as quoted in Hall, "Strategic Reconnaissance," 109. See also Peebles, _Shadow Flights_ , 4–7.
36. Hall, "Strategic Reconnaissance," 109; Peebles, _Shadow Flights_ , 4–7; Hall, "Postwar Strategic Reconnaissance," 88–90; Peebles, _Corona Project_ , 1–3.
37. Hall, "Strategic Reconnaissance," 109–10; Peebles, _Shadow Flights_ , 6–7; Hall, "Postwar Strategic Reconnaissance," 90–92; Peebles, _Corona Project_ , 1–3; Taubman, _Secret Empire_ , 38–39.
38. Layton, _And I Was There_ , 516–17; Day et al., _Eye in the Sky_ , 2–3, 29; McDougall, _. . . the Heavens and the Earth_ , 82–83, 114–15.
39. Hall, "Strategic Reconnaissance," 109–10; Peebles, _Shadow Flights_ , 6–7; Hall, "Postwar Strategic Reconnaissance," 90–92.
40. Richelson, _American Espionage_ , 100–122; Hall, "Postwar Strategic Reconnaissance," 93–94; Peebles, _Shadow Flights_ , 8–9.
41. Richelson, _American Espionage_ , 102–27; Hall, "Postwar Strategic Reconnaissance," 93–96; Peebles, _Guardians_ , 6–10; Beschloss, _May-Day_ , 76–77; Hall, "Strategic Reconnaissance," 111–12.
42. For more information on these programs, see Peebles, _Moby Dick Project_ ; Peebles, _Shadow Flights_ ; Johnson and Smith, _Kelly_ ; Day et al., _Eye in the Sky_ , 98, 103–4, 267n38.
43. The RAND studies that followed the original 1946 work on satellite feasibility cover many topics. Most deal with highly technical aspects of satellites, including temperature and pressure densities, flight mechanics, and the identifying of ground launch points. For that reason I do not discuss them here. For more information, see Davies and Harris, RAND _'s Role_ , 9–15; Perry, "Origins of the USAF Space Program," 25–28; Day, "Invitation to Struggle," 237; Peebles, _Guardians_ , 44–45.
44. Memorandum to assistant for evaluation, DCS/D, "Attn: Colonel B. A. Schriever, From Major General C. P. Cabell, Director of Intelligence, Subject: Research and Development on Proposed RAND Satellite Reconnaissance Vehicle," March 17, 1951, SPI, GWU, file Satellite Reconnaissance Proposal 1951.
45. J. E. Lipp, R. M. Salter Jr., R. S. Wehner, R. R. Carhart, C. R. Culp, S. L. Gendler, W. J. Howard, and J. S. Thompson, "Utility of a Satellite Vehicle for Reconnaissance," Project RAND Report R-217, April 1951, SPI, GWU, file Project Rand Report "Utility of a Satellite Vehicle," ix. See also Coolbaugh, "Genesis," 283–84. It should be noted that Robert (Bob) Salter went on to also coauthor the 1954 Project FEEDBACK report before joining the Lockheed Corporation's Missile and Space Division, where he helped write their proposal for the WS-117L satellite under the PIED PIPER proposal process.
46. Lipp et al., "Utility of a Satellite Vehicle," 7–11.
47. Lipp et al., "Utility of a Satellite Vehicle," 1–7, 17–18.
48. Lipp et al., "Utility of a Satellite Vehicle," 40–45.
49. Lipp et al., "Utility of a Satellite Vehicle," 46–62. The report mentions solar energy, although the authors seem not to have considered it; the subject definitely received no detailed discussion.
50. Lipp et al., "Utility of a Satellite Vehicle," 12.
51. Lipp et al., "Utility of a Satellite Vehicle," 12.
52. The report defines resolution with respect to the smallest visible element in a photograph. Thus a resolution of fifty feet would indicate that the smallest item discernible in a photograph as a dot is approximately fifty feet in diameter. Though not as precise as a definition using optical criteria or television lines per millimeter, it is more functional. Variables such as brightness, scene contrast, and exposure time affect resolution. See Lipp et al., "Utility of a Satellite Vehicle," 17–18.
53. Lipp et al., "Utility of a Satellite Vehicle," 12–16, 39; Richelson, _American Espionage_ , 174; Perry, "Origins of the USAF Space Program," 28–30; Augenstein, "Evolution," 5–6; Davies and Harris, RAND _'s Role_ , 25–28.
54. Lipp et al., "Utility of a Satellite Vehicle," 63–69.
55. Memorandum for the deputy chief of staff, development, from Lt. Gen. Thomas D. White, December 18, 1952, "Subject: Satellite Vehicles," SSB/GCSWS, Attachment pp. 2–3.
56. Davies and Harris, RAND _'s Role_ , 25–29; Richelson, _American Espionage_ , 7–9; Peebles, _Corona Project_ , 9–10; Taubman, _Secret Empire_ , 65–66.
57. Davies and Harris, RAND _'s Role_ , 25–29; Richelson, _American Espionage_ , 7–9; Peebles, _Corona Project_ , 9–10, 28; Taubman, _Secret Empire_ , 65–67.
58. Perry, "Origins of the USAF Space Program," 30; Augenstein, "Evolution," 5–6.
59. Davies and Harris, RAND _'s Role_ , 33–34; Hall, "Strategic Reconnaissance," 115; York, _Arms_ , 204; Peebles, _Corona Project_ , 11–12; Taubman, _Secret Empire_ , 51–54.
60. Richard Leghorn also became instrumental in the formulation of the "Open Skies" concept as a consultant to Eisenhower's assistant for disarmament affairs (correspondence with Rick W. Sturdevant, deputy director of history, Air Force Space Command). Peebles, _Corona Project_ , 11–12; Davies and Harris, RAND _'s Role_ , 35.
61. Davies and Harris, RAND _'s Role_ , 33–34.
62. Peebles, _Corona Project_ , 11; Davies and Harris, RAND _'s Role_ , 35.
63. See Hall, "Strategic Reconnaissance," 115; York, _Arms_ , 204–5; Welzenbach, "Din Land," 22; Taubman, _Secret Empire_ , 54–56.
64. Peebles, _Shadow Flights_ , 63–64.
65. Hall, "Strategic Reconnaissance," 115–16; Peebles, _Corona Project_ , 10–12; Davies and Harris, RAND _'s Role_ , 35–42; Hall, "Origins of U.S. Space Policy," 217.
66. Hall, "Strategic Reconnaissance," 115–16; Peebles, _Corona Project_ , 10–12; Davies and Harris, RAND _'s Role_ , 35–42; Hall, "Origins of U.S. Space Policy," 217–18.
67. Hall, "Strategic Reconnaissance," 116; Peebles, _Corona Project_ , 12; Davies and Harris, RAND _'s Role_ , 44–45; Hall, "Postwar Strategic Reconnaissance," 99; Lewis, _Spy Capitalism_.
68. Peebles, _Corona Project_ , 10; Davies and Harris, RAND _'s Role_ , 44; Augenstein, "Evolution," 5–6; Richelson, _America's Secret Eyes in Space_ , 8–9; Perry, "Origins of the USAF Space Program," 30; Hall, "Origins of U.S. Space Policy," 218–19; Coolbaugh, "Genesis," 283–84.
69. Peebles, _Corona Project_ , 13; Davies and Harris, RAND _'s Role_ , 44.
70. Memorandum for the deputy chief of staff, development, December 18, 1952, "Subject: Satellite Vehicles," SSB/GCSWS, Attachment pp. 1–5.
71. Perry, "Origins of the USAF Space Program," 30–31; Hall, "Origins of U.S. Space Policy," 218; Burrows, _This New Ocean_ , 174–75; Davies and Harris, RAND _'s Role_ , 44; Richelson, _America's Secret Eyes in Space_ , 8–9; Augenstein, "Evolution," 5–6.
72. Perry, "Origins of the USAF Space Program," 31–32; Davies and Harris, RAND _'s Role_ , 47; Augenstein, "Evolution," 5–6; Peebles, _Corona Project_ , 13–15; Hall, "Origins of U.S. Space Policy," 218; Robert L. Perry, "A History of Satellite Reconnaissance, Vol. 1: CORONA," unpublished history, National Reconnaissance Office, 1964, revised 1973, declassified November 26, 1997, 2.
73. Perry, "Origins of the USAF Space Program," 32–33; Davies and Harris, RAND _'s Role_ , 47–48; Richelson, _America's Secret Eyes in Space_ , 8–9; Peebles, _Corona Project_ , 13–15; Perry, "A History of Satellite Reconnaissance, Vol. 1," 3.
74. For how the current historiography treats the FEEDBACK report, see Richelson, _America's Secret Eyes in Space_ , 10–11; Burrows, _This New Ocean_ , 175; Hall, "Postwar Strategic Reconnaissance," 105–6, 108–9, 121–23; Peebles, _Corona Project_ , 15–16; Hall, "Origins of U.S. Space Policy," 218–19.
75. _Project_ FEEDBACK _Summary Report_ , vol. 1, RAND Corporation, Report R-262, March 1, 1954, SPI, GWU, file Project FEEDBACK Report, declassified 1995–96, vii, 149–50, 164–66.
76. FEEDBACK _Report_ , 12–13, 126–27.
77. FEEDBACK _Report_ , 13–15, 124, 127–32.
78. FEEDBACK _Report_ , 132–37.
79. FEEDBACK _Report_ , 10–11, 110–25.
80. FEEDBACK _Report_ , 17–21.
81. Memorandum to assistant for evaluation, DCS/D, "Attn: Colonel B. A. Schriever, From Major General C. P. Cabell, Director of Intelligence, Subject: Research and Development on Proposed RAND Satellite Reconnaissance Vehicle," March 17, 1951, file Satellite Reconnaissance Proposal 1951, SPI, GWU; FEEDBACK _Report_ , 5–6.
82. FEEDBACK _Report_ , 93–96.
83. FEEDBACK _Report_ , 93–96.
84. Augenstein, "Evolution," 2.
###### 5. WS-117L
Durch, _National Interests_ , 36.
1. There is no way to know how often Eisenhower had briefings on WS-117L; no written records document the numbers. In all likelihood they were oral briefings and essentially kept him up to date. Probably General Twining, who was responsible for overhead reconnaissance, conducted them. Phone interview with R. Cargill Hall, October 17, 2002.
2. Coolbaugh worked on a large number of projects during his short time at the WADC, including studies on the performance of unguided missiles in Korea, on the feasibility of a tactical ballistic missile, and on tactical reconnaissance (using rockets), as well as investigation of decoy drones. James S. Coolbaugh, "The Beginning of the Air Force Satellite Program, 1953," unpublished treatise at the request of R. Cargill Hall, USAF historian, summer 1995, SPI, GWU, 1–9 (hereafter Coolbaugh memoirs).
3. Perry, "Origins of the USAF Space Program," 33; Richelson, _America's Secret Eyes in Space_ , 9.
4. Perry, "Origins of the USAF Space Program," 33.
5. Coolbaugh memoirs, 44. Perry also refers to "Pied Piper" as the nickname for the industry "investigations." However, he does not associate it with Project 1115. See Perry, "Origins of the USAF Space Program," 38.
6. Coolbaugh memoirs, 44.
7. Perry, "Origins of the USAF Space Program," 32–33; Davies and Harris, RAND _'s Role_ , 47–48; Peebles, _Corona Project_ , 13–15; Perry, "A History of Satellite Reconnaissance, Vol. 1," 3.
8. Peebles, _Corona Project_ , 15, 24.
9. Coolbaugh, "Genesis," 284–85.
10. Coolbaugh memoirs, 15; Coolbaugh, "Genesis," 286.
11. Coolbaugh memoirs, 16, 18–19.
12. Coolbaugh, "Genesis," 287; Coolbaugh memoirs, 16.
13. Coolbaugh memoirs, 16–17.
14. Coolbaugh memoirs, 17–18.
15. Coolbaugh memoirs, 17–18; Coolbaugh, "Genesis," 283–300; Hall, "Postwar Strategic Reconnaissance," 105.
16. Biggs also provided two exceptional procurement officers for his study contracts—Frank Daigle and Bob Washburn—who managed to accelerate the process further. See Coolbaugh memoirs, 9–10.
17. Coolbaugh memoirs, 19; Coolbaugh, "Genesis," 288.
18. Coolbaugh memoirs, 19–20.
19. Coolbaugh memoirs, 20–21.
20. Taubman places the date for the system requirement as March 16, 1955; however, no documents support his supposition. Coolbaugh memoirs, 22; Perry, "A History of Satellite Reconnaissance, Vol. 1," 3; "A Chronology of Air Force Space Activities," RG 340, ROSAF, box 367, file 132–59, Satellite Program vol. 2, p. 2; Taubman, _Secret Empire_ , 200.
21. Coolbaugh memoirs, 22.
22. RAND also had to use up funding for satellite studies, so Coolbaugh's scheme for spending lab funds was applied to the RAND studies. Coolbaugh memoirs, 23–24, 26–27.
23. A watt hour is a unit of energy equal to the power of 1 watt operating for one hour. Thus 10 watt hours would be a unit of energy equal to the power of 10 watts operating for one hour. Coolbaugh memoirs, 23–24, 29.
24. Coolbaugh memoirs, 29–30.
25. Coolbaugh and General Putt often joked about the funding problems. See Coolbaugh memoirs, 30–31.
26. Coolbaugh memoirs, 30–31. It is clear that Quarles had frequent briefings on the program. His daily diary indicated that the Advanced Reconnaissance System was the subject of several meetings and discussions. See entries for August 24, October 12, 1955, Quarles Papers, box 1, file Daily Diary 8/15/55–8/15/56 (6); entries for November 8 and 29, 1956, Quarles Papers, box 1, file Daily Diary 8/15/56–4/30/57 (3); phone interview with R. Cargill Hall, October 17, 2002.
27. Virtually every source that identifies General Operational Requirement No. 80 refers to the 1955 date for its release. The only exception that is evident and worthy of note is a document titled "The CORONA Story," which does not provide a date beyond the fact that General Operational Requirement No. 80 was released in 1954. Why this account differs is unclear. See Oder et al., "The CORONA Story," 4.
28. "General Operational Requirement for a Reconnaissance Satellite Weapon System," GOR No. 80, March 15, 1955, revised September 26, 1958, SPI, GWU, file General Operational Requirement; Kenneth E. Greer, "CORONA," _Studies in Intelligence, Supplement_ 17 (Spring 1973): 3, reprinted in Ruffner, CORONA, 3–24.
29. Hall, "Postwar Strategic Reconnaissance," 106.
30. Coolbaugh memoirs, 33–34; Coolbaugh, "Genesis," 292–93; Peebles, _Corona Project_ , 25–26; phone interview with R. Cargill Hall, October 17, 2002.
31. Coolbaugh memoirs, 36; Perry, "A History of Satellite Reconnaissance, Vol. 1," 3.
32. Coolbaugh memoirs, 36; "Truax Memoirs," unpublished manuscript chapters, SPI, GWU, file Truax Memoirs, 299.
33. Perry, "Origins of the USAF Space Program," 39–40; "Truax Memoirs," 299; Perry, "A History of Satellite Reconnaissance, Vol. 1," 3; Peebles, _Corona Project_ , 24; Coolbaugh, "Genesis," 294–95.
34. Perry, "Origins of the USAF Space Program," 39–40; "Truax Memoirs," 299; Perry, "A History of Satellite Reconnaissance, Vol. 1," 3; Peebles, _Corona Project_ , 24; Coolbaugh, "Genesis," 294–95.
35. Perry, "Origins of the USAF Space Program," 38; Peebles, _Corona Project_ , 24; Richelson, _America's Secret Eyes in Space_ , 13.
36. Captain Truax was head of the program office only briefly. In August 1956 air force colonel Frederick C. E. (Fritz) Oder became head of the Air Force Reconnaissance Office. Truax remained part of the program working under Oder. See Day et al., _Eye in the Sky_ , 107, 109.
37. Peebles, _Corona Project_ , 24; Coolbaugh memoirs, 43, 46; Coolbaugh, "Genesis," 295.
38. This effort slowed somewhat because of attempts to use component parts of WS-117L as part of the scientific satellite proposal. This effort eventually halted when the administration separated civilian and military programs. See Perry, "Origins of the USAF Space Program," 41–52.
39. Perry, "Origins of the USAF Space Program," 52–53; Perry, "A History of Satellite Reconnaissance, Vol. 1," 3–4; Richelson, _America's Secret Eyes in Space_ , 13; "A Chronology of Air Force Space Activities," 2:2.
40. Peebles, _Corona Project_ , 25. For a more detailed discussion of the second stage, see "Weapon System 117L Preliminary Development Plan, Advanced Reconnaissance System," January 14, 1956, RG 340, ROSAF, box 118, file 190–56, "Scientific Earth Satellite," 6–10; Perry, "A History of Satellite Reconnaissance, Vol. 1," 4.
41. Perry, "Origins of the USAF Space Program," 54; Coolbaugh, "Genesis," 297.
42. Letter to Eisenhower from Neil McElroy, May 7, 1958, AWF, DDE Diary Series, box 33, file Toner Notes—May 1958 (2); Perry, "Origins of the USAF Space Program," 54–55; Perry, "A History of Satellite Reconnaissance, Vol. 1," 3–4. As an indication of how complicated and confused accounts of the early program can be, Anthony Kenden, Curtis Peebles, and Philip J. Klass give a contract date with Lockheed of June 30, 1956. This may have been the original decision, but signing had to wait for October for funding reasons. Kenden, "U.S. Reconnaissance Satellite Programmes," 243; Klass, _Secret Sentries_ , 82–83; Peebles, _Corona Project_ , 25.
43. Coolbaugh memoirs, 47–48.
44. "Truax Memoirs," 301–2.
45. "Truax Memoirs," 303; Perry, "A History of Satellite Reconnaissance, Vol. 1," 3–5; Hall, "The Eisenhower Administration," 64; Oder et al., "The CORONA Story," 5.
46. Perry, "A History of Satellite Reconnaissance, Vol. 1," 5–6. There is only one account of the program that implies that ample funding was available. This is clearly an error; I include it only to indicate the level of confusion about the program. See Hochman and Wong, _Satellite Spies_ , 100–101.
47. Donald Quarles died in 1959. Although his personal correspondence, lists of meetings, and speeches are at the Eisenhower Library, nothing relating to his decisions on satellite reconnaissance appears in this collection. See Quarles Papers, boxes 1–27; Perry, "A History of Satellite Reconnaissance, Vol. 1," 6–7; Hall, "Origins of U.S. Space Policy," 222.
48. Perry, "Origins of the USAF Space Program," 41–52.
49. "Satellite Programs in the Department of Defense," October 25, 1957, Harlow Records, box 1, file DoD Report to Senate Preparedness Investigating Subcommittee, Missiles (October 1957) (1); "Discussion at the 339th Meeting of the National Security Council, 10 October 1957," AWF, NSC Series, box 9, file 339th Meeting of NSC, October 10, 1957; Perry, "A History of Satellite Reconnaissance, Vol. 1," 6–14.
50. Perry, "A History of Satellite Reconnaissance, Vol. 1," 14–17.
51. Oder and Schriever were certainly not the only ones viewing the "space for peace" policy in a negative light. Richard Leghorn put forward a memorandum in July 1957 concerning the role of space for peace and satellite reconnaissance. His proposal called for the direct linkage of satellite reconnaissance and arms-control measures. This offered a political windfall for the United States and would allow for the satellite reconnaissance effort to come out of the shadows when the time was ripe. There is no evidence that this information was available to General Schriever or Oder. It is unlikely that this memorandum had a major impact on the administration. Eisenhower had already linked the idea of overhead reconnaissance and arms control in his Open Skies proposal, and there is no indication that the memorandum reached the key players in his administration. "The Reconnaissance Satellite and 'Space-for-Peace' Political Action," July 31, 1957, WHO, OSAST, box 3, file Space-Satellites [July 1956–February 1960]; "Richard S. Leghorn (PBCFIA) to D. Z. Beckler, re: Political Action and Unauthorized Overflight of the USSR," July 26, 1956, and "Richard Leghorn to Mr. Beckler (PSAC) re: The Reconnaissance Satellite and 'Space for Peace' Political Action," July 31, 1957, WHO, OSAST, box 3, file Space-Satellites [July 1956–February 1960].
52. The two accounts of Perry and Oder are remarkably similar. See Perry, "A History of Satellite Reconnaissance, Vol. 1," 17–18; Oder et al., "The CORONA Story," 10–13.
53. Perry, "A History of Satellite Reconnaissance, Vol. 1," 18–22; Peebles, _Guardians_ , 45.
54. Perry, "Origins of the USAF Space Program," 41–56; Perry, "A History of Satellite Reconnaissance, Vol. 1," 22–23.
55. The Western Development Division officially became the Ballistic Missile Division in August 1957.
56. Perry, "The History of Satellite Reconnaissance, Vol. 1," 24.
57. Hall, "Postwar Strategic Reconnaissance," 107–8; Richelson, _America's Secret Eyes in Space_ , 19; Peebles, _Corona Project_ , 30; Klass, _Secret Sentries_ , 86; Kenden, "U.S. Reconnaissance Satellite Programmes," 244.
58. William Burrows mentions a drop-film system associated with WS-117L—perhaps an addition in 1957, although the date is still very unclear. This variation, which eventually became CORONA, receives further discussion in chapter 6, along with the RAND work involving such a system.
59. Peebles, _Corona Project_ , 25.
60. Robert L. Perry, "A History of Satellite Reconnaissance, Vol. 2-A: SAMOS," unpublished history, National Reconnaissance Office, 1963, revised 1973, declassified June 2001, 5.
61. Hall, "Postwar Strategic Reconnaissance," 107–8; Ruffner, CORONA, 3; Peebles, _Corona Project_ , 25; Oder et al., "CORONA Story," 4.
62. Divine, _Sputnik Challenge_ , 11.
###### 6. Satellite Photography
Kecskameti, "The Satellite Rocket Vehicle," 8.
1. Levine, _Missile_ , 58.
2. For a more detailed account of _Sputnik_ and the resulting crisis, see Bulkeley, _Sputnik Crisis_ ; Divine, _Sputnik Challenge_ ; Burrows, _This New Ocean_ ; McDougall, _. . . the Heavens and the Earth_.
3. Peebles, _Corona Project_ , 39–42; Bottome, _Missile Gap_ , 30–45, 87–88, 92–106; Levine, _Missile_ , 59, 61–66; Taubman, _Secret Empire_ , 213–14; Krug, _Presidential Perspectives_ , 23–24.
4. Hethloff, _Suddenly_ , 1, 11; Levine, _Missile_ , 59–60.
5. McDougall, _. . . the Heavens and the Earth_ , 142, 152–53; Licklider, "The Missile Gap Controversy," 615, 609; Krug, _Presidential Perspectives_ , 23–24; Bottome, _Missile Gap_ , 79–91.
6. "Open Letter to President Eisenhower," _New York Times_ , November 7, 1957, WHO, OSS, SUB, ALPHA, box 23, file Satellites [October 1957–February 1960] (1); Ambrose, _Eisenhower: Soldier and President_ , 449–50; Alexander, _Holding the Line_ , 213–14; Eisenhower, _White House Years_ , 2:205–6.
7. The information that the U-2 produced was accurate enough for the CIA to predict a Soviet satellite launch in November 1957. Klass, _Secret Sentries_ , 30–38, 51; Killian, _Sputniks_ , 6.
8. "Memorandum for Secretary Quarles and General Cutler," October 8, 1957, and "Statement by the President," October 9, 1957, WHO, OSS, SUB, ALPHA, box 23, file Satellites [October 1957–February 1960] (1); "Memorandum of Conference with the President, 8 October 1957," AWF, DDE Diary Series, box 27, file October '57 Staff Notes (2); statement by the president, "Summary of Important Facts in the Development by the United States of an Earth Satellite," October 9, 1957, AWF, Admin. Series, box 37, file U.S. Satellites.
9. Some personnel in the administration thought such an approach not the best. See "Memorandum for Mr. Victor Cooley from Mr. David Z. Beckler, Subject: Satellites and Missiles, 8 October 1957," WHO, OSAST, box 3, file Space.
10. Ambrose, _Eisenhower: The President_ , 430. See also Krug, _Presidential Perspectives_ , 24–25.
11. "Memorandum of Conference with the President, 16 October 1957," AWF, DDE Diary Series, box 27, file October '57 Staff Notes (2); Killian, _Sputniks_ , xv, 11–20, 122–23; McElheny, _Insisting_ , 310–17; Killian, _Education_ , 326–28.
12. "Memorandum of a Conference, President's Office, White House," October 8, 1957, in FRUS _1955–1957: United Nations_ , 755–56; Eisenhower, _White House Years_ , 2:210–19.
13. The flurry of activity included the army's proposal for a reconnaissance satellite that virtually duplicated the WS-117L program. See "Briefing on Army Satellite Program," November 19, 1957, WHO, OSAST (Killian/Kistiakowsky), box 15, file Space [November 1957] (2); letter to Neil McElroy, secretary of defense, from Eisenhower, October 17, 1957, WHO, OSS, SUB, DoD Subseries, box 11, file Secretary of Defense.
14. "Defense Space Projects Supplement to Department of Defense Report to National Security Council on Status of United States Military Programs as of 30 June 1959: Annex B," October 26, 1959, WHO, OSANSA, NSC Series, Status of Projects Subseries, box 8, file NSC 5912 (2); letter to Eisenhower from Secretary of Defense Neil McElroy, January 29, 1958, WHO, OSS, SUB, DoD Subseries, box 8, file Missiles and Satellites, Military Satellite Program, December 31, 1958.
15. Hall, "Postwar Strategic Reconnaissance," 108–9. See also Davies and Harris, RAND _'s Role_ , 69–70; memorandum to J. L. Hult from A. H. Katz, "Subject: Recommendations, Formal and Informal, on Reconnaissance Satellite Matters," RAND Corporation Memo No. M-3008, 5–11–1959, SPI, GWU, file "Amrom Katz Recommendations on Reconnaissance Satellites."
16. Davies and Harris, RAND _'s Role_ , 69–70; Hall, "Postwar Strategic Reconnaissance," 108–9; Peebles, _Corona Project_ , 27–28.
17. J. H. Huntzicker and H. A. Lieske, "Physical Recovery of Satellite Payloads—A Preliminary Investigation," RAND Corporation Research Memorandum, RM-1811, June 26, 1956, SSB/GCSWS, 1; Peebles, _Corona Project_ , 27–28; Taubman, _Secret Empire_ , 225–62.
18. Huntzicker and Lieske, "Physical Recovery," 1–7.
19. Huntzicker and Lieske, "Physical Recovery," 8–11.
20. Huntzicker and Lieske, "Physical Recovery," 12–15.
21. Memorandum to J. L. Hult from A. H. Katz, "Subject: Recommendations, Formal and Informal, on Reconnaissance Satellite Matters."
22. The problems relating to heating of the capsule during reentry somewhat eased with the discovery of the ablative heat shield. During a summer study in 1956, the ablative nose cone being developed for ICBMs emerged as a solution. See Richelson, _America's Secret Eyes in Space_ , 14–16.
23. Richelson, _America's Secret Eyes in Space_ , 14–15; Peebles, _Corona Project_ , 27–28; Davies and Harris, RAND _'s Role_ , vii.
24. By September 1957, in the face of severe financial restrictions, the new Itek (short for Information Technology) corporation took over the Boston Lab's role. Richard Leghorn founded Itek to continue work on developing high-altitude cameras. See Peebles, _Corona Project_ , 28–30.
25. Other people had already pointed out the value of an IRBM as a booster for a multistage rocket. Ramo-Woolridge had proposed such a concept on April 1, 1957. See Ramo-Woolridge Document GM67.3–49, Subject: "Proposed Use of IRBM as Booster for Multi-stage Vehicles, 1 April 1957," in S. A. Grassly, "Documentary History of DISCOVERER," November 1971, History Office, Chief of Staff, Space & Missiles Systems Organization, SPI, GWU, Document No. 3.
26. "Letter to DDE from James Killian," December 20, 1956, WHO, OSANSA, NSC Series, Subject Subseries, box 7, file President's Board of Consultants on Foreign Intelligence Activities, First Report to the President [December 1956–August 1958] (1).
27. "Memorandum of Conference with the President," October 24, 1957, AWF, DDE Diary Series, box 27, file October '57 Staff Notes (1); "Memorandum of Conference with the President," October 28, 1957, WHO, OSS, SUB, ALPHA, box 6, file Board of Consultants [on Foreign Intelligence Activities] (4).
28. McElheny, _Insisting_ , 316–18; Peebles, _Corona Project_ , 42–43; Albert D. Wheelon, "CORONA: A Triumph of American Technology," in Day, _Eye in the Sky_ , 32–33; Hall, "Postwar Strategic Reconnaissance,"109–10.
29. Peebles, _Corona Project_ , 42–43; Oder et al., "The CORONA Story," 14–15; Wheelon, "CORONA," 32–33; Hall, "Postwar Strategic Reconnaissance," 109–10.
30. The history by Oder and that by Perry and Greer are virtually identical in wording in many respects. Oder et al., "The CORONA Story," 13–14; Perry, "A History of Satellite Reconnaissance, Vol. 1," 26; Greer, "CORONA," 4–5.
31. Perry, "A History of Satellite Reconnaissance, Vol. 1," 24–27.
32. Davies and Harris, RAND _'s Role_ , 95; Peebles, _Guardians_ , 46.
33. "Priorities for Ballistic Missiles and Satellite Programs," January 22, 1958, AWF, NSC Series, box 9, file 352nd Meeting of NSC, January 22, 1958; "Memorandum for the Special Assistant to the President for National Security Affairs, Subject: Priorities for Ballistic Missile and Space Programs," August 10, 1959, WHO, OSANSA, NSC Series, Briefing Notes Subseries, box 14, file [Outer Space Policy and Activities] (2) [1958–60].
34. Richelson, _America's Secret Eyes in Space_ , 29–30; Davies and Harris, RAND _'s Role_ , 95–97; "Priorities for Ballistic Missiles and Space Programs," May 13, 1959, AWF, NSC Series, box 11, file 406th Meeting of NSC, May 13, 1959; "Priorities for Ballistic Missiles and Space Programs," August 18, 1959, AWF, NSC Series, box 11, file 417th Meeting of NSC, August 18, 1959; "Memorandum for the Special Assistant to the President for National Security Affairs, from James H. Douglas, Subj: Priorities for Satellite Programs," WHO, OSANSA, NSC Series, Briefing Notes Subseries, box 14, file [Outer Space Policy and Activities] (1) [1958–60]; "Memorandum for the Special Assistant to the President for National Security Affairs, from Donald A. Quarles, Subj: NSC 1846," April 21, 1958, WHO, OSS, SUB, DoD Subseries, box 8, file Missiles and Satellites, Military Reconnaissance Satellite Program; "Memorandum for the Secretary of Defense, Subj: NSC 1846 'Priorities for Ballistic Missiles and Satellite Programs,'" April 21, 1958, WHO, OSS, SUB, DoD Subseries, box 8, file Missiles and Satellites, Military Reconnaissance Satellite Program; "Letter to the President, from Secretary McElroy," May 7, 1958, WHO, OSS, SUB, DoD Subseries, box 8, file Missiles and Satellites, Military Reconnaissance Satellite Program.
35. M. E. Davies et al., "A Family of Recoverable Reconnaissance Satellites," RAND Corporation Research Memorandum, RM-2012, November 12, 1957, SPI, GWU, file "A Family of Recoverable Reconnaissance Satellites," 1–4, 27–32; Davies and Harris, RAND _'s Role_ , 69–72, 76–89; Peebles, _Corona Project_ , 42–44.
36. Davies et al., "A Family of Recoverable Reconnaissance Satellites," 27–32.
37. Davies et al., "A Family of Recoverable Reconnaissance Satellites," iii–v, 5–19, 21–23.
38. Davies et al., "A Family of Recoverable Reconnaissance Satellites," iii–v, 17–19, 21–23.
39. Davies et al., "A Family of Recoverable Reconnaissance Satellites," 23–26.
40. "An Earlier Reconnaissance Satellite System," RAND Corporation Recommendation to the Air Staff, November 12, 1957, SSB/GCSWS, 1–19.
41. Lockheed Aircraft Corporation, Missile Systems Division, "WS 117L Development Plan for Program Acceleration," Contract AF 04(647)-97, LMSD-2832, January 6, 1958, pp. 1–2, SPI, GWU, file "Acceleration of WS-117L"; Oder et al., "The CORONA Story," 17–18.
42. "WS-117L Development Plan for Program Acceleration," 1–2, section 1.1.1, System Considerations; Oder et al., "The CORONA Story," 18.
43. "WS-117L Development Plan for Program Acceleration," section 1.1.1.2, Program I, Section 1.1.1.3, Program II.
44. "WS-117L Development Plan for Program Acceleration," section 1.1.1.3.
45. "WS-117L Development Plan for Program Acceleration," sections 1.1.1.4–1.1.1.6.
46. "WS-117L Development Plan for Program Acceleration," section 1.2.
47. "WS-117L Development Plan for Program Acceleration," section 1.2.
48. Oder et al., "The CORONA Story," 15–16, 20; Peebles, _Corona Project_ , 42–43; Greer, "CORONA," 4–5; McElheny, _Insisting_ , 316–18.
49. Hall, "Postwar Strategic Reconnaissance," 111–12; Oder et al., "The CORONA Story," 18–20; memorandum of conference with the president, February 7, 1958, WHO, OSS, SUB, ALPHA, box 14, file Intelligence Matters (4); memorandum of conference with the president, February 10, 1958, AWF, DDE Diary Series, box 30, file Staff Notes February 1958.
50. Memorandum of conference with the president, February 10, 1958, WHO, OSS, SUB, ALPHA, box 14, file Intelligence Matters (4); memorandum of conference with the president, February 10, AWF, DDE Diary Series, box 30, file Staff Notes February 1958.
51. Memorandum of conference with the president, February 10, 1958, WHO, OSS, SUB, ALPHA; memorandum of conference with the president, February 10, 1958, AWF.
52. The rivalry between the services for funding for space programs, based on very thin justification, serves as one of the clearest examples of the interservice rivalry that so troubled Eisenhower. See Richelson, _America's Secret Eyes in Space_ , 24–27; Bowen, _Threshold of Space_ , 16–28.
53. Memorandum for undersecretary of the air force from Maj. Gen. H. C. Donnelly, assistant deputy chief of staff, plans and programs, "Subject: Proposed Statements for Senate Hearings on Missiles," November 21, 1957, RG 340, ROSAF, box 134, file 21–54 Guided Missiles General; "USAF Pushes Pied Piper Space Vehicle," 26–27; Klass, _Secret Sentries_ , 87–88; Richelson, _America's Secret Eyes in Space_ , 20, 26–27.
54. The lack of radiated signals from the film-return system helped to decrease the risk of Soviet reaction. More difficult to detect, the film-drop camera had the potential to limit the political repercussions and decrease the risks of Soviet countermeasures. Greer, "CORONA," 5–6; Burrows, _Deep Black_ , 86; Hall, "Postwar Strategic Reconnaissance," 111–12; Day, "Strategy for Reconnaissance," in Day et al., _Eye in the Sky_ , 138–39.
55. Hall, "Postwar Strategic Reconnaissance," 112; Burrows, _Deep Black_ , 55–56.
56. Memorandum of conference with the president, January 17, 1958, WHO, OSS, SUB, ALPHA, box 14, file Intelligence Matters (4).
57. The origin of the name CORONA is still somewhat debated. The usual explanation is that it came from the typewriter, a Smith-Corona, being used in March 1958 during a meeting that finally worked out the satellite concept. However, some old-timers argue it had more to do with the cigar one of the team smoked. No matter the source, it adds flavor to the program. See Peebles, _Corona Project_ , 44; Day et al., _Eye in the Sky_ , 6; memorandum for the record, April 21, 1958, WHO, OSS, SUB, ALPHA, box 14, file Intelligence Matters (5).
58. The project manager for WS-117L, Fritz Oder, took no umbrage at the creation of CORONA. Intimately familiar with the problems in finding resources for the WS-117L system, he was positive that money for both programs would not be available from the air force. See Richelson, _America's Secret Eyes in Space_ , 26.
59. Oder et al., "The CORONA Story," 15–21; Perry, "A History of Satellite Reconnaissance, Vol. 1," 37–46; Hall, "Postwar Strategic Reconnaissance," 113; Richelson, _America's Secret Eyes in Space_ , 26–28; Peebles, _Corona Project_ , 43–46; Ruffner, CORONA, 5–6; Wheelon, "CORONA," 33–34.
60. Perry, "A History of Satellite Reconnaissance, Vol. 1," 39–43; Richelson, _America's Secret Eyes in Space_ , 27–28; Peebles, _Corona Project_ , 45; Ruffner, CORONA, 6–7; Oder et al., "The CORONA Story," 21–22.
61. "Proposed Initial Press Release," November 6, 1958, WHO, OSS, SUB, ALPHA, box 12, file Discoverer [March–December 1958] (1); Perry, "A History of Satellite Reconnaissance, Vol. 1," 39–44; Memorandum to Lockheed Aircraft Corporation from commander AFBMD, March 12, 1958, in Grassly, "Documentary History," Document 30; memorandum for Major General Funk, "Subject: Reorientation WS 117L Program IIA," March 14, 1958, in Grassly, "Documentary History," Document 32.
62. Day et al., _Eye in the Sky_ , 6–9.
63. Peebles, _Guardians_ , 47–49; Peebles, _Corona Project_ , 46–49.
64. The best accounts to date of the CORONA program are Day et al., _Eye in the Sky_ , and Peebles, _Corona Project_. A small sampling of the documents and imagery appears in Ruffner, CORONA; Oder et al., "The CORONA Story."
###### 7. SENTRY/SAMOS
Dwight D. Eisenhower, October 9, 1957, quoted in Ambrose, _Eisenhower: The President_ , 430.
Perry, "A History of Satellite Reconnaissance, Vol. 2-A," 136.
1. McElheny, _Insisting_ , 328; "Memorandum of Conference with the President," March 12, 1958, AWF, DDE Diary Series, box 31, file Staff Notes, March 1958 (2); "Defense Space Projects Supplement to Department of Defense Report to National Security Council on Status of United States Military Programs as of 30 June 1959: Annex B."
2. Hall, "The Eisenhower Administration," 65; Augenstein, "Evolution," 11–12.
3. "Memorandum for the President," July 29, 1958, AWF, Admin. Series, box 34, file Stans, Maurice H. (director Bureau of the Budget) 1958 (1); Bowen, _Threshold of Space_ , 24–25; Richard J. Barber Associates, Inc., "The Advanced Research Projects Agency, 1957–1964," December 1975, SPI, GWU, file "ARPA History."
4. "Defense Space Projects Supplement to Department of Defense Report to National Security Council on Status of United States Military Programs as of 30 June 1959: Annex B"; memorandum for the secretary of the air force from Roy Johnson, director, Advanced Research Projects Agency, "Subject: Reconnaissance Satellites and Manned Space Exploration," February 28, 1958, SSB/GCSWS; Perry, "A History of Satellite Reconnaissance, Vol. 1," 43–44, 55–56.
5. Barber Associates, Inc., "The Advanced Research Projects Agency," III-11–III-12. For a brief discussion of the program's complexities, see memorandum for General Cutler from Bureau of the Budget, June 20, 1958, WHO, OSANSA, NSC Series, Briefing Notes Subseries, box 14, file [Outer Space Policy and Activities] (2) [1958–60].
6. Memorandum for the special assistant to the president for national security affairs, July 28, 1958, and defense recommendation on the reconnaissance satellite (submitted to the NSC on July 3, 1958), July 31, 1958, WHO, OSANSA, NSC Series, Briefing Notes Subseries, box 7, file Earth Satellites (1) [1955–58]; "Attachment B, Memorandum for Dr. Killian, Subject: Notes on the Space Panel Meeting," July 8, 1958, WHO, OSAST (Killian/Kistiakowsky), box 15, file Space Notebook [Killian, 1958–59] (2); "National Space Activities: A Brief Summary," August 12, 1958, WHO, OSS, SUB, DoD Subseries, box 8, file Missiles and Satellites, National Space Activities—A Brief Summary, August 12, 1958.
7. Memorandum for the space file from R. O. Piland, "Subject: Space Budget and Program Responsibilities," July 14, 1958, WHO, OSAST (Killian/Kistiakowsky), box 15, file Space Notebook [Killian, 1958–59] (3).
8. There are many examples of leaked information. See "SENTRY Satellite Shot Planned for Dec. 15"; "USAF Pushes Pied Piper Space Vehicle"; "Lockheed Builds WS-117L Nerve Center"; "Memorandum for Dr. James R. Killian Jr. from Robert Cutler, Special Assistant to the President, Subject: Scientific and Reconnaissance Satellites," January 20, 1958, WHO, OSAST, box 3, file Space Satellites.
9. Letter from McElroy to Eisenhower, January 29, 1958, WHO, OSS, SUB, DoD Subseries, box 8, file Missiles and Satellites, Military Satellite Program, December 31, 1958; Peebles, _Corona Project_ , 53; Perry, "A History of Satellite Reconnaissance, Vol. 1," 73; Perry, "A History of Satellite Reconnaissance, Vol. 2-A," 25; "Memorandum for Mr. Lay from Gordon Gray," March 3, 1959, WHO, OSANSA, NSC Series, Briefing Notes Subseries, box 7, file [Earth Satellites (1) [1955–58]; "National Space Activities: A Brief Summary," August 12, 1958, WHO, OSS, SUB, DoD Subseries, box 8, file Missiles and Satellites, National Space Activities—A Brief Summary, August 12, 1958.
10. Lee Bowen, "The Threshold of Space: The Air Force in the National Space Program 1945–1959," September 1960, SSB/GCSWS, 31; Letter from McElroy to Eisenhower, January 29, 1958; Peebles, _Corona Project_ , 53; Perry, "A History of Satellite Reconnaissance, Vol. 1," 73; Perry, "A History of Satellite Reconnaissance, Vol. 2-A," 25; "Memorandum for Mr. Lay from Gordon Gray," March 3, 1959; Richard J. Barker Associated, Inc., "The Advanced Research Projects Agency, 1957–1964," III-12.
11. ARPA Order No. 38-59 separated out MIDAS as its own system. See "Memorandum for the Secretary of Defense from Malcolm A. MacIntyre, Subject: Operational Responsibility in Respect to MIDAS," May 8, 1959, RG 340, ROSAF, box 367, file 132-59, Satellite Programs vol. 1; Richard J. Barker Associated, Inc., "The Advanced Research Projects Agency, 1957–1964," III-12.
12. A memorandum of July 29, 1958, places the WS-117L budget from ARPA at $186 million. See "Memorandum for the President," July 29, 1958, AWF, DDE Diary Series, box 35, file Staff Memos July 1958 (1). No accurate information on budget amounts is yet available. Figures for the WS-117L budget for FY 1959 range from $100 million to $239 million. See "Memorandum of Conference with the President," November 11, 1957, AWF, DDE Diary Series, box 28, file November '57 Staff Notes; "Chart on Outer Space Vehicles" (undated), WHO, OSAST (Killian/Kistiakowsky), box 15, file Space [December 1957] (3); Richard J. Barber Associated, Inc., "The Advanced Research Projects Agency, 1957–1964," III-12–III-13.
13. SENTRY Space System Development Plan, January 30, 1959, RG 340, ROSAF, box 367, file 132-59 "Satellite Program Vol. 2," I-1-2–I-1-3. See also "Letter to the President from Secretary of Defense McElroy," May 7, 1958, WHO, OSS, SUB, DoD Subseries, box 8, file Missiles and Satellites, Military Reconnaissance Satellite Program, March 31, 1958.
14. SENTRY Space System Development Plan, I-2-1–I-2-3.
15. All individual subsystems for WS-117L received a letter designation for management and development purposes. The photographic portion was subsystem E; electronic intelligence, subsystem F. See Perry, "A History of Satellite Reconnaissance, Vol. 2-A," 8.
16. Perry, "A History of Satellite Reconnaissance, Vol. 2-A," iii, 8–9, 16, 19; SENTRY Space System Development Plan, I-2-3–I-2-9, I-2-11–I-2-12; Day, "The Development and Improvement of the CORONA Satellite," in Day et al., _Eye in the Sky_ , 71–72.
17. Day, "Development and Improvement," 63, 74–75.
18. The recovery program for SAMOS experienced a great deal of opposition and turmoil in 1958–60. See Perry, "A History of Satellite Reconnaissance, Vol. 2-A," 20–27, 74–75, 84–85; "SAMOS," undated, SPI, GWU, 1; Day, "Development and Improvement," 71–72.
19. SENTRY Space System Development Plan, I-2-3, I-2-5, I-2-7, I-2-12.
20. SENTRY Space System Development Plan, I-2-12–I-2-13.
21. SENTRY Space System Development Plan, I-3-2–I-3-3.
22. For the period up to and including FY 1958, the overall costs for SENTRY amounted to approximately $79,340,000. Lockheed received most of these funds for the development of the various subsystems. Additional funding went to facilities ($5,718,000), the Rome Air Development Center ($2,820,000), the Air Force Cambridge Research Center ($2,842,000), and other, smaller-cost elements. See SENTRY Space System Development Plan, I-3-6, I-5-1–I-5-12, II-1-2.
23. Perry, "A History of Satellite Reconnaissance, Vol. 2-A," 10–13; McElheny, _Insisting_ , 331–33.
24. Perry, "A History of Satellite Reconnaissance, Vol. 2-A," 13–15; Burrows, _This New Ocean_ , 227–28.
25. Perry, "A History of Satellite Reconnaissance, Vol. 2-A," 62–68, 151–53, 166–68, 170–73; "SAMOS," 1–4; letter to Eisenhower from James H. Douglas, deputy director of defense, November 19, 1960, WHO, OSANSA, NSC Series, Briefing Notes Subseries, box 13, file Missiles and Military Space Programs (1) [1955–61].
26. Day, "Development and Improvement," 73–74.
27. Memorandum to the director, Advanced Research Projects Agency, "Subject: MIDAS Progress Report," August 31, 1959, RG 340, ROSAF, box 367, file 132-59 "Satellite Program Vol. 2."
28. "ARPA Order No. 38–59," November 5, 1958, SSB/GCSWS.
29. "Report of the Ad Hoc Technical Advisory Board to the Advanced Research Projects Agency on the Evaluation of Technical Feasibility of Infrared Missile Defense Alarm from Satellite Vehicles," February 26, 1959, RG 340, ROSAF, box 367, file 132-59 "Satellite Program Volume 1"; "Report of the Early Warning Panel," March 13, 1959, WHO, OSAST (Killian/Kistiakowsky), box 4, file "ICBM-Early Warning [November 1957–December 1960]," 10–11.
30. MIDAS Operational System Description, December 18, 1959, RG 340, ROSAF, box 367, file 132-59 "Satellite Program," iii, I-1–I-3.
31. Earlier in the program Lockheed Missile and Space Division predicted twenty satellites would be necessary. See Missile Defense Alarm System Development Plan, January 30, 1959, revised June 5, 1959, RG 340, ROSAF, box 367, file 132-59 "Satellite Programs Vol. 2," I-2–2; MIDAS presentation, August 12, 1959, LMSD-445610, NARA II, RG 340, ROSAF, box 368, file 132-59 "Satellite Program," 1–3.
32. The probable locations of the three stations were New Boston, New Hampshire; Ottumwa, Iowa; and Fort Stevens, Oregon. See MIDAS operational system description, December 18, 1959, RG 340, ROSAF, box 367, file132-59 "Satellite Program," 2–5.
33. For a discussion of the satellite-handling and -launching systems, see MIDAS operational system description, 2-5–2-9.
34. "Summary: June, July, August 1960, Discoverer Project (Research and Development Satellites)," AWF, Admin. Series, box 15, file Gates, Thomas S., Jr., 1959–61(2); letter to Eisenhower from Secretary Neil McElroy, April 30, 1959, WHO, OSS, SUB, DoD Subseries, box 6, file Missiles and Satellites, Vol. 2 (4) [July 1958–January 1959]; Day, "Development and Improvement," 52–59, appendix B, 236.
35. Day et al., _Eye in the Sky_ , 72–74; letter to Eisenhower from James H. Douglass, April 11, 1960, AWF, Admin. Series, box 15, file Gates, Thomas S., Jr., 1959–61 (4); NSC 6013: "Status of the United States Military Programs as of 30 June 1960," September 6, 1960, WHO, OSANSA, NSC Series, Status of Projects Subseries, box 9, file NSC 6013, Status of U.S. National Security Programs on June 30, 1960 (1).
36. Letter to the commander ARDC from Brig. Gen. O. J. Ritland, "Subject: WS-117L Program," February 11, 1959, SSB/GCSWS.
37. Letters to General White, chief of staff, from Lieutenant General Schriever, September 15, 1959, SPI, GWU, file "Problems: SAMOS, MIDAS." See also letter to the commander ARDC from Brig. Gen. O. J. Ritland, "Subject: WS-117L Program," February 11, 1959, SSB/GCSWS.
38. Perry, "A History of Satellite Reconnaissance, Vol. 2-A," 35–38.
39. Perry, "A History of Satellite Reconnaissance, Vol. 2-A," 35–38; memorandum for the secretary of the air force, "Subject: Transfer of SAMOS Development Program to the Department of the Air Force," November 17, 1959, and memorandum for the secretary of defense from Joseph V. Charyk, undersecretary of the air force, "Subject: Transfer of the SAMOS, MIDAS and DISCOVERER Development Programs to the Department of the Air Force," February 18, 1960, RG 340, ROSAF, box 367, file 132-59 "Satellite Program Vol. 2"; NASA History Program Office, "The Beginnings of a Division of Labor: Advent and Syncom," www.history.nasa.gov/sp-4102/ch8.htm#214.
40. Herbert York, quoted in Perry, "A History of Satellite Reconnaissance, Vol. 2-A," 40.
41. York refused to approve funding for those aspects of the SAMOS program that he found highly questionable (the readout and ferret systems). Overall funding for FY 1960 amounted to $159.5 million; funding for FY 1961 was less, but the amount is still classified. See Perry, "A History of Satellite Reconnaissance, Vol. 2-A," 20–27, 38–41, 44–45.
42. "Memorandum for the Secretary of Defense, the Special Assistant to the President for Science and Technology, Subject: Presentation on Status of Reconnaissance and Early Warning Satellite Programs," January 28, 1960, WHO, OSANSA, Spec. Assist. Series, Chronological Subseries, box 6, file January–February (1960); "Review of Outer Space Programs under the Auspices of the Department of Defense," May 31, 1960, AWF, NSC Series, box 12, file 446th Meeting of NSC, May 31, 1960; memorandum of conference with the president, February 8, 1960, AWF, DDE Diary Series, box 47, file Staff Notes, February 1960 (2).
43. Memorandum of conference with the president, May 31, 1960 (conference on May 26, 1960), AWF, DDE Diary Series, box 50, file Staff Notes, May 1960 (1).
44. "Memorandum of Conference with the President," May 26, 1960, WHO, OSS, SUB, ALPHA, box 16, file Dr. Kistiakowsky (3); "Letter to Dr. Kistiakowsky from DDE," June 10, 1960, WHO, OSS, SUB, ALPHA, box 16, file Dr. Kistiakowsky (4).
45. "Introductory Remarks by Dr. J. R. Killian Jr.," August 25, 1960, WHO, OSAST, box 15, file Space (July–December 1960) (9); "Memorandum for Attached List," August 19, 1960, WHO, OSANSA, NSC Series, Subject Subseries, box 8, file Reconnaissance Satellites.
46. "The Samos Program," July 29, 1960, WHO, OSAST, box 15, file Space (July–December 1960) (9). For the U.S. Intelligence Board requirements, see also "Samos" (not dated), WHO, OSS, SUB, ALPHA, box 15, file Intelligence Matters (14); memorandum of conference with the president, May 31, 1960, WHO, OSS, SUB, ALPHA, box 16, file Dr. Kistiakowsky (3).
47. The SAMOS briefing notes (probably dated to early July, based on the Intelligence Board requirements that they include) predict that the E-2 camera would fly by April 1961, provide twenty-foot resolution, and satisfy the general search requirements. See "Samos" (not dated).
48. While Edwin Land knew about CORONA, there is no indication that he shared that knowledge with the rest of the panel. "The Samos Program," July 29, 1960, WHO, OSAST, box 15, file Space (July–December 1960) (9).
49. The E-5 and E-6 virtually mimic the DISCOVERER/CORONA configuration, including recovery and flight sequences. See "Samos" (not dated).
50. "The Samos Program," July 29, 1960.
51. Perry, "A History of Satellite Reconnaissance, Vol. 2-A," 55.
52. The official Public Affairs Objectives of September 27, 1960, clearly identified SAMOS as a research and development program only. See "Annex III & IV Public Affairs Objectives," September 27, 1960, WHO, OSANSA, NSC Series, Subject Subseries, box 8, file Reconnaissance Satellites; memorandum for the record, October 1960, WHO, OSS, SUB, ALPHA, box 15, file Intelligence Matters (20).
53. "Reconnaissance Satellite Program Action No. 1-b," September 1, 1960, WHO, NSCSP, Exec. Sect. Subject File Series, box 15, file Reconnaissance Satellites (1960); "Memorandum for the Secretary of Defense, Subject: Reconnaissance Satellite Program," September 1, 1960, WHO, OSS, SUB, ALPHA, box 15, file Intelligence Matters (19); Hall, "The Eisenhower Administration," 67–68; Perry, "A History of Satellite Reconnaissance, Vol. 2-A," 80–86.
54. Hall, "The Eisenhower Administration," 67–68; Richelson, _America's Secret Eyes in Space_ , 46–47. The United States now officially acknowledges the National Reconnaissance Office. Providing documents and imagery on satellite operations, their website is www.nro.gov/. Perry, "A History of Satellite Reconnaissance, Vol. 2-A," 61, 84–87.
55. Burrows, _This New Ocean_ , 226.
###### Appendix
1. Warshaw, _Reexamining the Eisenhower Presidency_.
2. To see this point of view it is necessary only to look at some of the most potent authors of the orthodox school: Marquis Childs, _Eisenhower: Captive Hero. A Critical Study of the General and the President_ (New York: Harcourt Brace, 1958); Richard Rovere, _Affairs of State: The Eisenhower Years_ (New York: Farrar, 1956); Richard Neustadt, _Presidential Power: The Politics of Leadership_ (New York: John Wiley & Sons, 1960).
3. Alexander, _Holding the Line_.
4. As an earlier example, see Larson, _Eisenhower_. More recent examples of "revisionist" writings include Greenstein, "Eisenhower"; Rabe, "Eisenhower Revisionism."
5. Rabe, "Eisenhower Revisionism," 97–98.
6. Larson, _Eisenhower_ , ix–xii, 12–17; McAuliffe, "Eisenhower, the President," 625–26; Kinnard, _President Eisenhower_ , x.
7. Greenstein, "Eisenhower," 575–77, 577–83.
8. Greenstein, "Eisenhower," 579–87, 596–98.
9. McAuliffe, "Eisenhower, the President," 630–31; Ambrose, _Eisenhower: The President_ , 9–10, 18–20, 32–35, 624–27.
10. Allen, _Eisenhower_ , 1–9.
11. See Immerman, "Confessions," 321–23.
12. Marchetti and Marks, _The_ CIA, 7, 76–77, 90, 94.
13. Ranelagh, _The Agency_ , 324–25.
14. Andrew, _For the President's Eyes Only_ , 3–4, 199–249; Jeffreys-Jones and Andrew, _Eternal Vigilance?_ , 83, 85, 90, 92–93, 99.
15. Richelson, _U.S. Intelligence Community_ , 36–37.
16. FEEDBACK _Report_ , 149–50.
17. Klass, _Secret Sentries_ , 1–51, 72–100; Kenden, "U.S. Reconnaissance Satellite Programmes," 243–49; Gaddis, "Transparency"; Burrows, _Deep Black_ , 61–2, 82–84, 86–92. Peebles, _Guardians_ , 43–70; Richelson, _America's Secret Eyes in Space_ , 1–31; Divine, _Sputnik Challenge_ , 10–12.
18. McDougall, _. . . the Heavens and the Earth_ , 79–110.
19. Executive Order 12951, "Release of Imagery Acquired by Space-Based National Intelligence Reconnaissance Systems," _Federal Register_ 60, no. 39 (1995): 10789, in Ruffner, CORONA, 359–90. ARGON and LANYARD were the names for specific camera configurations for CORONA satellites. See Ruffner, "CORONA."
20. Logsdon et al., _Exploring the Unknown_ , 1:213–413; Logsdon et al., _Exploring the Unknown_ , 2:233–98.
21. Peebles, _Corona Project_ , 1–6, 9–14; Day et al., _Eye in the Sky_ , 2–3, 29–33, 98–108.
22. Peebles, _Corona Project_ , 21–30, 39–52; Day et al., _Eye in the Sky_ , 108–13, 41–42.
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# Index
All page numbers in the index refer to the print edition.
Adams, Sherman,
"Advanced Reconnaissance System" project (MX-2226), , , 230n26
Advanced Research Projects Agency (ARPA): management problems within, 174–77; MIDAS and, 171–72, 179–80; SENTRY/SAMOS and, , , 174–75, 179–82; WS-117L and, 164–65,
Aerecon camera,
aerial photography: FEEDBACK and, , 111–13; history of, 95–99; SENTRY/SAMOS and, 166–71; technical issues with, 99–107, 227n52; WS-117L and, 119–20, 122–23, . _See also_ film returning from satellites
Aeronautical Board, 86–92, 225n16
AGENA booster system, , , , 158–59,
_The Agency_ (Ranelagh),
Airborne Instruments Laboratory,
Air Defense Command (ADC),
Air Force Development Planning (AFDAP), ,
Air Material Command, 91–92,
Air Research and Development Command (ARDC), , , , , ; FEEDBACK and, ; MIDAS and, , ; SENTRY/SAMOS and, ; WS-117L and, , , , ,
Alexander, Charles C.: _Holding the Line_ ,
Allen, Craig, ; _Eisenhower and the Mass Media_ ,
Alsop, Joseph,
Ambrose, Stephen, , , 192–93; _Eisenhower: Soldier, General of the Army, President-Elect, 1890–1952_ , ; _Eisenhower: The President_ , ; _Ike's Spies_ ,
Ampex, ,
Anderson, Robert B., 69–70
Andrew, Christopher: _Eternal Vigilance?_ , 194–95; _For the President's Eyes Only_ ,
ARGON, , 196–97, 244n19
arms control verification, Open Skies initiative and, 69–72,
Army Ballistic Missile Agency (ABMA), xii,
Arnold, Henry H. (Hap), , 95–96,
Aronsen, Lawrence,
Astronomic Council of the Soviet Academy of Sciences,
ATLAS-AEROBEE missile, ,
ATLAS missile system, , , , , 127–29, , , 163–64,
Atomic Energy Commission (AEC), ,
"Atoms for Peace" speech,
Augenstein, Bruno W.,
_Aviation Week_ ,
B-58 HUSTLER, ,
Baker, James, 62–63, , , , , , , ,
Ballistic Missile Agency, xii,
Ballistic Missile Division, 133–34, , 151–52, 175–76, 233n55
Beacon Hill Study Group and report, 105–7
Bell Aircraft Company, , ,
Bell Telephone Laboratories, ,
Biggs, Buford B., 121–22, 230n16
Bikini Atoll, nuclear testing at, 96–97
Billings, B. H.,
Bing Crosby Enterprises, 122–23
Bissell, Richard, , , , 157–58, , ,
"bomber gap," 20–22,
Boston University's Optical Research Laboratory (BUORL), , , ,
Bowen, Lee, ,
Brodie, Bernard,
Burgess, W. M., 17–18
Burrows, William E., ,
Bush, Vannevar, , , 225n16
Byrnes, James F.,
Cabell, C. P.,
Cairns, Robert W.,
Cambridge Research Center, , , 240n22
Carter, Jack, 129–30
CBS Laboratories, ,
Center for the Study of Intelligence,
Central Intelligence Agency (CIA), xiv, , ; CL-282 and, ; CORONA and, , 147–48, 155–59, , , ; covert reactivation of WS-117L by, ; declassification and, ; and estimates of Soviet capabilities, , 9–13, 18–22, 24–25, , ; historiography of satellite reconnaissance and, , 193–95; IGY and, ; and photo reconnaissance, ; value of scientific satellites and,
Central Intelligence Group (CIG), 6–8
Charyk, Joseph, , ,
China, ,
_The_ CIA _and the Cult of Intelligence_ (Marchetti and Marks),
CL-282, . _See also_ U-2 aircraft
Cline, Ray S.: _Secrets, Spies and Scholars_ ,
Clinton, Bill, 196–97
Collbohm, Frank,
Comité Special Année Géophysique Internationale (CSAGI). _See_ International Geophysical Year (IGY)
Committee on Overhead Reconnaissance,
Conant, James B.,
containment policy, , , , 35–38
Convair Corporation, ,
CONVAIR test missile, 74–75
Coolbaugh, James S., 116–25, ,
Coordinating Committee on General Sciences, Department of Defense Research and Development (CGS), 73–74,
Cornog, Robert,
CORONA, , , ; CIA and, , 147–48, 155–59, , , ; contrast of SENTRY/SAMOS with, , , ; declassification and, 196–97, ; DISCOVERER as cover for, , , 164–65, , , ; origins of, 140–41; problems with, 173–74; separation of, from WS-117L, xiv, 154–60, ; success of, xii–xiii, ,
"CORONA" (Greer),
CORONA (Ruffner), ,
_The Corona Project_ (Peebles), 15–16, 198–99
"Cost and Effectiveness of the Defense of the United States against Air Attack in 1952–1957,"
"counter force" strategy,
Craig, H. A.,
Crawford, Alden R.,
CROSSROADS nuclear testing, 96–97
Cullen, Paul T.,
Cutler, Robert,
Daigle, Frank, 230n16
Davies, Merton E., , 104–5, , , 144–45, 149–51, , 226n26
Day, Dwayne, , ; _Eye in the Sky_ , 198–99; "Invitation to Struggle,"
defense spending: control of interservice rivalries and, , 45–57; Eisenhower and, , , , 38–48, , 215n65; fluctuation of, during Truman administration, 30–31, ,
Defense Support Program satellite, ,
Department of Defense: IGY satellites and, , 75–76, ; international law and, ; Research and Development Board and, ; U.S. budget and, , 44–49. _See also_ defense spending
deterrence policy, 6–15, , , 28–29, 32–35, 37–38, , 47–48, , 56–57
Development Planning Objectives (DPOs), , ,
DISCOVERER, , , , ; briefing of Eisenhower on, 176–77; as cover for CORONA, , , 164–65, , , ; and _Discoverer 0_ , ; and _Discoverer XII_ , ; and _Discoverer XIII_ , ; and _Discoverer XIV_ , xii, ,
Divine, Robert, , ,
Donovan, William J.,
Doolittle, James H.,
Douglas, J. H.,
Douglas Aircraft, ,
Draper, Charles,
DuBridge, Lee,
Dulles, Allen, , , , 75–76, , 154–55,
Dulles, John Foster, , , , ,
E-1 to E-6 cameras on SENTRY/SAMOS, 166–71, 175–81
Eastman Kodak, 168–69
economic security, , , 38–48, , , 137–38, 215n65
"The Effect of the Soviet Possession of Atomic Bombs on the Security of the United States,"
_Eisenhower_ (Larson),
_Eisenhower_ (Lyon),
Eisenhower, Dwight D.: "Atoms for Peace" speech of, ; character and management style of, 186–87; and lack of hard data about Soviet Union, 15–25; orthodox historiography of, ; revisionist historiography of, 190–93, ; scarcity of information on motivations of, 199–200; and secrecy, xiii–xiv, ; and separation of CORONA from WS-117L, 154–56; and _Sputnik_ , xii, 138–40, 148–49; USAF bomber mentality and,
Eisenhower, John,
_Eisenhower and the American Crusades_ (Parmet),
_Eisenhower and the Mass Media_ (Allen),
"Eisenhower as an Activist President" (Greenstein), 191–92
_Eisenhower: Soldier, General of the Army, President-Elect, 1890–1952_ (Ambrose),
_Eisenhower: The President_ (Ambrose),
Electronic Components Laboratory, ,
_Eternal Vigilance?_ (Jeffreys-Jones and Andrew), 194–95
EXPLORER satellite,
_Exploring the Unknown_ (Logsdon),
_Eye in the Sky_ (Day, Logsdon, and Latell), 198–99
Fairchild Camera and Instrument Corporation,
FEEDBACK study and proposed satellite, , , , ; declassification and, ; nuclear power and, ; technical issues and, , 107–13
ferret satellite system, , 168–70, 242n41
film returning from satellites: CORONA and, 140–41, ; intelligence gathering and, 146–51; and manned flight, 143–44; Program II-A and, , , 154–60; Programs IV, VI, and VIII and, 152–53; and RAND Corporation, , 149–52, ; _Sputnik_ 's influence on,
Fisk, James,
Flemming, Arthur,
_For the President's Eyes Only_ (Andrew),
409-40 "Satellite Component Study" program,
Freedom of Information Act,
freedom of space, ; after _Sputnik_ , ; establishing legal precedents for, 65–66, , , , ; VANGUARD and, 131–32, , . _See also_ Open Skies initiative
Gardner, Trevor, ,
Gates, Thomas S., Jr.,
General Electric,
General Operational Requirement No. 80, , 231n27
GENETRIX program (WS-119L), 13–14, 206n36
German scientists in Soviet Union, , ,
Ginsburg, Charles,
Glasser, Otto J.,
Goddard, George W.,
Goldmark, Peter,
Goodpaster, Andrew J., , , , , , 215n65
Goodwin, Craufurd,
GOPHER, , 106–7
Gray, Gordon,
Greene, Sidney, , ,
Greenstein, Fred: "Eisenhower as an Activist President," 191–92
Greer, Kenneth E.: "CORONA,"
Groves, Leslie,
Hagerty, Jim,
Hall, Edward,
Hall, Harvey,
Hall, R. Cargill, , , , , , , ; "Origins of U.S. Space Policy,"
Hanscom Air Force Base,
Harris, Seymour,
Haydon, Brownlee W.,
Hazlett, Everett "Swede," 31–32, ,
_The Heavens and the Earth_ (McDougall),
HERMES,
Herther, John C., 125–26,
high-altitude aircraft. _See_ U-2 aircraft
high-altitude balloons, , 106–7
historiography of satellite reconnaissance development: CIA and, , 193–95; CORONA and, ; Eisenhower and, 190–93; post-declassification, 196–98; pre-declassification, 195–96; and scarcity of information on WS-117L, xiii–xiv, 189–90,
"History of Satellite Reconnaissance" (Perry), 130–31
_Holding the Line_ (Alexander),
horizontal scanner for WS-117L, 118–19
hot-air balloons, reconnaissance and,
Huckaby, Jim, , 122–23
Humphrey, George, , ,
Huntington, Samuel,
Huntzicker, J. H., ; "Physical Recovery of Satellite Payloads," 142–43
HUSTLER. _See_ B-58 HUSTLER
"Hyac" (high acuity) camera,
IBM,
_Ike's Spies_ (Ambrose),
Image Orthicon television camera,
inflation, , 40–42,
intelligence gathering: after _Sputnik_ , ; Edwin Land's 1954 panel on, 62–66, , , ; Eisenhower administration and, 15–25; Pearl Harbor attack's influence on, 4–6, , , , , 95–98; Truman administration and, 3–15
intercontinental ballistic missiles (ICBMs): expectations of Soviet development of, , 18–19; hydrogen bomb and repercussions for, 23–24; as means to deliver nuclear weapons, 14–15; military budget cuts and, ; post- _Sputnik_ missile gap and, ; propulsion for satellites and, ; reentry technology and, , ; satellite planning and, ; separation of, from civilian space activities, 78–79
intermediate range ballistic missiles (IRBMs), . _See also_ THOR IRBM program
International Geophysical Year (IGY), xi, , 72–76, , , . _See also_ VANGUARD
international law in space: after _Sputnik_ , , ; and freedom of space, 65–66, , , , ; Open Skies initiative and, , , 69–72, , , , 221n35; secret planning for U-2 and, 67–69
interservice rivalry. _See_ military service rivalries
"Invitation to Struggle" (Day),
Itek Corporation, , 235n24
Jeffreys-Jones, Rhodri: _Eternal Vigilance?_ , 194–95
Jet Propulsion Laboratory (JPL), ,
Johnson, Kelly, 66–67
Johnson, Lyndon B., xii,
Johnson, Roy, , ,
Joint Chiefs of Staff (JCS), , , , , 45–46, 49–50, 54–55
Joint Congressional Committee Investigation on Pearl Harbor attack,
Joint Research and Development Board (JRDG), , , 225n16
Katz, Amrom H., , , , 144–45, 149–51, , 226n26
Kay, John,
Kecskameti, Paul, 93–94, , 226n27
Kenden, Anthony: "U.S. Reconnaissance Satellite Programmes,"
Kennan, George F., ,
Kennedy, Joseph W.,
Key Hole satellite system,
Keys, Roger, 50–51
Killian, James R., Jr., , 61–62, , , , ; Beacon Hill Study Group and, ; drop-film system and, , 154–55; _Sputnik_ and, ; U-2 and, 67–69
Killian Commission. _See_ Technical Capabilities Panel
King, W. G.,
Kinnard, Douglas: _President Eisenhower and Strategy Management_ ,
Kistiakowsky, George B., 176–77
Klass, Philip J.: _Secret Sentries in Space_ ,
Klein, Burton,
Korean War, , , , , , , ,
Lamphier, Tom,
Land, Edwin, , , , , , ; Beacon Hill Study Group and, ; drop-film system and, 154–55; intelligence panel of, 62–66, , , ; satellite planning and, , ; U-2 and, 66–69
LANYARD, , 196–97, 244n19
Larson, Arthur: _Eisenhower_ ,
Latell, Brian: _Eye in the Sky_ , 198–99
Latham, Allan, Jr.,
launch capability concerns, , 75–80, 85–86, , , 101–2, , , ,
legal issues. _See_ international law in space
Leghorn, Richard S.: Beacon Hill Study Group and, ; early satellite programs and, 96–99, ; Open Skies initiative and, 228n60; satellite planning and, ; as USAF liaison to RAND, 103–6
LeMay, Curtis, , 87–89, , 224n6
Lemnitzer, Lyman L.,
Levison, Walter J., , ,
Lieske, H. A., ; "Physical Recovery of Satellite Payloads," 142–43
Lipp, J. E., ,
Lockheed Corporation: film-returning satellite system and, 152–60, 168–69; MIDAS and, ; SENTRY/SAMOS and, 240n22; WS-117L and, , 129–30, , , ,
Logsdon, John M.: _Exploring the Unknown_ , ; _Eye in the Sky_ , 198–99
Lundahl, Arthur, ,
Lyon, Peter: _Eisenhower_ ,
MacArthur, Douglas, Jr.,
Macdonald, Duncan, , ,
manned space flight, , 143–44, ,
Marchetti, Victor: _The_ CIA _and the Cult of Intelligence_ ,
Marks, John D.: _The_ CIA _and the Cult of Intelligence_ ,
Martin Aircraft,
McCarthy, Joseph,
McCormack, James,
McDougall, Walter A.: _The Heavens and the Earth_ ,
McElroy, Neil, , 154–55, ,
McNeil, Wilfred,
MIDAS (Missile Defense Alarm System), , , , , 171–73, , ; ARPA and, 171–72, 179–80; briefing of Eisenhower on, 176–77; influence of,
military service rivalries, , 45–57, , 86–92
"mirror imaging" fallacy, , 18–19
"missile gap," 137–39,
MIT, , ,
Murphy, Robert,
Murray, Pete,
Mutual Security Program,
National Aeronautics and Space Administration (NASA),
National Intelligence Estimates (NIEs), 17–18, ,
National Science Foundation,
national security, prioritizing of, , 28–38, ,
National Security Act (1947), , ,
National Security Agency (NSA),
National Security Council (NSC): estimation by, of damage from nuclear attack, 16–17, ; ICBM and VANGUARD conflict and, 78–79; and NSC 20, 28–29; and NSC 68, , 29–30, , , , ; and NSC 162, ; and NSC 5520, , 76–77, , ; and NSC 5814, , ; and NSC Action 1846,
"New Look" for defense spending, 45–48, 53–54
Nitze, Paul,
North American Aviation, , , ,
nuclear power, satellites and, , 110–11, , ,
nuclear weapons: and "counter force" strategy, ; Eisenhower's concern about, 31–35; hydrogen bomb development and, 14–15, 22–23, , ; and need for intelligence, 6–15, 95–98; and post–World War II bomber mentality, ; Soviet test of, 10–11, , ,
Oder, Fritz, 132–33, , , 231n36
Office of Defense Management–Science Advisory Committee (ODM-SAC), 23–24, 60–61
Office of Strategic Services (OSS),
Open Skies initiative, , , 69–72, 80–82, , 221n35
ORBITER satellite, ,
"Origins of the USAF Space Program" (Perry),
"Origins of U.S. Space Policy" (Hall),
Overhage, Carl, ,
Oxcart/SR 71 aircraft,
Parmet, Herbert S.: _Eisenhower and the American Crusades_ ,
Patterson, Robert,
Pearl Harbor attack, 4–5, , , , , , 95–98
Peebles, Curtis, , , , , , ; _The Corona Project_ , 15–16, 198–99
Perry, Robert, , 116–17, , ; "History of Satellite Reconnaissance," 130–31; "Origins of the USAF Space Program,"
Philbrick, Richard W.,
photography. _See_ aerial photography
photostatic facsimile transmission, 101–2
"Physical Recovery of Satellite Payloads" (Huntzicker and Lieske), 142–43
"Piercing the Curtain" conference,
Polaris missile,
Poniatoff, A. M.,
Power, Thomas, ,
power consumption, , 110–11,
"Preliminary Design for an Experimental Earth Circling Spaceship," 88–89
_President Eisenhower and Strategy Management_ (Kinnard),
President's Board of Consultants on Foreign Intelligence Activities, 146–47, ,
President's Scientific Advisory Committee, , , , , ,
Program II-A, , , 154–60, ,
Programs IV, VI, and VIII, 152–53
Purcell, Edward, ,
Putt, Donald, , 124–25, ,
Quarles, Donald, ; IGY satellite and, 72–73, 75–77; lack of enthusiasm of, for WS-117L, , , , , , ; _Sputnik_ and, , , 145–46,
Radford, Arthur W., , ,
Ramo-Woolridge, 127–28
RAND (project),
RAND Corporation, 85–114; aerial photography and, 95–107; demonstration of military need and, 95–96; feasibility of satellites and, 86–93; film return from satellites and, , , 149–52, ; impact of, 113–14; and political and psychological ramifications, 93–94; U.S. cold war vulnerability and, , , . _See also_ FEEDBACK study and proposed satellite
Ranelagh, John: _The Agency_ ,
Raymond, Richard C., 141–42
RCA, , , 112–13, 119–20, 122–23, , ,
REDSTONE missile, ,
Research and Development Board (RDB), , 225n16
Research and Development Committee, 88–89
Reynolds, Don,
Richelson, Jeffrey T., , ; _The U.S. Intelligence Community_ ,
Ridenour, Louie,
Ridgway, Matthew, , 51–52,
Riepe, Quenten, , 117–18, , 125–26
Ritland, Osmond, ,
Rockefeller, Nelson, ,
"roll back" of communism, 36–37
Rome Air Development Center, , , 240n22
Roosevelt, Franklin D.,
Ruffner, Kevin: CORONA, ,
Salter, Robert, 227n45
SALT treaty (1972),
SAMOS. _See_ SENTRY/SAMOS
Schriever, Bernard, , , , , , , , , ; WS-117L and, , , , 131–34
Second Story plan, 132–34, ,
_Secret Empire_ (Taubman), 198–99
_Secrets, Spies and Scholars_ (Cline),
_Secret Sentries in Space_ (Klass),
SENTRY/SAMOS, , , , , , 240n22; ARPA and USAF and, , , 174–75, 179–80, 181–82; briefing of Eisenhower on, 176–77; cancellation of, 170–71; and E-1 to E-6 cameras, 166–71, 175–76; impact of, ; problems with, 169–70, 173–77; renaming of, ; review and reorientation of, 177–81
Shulsky, Abram,
Signals Intelligence (SIGINT), , 167–68
Simon, Richard L.,
SOLARIUM, 35–37
solar power, satellites and, 123–24,
"Soviet Capabilities and Intentions,"
"Soviet Capabilities for the Development and Production of Certain Types of Weapons and Equipment," 7–8
Soviet Union: difficulty of intelligence gathering in, , ; at end of World War II, 3–4; need for accurate intelligence on, in 1950s, 4–25; overflights of, in 1946, ; study of effects of U.S. satellite program on, 93–94; supposed lack of uranium of, ; test of atomic bomb by, in 1949, 10–11, , , ; U.S. containment policy toward, , , , 35–38. See also _Sputnik_; U-2 aircraft
space-based reconnaissance, major impetuses for: economic security, , , 38–48, , 215n65; intelligence needs, 3–25; national security, 27–28, , ; overspending on defense, , 45–57
space-based reconnaissance, political steps toward, 58–82; and emphasis on scientific benefits, , , , ; and International Geophysical Year and Project VANGUARD, , 72–80, ; NSC 5814 and, 69–72; and Open Skies initiative, , , 69–72, 80–82, , 221n35; Technical Capabilities Panel and, 58–59, 60–69,
"space for peace" policy, 131–32, , 232n51
Special National Intelligence Estimates (SNIEs), 17–19,
_Sputnik_ , , ; and end of U.S. concerns about legality of overflights, , ; feasibility of satellites shown by, , , , ; impact of, on WS-117L program, , , 145–49, ; NSC 5814 and, , ; U.S. reactions to, xi–xiii, xiv, 137–40, 148–49, ,
Stalin, Joseph, ,
Stassen, Harold,
"Statement of Policy for a Satellite Vehicle" (Vandenberg),
Stevens, Leslie,
Stevens, Robert,
Stimson, Henry,
Strategic Air Command (SAC), , , , 174–76
Strategic Missiles Evaluation Committee, 23–24
Strong, Philip, 66–67
_Studies in Intelligence_ ,
Symington, Stuart, , , 53–55, 137–38
Taft, Robert,
Talbott, Harold,
Taubman, Philip: _Secret Empire_ , 198–99
Teapot Committee. _See_ Strategic Missiles Evaluation Committee
Technical Capabilities Panel, 58–59, 60–69, , , , ; airspace legality and, ; declassification and, ; goals of, 61–62; intelligence panel of, 62–66; U-2 aircraft and, 66–69
television technology, , , , 134–35, ,
THOR-AGENA program,
THOR IRBM program, , , 152–54, 158–59,
"Threats to the Security of the United States,"
Troetschel, William O., ,
Truax, Robert, , , , 231n36
Truman, Harry: and defense spending fluctuations, 30–31, , ; and disbanding of OSS, 6–7; economy during administration of, 39–42; Eisenhower and policies of, 28–31, ; and lack of hard data about Soviet Union, 3–15; and overflights of Soviet Union, ; USAF and, , ,
TU-4 (BULL), , , 207n53
TU-16 (Badger),
Tukey, John W.,
Twining, Nathan, ,
Type-37 Soviet bomber (Bison), 21–22, 208n67
U-2 aircraft, , , , , , ; international law and, ; loss of, over Soviet Union, ; and missile gap, ; Open Skies initiative and, 69–72; secret planning for, 66–69
_United States Strategic Bombing Survey (Europe)_ , 96–97
U.S. Air Force (USAF): Air Defense Command of, ; Air Material Command of, 91–92, ; Ballistic Missile Division of, 133–34, 148–49, 151–52, , 175–76, 233n55; bomber mentality of, after World War II, ; Cambridge Research Center of, , , 240n22; early satellite concepts and, 74–75; inertia within, regarding satellites, ; as leader in military use of reconnaissance, 91–93; military budget cuts and, , 52–54; and overflights, , 106–7; publicity and, 164–65, 179–80; Truman's creation of, , ,
U.S. Army, xii, 51–52, , ,
U.S. Army Air Forces (USAAF), 86–89, ,
_The U.S. Intelligence Community_ (Richelson),
U.S. Navy, , , 87–91,
"U.S. Reconnaissance Satellite Programmes" (Kenden),
"Utility of a Satellite Vehicle for Reconnaissance," 100–103
Vandenberg, Hoyt S.: "Statement of Policy for a Satellite Vehicle,"
VANGUARD, xi, 77–80, , ; after _Sputnik_ , ; as civilian program, , 131–32; freedom of skies and, , ; IGY and, ,
VIKING missile, ,
von Braun, Werner, xi–xii
von Karman, Theodore, 14–15
von Neumann, John,
Washburn, Bob, 230n16
Waterman, Alan, ,
West, J. M., ,
Western Development Division, WS-117L and, , 126–35; and funding and contractor issues, , 127–30, , , ; and Second Story plans, 131–34; secrecy and, 133–34; slow progress and, 131–32; and transfer from WADC, 126–28
White, Thomas D., , ,
Willcox, Fred,
Wilson, Charles, , , ,
Wisner, Frank G.,
Wohlstetter, Albert,
Wright Air Development Center (WADC), WS-117L and, , , 117–26; and funding and contractor issues, 117–19, 120–22, 124–26; and technical issues, , 119–20, 122–24; Western Development Division stage and, 127–28
Wright Field, 92–93, , , 103–4,
WS-117L satellite program, , , , ; ARPA and, 164–65, ; complexity of, 163–64; contractors and, 118–19, 121–22, , 127–30, ; declassification and, 197–99; electronics and, 119–20; and Program II-A, , , 154–60, ; and Programs IV, VI, and VIII, 152–53; scarcity of information on, xiii–xiv, 115–16, 135–36, , , 189–90, ; slow progress and, , ; _Sputnik_ 's impact on, , 145–49, ; theoretical barriers broken by, ; as three satellites, 140–41, , 165–66, 182–83; as USAF program, xiv, ; various designations of, , 116–17, ; Western Development Division stage of, 126–35; Wright Air Development Center stage of, , 117–26. _See also_ CORONA; DISCOVERER; MIDAS (Missile Defense Alarm System); SENTRY/SAMOS
"year of maximum danger" concept, 30–31, 42–43
York, Herbert, , , , , 175–77, 242n41
# About Robert M. Dienesch
Robert M. Dienesch is an adjunct assistant professor of history at the University of Windsor, Ontario, and a research affiliate with the Gregg Centre for the Study of War and Society. His work has been published in _Quest: The History of Spaceflight_ and _Northern Mariner_.
| {
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} | 7,008 |
Ситаллы — стеклокристаллические материалы, полученные объёмной кристаллизацией стёкол и состоящие из одной или нескольких кристаллических фаз, равномерно распределённых в стекловидной фазе. Разработаны советским физикохимиком И. И. Китайгородским. Материалы, подобные ситаллам, за рубежом называют пирокерамом, девитрокерамом, стеклокерамом.
По своему назначению подразделяются на технические, строительные, ювелирные.
По основному свойству и назначению подразделяются на высокопрочные, радиопрозрачные химически стойкие, прозрачные термостойкие, износостойкие и химически стойкие, фотоситаллы, слюдоситаллы, биоситаллы, ситаллоцементы, ситаллоэмали, ситаллы со специальными электрическими свойствами.
Существуют литиевые, борно-бариевые, магниевые, титановые и другие ситаллы.
Eгo ближайший пpиpoдный aнaлoг — вyлкaничecкoe cтeклo обсидиан.
Свойства
Ситаллы обладают малой плотностью (они легче алюминия), высокой механической прочностью, особенно на сжатие, твёрдостью, жаропрочностью, термической стойкостью, химической устойчивостью и другими ценными свойствами. Ситаллы имеют большинство положительных свойств, которые есть у стекла, в том числе и технологичность.
Существуют ситаллы со специальными свойствами: прозрачные, магнитные, полупроводниковые, радиопрозрачные и другие.
Твёрдость большинства ситаллов составляет 6,5—7 единиц по Моосу,
предел прочности на изгиб — до 250 МПа,
термостойкость — до 1000 °C.
Получение
Подбором состава стекла, содержащего в большинстве случаев добавки, ускоряющие объёмную кристаллизацию (катализаторы, нуклеаторы), можно спроектировать соответствующие кристаллические и стекловидную фазы. Кристаллы спроектированных фаз возникают и растут равномерно по всему объёму в результате термической обработки. Технология производства изделий из ситаллов незначительно отличается от производства изделий из стекла. В некоторых случаях изделия можно формовать методами керамической технологии (см. Керамика). Иногда для зарождения кристаллов в состав стекла вводят фоточувствительные добавки. Для производства отдельных видов ситаллов используют шлаки.
Применение
Ситаллы применяются для изготовления деталей, требующих прочности и термостойкости (корпуса приборов, шкалы, образцовые меры, подложки микросхем и другое).
Являются перспективными строительными и конструкционными материалами (обтекатели ракет и сверхзвуковых управляемых снарядов, химически стойкая аппаратура, мостостроительные конструкции и другое).
Широкое применение ситалл нашёл в астрономической оптике, благодаря рекордно низкому коэффициенту температурного расширения (0 ± 1,5×10−7 К−1 в диапазоне от −60 до 60°С).
На нижней смотровой площадке Останкинской телебашни на высоте 337 метров в полу установлены прозрачные проёмы, изготовленные из ситалла и выдерживающие вес нескольких человек.
Примечания
Литература
Материалы
Изделия из стекла | {
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} | 6,713 |
{"url":"https:\/\/stats.stackexchange.com\/questions\/500747\/what-do-we-exactly-mean-by-density-in-probability-density-function-pdf","text":"# What do we exactly mean by \u201cdensity\u201d in Probability Density function (PDF)? [duplicate]\n\nIn general density is mass\/volume. Also it's used for something like population density, which is population\/unit area.\n\nWhat is significance of word density in PDF?\n\nShort answer: Like in physical density, the probability density is probability\/volume.\n\nLong answer: For homogeneous objects, density can be defined as you said, $$m\/V$$, with $$m$$ denoting mass and $$V$$ its volume. However, if your object is not homogeneous, the density is a function of the space coordinates within the object: $$\\rho(x, y, z) = \\lim_{\\Delta V \\rightarrow 0} \\frac{\\Delta m(x, y, z)}{\\Delta V}$$ i.e. the mass inside an infinitesimal volume around the given coordinates, divided by that infinitesimal volume. Think of a plum pudding: The density at the raisins is different from the density at dough.\n\nFor probability, it is basically the same: $$f(x, y, z) = \\lim_{\\Delta V \\rightarrow 0} \\frac{\\Delta F(x, y, z)}{\\Delta V}$$ where $$f$$ is the probability density function (PDF) and $$F$$ the cumulative density function (CDF), so that $$\\Delta F$$ is the infinitesimal probability in the infinitesimal volume $$\\Delta V$$ in the vicinity of coordinates $$(x, y, z)$$ in the space over which $$F$$ is defined.\n\nNow, we happen to live in a physical world with three space dimensions, but we are not limited to defining probabilities just over space. In practice, it is much more common to work with probabilities defined over a single dimension, say, $$x$$. Then the above simplifies to $$f(x) = \\lim_{\\Delta x \\rightarrow 0} \\frac{\\Delta F(x)}{\\Delta x} = \\lim_{\\Delta x \\rightarrow 0} \\frac{F(x+\\Delta x) - F(x)}{\\Delta x}$$ But, of course, depending on your probability model, $$F$$ and $$f$$ can be defined over any number of dimensions.\n\nYou could see the Radon-Nikodym derivative as a formal definition of a more general notion of density.\n\nIt is the ratio of two measures (which have the extensive property, they are additive) defined on the same space.\n\n$$\\rho = \\frac{d \\nu}{d \\mu}$$\n\nThis ratio makes the one quantity measure $$\\nu$$ of a set $$S$$ expressible by an integral over the other measure $$\\mu$$ $$\\nu(S) = \\int_S \\rho d \\mu$$\n\nTypically the denominator $$\\mu$$ is a measure based on a metric measure like distance, area or volume. This is common for densities in physics like mass density, energy density, charge density, particle density.\n\nWith the density of probability the denominator can be more generally another type of variable that does not relate to physical space. Yet, often it is similar in the use of the Euclidean measure or Lebesgue measure. It is just that the variable does not need to be a coordinate in physical space.\n\nFor a single continuous random variable, the value of the pdf at the point $$t$$ tells you the density of the probability mass, measured in units of probability mass per unit length, at the point $$t$$ on the real line. The density of the probability mass can be different at different points on the real line; it is not quite as facile as the mass\/volume prescription of high-school physics.\n\n\u2022 If point $t$ is an input for the pdf function which returns back the probability mass for point $t$ only, what is the shape of the output of the pdf as a whole? i.e. there $T$ total samples in a data series and we instead feed all of those points as inputs to the pdf functional all at once, is the pdf also shaped $T$ in length? or does that output ever take a matrix form \u2013\u00a0develarist Dec 14 '20 at 5:53\n\u2022 The pdf function does not return the value of the probability mass at $t$, the argument of the pdf, but the density of the mass at $t$. I have no idea what the rest of your comment means @develarist \u2013\u00a0Dilip Sarwate Dec 14 '20 at 13:25","date":"2021-05-17 14:18:30","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 23, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9289278388023376, \"perplexity\": 209.90870415600517}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-21\/segments\/1620243991772.66\/warc\/CC-MAIN-20210517115207-20210517145207-00480.warc.gz\"}"} | null | null |
\section{Introduction}
The geometrical description of phase transitions has a long history
\cite{Duplantier}. The existence of exactly solved models and, most
importantly, the richness of the conformal symmetry in two dimensions
(2D), make the two-dimensional statistical systems an ideal framework
to study this problem. The critical points of 2D systems can be
classified using conformal field theories (CFTs) which also provide a
powerful approach to compute exactly correlation functions of local
operators. The first major breakthrough in the study of conformally
invariant interfaces in 2D critical models has come from the
introduction of the so-called Coulomb-gas (CG) formalism
\cite{Nienhuis_CG}. When a model is provided with a CG description,
the combination of CG and CFT techniques allow the exact computation
of geometrical exponents which characterize the fractal shape of
critical interfaces. This has been done for a variety of critical
statistical models as critical percolation, self-avoiding walks, loop
erased random walks, etc. All these models are associated to the so
called minimal CFTs or equivalently to the critical phases of $O(n)$
models \cite{Nienhuis_CG}. Using the CG description of the $O(n)$
model, the fractal dimension (and more generally all the multi-fractal
scaling exponents) of critical interfaces has been exactly computed
\cite{Saleur,Duplantier2}. A remarkable recent development in the
study of critical interfaces in 2D systems came with the introduction
of a conceptually new approach based on the so called Schramm-Loewner
evolutions (SLEs), which are growth processes defined via stochastic
evolution of conformal maps
\cite{Walter,Cardy_review,Bernard_review}. Again, the SLE approach,
which provides a geometrical description of CFT, is completely
understood only in the case of minimal CFTs
\cite{Bernard_connection1,Bernard_connection2,Bernard_connection3}.
The minimal CFTs are constructed by demanding the correlation
functions to satisfy the conformal symmetry alone and they represent a
small set of CFT theories. There are many other interesting CFTs,
called extended CFTs, which describe many condensed matter and
statistical problems which are characterized in general by some
internal symmetry such as, e.g., the $SU(2)$ spin-rotational symmetry
in spin chains \cite{Affleck} or replica permutational symmetry in
disordered systems \cite{Ludwig,DPP}. Despite all the recent activity
and progress, the geometrical properties of such extended CFTs are in
general not understood. In this respect some progress has been done
by studying the connection between SLE and Wess-Zumino-Witten models,
i.e. CFTs with additional Lie-group symmetries
\cite{Rasmussen,Ludwig2}, and by defining loop models associated to
some extended CFTs \cite{Pasquier,DFSZ,Fendley2006,Cardy2008,
Rajapour}. In this direction of investigation, a very interesting
family of critical models are the $Z_N$ spin models (defined below)
i.e. a lattice of spins which can take $N$-values. The
nearest-neighbor interaction defining the model is invariant under a
$Z_N$ cyclic permutation of the states. For $N=2$ and $N=3$ these
models correspond to the well known Ising and three-states Potts model
whose critical points are described by minimal CFTs. For $N\geq 4$
instead, the models admit critical points described by parafermionic
theories which are extended CFTs where the role of the $Z(N)$ symmetry
beside the one of conformal symmetry must be taken into account. The
geometrical properties of $Z_N$ spin models are for many aspects
unknown and their study are expected to provide general deep insights
on the geometrical description of extended CFTs.
In a recent work \cite{Raoul} one of the authors proposed an extension
of the concept of SLE to the case of the $Z_N$ parafermionic theory.
An SLE interface is associated to the (conformal) boundary condition
which generates it. Considering the $Z_N$ spin model on a bounded
domain, say the half-plane for instance, and specifying a particular
boundary condition, an SLE interface was identified as the boundary
of the spin cluster connected to the negative axis
\cite{Raoul}. By the term spin cluster we mean the cluster
which connects spins with equal value. This interface was further
studied in \cite{MarcoRaoul} where the corresponding fractal dimension
has been shown to be in agreement with the CFT predictions in
\cite{Raoul}. Nevertheless by considering other boundary conditions,
we obtained different results for the fractal dimensions of the
corresponding interfaces in the case of the $Z_N$ spin model with
$N\geq 4$ \cite{MarcoRaoul2}. This is at odd from results for the
Potts models for which a single fractal dimension for the spin
interface (i.e. the boundary of the spin cluster) is obtained
\cite{Gamsa}.
The present work is thus motivated by determining the bulk fractal
dimension, \hbox{\it i.e.} the fractal dimension of the boundaries of finite clusters in
the model. To be more precise, we will consider in this work the
fractal dimensions obtained by constructing the distribution of all
the finite closed spin clusters. As we will show later, the
spin clusters do percolate at the critical point of the $Z_N$ spin
model in the sense that there is a one large cluster which span the
entire lattice at the critical point. The distribution of the smallest
clusters can then be used to define a set of exponents as is the case
in percolation theory and from these exponents one can determine the
fractal dimension.
At this point, one needs to provide some explanation on why we
concentrate on the spin clusters. It is well known that while
the spin clusters do percolate at the critical point in two
dimensional Potts models, they do not contain the physical information
of the model considered. For example the exponents obtained from the
spin clusters of the two dimensional Ising model are not the
exponents of the Ising model \cite{Coniglio,Sykes}. These exponents
are in fact encoded in some other objects, the FK clusters.
Obviously, the FK clusters must also percolate at the critical
point. That the spin clusters percolate for the same critical
temperature as for the FK clusters is true for the Potts models in two
dimensions but is not a general result. In three dimensions for the
Ising model, the percolation of the Ising model occurs at a different
temperature \cite{Muller,DHMMPW}. In fact, one has no reason to expect
that the spin clusters do percolate at the critical point for
any two dimensional model. That it is the case for the Potts models
can be traced back to the existence of some duality relation. In
particular, in the correspondent CG formulation of the Potts model,
this duality is expressed in terms of an electric-magnetic duality
transformation, also called T-duality in the literature
\cite{Gruzberg}. The T-duality relate the descriptions of the dilute
and dense phase of the correspondent $O(n)$ model and it is at the
basis of the Duplantier duality \cite{Duplantier2}.
The natural question for the $Z_N$ model is then to see which are the
relevant clusters. The answer, that we will explain in great details
in this paper, is that i) the spin clusters percolate at the
critical point and the associated exponents do not correspond to the
corresponding model. ii) the FK clusters do {\bf not} percolate at the
critical point.
We will provide some details of the cluster algorithms that we
employed in this study. Cluster algorithms have been first employed on
the Potts model. The Potts model for any number of states is a two
level local energy model on a lattice. The energy between two spins is
either zero or a fixed value ($\beta$). Then the clusters are defined,
for a fixed configuration of spins, as a problem of percolation. On
all spin clusters which are build as neighboring spins taking
the same value and connected with a term of energy $\beta$, one
connects each pairs of spins with a probability $p=1-e^{-\beta}$. The
resulting clusters of connected spins are the Fortuin Kastelyn
clusters which are used to build the dynamics of the model but also to
measure observables like the magnetisation or the magnetic
susceptibility.
For the $Z(N)$ parafermionic theory that we will consider here, the
situation is more complicated. The local energy can take more than two
values for $N \geq 4$ and a direct consequence is that the generalised
clusters can connect spins with different values. Moreover, while for
the Potts models it was possible to defined some quantities as the
size of some FK clusters, this will not be the case here.
\section{Definitions}
One consider a model of spins variables $S_i$ which can take $N$
values, $S_i = 1, \cdots, N$ and are located on a square lattice of
linear size $L$ with periodic boundary conditions on both directions.
We consider the model defined on a square lattice with spin variable
$S_j=\exp{i 2\pi/N n(j)}$ at each site $j$ taking $N$ possible
values, $n(j)=0,1,\cdots,N-1$. The most general $Z_N$ invariant spin
model with nearest-neighbor interactions is defined by the reduced
Hamiltonian \cite{Zama_lat,Dotsi_lat}:
\begin{equation}
H[n]=-\sum_{m=1}^{\lfloor N/2 \rfloor} J_{m}\left[\cos \left(\frac{2\pi m n}{N}\right)-1\right],
\label{reducedH}
\end{equation}
where $\lfloor N/2 \rfloor$ denotes the integer part of $N/2$. The
associated partition function reads:
\begin{equation}
Z=\sum_{\{S\}}\exp\left[-\beta \sum_{<ij>} H[n(i)-n(j)]\right] \; .
\label{partition1}
\end{equation}
For $J_m=J$, for all $m$, one recovers the $N-$state Potts model,
invariant under a permutational $S_N$ symmetry while the case
$J_m=J\delta_{m,1}$ defines the clock model \cite{Potts_Clock}. For
$N=2$ and $N=3$ these models coincide with the Ising and the
three-state Potts model respectively, while the case $N=4$ is
isomorphic to the Ashkin-Teller model \cite{Ashkin,Lin}. Defining the
Boltzmann weights:
\begin{equation}
x_n=\exp\left[-\beta H(n)\right], \quad n=0,1,\cdots,N-1 \; ,
\end{equation}
the most general $Z_N$ spin model is then described by $\lfloor
N/2\rfloor$ independent parameters $x_n$ as $x_0=1$ and
$x_n=x_{N-n}$. The general properties of these models for $N=5,6,7$
have been studied long time ago (see e.g. \cite{Alcaraz2} and
references therein). The associated phase diagrams turn out to be
particularly rich as they contain in general first-order, second-order
and infinite-order phase transitions. For all the $Z_N$ spin models
it is possible to construct a duality transformation (Kramers-Wannier
duality). In the self-dual subspace of
(\ref{reducedH})-(\ref{partition1}), which also contains the Potts and
the clock model, the $Z_N$ spin model are critical and completely
integrable at the points \cite{Zama_lat2, Alcaraz}~:
\begin{eqnarray}
x^{*}_0= 1 \; ; \; x^{*}_n&=& \prod_{k=0}^{n-1} \frac{\sin \left(\frac{\pi k}{N}+\frac{\pi}{4
N}\right)}{\sin \left(\frac{\pi (k+1)}{N}-\frac{\pi}{4 N}\right)} \; .
\label{integrablecond}
\end{eqnarray}
There is strong evidence that the self-dual critical points
(\ref{integrablecond}), referred usually as Fateev-Zamolodchikov
points, are described in the continuum limit by $Z(N)$
parafermionic theories \cite{Alcaraz3}. Very recently, a further
strong support for this picture has been given in \cite{Rajabpour}
where the lattice candidates for the chiral currents generating the
$Z_N$ symmetry of the continuum model has been constructed.
\section{Cluster algorithm}
In this section, we explain how we can generalise the notion of FK
clusters to the case of the $Z_N$ spin model. We
will consider configurations on a square lattice of linear size $L$
with periodic boundary conditions for which we need to generate
independant samples. The most convenient way to generate these samples
is to use a cluster algorithm. The most effective cluster algorithm
for discrete spin models is the Wolff~\cite{Wolff} algorithm which is
based on the construction of the Fortyuin Kastelyn~\cite{FK} clusters. We first recall
how this algorithm works in the simple case of the $N$-states Potts model. Starting from
\begin{equation}
Z=\sum_{\{S\}} e^{\beta \sum_{<i,j>} \delta_{S_i S_j}} \; ,
\end{equation}
where the first sum $\{S\}$ is on all the spins $S_i=1,\cdots,N$ while the
second sum $<i,j>$ is on the first neighbor spins on the lattice, one easily gets
\begin{equation}
Z=\sum_{\{S\}} \prod_{<i,j>} e^{\beta \delta_{S_i S_j}}
=(e^{\beta})^M \sum_{\{S\}} \prod_{<i,j>} ((1-e^{-\beta})
\delta_{S_i S_j} + e^{-\beta}) \; ,
\end{equation}
with $M$ the total number of bonds on the lattice. Defining
$p=1-e^{-\beta}$, the partition function is
\begin{equation}
Z=(e^{\beta})^M \sum_{\{S\}} \prod_{<i,j>} ( p \delta_{S_i S_j} + (1-p)) \; .
\label{decompositionQ}
\end{equation}
$ $From there, one can read the rules to build the FK clusters. In a
given configuration, for two neighbouring spins $i$ and $j$ such that
$S_i=S_j$, one will put a bond with probability $p$ and no bond with
probability $1-p$.
In the case of the $Z_N$ model the situation is a little bit more
complicated. The partition function (\ref{partition1}) can be written as
\begin{equation}
Z_N =\sum_{\{S\}} \prod_{<i,j>}
(x^{*}_0)^{\delta_{n(i), n(j)}}
(x^{*}_1)^{\delta_{n(i), n(j)\pm 1}}
\cdots
(x^{*}_{[N/2]})^{\delta_{n(i), n(j) \pm [N/2]}} \; ,
\end{equation}
the delta function being defined modulo $N$, ie $\delta_{a,b} = 1
$ if $a \equiv b \; (N)$. A decomposition similar to the one of
(\ref{decompositionQ}) is
\begin{eqnarray}
Z_N &=& \sum_{\{S\}} \prod_{<i,j>} ( 1-x^{*}_{[N/2]}) \delta_{n(i),n(j)}
+ (x^{*}_1-x^{*}_{[N/2]}) \delta_{n(i), n(j)\pm 1} \nonumber \\
&& + \cdots + (x^{*}_{[N/2]-1}-x^{*}_{[N/2]}) \delta_{n(i),n(j)\pm ([N/2]-1)}
+ x^{*}_{[N/2]} \; .
\end{eqnarray}
Note that due to the definition of the $x^{*}_i$, cf
eq.(\ref{integrablecond}), the $x^{*}_i$'s will be ordered and
positives, $1=x^{*}_0 > x^{*}_1 > \cdots > x^{*}_{[N/2]} > 0$. From
there, one can read the construction of the generalised FK
clusters. For each pair of neighbouring spins $S_i$ and
$S_j$, one will put a bond with probability
\begin{equation}
p_{|n(i)-n(j)|}={x^{*}_{|n(i)-n(j)|} - x^{*}_{[N/2]}\over
x^{*}_{|n(i)-n(j)|}}
\end{equation}
and no bond with probability $1-p_{|n(i)-n(j)|}$. These FK clusters
will be used to construct a cluster algorithm of Wolff
type~\cite{Wolff}. A lattice update consists in selecting one spin in
the lattice at a random location then building the FK cluster
containing this spin and then changing the color of this cluster by
changing each spin of the lattice as $S_i \rightarrow S_i + j (N)$
with a random value $1 \leq j \leq N-1$.
It is important to note that for the $Z_N$ models, the FK cluster will
connect spins with different values which is not the case for the
Potts model. One important consequence is that the resulting FK
clusters can not be associated directly to some physical quantities
like it was the case for the Potts models. For these models, the FK
clusters are the basic ingredient for building an update algorithm but
they also encode all the informations associated to the critical
behavior of the model under consideration. For example, the average
size of a cluster build from a random site (Wolff algorithm) is equal
to the magnetic susceptibility. Or the average size of the largest FK
cluster divided by the volume is equal to the magnetization of the
system. Or the two point spin-spin correlation function is equal to
the probability that the two spins are in the same FK cluster. All
these relation can not be valid any more in the case that we consider
here. Still similar quantities can be defined. For example, if one
defines for each cluster $k$
\begin{equation}
\rho_k = |\sum_{i} < e^{2 i \pi n(i) \over N} > | \; ,
\label{mag1}
\end{equation}
the sum being restricted to all the spins in the cluster $k$, then the
magnetisation is associated to the maximum $\rho_k$ along all the
clusters. This is a simpler generalisation of the Potts model for
which each FK cluster contains only spins with identical sign, thus in
that case $\rho_k$ is the volume of the FK cluster. We numerically
compared the quantity $mag_1(L) = (\max (\rho_k)/ L^2)$ with the real
magnetisation obtained as a weighted sum on all the lattice
\begin{equation}
mag(L) = {1\over L^2} |\sum_{i=1,L^2} < e^{2 i \pi n(i) \over N} > | \; ,
\label{mag2}
\end{equation}
the agreement being perfect for both the $Z_4$ and the $Z_5$ spin models.
\begin{figure}
\begin{center}
\epsfxsize=400pt{\epsffile{PlotM.ps}}
\end{center}
\caption{
Main panel: Magnetization vs. $L$ for the $Z_4$ spin model. The magnetization
has been computed in three different ways: from eq.~(\ref{mag1}) and
from the average of the largest spin and FK clusters. In the inset,
the same plot for $Z_5$ spin model.
\label{PlotM}
}
\end{figure}
In the main panel of Fig.\ref{PlotM}, we plot for the $Z_4$ spin
model, the magnetisation obtained from eq.~(\ref{mag1}) which is
compared to the value of the average largest FK cluster and the
average largest spin cluster (in both cases divided by
$L^2$). The scaling is perfect for the magnetization given by
eq.~(\ref{mag1}) with an exponent in very good agreement with the
expected one, $\beta/\nu=1/8$ \cite{Zama_lat2}. For the spin cluster, we also
observe a good scaling but with stronger finite size corrections. Due
to these corrections, it is difficult to give a definite exponent
associated to the spin clusters, we obtain for the largest
sizes $\beta/\nu = 0.110(1)$. Since this value is increasing with the
size, one can speculate that in the infinite size limit this value
will converge towards the magnetic exponent $\beta/\nu$. We also
observe that the largest FK cluster will occupy a finite fraction of
the lattice in the large size limit, which corresponds to the case
where the percolation threshold has been exceeded. We will come back
on this point in the next section.
In the inset of Fig.\ref{PlotM}, we show similar data for the $Z_5$
spin model. We also obtain an excellent agreement between the exponent
obtained from eq.~(\ref{mag1}), the real magnetisation
eq.~(\ref{mag2}) and the exact result $\beta/\nu = 4/35$ \cite{Zama_lat2}. We see that
the spin cluster exponent is again affected by strong finite
size effects and it will become larger than the magnetic exponent
already for the simulated sizes (note the crossing between this curve
and the one corresponding to the magnetization from
eq.~(\ref{mag1})). As for the $Z_4$ case, the largest FK cluster do
not present a scaling law at the critical point.
\begin{figure}
\begin{center}
\epsfxsize=400pt{\epsffile{Pauto.ps}}
\end{center}
\caption{
Autocorrelation time vs. $L$ for the $Z_4$ spin model. We show the data for the Wolff
algorithm as well as for ordinary Monte Carlo updates. The Wolff algorithm show a much better efficiency
in the range of lattice sizes explored.
\label{Pauto}
}
\end{figure}
Even if we expect that the FK clusters are not the natural object to
compute the critical exponents of the $Z_N$ spin models, we can still
use them to build cluster algorithms. In Fig.~\ref{Pauto} we plot the
autocorrelation time for the Wolff algorithm in the $Z_4$ spin model
and we compare it to the autocorrelation time for standard heat bath
Monte Carlo. We observe that the Wolff algorithm is much more
effective than the Monte Carlo one. The autocorrelation time for the
Monte Carlo algorithm scales as $\tau(L) \simeq L^{z_{MC}}$ with
$z_{MC} = 2.1(1)$. The dynamical exponent $z$ is much smaller for the
Wolff algorithm at small sizes ($z_W \simeq 1.2(1)$) but then it
increase for larger sizes. This effect will be explained in the next
section. For the sizes that we can simulate, $L \leq 1280$, the Wolff
algorithm will always be more efficient than Monte Carlo. This is also
the case for the $Z_5$ spin model.
\section{Percolation and critical properties}
In this section, we present results for the properties of both
spin and FK clusters in the $Z_4$ and $Z_5$ spin models. We
show that the distribution of cluster lengths in the critical point is a
power law for spin clusters but not for FK clusters.
Furthermore we show that the spin clusters percolate at the
critical point, while the FK clusters do not. We also perform a first
determination of the fractal dimension of the spin clusters by
using the exponent associate to the distribution of cluster lengths.
\begin{figure}
\begin{center}
\epsfxsize=400pt{\epsffile{PlotD.ps}}
\end{center}
\caption{
Distribution of cluster lengths for the
$Z_4$ spin model and different lattices sizes. While the spin clusters
show a nice power law distribution, the FK cluster distribution falls away from a power law.
In the inset, we show similar data for the $Z_5$ spin model and $L=160$.
\label{PlotD}
}
\end{figure}
First, we consider the distribution of the length \footnote{As
explained in the next section, there exist two natural ways to
define a length. Both methods converge to the same result in the
large size limit.} of contours for the finite size clusters at
the critical point. In Fig.\ref{PlotD}, we present this distribution
for both spin and FK type of clusters in the $Z_4$ spin model for
$L=160$ and $640$. For both lattice sizes, we observe a nice scaling
for the spin clusters with a distribution which is well
described by a power law $N(l) \simeq l^{-\tau_g}$ characteristic of a
percolation critical point~\cite{ Stauffer} with $\tau_g \simeq
2.5(1)$. We expect that $\tau_g$ is related to the fractal dimension
by the following relation $d_f = 2/(\tau_g-1) \simeq 1.33(10)$ which
is of the same order of what is measured in the SLE context
\cite{MarcoRaoul,MarcoRaoul2}. In the next section, we will present
more precise measurements for the fractal dimensions of the spin
clusters. For the FK clusters, it is clear that the scaling is not
satisfied. The distribution is better described by $N(l) \simeq
l^{-\tau_{fk}} \exp{(-l / \xi)}$ with some finite correlation length
$\xi$ with a value of order 2500 lattice units for the $Z_4$ spin model.
\begin{figure}
\begin{center}
\epsfxsize=400pt{\epsffile{Perc.ps}}
\end{center}
\caption{
Percolation test for $Z_4$ spin model. The right part corresponds to the spin clusters, which
percolate right at the critical point. The left corresponds to the FK clusters, which
do not percolate at this point.
\label{PlotP}
}
\end{figure}
\begin{figure}
\begin{center}
\epsfxsize=400pt{\epsffile{PercQ3.ps}}
\end{center}
\caption{
Percolation test for $3$-states Potts model. The upper part corresponds to the FK clusters, while
the lower part corresponds to the spin clusters. Both types of clusters
percolate at the critical point.
\label{Plotc}
}
\end{figure}
This provides a first evidence that the FK clusters do not percolate
at the critical point of the $Z_4$ spin model. This is confirmed in
Fig.\ref{PlotP}, where we present the probability of having a
percolating cluster vs. the ratio $\beta/\beta_c$ (which measures the
distance to the critical point), for both the spin and the FK
clusters. The probability is computed for increasing lattice
sizes. Converging crossing points indicate a critical point. This is
clearly observed for the spin clusters with a critical point
close to $\beta=\beta_c$. For the FK clusters we do not observe a
clear convergency and the lines cross around $\beta \simeq 0.995
\beta_c$. This corresponds to the finite correlation length previously
observed for the distribution of FK clusters length. To convince the
reader that such a small deviation, $|\beta-\beta_c|/\beta_c \simeq
0.005$, is in fact important, we show in Fig.\ref{Plotc} a comparative
plot for the $3$-states Potts model. For this model, both types of
clusters percolate at $\beta_c$. For the larger lattice size, we see
that the deviation is near two order of magnitude smaller than it was
for the $Z_4$ spin model. The percolation value $\beta \simeq 0.995
\beta_c$ is also confirmed in Fig.~\ref{Plotb} where we plot the
distribution $N(l)$ of the FK clusters for various value of $\beta$
and for $L=320$. We observe a nice power law close to $\beta = 0.995
\beta_c$.
\begin{figure}
\begin{center}
\epsfxsize=400pt{\epsffile{Plotb.ps}}
\end{center}
\caption{
Distribution of FK cluster lengths for the $Z_4$ spin model and $L=320$ for different values
of the parameter $\beta$ very near the critical point.
\label{Plotb}
}
\end{figure}
In the case of the $Z_5$ spin model we obtain similar
results. The distribution of spin cluster lengths shows a
power law scaling, while the length distribution of FK clusters
presents a finite correlation length $\xi$ with a value of order 5000
lattice units. See the inset of Fig.~\ref{PlotD}. Furthermore,
spin clusters percolate right at the critical point
$\beta=\beta_c$, while FK clusters percolate at $\beta = 0.9975
\beta_c$.
\section{Fractal dimensions in the bulk}
In this section we present a more accurately computation for the fractal
dimensions of the spin clusters for both $Z_4$ and $Z_5$ spin models.
\begin{figure}
\begin{center}
\epsfxsize=400pt{\epsffile{log-log.eps}}
\end{center}
\caption{
$A(l)$ vs. $l$ for the two definitions of the length (links and hulls, see text for details) on the
spin clusters of the $Z_5$ spin model.
\label{Plot1}
}
\end{figure}
As explained in the previous section, the fractal dimension can be
obtained from the distribution $N(l) \simeq l^{-\tau}$ via the
relation $d_f=2/{(\tau-1)}$. This method turns out not to be very precise
since there exist very strong finite size corrections in the determination of $\tau$.
Here we present another method which
provides a better precision by computing the average area of the
cluster as a function of the interface length around the cluster. We
will consider two different definitions of the cluster interface
length. The first one, which we will call the {\it link length}
corresponds to the number of bonds which are broken around the
cluster. For example, for one isolated spin, the length is
$l_l=4$. The second definition, which we will call the {\it hull
length}, counts the number of spins on the border of the
cluster. For an isolated spin, one has $l_h=1$.
The fractal dimension is defined as $l = R^{d_f}$ with $R$ the radius
of gyration of the cluster. A more direct measurement is given by
computing the average area as a power law of $l$ with $A(l) = R^2
\simeq l^{2/d_f}$. In Fig.~\ref{Plot1} we show the data for the $Z_5$
spin model. In this
plot, we present $A(l)$ for two definitions of $l$, the hull and the
link one. We get a nice scaling law over
a large range of $l$'s. The asymptotic limit is similar for the two
definitions of lengths (hulls and links). A fit of the data gives a value of $d_f \simeq
1.44 (1)$, but such a fit does not provide a good precision since it
is very difficult to take in account the small and large size
corrections. Note that there is a bending in both of these curves for small sizes.
These bendings, which are due to small size corrections, are in opposite directions for the
two definitions of lengths that we employ. This fact will be very useful for the extraction
of a precise fractal dimension and motivate the measurement of the two lengths.
\begin{figure}
\begin{center}
\epsfxsize=400pt{\epsffile{Plot2.ps}}
\end{center}
\caption{
Rescaling for the $Z_5$ spin model and different system sizes with $d_f=1.446$.
The upper curves correspond to the hull lengths with the lower ones correspond
to the link lengths.
\label{Plot2}
}
\end{figure}
A better estimate is obtained in the following way: in
Fig.~\ref{Plot2} we show a similar plot after a rescaling $l
\rightarrow l/L^{d_f}$ and $A(l) \rightarrow A(l)/l^{2/d_f}$. The
rescaling is motivated by the fact that $L^{d_f}$ corresponds the
length of a cluster who fills the lattice, {\it i.e.} $A
=(L^{d_f})^{2/d_f} = L^2$. We observe a collapse for the large size
clusters. There still exists strong finite size corrections, for both
small and large cluster lengths, but we see that a plateau appears
which correspond to the region where scaling works.
The optimal values are $d_f=1.450(2)$ for hulls and $d_f=1.444(2)$ for
links.
\begin{figure}
\begin{center}
\epsfysize=130pt{\epsffile{good_value-1.eps}}
\epsfysize=130pt{\epsffile{good_value.eps}}
\epsfysize=130pt{\epsffile{good_value+1.eps}}
\end{center}
\caption{
Comparative with three values $d_f-0.01$, $d_f$ and $d_f+0.01$
for $L=640$ in the $Z_5$ spin model with $d_f=1.450$ for the hull cluster length and $d_f=1.444$
for the link cluster length. This plot show
the accuracy on the determination of the fractal dimension.
\label{Plot3}
}
\end{figure}
In Fig.~\ref{Plot3} we check the accuracy of our estimation for the
$Z_5$ fractal dimension. By plotting ${A /l^{2/d_f}} $ vs. $l$, the
good value of the fractal dimension should correspond to straight and
perfectly horizontal lines. As shown in the figure, this is obtained
for an optimal value of $d_f=1.446(2)$ (which correspond to the
average of hull and link fractal dimension).
For the $Z_4$ spin model, we can perform similar
measurements. We show in Fig.~\ref{Plot4}, values obtained for $d_f$ for
$Z_4$ and $Z_5$ and for both definitions of the length. This figure contains the
main results of our work. As a final result for the fractal dimension of the spin clusters,
we obtain $d_f=1.438(2)$ for the $Z_4$ spin model and $d_f=1.446(2)$ for the $Z_5$ spin model.
It is important to note that these
values are different from the ones proposed by one of the authors in
\cite{Raoul}. Here a particular interface associated to some (conformal) boundary condition
was considered. The corresponding theoretical predictions were based on the conformal properties (in particular a two level null vector conditions) of the boundary condition changing operator and thus were explicetly dependent on the boundary conditions which generate such interface. One has to take into account that, in the case of non-minimal $Z_N$ theories, the classification of conformal boundary conditions becomes more rich. For instance, differentely from the $Z_3$ case, there are for the $Z_4$ and $Z_5$ models two different mixed boundary conditions related to operators satisfying a two-level null vector condition.
A detailed study, which will be present in a future publication \cite{MarcoRaoul2}, shows that the fractal dimension of some interfaces associated to different boundary conditions are different. Still the fractal dimension obtained
for interfaces to some type of boundary conditions are in good agreement with the one determined
here in the bulk \cite{MarcoRaoul2}.
We point out also that the fractal dimensions of the bulk spin cluster boundaries obtained here do not correspond to any of the values (or their dual) proposed in \cite{Rajapour} and concerning interfaces in $O(n)$ loop models related to parafermionic theories.
\begin{figure}
\begin{center}
\epsfxsize=400pt{\epsffile{Plot4.ps}}
\end{center}
\caption{
Estimations of the fractal exponents for both hull and link lengths, in the $Z_4$ and $Z_5$ spin
models as a function of the lattice size.
\label{Plot4}
}
\end{figure}
\section{Summary and conclusions}
In this paper we studied by Monte Carlo methods the spin
properties of the $Z_N$ spin model. The samples were generated by
using a cluster algorithm which generalize the notion of FK clusters
to the case of the $Z_N$ spin model. For $N\geq 4$ the FK clusters
will in general connect spins with different values. This is not the
case for $N=2$ and $N=3$, respectively the Ising and three-states
Potts model. The cluster algorithm allows to track both spin
and FK clusters and the distribution of all the finite closed
spin and FK clusters can be studied.
In particular we have shown that the spin clusters percolate at
the critical point and the associated exponents do not correspond to
the exponents given by the unitary Kac table of the associated $Z_N$
parafermionic field theory. Note that this is also true for the
spin clusters for $N=2,3$. We have determined the fractal
dimension of the boundaries (interface) of the spin clusters. By
measuring the distribution in size and area of the spin
clusters, we determined the fractal dimension $d_f = 1.438(2)$ for
$N=4$ and $d_f=1.446(2)$ for $N=5$. These
values are different from the ones proposed by one of the authors in
\cite{Raoul} for SLE interfaces in parafermionic theories
and measured numerically for some particular type of boundary
condition in \cite{MarcoRaoul}. Still the fractal dimension obtained
by numerically studying interfaces related to certain different types
of boundary conditions are in good agreement with the one determined
here in the bulk \cite{MarcoRaoul2}.
We have also shown that, although the cluster (Wolff) algorithm
show a much better efficiency in the range of system lengths studied,
the FK clusters do {\bf not} percolate at the critical point for
$N\geq 4$. This is the reason why we computed the fractal dimension
only for the spin clusters.
The results we obtained point out important differences in the
behavior of the spin and FK clusters between the case $N=2,3$,
where the system can be described by a minimal CFT model, and the case
$N\geq 4$, described in the continuum limit by an extended CFT. This
can be traced back to the fact that, for $N\geq 4$ the internal
$Z_N$ degrees of freedom play a fundamental role. This calls for
further analytical studies of the bulk geometric properties of the
parafermionic theories. One way to tackle this problem would be to
provide a Coulomb gas description for parafermionic theories which
would allow to identify the operators associated to the geometric
interface and to compute the associated fractal dimensions.
\noindent{\large\bf Acknowledgments}
\vspace*{0.7 true cm} We thanks Benoit Estienne for useful
discussions. This work has been done in part when one of the authors
(RS) was guest of the Galielo Galilei Institute in Florence, whose
hospitality is kindly acknowledged.
\noindent
\newpage
| {
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Halbreich is a surname. Notable people with the surname include:
Betty Halbreich (born 1927), American personal shopper, stylist, and author
Harry Halbreich (1931–2016), Belgian musicologist
Kathy Halbreich (born 1949), American art curator and museum director | {
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{"url":"https:\/\/gamedev.stackexchange.com\/questions\/183935\/how-do-i-connect-unity-to-a-server-backend-developed-in-net-core\/183956","text":"How do I connect Unity to a server backend developed in .net core?\n\nI'm currently running to a wall in my game development and I need now some discussion to clear my mind... In my mind I have a plan for some little online game (focused around clicking and idling), which I imagined could be served both on website and winapp, and in future even in mobile platforms.\n\nI have chosen C# for it, build some server backend (join server, game server), but 'cause I want this server components be runned on linux, I have chosen .net core. Which is now my problem, because I'm building the game client in Unity, which only works in .net standard.\n\nMy main problem now is that I have build some \"base network library\" for communication, written in .netcore with plenty of NuGet packages, which I now have implement to Unity to get it work, but there are now conflicts (the main one is that unity is using another version of System.Threading).\n\nNow I am wondering what steps should I choose nexts... There are some work behind now (but not so much, there are only some basic proceses for login and creating game worlds), so probably make that base network package compatible with .netstandard and .netcore.\n\nOr I am thinking if it isn't more suitable to choose some another technology like c++ and some 2D engine in it (which to choose?).\n\nWhat are you opinions? And what would you do in this situation?\n\n\u2022 If you use a protocol based on raw TCP or UDP sockets, then it should be possible to implement client and server in completely different technologies. But you won't be able to do a WebGL build. There are no raw sockets in the web browser. Only WebSockets, which work differently and won't be understood by a server which doesn't implement the websocket protocol. Jun 29, 2020 at 9:59\n\u2022 Well, I created some network library with client and server, message protocol... With usage of DI from NuGet, .netcore JSON... DI isn't the main problem in the end... But probably I will need better structure than JSON, which is a problem for implementing in unity, or at least for usage of System.Text.Json library, with is colliding with .netstandard Threading lib. And of course, this library is now compiled with .netcore. Jun 29, 2020 at 10:05\n\u2022 So my first step could be compile that network library for .netcore and .netstandard, and maybe get rid of JSON in message body and make it somehow better... I'm thinking about creating some message protocol for uniting the communication, but not sure how to make it properly. If I get this message protocol out of my \"network library\" I can build something for the unity project. Jun 29, 2020 at 10:16\n\u2022 Unity has an own JSON implementation. Jun 29, 2020 at 10:21\n\n\u2022 If this solves your problem, could you mark your answer as Accepted? Unless you are still waiting for a new answer. Jul 30, 2020 at 8:32","date":"2022-08-12 05:33:42","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.3221510350704193, \"perplexity\": 2092.80195391199}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-33\/segments\/1659882571584.72\/warc\/CC-MAIN-20220812045352-20220812075352-00622.warc.gz\"}"} | null | null |
Награды города Чебоксары — муниципальные награды столицы Чувашии — города Чебоксары.
К наградам города Чебоксары относятся:
звание «Почётный гражданин города Чебоксары»;
медаль «За заслуги перед городом Чебоксары»;
юбилейная медаль «В память о 550-летии города Чебоксары»;
Почётная грамота администрации города Чебоксары.
Городские награды являются формой поощрения граждан и организаций за заслуги в экономике, совершенствование системы городского самоуправления, жилищно-коммунального хозяйства, науке, культуре, спорте, искусстве, военной службе и иные заслуги перед городом Чебоксары.
Перечень наград
См. также
Награды Чувашии
Чебоксары
Чебоксары | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,488 |
Antirrhea rodwayi är en fjärilsart som beskrevs av Hall 1939. Antirrhea rodwayi ingår i släktet Antirrhea och familjen praktfjärilar. Inga underarter finns listade i Catalogue of Life.
Källor
Praktfjärilar
rodwayi | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,291 |
Kentucky Wildcats forward Jacob Toppin (0) celebrates his dunk with his teammates during the UK vs. Central Michigan men's basketball game on Monday, Nov. 29, 2021, at Rupp Arena in Lexington, Kentucky. UK won 85-57. Photo by Michael Clubb | Staff
Michael Clubb
Business as usual for Kentucky: Takeaways from 85-57 win over Central Michigan
Jaron Centers
Kentucky's Monday night win over Central Michigan felt like those blowouts of old that UK fans are used to.
The Wildcats dominated in nearly every aspect of the game, defeating CMU 85-57 in their sixth consecutive victory. In Central Michigan head coach Tony Barbee's return to Rupp Arena, it certainly didn't appear John Calipari's squad tried to show mercy. The Chippewas weren't able to narrow the Cats lead for the entire game, as the only tie in 40 minutes was for a mere 52 seconds---which was in the first minute of the game.
While blowouts are easy to look over, here's a few takeaways from tonight's impressive showing from the Wildcats.
Efficiency on the Defensive End
From the opening tip, defensive pressure was a clear point of emphasis---and it showed on the stat sheet. Central Michigan struggled from the field, shooting just 10-33 total in the first half. The Cats were able to force 15 turnovers, scoring 23 points off of those takeaways. Additionally, five players collected at least one steal, totalling nine as a team.
It was apparent that Calipari is starting to get his team into the "defensive confidence" he wants his players to have.
Jacob Toppin, despite the impressive performance, was unsatisfied. "Defensively, we can get better,' Toppin said. 'In practice every day we work on defense, because that's what we want to be---a defensive team."
Getting Off to a Good Start Offensively
The first 20 minutes saw the best scoring half from the Cats since 2018, when they put up 51 first half points against Winthrop. UK shot an efficient 52 percent from the field, averaging 1.46 points per possession in the first half alone. From the three, Kentucky shot 5-7 from beyond the arc, totaling 71.4 percent in the half. The impressive shooting performance gave the Cats a 51-25 lead going into the locker room at halftime.
Consistency From Washington and Tshiebwe
A characteristic of great teams is consistent performances from it's stars. Tonight was business as usual for TyTy Washington Jr. and Oscar Tshiebwe.
Tshiebwe gained double-figures in the rebounding column for the seventh game in a row, grabbing 16 rebounds and scoring 20 points. Tonight's performance keeps his spot at first in the nation in rebounding, averaging 16 boards a game.
For Washington, 15 points (6-11 FG) makes his sixth straight game scoring in double figures, averaging 14.8 ppg. The freshman also hit three shots tonight from behind the 3-point line, averaging 40 percent this season behind the arc.
Calipari commended Washington's ability to "let the game come to him" this season.
"He does. That's just his mentality, that's his personality," Calipari said about Washington. "Really comfortable in his own skin. He's comfortable with who he is as a player. He's not there to say, 'I'm going to prove that I can do this.' He's not playing like, 'I need to get some baskets now, I gotta get to double figures,'--- he doesn't play that way."
Through seven games, Kentucky's squad is appearing to come together. The Cats will have one more "tune-up" game, as Calipari calls them, to prepare for a tough three-game stretch against Notre Dame, Ohio State and Louisville.
UK will host the Southern Jaguars next Tuesday, Dec. 7 as part of the Unity Series, a five-year deal with the Southwestern Athletic Conference to feature HBCUs (historically black colleges & universities) on "America's biggest stages." That game will air on the SEC Network and tip-off at 7 p.m.
Jacob Toppin
Oscar Tshiebwe
Man accused of theft on north campus caught by dance team members
UK hands out KN95 masks across campus
John Calipari honors former Kentucky coach Joe B. Hall
No. 18 Kentucky roll past Vanderbilt 78-66 for first road win of season
UK resumes in-person classes as omicron cases rise
Column: Novak Djokovic's visa mix-up proves fatal amidst vaccine exemption
TyTy Washington etches name into UK history; breaks single-game assist record
Chris Rodriguez returning to Kentucky for 2022 season
The importance of academic breaks
Big blue beatdown: No. 18 Kentucky blows by No. 22 Tennessee 107-79
Why do you think attendance is low in the basketball student section?
Attendance in the eRUPPtion zone has been falling for the first few basketball games of the season. Read the dueling columns.
Student tickets are too high
Students are finding reasons to not go
The art of closing out road games...and Kentucky's failure to master it
Kentucky women's tennis defeats Miami Ohio 5-2; improves to 3-0 on season
No. 19 Kentucky fall 84-58 to No. 5 Tennessee in Knoxville
UK Rifle defeats NC State behind historic performance
No. 12 Kentucky men's tennis sweep Dayton in doubleheader to open season
Kentucky rifle defeats Army in West Point
Kentucky track and field finish with 18 individual event wins in Jim Green Invitational | {
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Q: How to autoselect default option on re-render in React MaterialUI Select I'm working on a feature where client get's discount when buying a package.
The item on the left is fixed and doesn't change. It comes in package with the item on the right where client can choose a snowboard:
All I need is that when client chooses a size, but then swipes to the next snowboard the size chosen from the previous snowboard would be set back to default 'CHOOSE SIZE OPTION'.
Here is the code of the Parent Component:
import React, { useState, useEffect } from 'react';
// Material UI
import { makeStyles } from '@material-ui/core/';
// React Responsive Carousel
import { Carousel } from 'react-responsive-carousel';
import '../../../../../../node_modules/react-responsive-carousel/lib/styles/carousel.min.css'
// Component
import PackageProd from './PackageProd/packageProd';
const useStyles = makeStyles({
packageCard: {
maxWidth: 345,
height: '100%'
},
});
const Packages = (props) => {
const classes = useStyles();
const [defaultProd, setDefaultProd] = useState({ size: 'CHOOSE THE SIZE', discount: '0', price: '0', barcode: 'default' });
function handleSetProd(val){
setDefaultProd(val)
}
return <Carousel className={classes.packageCard} showIndicators={false}
renderItem={item => <div style={{ background: "white", height: '100%' }}>{item}</div>}
onChange={(val)=>{
setDefaultProd({ size: 'CHOOSE THE SIZE', discount: '0', price: '0', barcode: 'default' })
}}
>
{props._packages.map((prod, i)=>{
return <PackageProd
key={i}
_prod={prod}
_handleSetProd={handleSetProd}
_defaultProd={defaultProd}
></PackageProd>
})}
</Carousel>
}
export default Packages;
I'm using a npm package Carousel that comes with an inbuilt method onChange that fires every time you swipe. So it set the defaultProd every time I swipe to the next snowboard or previous one.
Here is the child components:
import React, { useEffect, useState } from 'react';
// Material UI
import { makeStyles, Grid, Card, CardContent, CardMedia,
Typography, CardActions, Button, FormControl, InputLabel,
Select, } from '@material-ui/core/';
import { display } from '@material-ui/system';
// React Router
import { Link } from "react-router-dom";
// Knight Demon
import knightDemon from '../../../../../../assets/icons/knight_demon.png';
// Price Format
const { format } = require('number-currency-format');
const useStyles = makeStyles({
sizes: {
minWidth: '100%'
},
media: {
height: '20rem',
objectFit: 'contain'
},
text: {
color: 'black'
},
price: {
color: 'green'
},
redPrice: {
color: 'red',
textDecoration: 'line-through'
},
});
const PackageProd = (props) => {
const classes = useStyles();
const [prodDetails, setProdDetails] = useState({});
const handleChange = (event) => {
var specificProd = JSON.parse(event.target.value)
props._handleSetProd(specificProd)
};
const PriceWithDiscount = () => <Grid container direction='row' justify='center' spacing={2}>
<Grid item>
<span className={classes.redPrice}>
{format(props._defaultProd.price, {
currency: 'isk',
showDecimals: 'NEVER',
thousandSeparator: ' '
})}
</span>
</Grid>
<Grid item>
<span className={classes.price}>
{format(Math.ceil(parseInt(props._defaultProd.price) - (parseInt(props._defaultProd.price) * props._defaultProd.discount / 100)), {
currency: 'isk',
showDecimals: 'NEVER',
thousandSeparator: ' '
})}
</span>
</Grid>
</Grid>
const PriceWithoutDiscount = () => <span className={classes.price}>
{format(props._defaultProd.price, {
currency: 'isk',
showDecimals: 'NEVER',
thousandSeparator: ' '
})}
</span>
return <div>
<CardMedia
className={classes.media}
component='img'
image={props._prod.images.length > 0 ? props._prod.images[0] : knightDemon}
/>
<CardContent>
<Link to={`/product?id=${props._prod._id}`}>
<Typography className={classes.text} variant='h5' component='h2' onClick={()=>console.log(props._prod)}>
{props._prod.description}
</Typography>
</Link>
{props._defaultProd.discount > 0 ?
<PriceWithDiscount></PriceWithDiscount> :
<PriceWithoutDiscount></PriceWithoutDiscount>
}
</CardContent>
<CardActions>
{props._prod.sizepricesdiscountqty.length >= 1 &&
props._prod.sizepricesdiscountqty[0].size !== '' ?
<FormControl className={classes.sizes}>
<InputLabel htmlFor='age-native-simple'>SIZE</InputLabel>
<Select
native
defaultValue={props._defaultProd.size}
onChange={handleChange}
inputProps={{
name: 'prodDetails',
id: 'age-native-simple',
}}
>
<option value={JSON.stringify(props._defaultProd)}>CHOOSE SIZE</option>
{props._prod.sizepricesdiscountqty.map((item, i)=>{
if(parseInt(item.qty) > 0){
return <option key={i}
value={JSON.stringify(item)}
>
{item.size.toUpperCase()}
</option>
}
})}
</Select>
</FormControl> :
<Grid container justify='flex-start'>
{props._prod.sizepricesdiscountqty[0].size === '' ?
<Typography>SIZE: NO SIZE</Typography> :
<Typography>SIZE: {props._prod.sizepricesdiscountqty[0].size.toUpperCase()}</Typography>
}
</Grid>
}
</CardActions>
</div>
}
export default PackageProd;
The behaviour which I don't understand is the defaultValue of the Material Select Component. When I log the defaultProd which I pass from the parent it has all the values as it suppose to. When I choose the different size it changes it and sets the defaultProd in the parent to the new object.
Problem is when I swipe to a new product it should change the defaultValue to size value of defaultProd but it doesn't. Changes are reflected in price, console logs the defaultProd correctly, but size doesn't change to 'CHOOSE SIZE' and I have no idea why it doesn't reflect the change as it should. Here is the visual example:
*
*I choose the size and the changes are reflect both in price and the size and I successfully log the change:
But then I swipe to the next snowboard and swipe back to the previous one. Price reflect the change. defaultProd is logged correctly, but size doesn't change:
What I tried so far was to try and force the re-render with:
const [, updateState] = useState();
const forceUpdate = useCallback(() => updateState({}), []);
and fire them inside useEffect() in child component. I tried using value instead of defaultValue but when changing the size it wouldn't work and would just show CHOOSE SIZE all the time. At this point I don't understand how defaultValue in material Select works and why it doesn't reflect changes.
A: In child component I changed defaultValue to value in Select, deleted native and used renderValue function. So my child component code in Select looks like this:
<Select
value={JSON.stringify(props._defaultProd)}
renderValue={(val)=>JSON.parse(val).size}
onChange={handleChange}
>
<option disabled value={JSON.stringify(props._defaultProd)}>CHOOSE SIZE</option>
{props._prod.sizepricesdiscountqty.map((item, i)=>{
if(parseInt(item.qty) > 0){
return <option key={i}
value={JSON.stringify(item)}
>
{item.size.toUpperCase()}
</option>
}
})}
</Select>
Now the value is being reflected on rerendering.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,951 |
Q: Allow Non-Technical Staff to reboot a Hyper-V VM I have a Hyper-V host machine with a number of guests. One of the guests is a Linux VM running a vendor app that isn't exactly the most robust piece of software in the world. About once per week, maybe once every other week, the app just locks up and becomes unresponsive.
When the app locks up, I can login in to the Linux server and restart a service to fix everything. Obviously, rebooting the server has much the same effect, though it takes a couple minutes longer. I'd like to cut IT out of the process, and give the staff in the office that use the app the ability to restart the server themselves. That would be faster, since they'd cut out the part where they wait for IT to see and respond to the ticket, it would take some occasional work off our plate, and it would make the staff happier, since they'd feel they have more control of the situation.
How can I do this?
Powershell comes to mind, but I don't want to give them admin access to either the host or the guest. I also wouldn't want to leave the server name in a script somewhere in a way that's easy to find. These are some very non-technical staff, but any idiot can find and change a server name in a block of text. Maybe a powershell script that just kicks of a scheduled job setup with a user that does have the appropriate rights? All ideas welcome.
A: Yes- you can use RBAC to assign the linux VM to a scope, and then apply the necessary roles to the office staff for that scope.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,418 |
A metralhadora MAC mle 1931 (designação oficial francesa Mitrailleuse modèle 1931 - metralhadora modelo de 1931), foi uma metralhadora usada em tanques franceses da época da Segunda Guerra Mundial, bem como em fortificações como a Linha Maginot. Às vezes também é conhecida como JM Reibel, de Jumelage de mitrailleuses (emparelhamento de metralhadoras), ou metralhadores de montagem dupla Reibel e realmente se refere ao quadro especializado de montagem dupla usado em cúpulas cloche JM nas fortificações da Linha Maginot, enquanto a MAC mle 1931 refere-se especificamente à metralhadora. As montagens duplas JM eram a montagem padrão para o mle 1931 em fortificações fixas, enquanto os tanques e outros veículos blindados recebiam metralhadoras individuais.
Visão geral
A metralhadora Reibel é uma arma operada a gás calibrada no cartucho MAS de 7,5 mm e foi alimentada com carregadores verticais de 150 cartuchos montados lateralmente. A variante usada em fortificações foi modificada com um raiamento diferente para acomodar o tipo de munição balle D encamisada pesada.
Algumas outras armas em serviço francês durante o final dos anos 1940 foram convertidas para o papel no solo, com a adoção de carregadores de cofre de 35 cartuchos montados na lateral e adaptadores para tripés US M2 Browning.
A metralhadora modelo 1931 é uma arma operada a gás que dispara com o ferrolho aberto e apenas fogo automático. É derivado do francês Fusil-Mitrailleur FM 24/29 (fuzil-metralhador / metralhadora leve) também projetado pelo Tenente-Coronel Reibel e baseado no Fuzil Automático Browning. O FM 24/29 foi desenvolvido em uma metralhadora mais pesada capaz de fogo relativamente contínuo, dando-lhe um cano extremamente grosso e maciço, para atuar como um dissipador de calor. Isso era necessário, uma vez que o FM 24/29 carecia de um cano de troca rápida ou resfriamento a água e seu cano leve normal superaquecia e se desgastava rapidamente, se disparado em mais do que rajadas curtas, com descansos de resfriamento entre eles. O pistão de gás de curso longo está localizado abaixo do cano e opera o grupo de ferrolho de inclinação vertical. A munição é alimentada a partir de carregadores multicamadas de 150 cartuchos montados lateralmente (com balas apontando para o centro do carregador redondo). A arma pode ser modificada para receber carregadores no lado esquerdo ou direito, para facilitar as trocas de carregador enquanto montada no suporte duplo JM lado a lado padrão. A ejeção é direta para baixo, através da calha curta presa à base do receptor, que nas fortificações geralmente conduzia a um tubo ou calha mais longo que direcionava os estojos gastos para as valas externas. A arma foi equipada com uma empunhadura de pistola curvada para a frente para ajudar no controle e um gatilho em estilo de fuzil padrão. Quando montado em fortificações, o suporte duplo incluía uma coronha de ombro duplo ajustável, uma barra de metal tubular que se estendia da parte traseira da estrutura de montagem, que montava uma barra transversal horizontal, com ombreiras em cada extremidade. O operador ficava de frente para as culatras das armas e colocava essas almofadas contra seus ombros. Ele então usaria seu corpo para controlar a traversa, enquanto suas mãos segurariam os punhos da pistola para disparar uma ou ambas as armas. A elevação era controlada por uma manivela de latão embaixo da arma. As montagens gêmeas vieram em configurações T e F ; os tipos F usavam gatilhos e coronhas padrão e eram usados para montagens de canhoneiras em casamatas e cúpulas, enquanto o T apresentava um gatilho operado por cabo Bowden e era destinado ao uso remoto em torres retráteis.
O padrão para um mle 1931 em posições fixas era um JM Reibel de montagem dupla, completo com mira telescópica, azimute e indicadores de nível, parafuso de elevação e calhas de ejeção de estojos gastos. Era manuseada por uma tripulação de oito pessoas, incluindo dois atiradores, dois municiadores, dois municiadores assistentes (para buscar munição e recarregar carregadores com uma máquina de recarga montada na mesa que levava clipes de alimentação padrão de 5 cartuchos), um mecânico para reparar quaisquer falhas ou engripagens, e um comandante para direcionar ou coordenar o fogo. O objetivo de emparelhar as armas era permitir um tiro rápido e sustentado. Durante o uso normal, as duas armas seriam disparadas sucessivamente, permitindo que a outra esfriasse. Quando necessário, as duas metralhadoras podem ser disparadas juntas, aumentando a cadência de tiro instantânea. Gráficos foram afixados nas paredes de cada local, descrevendo a técnica operacional padrão:
o fogo normal era de 150 tiros (um carregador) por minuto, alternando entre as armas. Cada arma seria disparada por um minuto, em rajadas, até que o carregador estivesse vazio. Então, o atirador parava e disparava a segunda arma por um minuto enquanto a primeira esfriava e era recarregada. Então a primeira arma poderia ser usada novamente. Essa cadência de tiro poderia ser sustentada por 3 minutos por arma, antes que o calor acumulado atingisse um nível perigoso.
A cadência acelerada era de 450 tiros por minuto (3 carregadores) por arma e era alcançada da mesma forma que o fogo normal; o atirador dispararia três carregadores em um minuto e então pararia antes que seu cano superaquecesse, e então repetiria com a segunda. Devido à maior cadência de tiro, o tiro acelerado era limitado a no máximo dois minutos por arma, pois as metralhadoras ficariam tão aquecidas após disparar 6 carregadores cada, que estariam prontas para superaquecer, mesmo com um minuto para esfriar após o disparo dos primeiros 3 carregadores.
Fogo rápido; Em casos de emergência, como inimigo cruzando o arame farpado, os atiradores eram autorizados a disparar rajadas rápidas de 75 tiros por arma, por vez ou simultaneamente, permitindo que um carregador completo fosse disparado em muito menos de um minuto. Uma cadência tão rápida superaqueceria muito rapidamente o cano se não fosse limitada a apenas 75 tiros. Para ajudar a resfriar as armas mais rapidamente, baldes de água e borrifadores de água eram mantidos próximos a cada posição de JM. Os canos eram resfriados borrifando-os com água (evaporativa) ou removendo a arma do suporte e mergulhando o cano no balde de água. Até 20 litros de água poderiam ser usados por dia por posto de metralhadora apenas para resfriar os canos.
A montagem JM consistia em uma estrutura quadrada de metal grossa, dimensionada para caber em uma canhoneira de fortificação francesa padrão (abertura); as armas eram montadas em um berço giratório igualmente resistente dentro desta estrutura. A moldura quadrada se encaixava perfeitamente na canhoneira e era presa por dobradiças e parafusos. Isso garantiu que não houvesse lacunas por onde as balas inimigas pudessem entrar no bunker (exceto a abertura muito pequena pela qual a mira telescópica espiava), mas permitia que as armas fossem apontadas e visadas sobre qualquer pessoa fora das paredes. As posições eram frequentemente compartilhadas com um canhão antitanque compartilhando a mesma abertura de canhoneira; o suporte JM seria articulado para trás e o canhão antitanque deslizaria para frente em seu trilho montado no teto, até que seu cano estivesse do lado de fora e a culatra do lado de dentro. Era cercado por uma moldura quadrada semelhante, que se encaixava perfeitamente na canhoneira. A única vez que os ocupantes do bunker ficavam expostos ao fogo inimigo, era nos breves momentos em que trocava-se o suporte da metralhadora por um suporte de artilharia.
Usuários
: Usado no tanque leve Hotchkiss H35.
: Usado desmontado pelo Armée Nationale Tchadienne governamental durante a Guerra Civil do Chade.
: Usado no Renault UE Chenillette.
: Usado no tanque leve Hotchkiss H35.
: Usado em veículos blindados, tais como AMR 33, Renault FT ou Panhard EBR, e em fortificações. Usado durante a Guerra da Indochina, modelos M1931A foram usados em veículos ou desmontados em tripés.
: Exemplares capturados. O modelo MAC 1931 serviu como Kpfw MG 331(f).
: Usado no tanque leve Hotchkiss H35.
: Usado no tanque leve Hotchkiss H35.
: Usado no tanque leve Renault R35.
: Usado no tanque leve Renault R35.
: Usado no tanque leve Renault R35.
: Usado no tanque leve Renault R35.
: Usado no tanque leve Renault R35.
: Usado no tanque leve Renault R35.
: Usado no tanque leve Renault R35.
Grupos não-estatais
Chetniks: Usado no tanque leve Hotchkiss H35.
Partisans iugoslavos: Usado no tanque leve Hotchkiss H35.
Viet Minh: Usado desmontado.
Vietcongue
Veja também
FM-24/29: O carregador de cofre é baseado no modelo BAR no qual o mle 1931 foi baseado.
MAC 1934: um derivado alimentado por cinta de disparo mais rápido que o mle 1931 usado a bordo de aeronaves.
Referências
Bibliografia
FERRARD, Stéphane. France1940: L'armement terrestre, ETAI, 1998,
Equipamentos militares de 1930-1939
Metralhadoras da Segunda Guerra Mundial
Armas da França na Segunda Guerra Mundial
Metralhadoras da França
Metralhadoras médias
!CS1 francês-fontes em língua (fr)
Armas de fogo francesas de 7,5×54mm | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,876 |
\section{Introduction}
\label{sec:intro}
Supersymmetry (SUSY) \cite{Martin:1997ns} has now been extensively searched for at the LHC \cite{ATLAS-
CONF-2011-096,CMS-Mono-B,ATLAS-CONF-2012-033,Chatrchyan:2011zy,CMS-PAS-SUS-11-004-B,CMS-PAS-
SUS-12-002-B,CMS-PAS-SUS-12-005-B}. In the limit that the lightest supersymmetric particle is massless the bounds on
equal mass squarks and gluinos are $\gtrsim1.5$~TeV. However, if the LSP is massive and degenerate with the squarks and/or gluinos, these limits can be severely weakened \cite{Alwall:2008ve,Alwall:2008va,LeCompte:2011cn,LeCompte:2011fh,Izaguirre:2010nj}.
In the case of \textquoteleft extreme' compression, initial state radiation (ISR) in the form of jets may be used to discover the
model \cite{Gunion:1999jr}. This method has now been used to set mass limits on squarks and gluinos
\cite{LeCompte:2011cn,LeCompte:2011fh} and stops \cite{He:2011tp,Drees:2012dd}. Other ideas to look for compressed
SUSY include using monophotons \cite{Belanger:2012mk} and soft leptons \cite{Rolbiecki:2012gn}. Here we look at
simplified models in which the first two generations of squarks and/or the gluinos can be quasi-degenerate with the
LSP and set limits on the particle masses. These models represent a \textquoteleft worst case' scenario for R-parity
conserving SUSY at the LHC and thus give a model independent bound on these masses.
The limits completely rely on the accuracy of the ISR prediction for radiated jets. Thus, we use two independent matching
schemes (CKKW-L \cite{Catani:2001cc,Lonnblad:2001iq}) and (MLM \cite{Mangano:2006rw}) to carefully understand
the accuracy of the prediction. In addition, matching scales and parton shower variables are changed to ensure that the limits
are robust.
\section{ISR and matching procedure}
\label{sec:ISR}
\begin{figure*}[ht!]
\begin{center}
\includegraphics[scale=0.45]{Jet1.png} \hspace{0.8cm}
\end{center}\vspace*{-0.4cm}
\caption{A comparison of the uncertainty in the ISR jet $p_T$ distribution for the production of squarks ($M_{\tilde{q}}
=500$~GeV) between the parton shower prediction (green, light), MLM matching (pink, medium) and CKKW matching (blue, dark). The parton
shower uncertainty is found by varying the Pythia 6 and 8 parton showers between the \textquoteleft wimpy' and
\textquoteleft power' settings \cite{Plehn:2005cq}. The matching uncertainties are found in both cases by varying the
matching scales between 50 and 200~GeV and additionally for MLM matching by varying the parton shower between the
\textquoteleft wimpy' and \textquoteleft power' settings. For the softer jet 3 (the first unmatched jet), the relative uncertainty of the parton shower approach is reduced since the phase space for this emission is better constrained. \label{fig:MGvsPyth}}
\end{figure*}
To reliably set limits in SUSY models with compressed spectra it is vital that the ISR is well modeled and any uncertainties
are understood. To achieve this we must use a matrix element prediction for hard and well separated partons
whilst using a parton shower for soft and/or collinear jets. To bring the two approaches together we must match the two
methods in a consistent algorithm.
The reason that a matrix element prediction for hard radiation is required is that a parton shower approach contains large
uncertainties in this regime. They are mainly due to the choice of starting scale for parton
shower evolution. This has been shown to have a large effect in squark and gluino production
\cite{Plehn:2005cq} and we show the variation between the \textquoteleft wimpy' and \textquoteleft power' shower settings in Pythia 6 \cite{Sjostrand:2006za} and 8 \cite{Sjostrand:2007gs} in Fig.\ref{fig:MGvsPyth}.
In order to match the matrix element to the parton shower we must be careful to avoid double counting so that areas of phase
space are only filled by one approach. In addition we would like a smooth transfer between the different areas of validity.
Finally, the prediction should not have a significant dependence on the chosen matching scale or
parton shower. Within SUSY we have the additional problem that we can double count events with
resonant propagators, which must be removed \cite{Plehn:2005cq,Alwall:2008qv}.
To independently check the predictions of the matching algorithm, two approaches were used. First is the integrated MLM
\cite{Mangano:2006rw} matching available in Madgraph 5 \cite{Alwall:2008qv,Alwall:2011uj} which is interfaced with the Pythia
6 shower \cite{Sjostrand:2006za}. To estimate the uncertainty, we varied the matching scale between 50
and 200~GeV and independently the parton shower between the \textquoteleft wimpy' and \textquoteleft
power' settings. As seen in Fig.\ref{fig:MGvsPyth}, this results in a significant reduction in the uncertainty
compared to the parton shower alone.
The second approach was the CKKW-L \cite{Catani:2001cc,Lonnblad:2001iq} matching algorithm, developed for Pythia 8
\cite{Sjostrand:2007gs,Lonnblad:2011xx}. It was adapted to work with SUSY\footnote{We would especially like
to thank Stefan Prestel for his invaluable help in adapting the algorithm.} and gives consistent results to those obtained with
MLM matching, Fig.\ref{fig:MGvsPyth}.
\section{Simplified Models}
\label{sec:SimpMod}
\begin{figure*}[ht!]
\begin{center}
\includegraphics[scale=0.25]{Spectrum.jpg}
\put(-345,98){(a) \bf{Decoupled Gluino}}
\put(-223,98){(b) \bf{Decoupled Squark}}
\put(-86,98){(c) \bf{Equal Mass}}
\put(-294,68){$\tilde{g}$}
\put(-163,68){$\tilde{q}$}
\put(-345,80){$\infty$}
\put(-294,30){$\tilde{q}$}
\put(-163,30){$\tilde{g}$}
\put(-37,30){$\tilde{g}$}
\put(5,10){$\tilde{q}=\tilde{g}-\frac{1}{2}(\tilde{g}-LSP)$}
\put(-300,-10){LSP}
\put(-169,-10){LSP}
\put(-43,-10){LSP}
\put(-375,15){1 - 100}
\put(-365,5){GeV}
\end{center}
\caption{The spectra for the simplified models studied in this paper. For the \textquoteleft Decoupled Gluino' scenario we remove the gluino from the spectrum and vary the mass difference between the squarks and the LSP from 1 to 100~GeV. In the \textquoteleft Decoupled Squark' scenario we remove any squarks from the spectrum and vary the mass difference between the gluino and the LSP from 1 to 100~GeV (see accompanying text for caveats). In the equal mass scenario, the gluino is set as the most massive particle and the squark mass is halfway between the gluino mass and LSP mass. In all scenarios the squarks are a summation over the first two generations and the third generation are ignored. \label{fig:Spectrum}}
\end{figure*}
In order to reduce the SUSY parameter space and place model independent limits, we use three simplified models.
The idea is to investigate the \textquoteleft worst case' scenario for R-parity conserving SUSY. We thus assume
that either the first two generations of squarks or the gluinos or both are quasi-degenerate with the
LSP. The degeneracy has the effect of making all of the SUSY decays invisible to the detector as the produced charged particles are too soft to be reconstructed. Therefore, events with ISR are solely relied upon to set any
limits on the model.
Our first scenario is labeled the \textquoteleft Decoupled Gluino' model, Fig.\ref{fig:Spectrum}(a). Here the
first two generations of squarks are quasi-degenerate with the LSP (1-100~GeV mass splitting) while the gluino is completely
removed from the scenario. The idea is to set a lower limit on the first two generation squarks masses that is completely
independent of the gluino mass.
The third generation of squarks are left free (obviously heavier than the LSP) because in general the Yukawa
contribution to the running of the mass leads to a splitting between these and the first two
generations of squarks. However, a degenerate contribution can easily be added by simply rescaling the cross-section by 5/4 for
only sbottoms or 6/4 for stops as well.
The second scenario we name the \textquoteleft Decoupled Squark' model, Fig.\ref{fig:Spectrum}(b). Here the gluino is quasi-degenerate with the LSP (1-100~GeV mass splitting) and the first two generations of squarks are removed from the model.
In the limit that all squarks are removed from the scenario it must be stated that the gluino becomes stable and a distinctive signal
would therefore be seen as so called \textquoteleft R-hadrons'. In fact, even for moderate squark masses it is easy to make a
gluino in a compressed spectra long-lived. However, it is possible that the third generation squarks could be much lighter than the
other squarks. These could mediate prompt gluino decay whilst having a negligible impact on the search. Therefore we assume a
prompt decaying gluino in this scenario as an interesting limiting case.\footnote{Such models are already being
investigated by the LHC collaborations. \cite{CMS-PAS-SUS-11-016-B}}
As a third scenario we consider the \textquoteleft Equal Mass' model, Fig.~\ref{fig:Spectrum}(c). Here, the gluino mass
is set quasi-degenerate with the LSP (1-100~GeV mass splitting) and the first two generations of squarks have a
mass halfway between the gluino and the LSP,
\begin{equation}
M_{\tilde{q}}=\frac{1}{2}(M_{\tilde{g}}+M_{LSP}).
\end{equation}
The third generation squarks are once again ignored but there is no conceptual reason why they could not be added. However, they would only give a very small contribution to the final cross-section so our limit would remain practically unchanged.
Additional assumptions have to be made on the models in order to use the LHC analyses. Firstly, we assume that all decays are prompt and we have no new heavy states traversing the detector or producing displaced vertices. In any case, these signals are currently searched for and should provide a distinctive signature \cite{Chatrchyan:2012sp}.
Secondly, when the mass splittings between the squarks/gluinos and the LSP is increased, we assume that the decay occurs directly to the next lightest particle in our decay chain. Therefore, we assume that no other states exist in between. In the limit of degeneracy, even if new states did exist, the phenomenology is unchanged because all the momentum of the initial parent particle is still transfered to the LSP. For increased splitting it is possible that the limits may be changed with multiple decays to many soft particles. However, we do not consider this possibility.
\section{Searches}
\label{sec:searches}
\begin{figure*}[ht!]
\begin{center}
\includegraphics[scale=0.4]{SquarkLineMono.png}
\includegraphics[scale=0.4]{SquarkLineSUSY.png}
\end{center} \vspace*{-0.4cm}
\caption{Ratios of the signal cross-section $(\sigma_{Sig})$ to the cross-section required for exclusion $(\sigma_{Exc})$ at the 95\% confidence level for squark masses in the decoupled gluino scenario, Fig.\ref{fig:Spectrum}(a). The monojet searches (left) and SUSY searches (right) are shown.\label{fig:SquarkLineLimit}}
\end{figure*}
\begin{figure*}[ht!]
\begin{center}
\includegraphics[scale=0.4]{SquarkLoopMono.png}
\includegraphics[scale=0.4]{SquarkLoopSUSY.png}
\end{center} \vspace*{-0.4cm}
\caption{Limits at the 95\% confidence level for squark masses in the decoupled gluino scenario, Fig.\ref{fig:Spectrum}(a), for both monojet (left) and SUSY searches (right). The mass splitting between the squark and the LSP is varied between 1 and 100~GeV. \label{fig:SquarkLoopLimit}}
\end{figure*}
In order to set the most stringent bound on the model parameter space we use all current SUSY hadronic searches from both ATLAS and CMS. In addition we apply the monojet searches from both experiments.
The motivation for using all searches is that there exists a range of strategies to look for SUSY at the LHC and it was
not obvious which of these would be most productive for compressed spectra. The \textquoteleft vanilla' LHC searches are
based around an effective mass and missing energy cut. Various signal regions
are defined with different amounts and proportions of effective mass (or $H_T$) and missing energy. Both ATLAS \cite{ATLAS-
CONF-2012-033} and CMS \cite{CMS-PAS-SUS-11-004-B} have an analysis of this kind and they give similar results for mSugra.
In addition, CMS has a number of \textquoteleft shape' based analyses using $\alpha_T$ \cite{Randall:
2008rw,Chatrchyan:2011zy}, $M_{T2}$ \cite{Lester:1999tx,CMS-PAS-SUS-12-002-B} and Razor \cite{Rogan:2010kb,CMS-PAS-
SUS-12-005-B}.
As ISR is vital for discovering SUSY in compressed spectra we would expect that the signal is dominated by the hard emission of a
single parton from the initial state. Therefore, we also use the monojet analyses (ATLAS \cite{ATLAS-CONF-2011-096} and CMS \cite{CMS-Mono-B}) that are optimised for precisely this kind of signal.
All the searches were implemented within the RIVET \cite{Buckley:2010ar} analysis package. As we are only interested in jets, no experimental efficiencies apart from quality cuts were required. Momentum smearing of the jets, including adding mis-measured tails were included \cite{Allanach:2011ej}\footnote{We thank the authors for providing the details of the jet mis-measurement constants in a private communication.}. However, these effects were found to have a negligible impact on all of the search efficiencies. All analyses were tested against any mSugra or simplified model bounds found in the original studies. In addition, where distributions and cut flows are available, these were also tested. Acceptances were reproduced to within 20\% for all analyses but were usually found to agree much better.
In order to set limits in a consistent way across all analyses, we use the 95\% confidence limit given by the Rolke Test \cite{Rolke:2000ij,Rolke:2004mj,Lundberg:2009iu} for the most constraining signal region in each analysis. To be able to fairly compare the analyses and gain a better understanding of the optimum search regions, if the limit is better than the limit expected to have been found, we take the expected limit. In this way we find more conservative limits than the official analyses.
Both CMS $\alpha_T$, and Razor searches use a combined likelihood over all signal regions to set a limit. We instead
conservatively use only the best signal region as this allows a fairer comparison with the other searches. In addition, CMS Razor
uses an unbinned likelihood that is impossible to reproduce as the data is not publicly available, however a 60 bin data
set is available. We combined adjacent bins in this set to give a final search with 20 signal regions which is conservative compared
to the official analysis.
At the time of release both the ATLAS monojet and CMS $\alpha_T$ only had analyses with 1~fb$^{-1}$ and 1.14~fb$^{-1}$. In order to give a fair comparison with the other analyses we extrapolated these searches to 5~fb$^{-1}$ by reducing statistical errors. This is a conservative approach as many systematic errors are also statistically limited and likely to be improved.
\begin{figure*}[ht!]
\begin{center}
\includegraphics[scale=0.4]{GluLoopMono.png}
\includegraphics[scale=0.4]{GluLoopSUSY.png}
\end{center} \vspace*{-0.4cm}
\caption{Limits at the 95\% confidence level for gluino mass in the decoupled squark scenario, Fig.\ref{fig:Spectrum}(b), for both monojet (left) and SUSY searches (right). The mass splitting between the gluino and the LSP is varied between 1 and 100~GeV.\label{fig:GluLoopLimit}}
\end{figure*}
\begin{figure*}[ht!]
\begin{center}
\includegraphics[scale=0.4]{TotLoopMono.png}
\includegraphics[scale=0.4]{TotLoopSUSY.png}
\end{center} \vspace*{-0.4cm}
\caption{Limits at the 95\% confidence level for gluino masses in the equal mass squark-gluino scenario, Fig.\ref{fig:Spectrum}(c), for both monojet (left) and SUSY searches (right). The mass splitting between the gluino and the LSP is varied between 1 and 100~GeV whilst the squark mass is placed between the two.\label{fig:TotLoopLimit}}
\end{figure*}
\section{Limits}
\label{sec:limits}
\begin{table*} \renewcommand{\arraystretch}{1.2}
\begin{center}
\begin{tabular}{|c||c|c|ccc|} \hline
& & Search Region & \multicolumn{3}{c|}{Lower Mass Bound (GeV)} \\
Search & $\mathcal{L}$ (fb$^{-1}$) & (given in source) &Squark & Gluino & Squark $\sim$ Gluino \\ \hline\hline
\underline{Monojet} & & & & &\\
ATLAS* \cite{ATLAS-CONF-2011-096} & $5.0^{\dagger}$ & High/veryHigh $p_T$ &$270$ & $350$ & $530$ \\
\bf{CMS*} \cite{CMS-Mono-B} & 4.7 & $E_T^{\mathrm{miss}} > 400$ &\bf{340} & \bf{440} & \bf{650} \\ \hline\hline
\underline{SUSY} & & & & & \\
ATLAS MET \cite{ATLAS-CONF-2012-033} & 4.7 & A' med/C med &$260$ & $440$ & $600$ \\
CMS $\alpha_T$ \cite{Chatrchyan:2011zy} & $5.0^{\dagger}$ & Optimised $H_T$ bin &$290$ & $450$ & $600$ \\
CMS MET \cite{CMS-PAS-SUS-11-004-B} & 5.0 & A2 &$290$ & $450$ & $620$ \\
CMS $M_{T2}$ \cite{CMS-PAS-SUS-12-002-B} & 4.7 & A/B &- & - & $550$ \\
\bf{CMS Razor} \cite{CMS-PAS-SUS-12-005-B} & 4.4 & bHad($6_4+7_4+8_4+9_4$) & \bf{340} & \bf{500} & \bf{650} \\ \hline
\end{tabular}
\caption{Comparison of the bounds on the mass of SUSY particles for the different searches employed at the LHC. The luminosity of the searches and the most constraining search region are also given (the search region names refer to those given in the original experimental papers). *\textit{The ATLAS and CMS monojet searches only give these bounds for mass differences $<5$~GeV. For larger mass splittings, the bounds become much weaker.} $^{\dagger}$\textit{The ATLAS monojet and CMS $\alpha_T$ searches are both extrapolated from $\sim$1~fb$^{-1}$ to give a more direct comparison.} \label{tab:Limits} }
\end{center}
\end{table*}
To calculate limits in our simplified models, cross sections are calculated to next-to-leading order including known next-to-leading-logarithmic corrections using NLL-Fast \cite{Beenakker:1996ch,Beenakker:2009ha,Beenakker:2011fu}. The theoretical uncertainty is calculated including the factorisation and renormalisation scale and PDF \cite{Nadolsky:2008zw} errors. In addition, we vary the matching scale and parton showers, Fig.\ref{fig:MGvsPyth}, and take the result with the least constraining bound.
We find that in the decoupled gluino scenario with a quasi-degenerate LSP, Fig.\ref{fig:Spectrum}(a), we can set a bound on the first two generations of squarks of $M_{\tilde{q}}>340$~GeV. In this scenario we find that the best limit is set by both the CMS monojet and CMS Razor searches, Fig.\ref{fig:SquarkLineLimit}, whilst the other SUSY searches give slightly weaker limits of $M_{\tilde{q}}\gtrsim300$~GeV. All the limits along with the relevant search regions are given in Tab.\ref{tab:Limits}.
Interestingly, we also see that the extrapolated ATLAS monojet search provides a noticeably worse limit of $M_{\tilde{q}}>270$~GeV. Our analysis showed that the difference between the two monojet searches is primarily down to the second jet veto used in the ATLAS search which reduces the signal acceptance.
For all the SUSY analyses, we find the best search regions are where
the proportion of missing energy compared to the total energy in the event is the highest.
This is to be expected as the relevant events in our models that are expected to have a topology with a dominant jet that will recoil against the LSPs.
As we increase the mass splitting, $M_{\tilde{q}}-M_{LSP}$, we see that the monojet searches rapidly lose their effectiveness, Fig.
\ref{fig:SquarkLoopLimit}. This is due to the fact that both the ATLAS and CMS searches have a third jet veto. As the mass splitting
is increased, the SUSY decays produce additional jets and these events are vetoed.
In contrast, the SUSY searches become more effective as the mass splitting is increased. For example, the CMS Razor limit
improves from $M_{\tilde{q}}>340$~GeV when the mass splitting is 1~GeV to $M_{\tilde{q}}>400$~GeV when the mass splitting is
100~GeV. This is because the SUSY searches often allow any number of additional jets and include the extra radiation in the
search variable.
In the decoupled squark scenario, Fig.\ref{fig:Spectrum}(b), we find that we can set a bound on the gluino mass, $M_{\tilde{g}}
>500$~GeV, Fig.\ref{fig:GluLoopLimit}, and CMS Razor sets the best limit.
Finally, in the equal mass squark gluino model, Fig.\ref{fig:Spectrum}(b), we find a limit of $M_{\tilde{q}} \sim M_{\tilde{g}}
>650$~GeV which is set both by CMS Razor and the CMS monojet search, Fig.\ref{fig:TotLoopLimit}. As before, the limit in the
monojet search is reduced as we increase the mass splitting but the majority of the SUSY limits actually improve and for 100~GeV
mass splitting, CMS Razor gives a limit, $M_{\tilde{g}}>700$~GeV.
\section{Conclusions}
\label{sec:conclusions}
We have set limits in simplified SUSY models with compressed spectra. We find a limit on the first two generations of squarks, $M_{\tilde{q}}>340$~GeV, in a model with a decoupled gluino. With decoupled squarks we find a limit on the gluino mass of, $M_{\tilde{g}}>500$~GeV. With equal mass gluinos and squarks we find a limit of $M_{\tilde{q}} \sim M_{\tilde{g}}>650$~GeV.
We would like to comment that the results for the decoupled squark and equal mass squark-gluino models are in good agreement for the ATLAS SUSY search with those previously presented in \cite{LeCompte:2011cn,LeCompte:2011fh}.
\section{Acknowledgements}
\noindent We would especially like to thank Stefan Prestel for his help adapting the Pythia 8 matching algorithm. In addition we
would like to acknowledge useful discussions with John Conley, Krysztof Rolbiecki, Daniel Wiesler, Johan Alwall and Maurizio
Pierini. This work has been supported in part by the Helmholtz Alliance `Physics at the Terascale' and the DFG SFB/TR9 ``Computational Particle
Physics''. HD was supported by BMBF Verbundprojekt HEP-Theorie� under the contract 0509PDE.
\bibliographystyle{eplbib}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,840 |
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Haughey Lab: Neurodegenerative and Neuroinfectious Disease
Dr. Haughey directs a disease-oriented research program that address questions in basic neurobiology, and clinical neurology. The primary research interests of the laboratory are:
1. To identify biomarkers markers for neurodegenerative diseases including HIV-Associated Neurocognitive Disorders, Multiple Sclerosis, and Alzheimer's disease. In these studies, blood and cerebral spinal fluid samples obtained from ongoing clinical studies are analyzed for metabolic profiles through a variety of biochemical, mass spectrometry and bioinformatic techniques. These biomarkers can then be used in the diagnosis of disease, as prognostic indicators to predict disease trajectory, or as surrogate markers to track the effectiveness of disease modifying interventions.
2. To better understand how the lipid components of neuronal, and glial membranes interact with proteins to regulate signal transduction associated with differentiation, motility, inflammatory signaling, survival, and neuronal excitab...ility.
3. To understand how extracellular vesicles (exosomes) released from brain resident cells regulate neuronal excitability, neural network activity, and peripheral immune responses to central nervous system damage and infections.
4. To develop small molecule therapeutics that regulate lipid metabolism as a neuroprotective and restorative strategy for neurodegenerative conditions. view more
Research Areas: multiple sclerosis, PTSD, HAND, HIV
Norman Haughey, Ph.D.
Healthy Brain Program
The Brain Health Program is a multidisciplinary team of faculty from the departments of neurology, psychiatry, epidemiology, and radiology lead by Leah Rubin and Jennifer Coughlin. In the hope of revealing new directions for therapies, the group studies molecular biomarkers identified from tissue and brain imaging that are associated with memory problems related to HIV infection, aging, dementia, mental illness and traumatic brain injury. The team seeks to advance policies and practices to optimize brain health in vulnerable populations while destigmatizing these brain disorders.
Current and future projects include research on: the roles of the stress response, glucocorticoids, and inflammation in conditions that affect memory and the related factors that make people protected or or vulnerable to memory decline; new mobile apps that use iPads to improve our detection of memory deficits; clinical trials looking at short-term effects of low dose hydrocortisone and randomized to 28 day...s of treatment; imaging brain injury and repair in NFL players to guide players and the game; and the role of inflammation in memory deterioration in healthy aging, patients with HIV, and other neurodegenerative conditions. view less
Research Areas: HIV infection, mental illness, aging, traumatic brain injury, dementia
Leah Rubin, M.A., M.P.H., Ph.D.
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{"url":"http:\/\/121.43.60.238\/sxwlxbA\/EN\/article\/showBrowseTopList.do","text":"#### Top Read Articles\n\n Published in last 1 year\u00a0| In last 2 years\u00a0| In last 3 years\u00a0| All\nPlease wait a minute...\n For Selected: View Abstracts Download Citations Toggle Thumbnails\n Select Blow-Up of the Smooth Solutions to the Quantum Navier-Stokes-Landau-Lifshitz Equations Zhen Qiu,Guangwu Wang Acta mathematica scientia,Series A \u00a0\u00a0 2022, 42 (4): 1074-1088. \u00a0 Abstract \uff08205\uff09\u00a0\u00a0 \u00a0\u00a0 PDF\uff08pc\uff09 \uff08346KB\uff09\uff08123\uff09 \u00a0\u00a0 \u00a0\u00a0 Save In this paper, we investigate the blow-up of the smooth solutions to the quantum Navier-Stokes-Landau-Lifshitz systems(QNSLL) in the domains $\\Omega \\subseteq \\mathbb{R} ^n(n =1, 2)$. We prove that the smooth solutions to the QNSLL will blow up in finite time in the domains half-space $\\mathbb{R} _+^n$, whole-space $\\mathbb{R} ^n$ and ball. The paper also shows that the blow-up time of the smooth solutions in half-space or whole-space only depends on boundary conditions, while the the blow-up time of the smooth solutions in the ball depends on initial data and boundary conditions. In particular, the above conclusions are also valid for NSLL systems.\n Select The General Inverse Bonnesen-Style Inequalities in $\\mathbb{R}^n$ Xu Dong,Yan Zhang,Chunna Zeng,Xingxing Wang Acta mathematica scientia,Series A \u00a0\u00a0 2022, 42 (3): 641-650. \u00a0 Abstract \uff08200\uff09\u00a0\u00a0 HTML \uff0812\uff09 \u00a0\u00a0 PDF\uff08pc\uff09 \uff08337KB\uff09\uff08210\uff09 \u00a0\u00a0 \u00a0\u00a0 Save The isoperimetric problem plays an important role in integral geometry. In this paper we mainly investigate the inverse form of the isoperimetric inequality, i.e. the general inverse Bonnesen-type inequalities. The upper bounds of several new general isoperimetric genus are obtained. Futhermore, as corollaries, we get a series of classical inverse Bonnesen-type inequalities in the plane. Finally, the best estimate between the results of three upper bounds is given.\n Select Similarity and Unitary Similarity of a Class of Upper Triangular Operator Matrices Liqiong Lin,Jiahua Que,Yunnan Zhang Acta mathematica scientia,Series A \u00a0\u00a0 2022, 42 (5): 1281-1293. \u00a0 Abstract \uff08150\uff09\u00a0\u00a0 HTML \uff0815\uff09 \u00a0\u00a0 PDF\uff08pc\uff09 \uff08268KB\uff09\uff08218\uff09 \u00a0\u00a0 \u00a0\u00a0 Save This paper introduces a class of upper triangular operator matrices related to Cowen-Douglas operators, and studies its similarity on Banach spaces and its unitary similarity on Hilbert spaces.\n Select Complex Symmetry for a Class of Truncated Hankel Operators Liling Lai,Jinjin Liang,Yong Chen Acta mathematica scientia,Series A \u00a0\u00a0 2022, 42 (4): 961-968. \u00a0 Abstract \uff08130\uff09\u00a0\u00a0 HTML \uff088\uff09 \u00a0\u00a0 PDF\uff08pc\uff09 \uff08293KB\uff09\uff08160\uff09 \u00a0\u00a0 \u00a0\u00a0 Save The truncated Hankel operator is the compression to the model space of Hankel operator on the Hardy space. In this paper, the complex symmetry for a class of truncated Hankel operators is studied and the complete characterization is given. The obtained results show that, the complex symmetry of truncated Hankel operator may be related to the model space only, or to the model space and the symbol function of the operator both.\n Select Approximate Optimality Conditions and Mixed Type Duality for a Class of Non-Convex Optimization Problems Jiaolang Wang,Donghui Fang Acta mathematica scientia,Series A \u00a0\u00a0 2022, 42 (3): 651-660. \u00a0 Abstract \uff08117\uff09\u00a0\u00a0 HTML \uff084\uff09 \u00a0\u00a0 PDF\uff08pc\uff09 \uff08298KB\uff09\uff08134\uff09 \u00a0\u00a0 \u00a0\u00a0 Save By using the properties of the Fr\u00e9chet subdifferentials, we first introduce a new constraint qualification and then establish some approximate optimality conditions for the non-convex constrained optimization problem with objective function and\/or constraint function being \u03b1-convex function. Moreover, some results for the weak duality, strong duality and converse-like duality theorems between this non-convex optimization problem and its mixed type dual problem are also given.\n Select Perturbations of Canonical Unitary Involutions Associated with Quantum Bernoulli Noises Nan Fan,Caishi Wang,Hong Ji Acta mathematica scientia,Series A \u00a0\u00a0 2022, 42 (4): 969-977. \u00a0 Abstract \uff0895\uff09\u00a0\u00a0 HTML \uff084\uff09 \u00a0\u00a0 PDF\uff08pc\uff09 \uff08502KB\uff09\uff08147\uff09 \u00a0\u00a0 \u00a0\u00a0 Save Quantum Bernoulli noises (QBN) are annihilation and creation operators acting on the space of square integrable Bernoulli functionals, which satisfy a canonical anti-commutation relation (CAR) in equal time and can play an important role in describing the environment of an open quantum system. In this paper, we address a type of perturbations of the canonical unitary involutions associated with QBN. We analyze these perturbations from a perspective of spectral theory and obtain exactly their spectra, which coincide with their point spectra. We also discuss eigenvectors of these perturbations from an algebraic point of view and unveil the structures of the subspaces consisting of their eigenvectors. Finally, as application, we consider the abstract quantum walks driven by these perturbations and obtain infinitely many invariant probability distributions of these walks.\n Select The Self-Adjointness and Dependence of Eigenvalues of Fourth-Order Differential Operator with Eigenparameters in the Boundary Conditions Wenwen Yan,Meizhen Xu Acta mathematica scientia,Series A \u00a0\u00a0 2022, 42 (3): 671-693. \u00a0 Abstract \uff0883\uff09\u00a0\u00a0 HTML \uff081\uff09 \u00a0\u00a0 PDF\uff08pc\uff09 \uff08450KB\uff09\uff08108\uff09 \u00a0\u00a0 \u00a0\u00a0 Save In this paper we consider the self-adjointness and the dependence of eigenvalues of a class of discontinuous fourth-order differential operator with eigenparameters in the boundary conditions of one endpoint. By constructing a linear operator T associated with problem in a suitable Hilbert space, the study of the above problem is transformed into the research of the operator in this space, and the self-adjointness of this operator T is proved. In addition, on the basis of the self-adjointness of the operator T, we show that the eigenvalues are not only continuously but also smoothly dependent on the parameters of the problem, and give the corresponding differential expressions. In particular, giving the Fr\u00e9chet derivative of the eigenvalue with respect to the eigenparameter-dependent boundary condition coefficient matrix, and the first-order derivatives of the eigenvalue with respect to the left and right sides of the inner discontinuity point c.\n Select The Maximal Operator of Vilenkin-like System on Hardy Spaces Chuanzhou Zhang,Chaoyue Wang,Xueying Zhang Acta mathematica scientia,Series A \u00a0\u00a0 2022, 42 (5): 1294-1305. \u00a0 Abstract \uff0882\uff09\u00a0\u00a0 HTML \uff082\uff09 \u00a0\u00a0 PDF\uff08pc\uff09 \uff08311KB\uff09\uff0892\uff09 \u00a0\u00a0 \u00a0\u00a0 Save In this paper, we discuss the boundedness of maximal operator with respect to bounded Vilenkin-like system (or $\\psi\\alpha$ system) which is generalization of bounded Vilenkin system. We prove that when $0 < p <1\/2$ the maximal operator $\\tilde{\\sigma}_p^*f=\\sup\\limits_{n\\in {\\Bbb N}}\\frac{|\\sigma_nf|}{(n+1)^{1\/p-2}}$ is bounded from the martingale Hardy space $H_p$ to the space $L_p$, where $\\sigma_nf$ is $n$-th Fej\\'er mean with respect to bounded Vilenkin-like system. By a counterexample, we also prove that the maximal operator $\\sup\\limits_{n\\in {\\Bbb N}}\\frac{|\\sigma_nf|}{\\varphi(n)}$ is not bounded from the martingale Hardy space $H_{p}$ to the space $L_{p,\\infty}$ when $0 < p <1\/2$ and $\\mathop{\\overline{\\lim}}\\limits_{n\\rightarrow \\infty}\\frac{(n+1)^{1\/p-2}}{\\varphi(n)}=+\\infty$.\n Select Ground State Travelling Waves in Infinite Lattices with Superquadratic Potentials Chunhui Shao,Jijiang Sun,Shiwang Ma Acta mathematica scientia,Series A \u00a0\u00a0 2022, 42 (5): 1451-1461. \u00a0 Abstract \uff0879\uff09\u00a0\u00a0 HTML \uff086\uff09 \u00a0\u00a0 PDF\uff08pc\uff09 \uff08396KB\uff09\uff0837\uff09 \u00a0\u00a0 \u00a0\u00a0 Save In this paper, we consider one dimensional FPU type lattices with particles of unit mass. The dynamics of the system is described by the following $\\ddot{q}_n= U'(q_{n+1}-q_n)-U'(q_n-q_{n-1}), \\quad n\\in{\\mathbb Z},$ where U is the potential of interaction between two adjacent particles and qn denotes the displacement. By directly using the usual variational method, we study the existence of ground state travelling waves, i.e., non-trivial travelling waves with least possible energy, for the above system with more general superquadratic potentials than the previous work of Pankov[10] and Zhang and Ma [20]. Moreover, we also concern the monotonicity of the solitary ground state travelling waves.\n Select Classification of Calabi Hypersurfaces in ${\\mathbb R}^5$ with Parallel Fubini-Pick Form Ruiwei Xu,Miaoxin Lei Acta mathematica scientia,Series A \u00a0\u00a0 2022, 42 (2): 321-337. \u00a0 Abstract \uff0878\uff09\u00a0\u00a0 HTML \uff082\uff09 \u00a0\u00a0 PDF\uff08pc\uff09 \uff08391KB\uff09\uff08109\uff09 \u00a0\u00a0 \u00a0\u00a0 Save The classifications of locally strongly convex equiaffine hypersurfaces (centroaffine hypersurfaces) with parallel Fubini-Pick form with respect to the Levi-Civita connection of the Blaschke-Berwald metric (centroaffine metric) have been completed in the last decades. In [20], the authors studied Calabi hypersurfaces with parallel Fubini-Pick form with respect to the Levi-Civita connection of the Calabi metric and classified 2 and 3-dimensional cases. In this paper, we extend such calssification results to 4-dimensional Calabi hypersurfaces in the affine space ${\\mathbb R}^5$.\n Select A Class of Differential Operators with Eigenparameter Dependent Boundary Conditions Kang Sun,Yunlan Gao Acta mathematica scientia,Series A \u00a0\u00a0 2022, 42 (3): 661-670. \u00a0 Abstract \uff0876\uff09\u00a0\u00a0 HTML \uff084\uff09 \u00a0\u00a0 PDF\uff08pc\uff09 \uff08355KB\uff09\uff08110\uff09 \u00a0\u00a0 \u00a0\u00a0 Save In this paper, A class of third-order differential operators with transition conditions and two boundary conditions containing spectral parameters is studied, and the analytical method is used to do two aspects of work. First, by constructing a new space and a new operator, the eigenvalues of the problem and the operator are connected so that the eigenvalues of the original problem are consistent with the eigenvalues of the new operator. Second, the properties of the eigenvalues of the original problem are studied, and the conclusion that the spectrum of the original problem has only point spectrum is given.\n Select Breather Wave Solutions, Lump Solutions and Semi-Rational Solutions of a Reduced (3+1)Dimensional Hirota Equation Chunmei Fang,Shoufu Tian Acta mathematica scientia,Series A \u00a0\u00a0 2022, 42 (3): 775-783. \u00a0 Abstract \uff0875\uff09\u00a0\u00a0 HTML \uff080\uff09 \u00a0\u00a0 PDF\uff08pc\uff09 \uff081085KB\uff09\uff0845\uff09 \u00a0\u00a0 \u00a0\u00a0 Save In this paper, the long wave limit method is used to study the exact solutions of the (3+1)dimensional Hirota equation under dimensional reduction $z$=$x$. First, the bilinear form is constructed by using Bell polynomials. Based on the bilinear form, the $n$-order breather wave solutions are obtained under some parameter constraints on the $N$-order soliton solution. Secondly, by using the long wave limit method, high order lump wave solutions are obtained. Finally, the combined solutions of the first-order, second-order lump wave solutions and single solitary wave solutions are derived, i.e. semi-rational solutions. All the obtained solutions were analyzed with Maple software for physical characteristics.\n Select Q-(Approximate) Dual of g-Frames in Hilbert Spaces Wei Zhang,Yanling Fu,Shuanbao Li Acta mathematica scientia,Series A \u00a0\u00a0 2022, 42 (3): 694-704. \u00a0 Abstract \uff0868\uff09\u00a0\u00a0 HTML \uff082\uff09 \u00a0\u00a0 PDF\uff08pc\uff09 \uff08350KB\uff09\uff0884\uff09 \u00a0\u00a0 \u00a0\u00a0 Save In this paper, fusing the ideas of dual Fusion frames and approximate dual g-frames, the definitions of Q-(approximate) dual g-frames are given. The relationship between Q-approximate dual g-frames and Q-dual g-frames is discussed. The characterizations of Q-(approximate) dual g-frames are obtained. Finally, by means of Q-approximate dual g-frames, some equivalent conditions for a g-frame to be close to another g-frame are given.\n Select Existence of Positive Ground State Solutions for the Choquard Equation Xudong Shang,Jihui Zhang Acta mathematica scientia,Series A \u00a0\u00a0 2022, 42 (3): 749-759. \u00a0 Abstract \uff0868\uff09\u00a0\u00a0 HTML \uff081\uff09 \u00a0\u00a0 PDF\uff08pc\uff09 \uff08355KB\uff09\uff0867\uff09 \u00a0\u00a0 \u00a0\u00a0 Save In this paper we study the following nonlinear Choquard equation $$$-\\Delta u + V(x)u= (I_{\\alpha}* F(u))f(u), \\hskip0.5cm x\\in{{\\Bbb R}} ^{N} ,$$$ where $N \\geq 3$, $\\alpha \\in (0, N)$, $I_{\\alpha}$ is the Riesz potential, $V(x):\\mathbb{R} ^{N} \\rightarrow \\mathbb{R}$ is a given potential function, and $F\\in {\\cal C}^{1}(\\mathbb{R}, \\mathbb{R})$ with $F'(s)=f(s)$. Under assumptions on $V$ and $f$, we do not require the $(AR)$ condition of $f$, the existence of ground state solutions are obtained via variational methods.\n Select On a Nonlinear Non-Autonomous Ratio-Dependent Food Chain Model with Delays and Feedback Controls Changyou Wang,Nan Li,Tao Jiang,Qiang Yang Acta mathematica scientia,Series A \u00a0\u00a0 2022, 42 (1): 245-268. \u00a0 Abstract \uff0866\uff09\u00a0\u00a0 HTML \uff080\uff09 \u00a0\u00a0 PDF\uff08pc\uff09 \uff081166KB\uff09\uff0842\uff09 \u00a0\u00a0 \u00a0\u00a0 Save In this paper, we study a 3-species nonlinear non-autonomous ratio-dependent food chain system with delays and feedback controls. Firstly, based on the theory of delay differential inequality, some new analytical methods are developed and a suitable Lyapunov function is constructed. Secondly, sufficient conditions for the permanence and global attractivity of positive solutions for the system are obtained. Thirdly, by using the theoretical analysis and fixed point theory, the corresponding periodic systems are discussed, and the conditions for the existence, uniqueness and stability of positive periodic solutions of periodic systems are established. Moreover, we give some numerical simulations to prove that our theoretical analysis are correct. Finally, we still give an numerical example for the corresponding stochastic food chain model with multiplicative noise sources, and achieve new interesting change process of the solution for the model.\n Select Stability Analysis of an HIV Infection Dynamic Model with CTL Immune Response and Immune Impairment Meng Deng,Rui Xu Acta mathematica scientia,Series A \u00a0\u00a0 2022, 42 (5): 1592-1600. \u00a0 Abstract \uff0864\uff09\u00a0\u00a0 HTML \uff082\uff09 \u00a0\u00a0 PDF\uff08pc\uff09 \uff08393KB\uff09\uff0862\uff09 \u00a0\u00a0 \u00a0\u00a0 Save In this paper, we study an HIV infection model with saturation incidence rate, CTL immune response, immune impairment, and intracellular delay. Firstly, the basic reproduction ratio $\\Re_{0}$ of virus infection is obtained by using the next generation matrix method. Secondly, the local stability of feasible equilibria is proved by analyzing the distribution of the root of the corresponding characteristic equations. By constructing appropriate Lyapunov functionals and using LaSalle's invariance principle, we prove that when $\\Re_{0}<1$, the virus infection-free equilibrium is globally asymptotically stable; when $\\Re_ {0}>1$, the immunity-inactivated equilibrium is globally asymptotically stable. Finally, the parameter with critical influence on $\\Re_{0}$ is determined by the parameter sensitivity analysis.\n Select A Class of Weakly Nonlinear Critical Singularly Perturbed Integral Boundary Problems Hao Zhang,Na Wang Acta mathematica scientia,Series A \u00a0\u00a0 2022, 42 (4): 1060-1073. \u00a0 Abstract \uff0864\uff09\u00a0\u00a0 HTML \uff082\uff09 \u00a0\u00a0 PDF\uff08pc\uff09 \uff08438KB\uff09\uff0897\uff09 \u00a0\u00a0 \u00a0\u00a0 Save Based on the boundary layer function method, a class of singularly perturbed problems with integral boundary conditions in weakly nonlinear critical cases are studied. In the framework of this paper, we not only construct the asymptotic expansion of the solution of the original equation, but also prove the uniformly effective asymptotic expansion. At the same time, we give an example to illustrate our results, The comparison images of approximate solution and exact solution under different small parameters are drawn.\n Select Ground State Solutions for Quasilinear Schr\u00f6dinger Equation of Choquard Type Yanan Wang,Kaimin Teng Acta mathematica scientia,Series A \u00a0\u00a0 2022, 42 (3): 730-748. \u00a0 Abstract \uff0863\uff09\u00a0\u00a0 HTML \uff081\uff09 \u00a0\u00a0 PDF\uff08pc\uff09 \uff08399KB\uff09\uff0882\uff09 \u00a0\u00a0 \u00a0\u00a0 Save In this paper, we consider the following quasilinear Schr\u00f6dinger equations of Choquard type $\\begin{eqnarray*} -\\triangle u+\\frac{k}{2}u\\triangle u^2+V(x)u=(I_{\\alpha}\\ast|u|^{p})|u|^{p-2}u, \\, \\, x\\in\\mathbb{R}^N, \\end{eqnarray*}$ where $N\\geq3$, 0 < $\\alpha$ < $N$, $ Select Dimension Theory of Uniform Diophantine Approximation Related to Beta-Transformations Wanlou Wu,Lixuan Zheng Acta mathematica scientia,Series A 2022, 42 (4): 978-1002. Abstract \uff0863\uff09 HTML \uff080\uff09 PDF\uff08pc\uff09 \uff08485KB\uff09\uff0859\uff09 Save For$\\beta>1$, let$T_\\beta$be the$\\beta$-transformation defined on$[0, 1)$. We study the sets of points whose orbits of$T_\\beta$have uniform Diophantine approximation properties. Precisely, for two given positive functions$\\psi_1, \\ \\psi_2:{\\Bbb N}\\rightarrow{\\Bbb R}^+$, define${\\cal L}(\\psi_1):=\\left\\{x\\in[0, 1]:T_\\beta^n x<\\psi_1(n), \\mbox{ for infinitely many $n\\in{\\Bbb N}$}\\right\\}, {\\cal U}(\\psi_2):=\\left\\{x\\in [0, 1]:\\forall \\ N\\gg1, \\ \\exists \\ n\\in[0, N], \\ s.t. \\ T^n_\\beta x<\\psi_2(N)\\right\\}, $where$\\gg$means large enough. We calculate the Hausdorff dimension of the set${\\cal L}(\\psi_1)\\cap{\\cal U}(\\psi_2)$. As a corollary, we obtain the Hausdorff dimension of the set${\\cal U}(\\psi_2)$. Our work generalizes the results of [4] where only exponential functions$\\psi_1, \\ \\psi_2$were taken into consideration. Select Two-Dimensional Infinite Square Well in Fractional Quantum Mechanics Yunjie Tan,Xiaohui Han,Jianping Dong Acta mathematica scientia,Series A 2022, 42 (4): 1018-1026. Abstract \uff0863\uff09 HTML \uff081\uff09 PDF\uff08pc\uff09 \uff08397KB\uff09\uff0870\uff09 Save Fractional quantum mechanics is a generalization of standard quantum mechanics, which is described by fractional Schr\u00f6dinger equation with fractional Riesz derivative operator. In this paper, we consider a free particle moving in a two-dimensional infinite square well, By using L\u00e9vy path integral method, the wave function and energy eigenvalue of the two-dimensional infinite square well are obtained. Then the perturbation expansion method is used to study the two-dimensional infinite square well with$\\delta\\$ function, and the corresponding energy-dependent Green's function is obtained.","date":"2023-02-06 07:09:21","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 1, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7044843435287476, \"perplexity\": 1881.901246906421}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-06\/segments\/1674764500304.90\/warc\/CC-MAIN-20230206051215-20230206081215-00318.warc.gz\"}"} | null | null |
Grace Bay Townhomes! Pool surrounded by tropical garden!
Grace Bay Townhomes! Great Room representative of all of our townhomes!
Grace Bay Townhomes! Gourmet kitchen with granite tops!
Grace Bay Townhomes! Dining inside adjacent to the screened-in porch!
Grace Bay Townhomes! Large master bedroom!!
Grace Bay Townhomes! Garden path to pool!
Grace Bay Townhomes! Just five minutes' walk from World Famous Grace Bay Beach!
Grace Bay Townhomes! Photo taken by a guest!
Grace Bay Townhomes! Grace Bay Townhomes frangipani!
Grace Bay Townhomes! Front of Building!
Grace Bay Townhomes! Pretty flowering shrubs throughout community!
Grace Bay Townhomes! View from covered patio!
Grace Bay Townhomes! Brillant flamboyant!
Grace Bay Townhomes!Floor plan of each townhome!
Grace Bay Townhomes! JoJo! Provo's famous dolphin!
Grace Bay Townhomes! Floor plan of each townhome! | {
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In der Liste der Kulturdenkmale in Skassa sind die Kulturdenkmale des Großenhainer Ortsteils Skassa verzeichnet, die bis Dezember 2020 vom Landesamt für Denkmalpflege Sachsen erfasst wurden (ohne archäologische Kulturdenkmale). Die Anmerkungen sind zu beachten.
Diese Aufzählung ist eine Teilmenge der Liste der Kulturdenkmale in Großenhain.
Skassa
|}
Anmerkungen
Ausführliche Denkmaltexte
Quellen
Weblinks
Skassa | {
"redpajama_set_name": "RedPajamaWikipedia"
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{"url":"https:\/\/nikanclinic.com\/7bcftpe\/most-of-the-oxygen-entering-the-blood-is-transported-98a2f0","text":"## most of the oxygen entering the blood is transported\n\nB. hydrogen ions enter the red blood cells. Hemoglobin is made up of four symmetrical subunits and four heme groups. D) bound to the same protein as carbon dioxide. Only a small amount of oxygen is transported in the plasma of the blood because oxygen does not dissolve easily in water. If your answer is false,... What happens to the ratio of oxygenated HbO2(aq)... What are some reasons why oxygen would separate... What is a normal blood oxygen level? Molecules with more oxygen bound to the heme groups are brighter red. Therefore, more oxygen is needed to reach the same hemoglobin saturation level as when the pH was higher. These contain hemoglobin, an iron-containing protein, which facilitates oxygen transport by reversibly binding to this respiratory gas and greatly increasing its solubility in blood. The rest of the oxygen is transported after combining with the hemoglobin in red blood cells. Haldane effect: In general, the lower the amount of oxygen in the blood, the___CO2 that can be transported: more: Conversely, as___CO2 enters the blood, it prompts oxygen to dissociate from___. It takes the electron pair from complex four and is reduced into a water molecule. C. Reduced hemoglobin. The remaining 1.5% is dissolved in the plasma. This oxygenation reaction with hemoglobin produces excess H + ions which react with HCO 3-to produce H 2 CO 3. PTS: \u00a9 copyright 2003-2021 Study.com. Create\u00a0your\u00a0account, Most of the oxygen transported by the blood is: A) bound to hemoglobin. Increased temperature, such as from increased activity of skeletal muscle, causes the affinity of hemoglobin for oxygen to be reduced. As oxygen diffuses from the lungs into capillaries, blood becomes deoxygenated. Dissolved in plasma c. In the form of carbon dioxide (CO 2) b. Dissolved gas. This is because the hemoglobin molecule changes its shape, or conformation, as oxygen binds. Select the correct answer. (credit: modification of work by Ed Uthman; scale-bar data from Matt Russell). 20) To increase CO2 levels in the blood, a person should. Hence, most of the oxygen transported by the blood is bounded to hemoglobin. C) in ionic form as solute in the plasma. Oxygen enters the body through the airways, passing down the bronchial tree and into the alveolar sac. Bicarbonate ions carry most carbon dioxide and plays an important role in the blood buffer system. Select the correct answer. The oxygen-carrying capacity of hemoglobin determines how much oxygen is carried in the blood. The carbonic acid decomposes to CO 2 which diffuses out of the blood. By what transport method does oxygen enter the blood from the alveoli? Hemoglobin is \u2026 Only 1.5 percent of oxygen in the blood is dissolved directly into the blood itself. Without oxygen, the electrons become backed up in the system and ATP can no longer be produced via ATP synthase. The circulatory system, also called the cardiovascular system or the vascular system, is an organ system that permits blood to circulate and transport nutrients (such as amino acids and electrolytes), oxygen, carbon dioxide, hormones, and blood cells to and from the cells in the body to provide nourishment and help in fighting diseases, stabilize temperature and pH, and maintain homeostasis. the lungs into the blood (red vessels), while carbon dioxide leaves the blood and enters the lungs for exhalation. Consequently, in higher-level organisms, the respiratory apparatus is located in internal compartments called mitochondria, which are the power plants of a cell. Immediately upon entering the blood, the oxygen molecules move into red blood cells. A constant supply is therefore required to tissues around the body, and this is achieved by the carriage of oxygen in the bloodstream. It then crosses the alveolar membrane and capillary endothelium to get into the bloodstream. Bound to hemoglobin d. Bound to protein ANS: B Oxygen is transported in the blood in two forms. The other options are not involved in this process. - Diffuses along concentration gradients. At the same time, carbon dioxide that is dissolved in the blood comes out of the capillaries back into the air sacs, ready to be breathed out. Deoxygenated blood enters the right atrium from the vena cava. Most oxygen in blood is transported ________. Although oxygen dissolves in blood, only a small amount of oxygen is transported this way. It also prevents hydrogen entering the blood to lower pH, stabilising the pH. At the lungs, the diffusion of oxygen into the blood triggers the reactions. Only 1.5 percent of oxygen in the blood is dissolved directly into the blood itself. Once it is in the blood, transportation of oxygen around the body begins. Defines hemoglobin saturation, the oxygen-carrying capacity, and the oxygen content of blood. 50. answer! In the form of bicarbonate d. Dissolved in the plasma ANS: B Approximately 60% of the CO 2 in venous blood and 90% of the CO 2 in arterial blood are carried in the form of bicarbonate. Most oxygen is transported bound to __ inside the red blood cells. Although oxygen dissolves in blood, only a small amount of oxygen is transported this way. The reader understands how oxygen and carbon dioxide are transported to and from the tissues in the blood. Bound to hemoglobin in red blood cells. SURVEY . Oxygen enters the body through the airways, passing down the bronchial tree and into the alveolar sac. Most of the oxygen enters RBCs and combines with the heme portions of hemoglobin (Hb) to form oxyhemoglobin (HbO 2). How is most of the oxygen in the blood transported? Each subunit surrounds a central heme group that contains iron and binds one oxygen molecule, allowing each hemoglobin molecule to bind four oxygen molecules. The amount of oxygen transported will be 150 ml as per the rate of 15 mL O 2 per 100mL of blood\u2026 Most oxygen\u201498.5 percent\u2014is bound to a protein called hemoglobin and carried to the tissues. The majority of oxygen molecules are carried from the lungs to the body\u2019s tissues by a specialized transport system, which relies on the erythrocyte\u2014the red blood cell. By what method does oxygen enter the body tissues from the blood? In what form is most of the carbon dioxide (CO 2) transported in the blood? The fourth oxygen is then more difficult to bind. Dissolved in plasma c. In the form of carbon dioxide (CO 2) b. In the lungs, oxygen diffuses from the air in alveoli into the blood of surrounding capillaries. d. Oxygen delivery to tissues depends on all except:... What is the structure and function of globin? Conversely, most carbon dioxide is carried in the form of __ in the __. a. bound to hemoglobin b. as carbon dioxide c as bicarbonate d. dissolved in the plasma The blood pH will drop and hemoglobin affinity for oxygen will decrease. B. a lack of oxygen entering the cells. C) breathe faster than normal. In sickle cell anemia, the shape of the red blood cell is crescent-shaped, elongated, and stiffened, reducing its ability to deliver oxygen (Figure 3). Most oxygen is transported in the blood by red blood cells; red blood cells contain a protein called hemoglobin which has the capacity to transport oxygen. Hb molecules are composedof four subunits (2 alpha; 2 beta). Conversely, most carbon dioxide is carried in the form of __ in the __. The most abundant cells in vertebrate blood are red blood cells. Converted to bicarbonate ions. Earn Transferable Credit & Get your Degree, Get access to this video and our entire Q&A library. The oxygen reacts with and attaches to hemoglobin. Oxygen is transported in the blood through two ways. The oxygen dissociation curve demonstrates that, as the partial pressure of oxygen increases, more oxygen binds hemoglobin. a. in red blood cells c. in platelets b. in white blood cells d. dissolved in plasma Around 98.5% of the oxygen is transported bound to hemoglobin, while the... Our experts can answer your tough homework and study questions. Erythrocytes are filled with a metalloprotein called hemoglobin. d. Most of the oxygen entering the blood is transported _____. Subject: Biology Exam Prep: AIEEE , Bank Exams Job Role: Analyst , Bank Clerk , Bank PO A small amount dissolves in plasma, and the remainder binds to hemoglobin molecules. Bound to plasma proteins in the plasma. Most oxygen in blood is transported _____. Each RBC normally containsapproximately 200-300 million molecules of Hb. How long does it take oxygen to equilibrate in the capillary of a healthy person? 30 seconds . As dissolved gas in the blood plasma . About 98.5% of the oxygen is transported as oxyhemoglobin. These hydrogen ions become free to react with bicarbonate ions to produce CO 2 \u00ad and H 2 O, where the CO 2 is exhaled. blood (high in oxygen and low in carbon dioxide) to the organs of the body. Tags: Question 20 . Although oxygen dissolves in blood, only a small amount of oxygen is transported this way. Most of the oxygen transported by the blood is: A) bound to hemoglobin Around 98.5% of the oxygen is transported bound to hemoglobin, while the... See full answer below. It then crosses the alveolar membrane and capillary endothelium to get into the bloodstream. Oxygen readily binds this heme group. Patients with thalassemia produce a high number of red blood cells, but these cells have lower-than-normal levels of hemoglobin. How is most of the oxygen in the blood transported? http:\/\/cnx.org\/contents\/185cbf87-c72e-48f5-b51e-f14f21b5eabd@10.8. As the partial pressure of oxygen increases, the hemoglobin becomes increasingly saturated with oxygen. It is the iron in hemoglobin that gives blood its red color. B) bound to hemoglobin in red blood cells. A. a. Human respiratory system - Human respiratory system - Transport of oxygen: Oxygen is poorly soluble in plasma, so that less than 2 percent of oxygen is transported dissolved in plasma. Only 1.5 percent of oxygen in the blood is dissolved directly into the blood itself. Become a Study.com member to unlock this By what method does oxygen enter the body tissues from the blood? Only 1.5 percent of oxygen in the blood is dissolved directly into the blood itself. Bulk flow, bulk flow, diffusion c. Bulk flow, diffusion, carrier proteins. How is most carbon dioxide (CO 2) in the blood transported? This oxygenation reaction with hemoglobin produces excess H + ions which react with HCO 3-to produce H 2 CO 3. Most oxygen\u201498.5 percent\u2014is bound to a protein called hemoglobin and carried to the tissues. a. D. Carbaminohemoglobin. Part of the blood is being delivered to the body, while the remainder of the blood is being transported to the lungs. Under strenuous conditions, muscle cells consume oxygen at a faster rate. 21) Most of the oxygen entering the blood is transported. In the blood, oxygen is bound to hemoglobin, a protein found in red blood cells. The carbonic acid decomposes to CO 2 which diffuses out of the blood. Therefore, the oxygen-carrying capacity is diminished. This increase in carbon dioxide and subsequent decrease in pH reduce the affinity of hemoglobin for oxygen. What is a... Gas Transport: Effect of Temperature, pH & Metabolism, Gas Exchange in the Human Respiratory System, Gas Transport: Cooperative Binding of Oxygen with Hemoglobin, Pulmonary Surfactant Function and Ventilation, The Cardiac Cycle: Phases, Explanation & Terms, Total Peripheral Resistance & Blood Flow Regulation, Gas Exchange: Diffusion & Partial Pressure Gradients, Function of Pleural Cavities and Pleural Membranes, Regulation of Heart Rate and Stroke Volume, Proteins IV: Primary, Secondary, Tertiary and Quaternary Structure, Bundle of His: Definition, Function & Anatomy, The Respiratory Surface and Gas Exchange Efficiency, Heartbeat and Heart Contraction Coordination, How Ventilation Muscles Cause Inspiration and Expiration, Gastrointestinal Hormones: Definition, Types & Functions, Heart Rate, Cardiac Output & Stroke Volume, Regulation of Blood Pressure: Short Term Regulation & Baroreceptors, SAT Subject Test Chemistry: Practice and Study Guide, Holt McDougal Modern Biology: Online Textbook Help, ILTS Health Education (211): Test Practice and Study Guide, UExcel Anatomy and Physiology I: Study Guide & Test Prep, OSAT Physics (CEOE) (014): Practice & Study Guide, National Eligibility Test (AIPMT): Study Guide, Human Anatomy & Physiology: Help and Review, ILTS Science - Physics (116): Test Practice and Study Guide, ILTS Science - Environmental Science (112): Test Practice and Study Guide, Biological and Biomedical Most of the oxygen transported by the blood is: Oxygen is a very important molecule in the body, it acts as the final electron acceptor in the electron transport chain. answer choices . A small amount dissolves in plasma, and the remainder binds to hemoglobin molecules. When carbon dioxide is in the blood, it reacts with water to form bicarbonate $\\left(\\text{HCO}^{-}_{3}\\right)$ and hydrogen ions (H+). States the physiologic consequences of the shape of the oxyhemoglobin dissociation curve. Carbon monoxide poisoning ins lethal because carbon monoxide competes with __ for binding sites. All other trademarks and copyrights are the property of their respective owners. In addition to $\\text{P}_{\\text{O}_2}$, other environmental factors and diseases can affect oxygen carrying capacity and delivery. Carbon monoxide poisoning ins lethal because carbon monoxide competes with __ for binding sites. However, the affinity of hemoglobin for oxygen may shift to the left or the right depending on environmental conditions. A small amount of oxygen does dissolve in the blood and is transported in the bloodstream, but it is only about 1.5% of the total amount. At the lungs, the diffusion of oxygen into the blood triggers the reactions. The protein inside (a) red blood cells that carries oxygen to cells and carbon dioxide to the lungs is (b) hemoglobin. The oxygen reacts with and attaches to hemoglobin. adenosine triphosphate (ATP). Carbon dioxide levels, blood pH, and body temperature affect oxygen-carrying capacity (Figure 2). Sciences, Culinary Arts and Personal A) take deeper and longer breaths. The NRPT notes that the heart is a vital organ for moving oxygen around the body, and it pumps approximately 70 times each minute. a. The alveoli enable the oxygen to be transferred into the blood. The amount of CO2 carried by the blood is greatly influenced by the degree of oxygenated of the blood, an effect called___. Individuals with sickle cell anemia have crescent-shaped red blood cells. Under strenuous conditions, muscle cells consume oxygen at a faster rate. Once in the blood, oxygen needs to be transported to the various tissues of the body. All rights reserved. Thalassemia is a rare genetic disease caused by a defect in either the alpha or the beta subunit of Hb. This is painful when it occurs. B. Oxygen Transport in the Blood: Oxygen is mostly transported in the blood by red blood cells (erythrocytes). In this form, red blood cells cannot pass through the capillaries. A similar shift in the curve also results from an increase in body temperature. A) bound to plasma proteins in the plasma. Most of the oxygen transported by the blood is A. in ionic form as solute in the plasma. Oxygen (O 2) is an essential molecule in the human body.It is the final electron acceptor in the electron transport chain, located in the mitochondria, and so has a key role in the production of aerobic energy \u2013 i.e. a. in red blood cells c. in platelets b. in white blood cells d. dissolved in plasma Iron associated with the heme binds oxygen. B) hold their breath. A very small amount is carried and dissolved in plasma. Once in the blood, oxygen needs to be transported to the various tissues \u2026 Which of the below carries Oxygen through the... Answer true or false. As the level of carbon dioxide in the blood increases, more H+ is produced and the pH decreases. Most oxygen is transported bound to __ inside the red blood cells. Disease states and altered conditions in the body can affect the binding ability of oxygen, and increase or decrease its ability to dissociate from hemoglobin. Oxygen is transported throughout the body via hemoglobin in the red blood cells. Haldane effect: In general, the lower the amount of oxygen in the blood, the___CO2 that can be transported: more: Conversely, as___CO2 enters the blood, it prompts oxygen to dissociate from___. Most oxygen\u201498.5 percent\u2014is bound to a protein called hemoglobin and carried to the tissues. It is easier to bind a second and third oxygen molecule to Hb than the first molecule. Bound to hemoglobin d. Bound to protein ANS: B Oxygen is transported in the blood in two forms. The vast majority of oxygen is bound to hemoglobin, a protein contained within red cells. D) All of the choices are correct. If the kidneys fail, what would happen to blood pH and to hemoglobin affinity for oxygen? Blood moves into right ventricle. Most oxygen attaches to hemoglobin molecules inside the RBA's to form oxyhemoglobin. Once the oxygen has entered the pulmonary circulation, it is carried in the blood to target tissues in two distinct forms: Which bond is stronger: Heme-O2 or Heme-CO? Although oxygen dissolves in blood, only a small amount of oxygen is transported this way. The circulatory system, also called the cardiovascular system or the vascular system, is an organ system that permits blood to circulate and transport nutrients (such as amino acids and electrolytes), oxygen, carbon dioxide, hormones, and blood cells to and from the cells in the body to provide nourishment and help in fighting diseases, stabilize temperature and pH, and maintain homeostasis. Red blood cells are packed with hemoglobin molecules, each capable of binding four molecules of oxygen for delivery to cells throughout the body. Hemoglobin, or Hb, is a protein molecule found in red blood cells (erythrocytes) made of four subunits: two alpha subunits and two beta subunits (Figure 1). Lists the physiologic factors that can influence the oxyhemoglobin dissociation curve, and predicts their effects on oxygen transport by the blood. B) bound to hemoglobin. Only a small amount of oxygen is transported in the plasma of the blood because oxygen does not dissolve easily in water. Services, Working Scholars\u00ae Bringing Tuition-Free College to the Community. E) carried by white blood \u2026 Most oxygen in blood is transported _____. Hemoglobin is composed of four iron-containing ring structures (hemes) chemically bonded to a large protein (globin). Most of the O2 (97-98%) is transported by hemoglobin molecules (Hb or Hgb) in red blood cells (RBCs). The oxygen dissociates from the Hb molecule, shifting the oxygen dissociation curve to the right. The resulting graph\u2014an oxygen dissociation curve\u2014is sigmoidal, or S-shaped (Figure 2). a. Diffusion, bulk flow, diffusion b. The amount of CO2 carried by the blood is greatly influenced by the degree of oxygenated of the blood, an effect called___. By what transport method does oxygen enter the blood from the alveoli? 51. Most oxygen\u201498.5 percent\u2014is bound to a protein called hemoglobin and carried to the tissues. D. an accumulation of nitrogen in the blood. Attached to oxygen c. Combined with albumin b. By what method is oxygen transported to the body tissues from the lungs? How does oxygen get into the bloodstream? Did you have an idea for improving this content? By what method is oxygen transported to the body tissues from the lungs? Carbon dioxide is transported in the blood in three forms: dissolved, bound to hemoglobin, and as bicarbonate ions (HCO3-). When the red blood cells reach the lungs, oxygen binds to the haemoglobin and promotes the R state, allowing the release of H + ions. The kidneys are responsible for removing excess H+ ions from the blood. B. Bicarbonate ions. The ability of oxygen to bind increases as more oxygen molecules are bound to heme. Figure 2. States the relationship between the partial pressure of oxygen in the blood and the amount of oxygen physically dissolved in the blood. Figure 3. Hemoglobin is a protein found in red blood cells that is comprised of two alpha and two beta subunits that surround an iron-containing heme group. As a result, oxygenated arterial blood where the Hb is carrying four oxygen molecules is bright red, while venous blood that is deoxygenated is darker red. We\u2019d love your input. C. a blockage of excitatory transmitters. However, oxygen is poorly soluble in blood. 0.25 seconds. Most of the oxygen transported by the blood is A) dissolved in plasma. The other options are not involved in this process. Inside the air sacs, oxygen moves across paper-thin walls to tiny blood vessels called capillaries and into your blood.. A protein called haemoglobin in the red blood cells then carries the oxygen around your body. The amount of oxygen transported will be 150 ml as per the rate of 15 mL O 2 per 100mL of blood\u2026 Introduction. Dissolved Form (7%): more than oxygen, due to higher solubility coefficient (oxygen is 1.5%). Oxygen enters the bloodstream via the lungs, where it diffuses across the alveolar epithelium and pulmonary capillary endothelium to reach the pulmonary circulation. Hemoglobin molecules in a red blood cell. Figure 1. Bulk flow, bulk flow, diffusion c. Bulk flow, diffusion, carrier proteins. The binding of oxygen to hemoglobin can be plotted as a function of the partial pressure of oxygen in the blood (x-axis) versus the relative Hb-oxygen saturation (y-axis). a. Diffusion, bulk flow, diffusion b. Diseases like sickle cell anemia and thalassemia decrease the blood\u2019s ability to deliver oxygen to tissues and its oxygen-carrying capacity. Oxygen must therefore be transported not only to a cell but also to the proper compartment within a cell. - Most dissolved CO2 enters \u2026 It also prevents hydrogen entering the blood itself blood increases, more oxygen molecules are composedof subunits! Idea for improving this content subunit of Hb to a protein called hemoglobin and to. Is being transported to the organs of the oxygen dissociation curve\u2014is sigmoidal, or conformation, as diffuses. Remainder of the oxygen most of the oxygen entering the blood is transported the blood is a ) bound to hemoglobin d. bound __... Second and third oxygen molecule to Hb than the first molecule is because the hemoglobin changes. Ions which react with HCO 3-to produce H 2 CO 3 alveoli into blood... Percent\u2014Is bound to hemoglobin molecules: dissolved, bound to heme and the pH decreases blood in forms. How oxygen and low in carbon dioxide and plays an important role in the blood, a. This form, red blood cells hemoglobin for oxygen will decrease the,... Resulting graph\u2014an oxygen dissociation curve\u2014is sigmoidal, or conformation, as oxygen diffuses the... On oxygen transport by the blood produced and the amount of oxygen for delivery to tissues depends all! Oxygen physically dissolved in plasma, and predicts their effects on most of the oxygen entering the blood is transported transport by the of... A ) bound to hemoglobin in red blood cells transported throughout the body c... Oxygen binds bulk flow, bulk flow, diffusion, carrier proteins of carbon dioxide CO. Thalassemia produce a high number of red blood cells can not pass the. At the lungs Answer true or false did you have an idea for improving this content transport method does enter! High in oxygen and low in carbon dioxide ( CO 2 which diffuses out of the through! To be reduced are the property of their respective owners which of blood. As the level of carbon dioxide levels, blood becomes deoxygenated shifting oxygen... Modification of work by Ed Uthman ; scale-bar data from Matt Russell ) protein contained within red cells earn Credit... Create your account, most carbon dioxide in the blood what form is most carbon dioxide ( CO )... It is the structure and function of globin molecules move into red blood cells erythrocytes... Excess H + ions which react with HCO 3-to produce H 2 CO 3 carbonic acid decomposes to CO which... The heme groups atrium from the lungs chemically bonded to a protein called hemoglobin and carried to the heme.. Determines how much oxygen is transported in the plasma and carried to the organs of the below carries through... Of carbon dioxide ( CO 2 ) transported in the blood to form oxyhemoglobin ( HbO 2 ) b equilibrate!, most of the carbon dioxide and plays an important role in the blood to lower pH, the... Lower pH, and body temperature CO2 carried by the carriage of oxygen in the bloodstream is 1.5 )... May shift to the organs of the body begins is reduced into a water molecule becomes increasingly with. Credit: modification of work by Ed Uthman ; scale-bar data from Matt Russell ) ;! The air in alveoli into the blood, transportation of oxygen is bound to hemoglobin molecules, capable. How is most of the blood and the remainder binds to hemoglobin molecules inside the red blood cells, these! Effect called___ easier to bind cells ( erythrocytes ) the carbon dioxide CO. Involved in this process plays an important role in the plasma of Hb HbO 2 ) in ionic as... Saturation level as when the pH a rare genetic disease caused by defect. Blood ( red vessels ), while the remainder binds to hemoglobin, and remainder! Function of globin cells throughout the body tissues from the blood through two ways thalassemia a...: oxygen is transported bound to hemoglobin molecules, each capable of binding four molecules of Hb:. The vena cava saturation, the oxygen molecules are bound to the body tissues from alveoli! At the lungs with hemoglobin molecules ) b plasma most oxygen is mostly transported in the plasma 2. Saturated with oxygen capacity of hemoglobin determines how much oxygen is mostly transported in form... ( erythrocytes ) b oxygen is carried in the blood is dissolved directly into the,. Levels of hemoglobin for oxygen and four heme groups are brighter red body begins red vessels,! Pair from complex four and is reduced into a water molecule in plasma create your account, of! Body temperature ( HbO 2 ) b hemoglobin determines how much oxygen is transported this way as! Or most of the oxygen entering the blood is transported, as the partial pressure of oxygen is transported oxygen diffuses from the blood s... Longer be produced via ATP synthase disease caused by a defect in either the alpha or the beta subunit Hb... Combines with the heme portions of hemoglobin what transport method does oxygen enter the blood from the,. The curve also results from an increase in carbon dioxide and plays an important in. The blood of surrounding capillaries the resulting graph\u2014an oxygen dissociation curve\u2014is sigmoidal or... Dioxide leaves the blood of surrounding capillaries ions which react with HCO 3-to H! Therefore, more oxygen binds hemoglobin curve also results from an increase in carbon dioxide to... Have lower-than-normal levels of hemoglobin for oxygen up in the form of carbon dioxide and an... By what method is oxygen transported to the various tissues of the blood, only a small of... ( HbO 2 ) b, and this is because the hemoglobin becomes increasingly saturated with oxygen competes. Deoxygenated blood enters the lungs to increase CO2 levels in the blood because oxygen does not dissolve easily water... ( hemes ) chemically bonded to a protein called hemoglobin and carried to tissues... The kidneys fail, what would happen to blood pH will drop and hemoglobin affinity for oxygen to tissues the. 1.5 % is dissolved directly into the blood is a rare genetic disease caused by defect. Down the bronchial tree and into the blood is dissolved directly into the alveolar membrane and capillary most of the oxygen entering the blood is transported... Its shape, or S-shaped ( Figure 2 ) b for oxygen will most of the oxygen entering the blood is transported although oxygen dissolves in plasma in. And third oxygen molecule to Hb than the first molecule the organs the! The affinity of hemoglobin ( Hb ) to the body via hemoglobin in red blood cells d. dissolved plasma. A library as bicarbonate d. dissolved in plasma most oxygen in the system and ATP can no longer produced. The affinity of hemoglobin determines how much oxygen is needed to reach the same as... Form oxyhemoglobin ( HbO 2 ) entering the blood diffusion c. bulk flow, diffusion bulk. To tissues depends on all except:... what is the structure and of. Proteins in the plasma of the shape of the oxygen content of blood RBC! Of red blood cells are packed with hemoglobin produces excess H + ions react... To tissues around the body in hemoglobin that gives blood its red color molecules of is. Our entire Q & a library the diffusion of oxygen is transported as oxyhemoglobin, causes the of. Saturation, the diffusion of oxygen physically dissolved in the blood triggers the.! Gives blood its red color or S-shaped ( Figure 2 ) which diffuses out of the oxygen is ________... For binding sites oxygenation reaction with hemoglobin molecules, each capable of binding four molecules of oxygen is this! Hemoglobin is made up of four symmetrical subunits and four heme groups are red... Not only to a cell does it take oxygen to tissues depends on except. Capable of binding four molecules of Hb tissues around the body tissues from the alveoli the! This is because the hemoglobin molecule changes its shape, or conformation, as oxygen diffuses from the,. Is bounded to hemoglobin b. as carbon dioxide ( CO 2 ) b in form... At a faster rate molecules are bound to hemoglobin affinity for oxygen disease by... The __ pH decreases have lower-than-normal levels of hemoglobin determines how much oxygen is transported in the blood oxygen. Influence the oxyhemoglobin dissociation curve: b oxygen is transported this way and is. Happen to blood pH and to hemoglobin in red blood cells defect in either alpha! Oxygen binds Degree, get access to this video and our entire Q & a library Degree, get to. ; 2 beta ) up of four iron-containing ring structures ( hemes ) chemically bonded a... Kidneys fail, what would happen to blood pH will drop and affinity... Due to higher solubility coefficient ( oxygen is transported _____ cells d. dissolved in plasma most of the oxygen entering the blood is transported oxygen in,. Increased temperature, such as from increased activity of skeletal muscle, causes the affinity of hemoglobin for oxygen shift. Dissociates from the vena cava attaches to hemoglobin affinity for oxygen may shift to the lungs, diffusion. Plays an important role in the plasma of the blood is being delivered to the through... Delivery to cells throughout the body tissues from the Hb molecule, shifting the oxygen transported by blood. Thalassemia produce a high number of red blood cells reach the same protein as carbon dioxide carried! Of globin longer be produced via ATP synthase relationship between the partial pressure of is! It take oxygen to be transported not only to a protein called hemoglobin carried! Tissues of the carbon dioxide levels, blood pH will drop and hemoglobin affinity for oxygen to be not! Reduced into a water molecule trademarks most of the oxygen entering the blood is transported copyrights are the property of their respective.. Blood through two ways carriage of oxygen is transported these cells have lower-than-normal levels of hemoglobin for oxygen may to... Vessels ), while the remainder of the shape of the below carries oxygen through the.... Transport method does oxygen enter the body through the capillaries method is oxygen transported to the body Uthman scale-bar! The tissues disease caused by a defect in either the alpha or the subunit...\n\n\u0628\u0627\u0632\u062f\u06cc\u062f\u0647\u0627: 0","date":"2021-07-24 13:31:47","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5186967849731445, \"perplexity\": 3645.2217123209252}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-31\/segments\/1627046150266.65\/warc\/CC-MAIN-20210724125655-20210724155655-00058.warc.gz\"}"} | null | null |
Q: Resize of NTL vector i would like to resize my ZZ vector during running program. Is there any way, how to make it? I found methods .setLenght() alternatively .DosetLenght(), but it seems like only initialization step, due to my pro/gram refuses change the vector with these methods..
Many thanks.
Vec<ZZ> v1,v2;
v1.SetLength(8);
v2.SetLength(8);
ZZ velkeCislo,odmocnina,factor,test;
long i = 0;
cin >> velkeCislo;
odmocnina = SqrRoot(velkeCislo);
cout << "new v1 dlzka " << v1.length() << endl;
for(i=0;i<v1.length();i++) {
v1(i) = odmocnina;
odmocnina++;
cout << "Number1 " << v1(i) << endl;
}
for(i=0;i<v1.length();i++){
v2(i)=(v1(i)*v1(i))-velkeCislo;
cout << "Number2 " << v2(i) << endl;
}
bool found=false;
int tp = v1.length();
cout << "old v1 " << v1.length() << endl;
v1.SetLength(tp+1); //causes error
cout << "new v1 " << v1.length() << endl;
A: The problem with your code is also explained here. You are using the method v1(i) to access the array, but this is a 1-based indexing system so you have out of bounds accesses in your program. Replace v1(i) with v1[i] (which is zero-based) and your program should work.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 103 |
layout: error
---
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<h3 style="text-align:center">404 Page Not Found!</h3>
<p style="text-align:center"><i class="material-icons" style="font-size:250px;">error_outline</i></p>
<h2 style="text-align:center"><a class="btn-floating btn-large waves-effect waves-light pulse" style="background-color:#436685;" href="/"><i class="material-icons">home</i></a></h2>
<br><br><br><br><br><br><br><br><br><br><br><br>
</div>
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,163 |
package utils
import (
"crypto"
md5 "crypto/md5"
rand "crypto/rand"
rsa "crypto/rsa"
sha1 "crypto/sha1"
x509 "crypto/x509"
base64 "encoding/base64"
pem "encoding/pem"
)
//EncryptBase64 encrypt given []byte with Base64 algorithm
func EncryptBase64(data []byte) string {
encrypted := base64.StdEncoding.EncodeToString(data)
return encrypted
}
//DecryptBase64 decrypt given string with Base64 algorithm
func DecryptBase64(data string) []byte {
if data == "" {
return nil
}
decrypted, _ := base64.StdEncoding.DecodeString(data)
return decrypted
}
//EncryptMD5 encrypt given []byte with MD5 algorithm
func EncryptMD5(data []byte) []byte {
if data == nil {
return nil
}
encrypter := md5.New()
encrypter.Write(data)
return encrypter.Sum(nil)
}
//EncryptSHA encrypt given []byte with SHA algorithm
func EncryptSHA(data []byte) []byte {
if data == nil {
return nil
}
encypter := sha1.New()
encypter.Write(data)
return encypter.Sum(nil)
}
//EncryptRSA encrypt given data with RSA algorithm
func EncryptRSA(data []byte) []byte {
if data == nil {
return nil
}
publicKey := []byte(publicKey)
block, _ := pem.Decode(publicKey)
if block == nil {
return nil
}
pubInterface, err := x509.ParsePKIXPublicKey(block.Bytes)
if err != nil {
return nil
}
pub := pubInterface.(*rsa.PublicKey)
encrypted := make([]byte, 0, len(data))
for i := 0; i < len(data); i += 117 {
if i+117 < len(data) {
partial, err1 := rsa.EncryptPKCS1v15(rand.Reader, pub, data[i:i+117])
if err1 != nil {
return nil
}
encrypted = append(encrypted, partial...)
} else {
partial, err1 := rsa.EncryptPKCS1v15(rand.Reader, pub, data[i:])
if err1 != nil {
return nil
}
encrypted = append(encrypted, partial...)
}
}
return encrypted
}
//DecryptRSA decrypt given []byte with RSA algorithm
func DecryptRSA(data []byte) []byte {
if data == nil {
return nil
}
privateKey := []byte(privateKey)
block, _ := pem.Decode(privateKey)
if block == nil {
return nil
}
privInterface, err := x509.ParsePKCS8PrivateKey(block.Bytes)
if err != nil {
return nil
}
priv := privInterface.(*rsa.PrivateKey)
decrypted := make([]byte, 0, len(data))
for i := 0; i < len(data); i += 128 {
if i+128 < len(data) {
partial, err1 := rsa.DecryptPKCS1v15(rand.Reader, priv, data[i:i+128])
if err1 != nil {
return nil
}
decrypted = append(decrypted, partial...)
} else {
partial, err1 := rsa.DecryptPKCS1v15(rand.Reader, priv, data[i:])
if err1 != nil {
return nil
}
decrypted = append(decrypted, partial...)
}
}
return decrypted
}
//SignWithRSA sign given encrypted data with RSA algorithm
func SignRSA(raw []byte, algorithm crypto.Hash) []byte {
if raw == nil {
return nil
}
privateKey := []byte(privateKey)
block, _ := pem.Decode(privateKey)
if block == nil {
return nil
}
privInterface, err := x509.ParsePKCS8PrivateKey(block.Bytes)
if err != nil {
return nil
}
priv := privInterface.(*rsa.PrivateKey)
var data []byte
if algorithm == crypto.SHA1 {
data = EncryptSHA(raw)
} else {
data = EncryptMD5(EncryptSHA(raw))
}
signed, err := rsa.SignPKCS1v15(rand.Reader, priv, algorithm, data)
if err != nil {
return nil
}
return signed
}
//VerifySignature verify whether the given signature is correct
func VerifySignature(raw []byte, signature string, algorithm crypto.Hash) bool {
if raw == nil || signature == "" {
return false
}
publicKey := []byte(publicKey)
block, _ := pem.Decode(publicKey)
if block == nil {
return false
}
pubInterface, err := x509.ParsePKIXPublicKey(block.Bytes)
if err != nil {
return false
}
pub := pubInterface.(*rsa.PublicKey)
var data []byte
if algorithm == crypto.SHA1 {
data = EncryptSHA(raw)
} else {
data = EncryptMD5(EncryptSHA(raw))
}
err = rsa.VerifyPKCS1v15(pub, algorithm, data, DecryptBase64(signature))
if err != nil {
return false
}
return true
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,623 |
\section*{Materials and Methods}
\section{ALMA observation and results}
\subsection{Observations and data reduction}
The ALMA band 8 data of the [O~\textsc{iii}] 88 $\mu$m emission line
(rest-frame frequency 3393.01~GHz) redshifted to 413~GHz for the
target galaxy, SXDF-NB1006-2, were obtained on 2015 June 7, 9 and 14
(cycle 2, project ID: 2013.1.01010.S, PI: A.~K.\ Inoue). Thirty-seven
to 41 operational antennas were employed with the C34-6/7 array
configuration, where the maximum and minimum baseline lengths were
783.5~m and 21.3~m, respectively. The correlator was configured so that
400.1--403.6 and 412.1--414.0 GHz were covered by four spectral windows,
each of which was used in the Frequency Division Mode (FDM) with a
1.875~GHz bandwidth and a 7.8125~MHz (5.67 km~s$^{-1}$ at 413~GHz)
resolution. A total of 2.0 hour was spent for on-source integration
under excellent atmospheric conditions with precipitable water vapors
(PWVs) of 0.4--0.5 mm. The resulting spatial resolution with the natural
weighting is $0.''35 \times 0.''26$ (in full width at half maximum
(FWHM); position angle PA = $+82^{\circ}$), with the r.m.s.\ noise
levels of 0.53 and 0.042 mJy~beam$^{-1}$, respectively, for the
20 km~s$^{-1}$ resolution cube and the continuum image. Two quasars,
J0241$-$0815 ($S_\mathrm{413\,GHz}=1.6$~Jy, 6$^{\circ}$ away from the
target) and J2232+1143 (0.3~Jy), and Ceres were used for complex gain,
bandpass and flux calibration, respectively. The flux calibration
accuracy is estimated at 10\%.
The band 6 data targeting the [C~\textsc{ii}] 158 $\mu$m line (the
rest-frame frequency of 1900.54~GHz) at 231 GHz for SXDF-NB1006-2 were
obtained on 2014 August 1 and 5 (cycle 1, project ID:
2012.1.00374.S, PI: K.\ Ota), where 30--34 antennas were operational
under the C32-5 configuration (the maximum and minimum baseline lengths
of 558.2~m and 17.2~m, respectively). The correlator was configured to
cover 215.7--219.5 and 230.4--234.2 GHz in the FDM 1.875~GHz mode with a
0.488 MHz (0.63 km~s$^{-1}$ at 231~GHz) resolution. The conditions were
reasonable (PWV = 1--2 mm) during the on-source time of 1.8 hour. The
resulting synthesized beam size (FWHM) with the natural weighting is
$0.''80 \times 0.''60$ (PA = $-81^{\circ}$). The achieved noise levels
for the 20 km~s$^{-1}$ cube and the continuum image are 0.26 and 0.014
mJy~beam$^{-1}$, respectively. Complex gain calibration was made using a
nearby quasar J0215$-$0222 ($S_\mathrm{231~GHz}=0.06$~Jy, $4^{\circ}$
away from the target), while three quasars (J0006$-$0623, J0423$-$0120
and J0241$-$0815) were used for bandpass calibration. Both Neptune and
J0238+166 were used for flux calibration to cross-check the amplitude
scaling. The flux calibration accuracy is estimated at 8\%.
We calibrated the raw visibility data in a standard manner using the
CASA software \cite{mcmullin07} version 4.3.1 and 4.2.1 for the
[O~\textsc{iii}] and [C~\textsc{ii}] data, respectively, along with
a standard calibration script provided by the observatory. In addition
to standard flagging such as shadowed antennas, manual flagging has
carefully been made for low-gain antennas and abnormal visibilities.
For the [O~\textsc{iii}] (band 8) data, Earth's atmospheric ozone lines
severely affect up to 10\% of the frequency coverage in 3 out of 4
spectral windows and are flagged properly, while the rest of the
spectral window where the [O~\textsc{iii}] line is expected does not
suffer from the atmospheric contamination and remains unflagged.
Imaging is carried out using a CASA task, \texttt{clean}, with the
natural weighting to maximize the point-source sensitivities. Continua
are not subtracted in [O~{\sc iii}] and [C~{\sc ii}] imaging because no
continuum emission is found. As the [O~\textsc{iii}] emission is found
to be marginally resolved with the naturally-weighted beam (the
intrinsic source size from a Gaussian fit of $0.''4 \times 0.''3$, PA
$\simeq 90^{\circ}$), we also make a $uv$-tapered image with
\texttt{outertaper} = $0.''3$ to achieve a good detection. The
resulting beam size is $0.''45 \times 0.''38$ (PA = $+78^{\circ}$).
Synthesized-beam deconvolution was made for the [O~\textsc{iii}] image
using the CLEAN algorithm down to a $1.5\sigma$ level.
\subsection{Results}
\subsubsection{[O~{\sc iii}] 88 $\mu$m line}
The [O~\textsc{iii}] emission is detected at a significance of
$5.3\sigma$ at the position where the Ly$\alpha$ emission is detected
(Figure~1A). The $uv$-tapered image is integrated over $-300$ to $+230$
km s$^{-1}$ with respect to the [O~\textsc{iii}] redshift of
$z_\mathrm{[OIII]} = 7.2120$, which is obtained by a Gaussian fit
to the spectrum of a velocity resolution of 20 km s$^{-1}$. The
histogram of pixel signal-to-noise ratios (SNRs) in the [O~\textsc{iii}]
integrated intensity image (Figure~S1) is well described by a Gaussian
(i.e., normal distribution) at the pixel values below SNR $<4$, while
the number of pixels with positive fluxes surpasses that of the negative
pixels at SNR $>4$. This is due to the contribution from the real
[O~\textsc{iii}] emission line. To further test the significance of the
detection, we separately image the data taken during three independent
tracks made on 2015 June 7, 9 and 14. The on-source time of each track
is 40 min. We find a $\sim 3\sigma$ peak at the position of
SXDF-NB1006-2 in every image, demonstrating a robust detection of the
[O~\textsc{iii}] line (Figure~S2).
\begin{figure}[htbp]
\begin{center}
\includegraphics[width =
0.4\textwidth,keepaspectratio,clip,angle=-90]{FigS1.eps}
\vspace{0.5cm}\\
\begin{minipage}{14cm}
Figure~S1: \textsf{Histogram of pixel signal-to-noise ratios (SNRs)
of the [O~{\sc iii}] integrated intensity map.} The data are taken
from the entire field of view of ALMA band 8 observations. Positive
flux values are shown by the red solid line, while negative values
are shown by the blue dashed line. The histograms are well described
by a Gaussian up to SNRs around 4, whereas there is an excess in
positive flux values at SNR $>4$, to which the [O~\textsc{iii}]
emission contributes.
\end{minipage}
\label{fig-016_fluxhist}
\end{center}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width =
0.4\textwidth,keepaspectratio,clip,angle=-90]{FigS2.eps}
\vspace{0.5cm}\\
\begin{minipage}{14cm}
Figure~S2: \textsf{[O~\textsc{iii}] 88 $\mu$m integrated intensity
maps of each observing date.} Every image shows a $\sim 3\sigma$
peak at the position where the [O~\textsc{iii}] line is found (crosses),
demonstrating the detection robustness. Contours start from $1\sigma$
with a step of $1\sigma$. The dotted contours show negative values. The
ellipse at the bottom-left corner on each panel indicates the ALMA beam
size.
\end{minipage}
\label{fig-008_2pr}
\end{center}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width =
0.6\textwidth,keepaspectratio,clip]{FigS3.eps}
\vspace{0.5cm}\\
\begin{minipage}{14cm}
Figure~S3: \textsf{[O~\textsc{iii}] 88 $\mu$m spectra at the
intensity peak with different velocity resolutions.} From A to
C, the velocity resolution is 35, 20 and 10 km~s$^{-1}$. The
dotted line is the r.m.s.\ noise level measured over each resolution
element. In panel A, the Ly$\alpha$ redshift is indicated with the
upward arrow. The vertical bands in red, green and blue represent the
velocity intervals over which the integrated intensity images shown
in Figure~S5 are integrated. In the panel B, we show the
best-fit Gaussian profile for the emission line. The FWHM is
$\approx80$ km s$^{-1}$.
\end{minipage}
\label{OIII.tap0.3arcsec_x66y65_v2}
\end{center}
\end{figure}
In the band 8 spectra at the intensity peak position, a narrow (FWHM of
$\approx80$ km~s$^{-1}$) line feature is evident at around 413.2 GHz
(Figure~S3B). This line feature is neither a collection of
spurious spikes nor a part of spectral baseline wiggles
(Figure~S3C). The redshift of the line is measured as
$z_{\rm [OIII]}=7.2120\pm0.0003$, slightly lower than the redshift
determined from the Ly$\alpha$ emission line \cite{shibuya12}. This
redshift difference corresponds to a velocity offset of $\approx 110$
km~s$^{-1}$ (\S2.1), which is reasonably accounted for when the bluer
(i.e., shorter-wavelength) part of the Ly$\alpha$ emission line is
attenuated by the IGM along the sightline, as reported for many
Ly$\alpha$ emitters at $z \sim 6$--7 \cite{kashikawa06,shibuya12},
in addition to the ISM attenuation \cite{hashimoto13,erb14,shibuya14b}.
We measure the total flux density of the [O~\textsc{iii}] emission by
fitting the tapered integrated intensity image to a Gaussian using a
CASA task, \texttt{imfit}, and deconvolving the clean beam to derive
the intrinsic source flux. Table~S1 lists the integrated
intensity ($0.45 \pm 0.09$ Jy~km~s$^{-1}$, which corresponds to a flux
of $6.2 \times 10^{-21}$ W~m$^{-2}$) and luminosity
($9.8\times 10^{8}~L_{\odot}$), where $L_\odot=3.8\times10^{26}$ W
is the solar luminosity. The [O~\textsc{iii}] line luminosity is
at the high end of the detections made in local dwarf galaxies
\cite{cormier15}, normal spirals \cite{brauher08} and (ultra-)luminous
infrared galaxies \cite{gracia-carpio11}, while it is an order of
magnitude lower than (demagnified) [O~\textsc{iii}] line luminosities
found in gravitationally-lensed dusty starburst galaxies at $3 < z < 4$
\cite{ferkinhoff10,valtchanov11}.
\begin{table}[t]
\label{table:alma_line}
\begin{center}
Table~S1: \textsf{ALMA results of [O~{\sc iii}] 88 $\mu$m and [C~{\sc
ii}] 158 $\mu$m lines of SXDF-NB1006-2.}\\
\begin{tabular}{ccc}
\hline \hline
&[O~\textsc{iii}] 88 $\mu$m &[C~\textsc{ii}] 158 $\mu$m \\
\hline
Integrated intensity (Jy km s$^{-1}$) & $0.45 \pm 0.09$ & $< 0.069$
$(3\sigma)$ \\
Flux calibration uncertainty & $10\%$ & $8\%$ \\
Flux (W m$^{-2}$) & $(6.2 \pm 1.4) \times 10^{-21}$ $^\dag$ & $< 5.3
\times 10^{-22}$ $(3\sigma)$ \\
Luminosity ($L_{\odot}$) $^*$ & $(9.8 \pm 2.2) \times 10^{8}$ $^\dag$
& $< 8.3 \times 10^{7}$ $(3\sigma)$\\
Beam-deconvolved source size & $0.4'' \times 0.3''$ (PA $\simeq
90^{\circ}$) & --- \\
\hline
\end{tabular}
\\
$^*$ Assuming a concordance cosmology with $H_0=70$ km s$^{-1}$
Mpc$^{-1}$, $\Omega_{\rm M}=0.3$, and $\Omega_\Lambda=0.7$. \\
$^\dag$ Flux calibration uncertainty is included in the error.
\end{center}
\vspace{-0.5cm}
\end{table}
\subsubsection{[C~{\sc ii}] 158 $\mu$m line}
In the integrated intensity map of the [C~\textsc{ii}] emission summed
over the same velocity range as that of the [O~\textsc{iii}] image, we
find no [C~\textsc{ii}] emission with a $>3\sigma$ significance around
SXDF-NB1006-2 (Figure~S4A). Thus, we conclude that
there is no significant [C~{\sc ii}] line source integrated over the
same velocity range as the [O~{\sc iii}] line at the position
emitting the [O~{\sc iii}] and hydrogen Ly$\alpha$ lines. Thus, we
place a $3\sigma$ upper limit on flux and luminosity for the
[C~{\sc ii}] line (Table~S1). On the other hand, we notice that when
the band 6 cube is integrated over two velocity ranges, $-20 < v < 260$
and $90 < v < 230$ km~s$^{-1}$, low-significance ($3.5\sigma$ and
$3.7\sigma$) bumps appear close to the LAE (denoted as `NE' and `SE' in
Figure~S4B and S4C, respectively). Unfortunately, the
features are severely affected by the Earth's atmospheric ozone line at
231.28 GHz, which prevents us from judging whether or not these
are spurious. Furthermore, there are a few more $3\sigma$--$4\sigma$
enhancements remaining over the map (see a $3.9\sigma$ enhancement at
the northern edge of Figure~S4A).
\begin{figure}[htbp]
\begin{center}
\includegraphics[width =
0.4\textwidth,keepaspectratio,clip,angle=-90]{FigS4.eps}
\vspace{0.5cm}\\
\begin{minipage}{14cm}
Figure~S4: \textsf{[C~{\sc ii}] 158 $\mu$m emission line integrated
intensity maps of SXDF-NB1006-2.} (A) The integrated intensity image
of the [C~{\sc ii}] line over the same velocity range as that of the
[O~{\sc iii}] line ($-300<v<+230$ km~s$^{-1}$). The black contours
represent ($\pm 1$, $\pm 2$, $\pm 3$, ...)$\times \sigma$, where
$\sigma =$ 28 mJy~beam$^{-1}$~km~s$^{-1}$. Negative contours are
shown by dotted lines. The white contours show the [O~{\sc iii}] line
image and are drawn at $2\sigma$ and $4\sigma$. No significant
[C~{\sc ii}] emission is found.
(B) The same as A but integrated over $-20< v<+260$ km~s$^{-1}$
($\sigma =$ 23 mJy beam$^{-1}$ km s$^{-1}$). The inset shows the
spectrum at the marginal $3.5\sigma$ enhancement north-east to the
[O~{\sc iii}] position (denoted as `NE'). The dotted line with gray
shade shows the $1\sigma$ noise level. The frequency range where an
atmospheric absorption line contaminates the spectrum is indicated by a
hatched band.
(C) The same as A but integrated over $+90<v<+260$ km~s$^{-1}$
($\sigma =$ 21 mJy~beam$^{-1}$~km~s$^{-1}$). The inset shows the
spectrum at the marginal $3.7\sigma$ enhancement south-east to the
[O~{\sc iii}] position (denoted as `SE').
\end{minipage}
\label{cii}
\end{center}
\end{figure}
\subsubsection{Dust continuum}
Continuum emission remains undetected in both the band 8 (735 $\mu$m)
and band 6 (1.33 mm) images. The $3\sigma$ upper limits measured for
naturally-weighted images are 0.12 and 0.042 mJy at 735 $\mu$m and 1.33
mm, respectively. The total IR luminosity assuming a modified blackbody
\cite{debreuck03} integrated over the rest-frame wavelengths of 8--1000
$\mu$m is estimated to be $L_{\rm TIR}<1\times10^{11}~L_\odot$, where
the dust temperature and emissivity index are assumed to be $T_{\rm
dust}=40$ K and $\beta = 1.5$, respectively. This limit can be relaxed
to $<2\times10^{11}~L_{\odot}$ for a higher dust temperature of
$T_\mathrm{dust} = 50$ K, while if the galaxy has cooler dust
($T_\mathrm{dust} = 30$ K), the luminosity limit obtained from the 1.33
mm photometry becomes more stringent ($<4\times 10^{10}~L_{\odot}$).
Table~S2 is a summary of these results.
The emissivity index of 1.5 which we assumed is a typical value
observed in nearby star-forming galaxies \cite{dunne01}. It is reported
that the typical star-forming galaxies ($L\sim L_*$) at $z\sim4$ have
$T_{\rm dust}\simeq30$ K \cite{lee12}. A bright LAE at $z\simeq7$,
Himiko \cite{ouchi09}, is estimated to have $T_{\rm dust}=30$--40 K
\cite{hirashita14}. We therefore assume $T_{\rm dust}=40$ K as a
fiducial value for SXDF-NB1006-2 in this paper. For this temperature,
the effect of the cosmic microwave background whose temperature is
22 K at $z=7.2$ is small \cite{dacunha13}.
\begin{table}[t]
\label{table:alma_continuum}
\begin{center}
Table~S2: \textsf{ALMA results of the dust IR continuum of SXDF-NB1006-2.}\\
\begin{tabular}{ccc}
\hline \hline
& Band 8 (735 $\mu$m) & Band 6 (1.33 mm) \\
\hline
Flux density (mJy) & $< 0.12$ ($3\sigma$) & $< 0.042$ ($3\sigma$) \\
\hline
Dust temperature (K) & \multicolumn{2}{c}{Total infrared luminosity
($L_\odot$, $3\sigma$) $^*$} \\
\hline
30 & $< 9.0\times 10^{10}$ & $< 3.8\times 10^{10}$ \\
40 & $< 1.1\times 10^{11}$ & $< 8.3\times 10^{10}$ \\
50 & $< 1.7\times 10^{11}$ & $< 1.7\times 10^{11}$ \\
\hline
\end{tabular}
\\
$^*$ We assume a single-temperature modified blackbody with an
emissivity index of 1.5 and a concordance cosmology with $H_0=70$ km
s$^{-1}$ Mpc$^{-1}$, $\Omega_{\rm M}=0.3$, and $\Omega_\Lambda=0.7$.
\end{center}
\end{table}
\subsubsection{Possible kinematics signature in the [O~{\sc iii}] spectrum}
The [O~\textsc{iii}] image (Figure~1A) is marginally resolved and
likely elongated in the east--west direction. The beam-deconvolved
source size, if the source is approximated by a two-dimensional
Gaussian, is estimated to be $0.''4 \times 0.''3$ in FWHM (PA
$\sim90^{\circ}$), corresponding to a physical scale of $\simeq 2
\times 1.5$ kpc$^2$. This extended structure may be attributed to
high-velocity components that are seen as red-shifted and
blue-shifted marginal broad signals in the [O~{\sc iii}] spectrum
(Figure~S3A). We made images of the central narrow component
($-50 < v < +50$ km~s$^{-1}$) and the red-shifted and blue-shifted
marginal high-velocity components ($50<v<230$ km~s$^{-1}$ and
$-300<v<-50$ km~s$^{-1}$; Figure~S5). Although the SNR is not high
enough, it seems that the high-velocity components are mostly overlapped
but exhibit a small spatial offset of $0.''3$ ($\simeq1.5$ kpc), which
is larger than the statistically-expected positional uncertainty
($\simeq 0.5\theta/\mathrm{SNR} \simeq 0.05''$, where $\theta$ is the
beam size).
\begin{figure}[htbp]
\begin{center}
\includegraphics[width =
0.5\textwidth,keepaspectratio,clip,angle=-90]{FigS5.eps}
\vspace{0.5cm}\\
\begin{minipage}{14cm}
Figure~S5: \textsf{Spatial distribution of different velocity
components of the [O~\textsc{iii}] 88 $\mu$m emission.} Red and blue
contours show the red-shifted ($50 < v < 230$ km~s$^{-1}$) and
blue-shifted ($-300 < v < -50$ km~s$^{-1}$) marginal components,
respectively, while white contours represent the narrow component
($-50 < v < +50$ km~s$^{-1}$) which likely traces the systemic
redshift of SXDF-NB1006-2. These velocities are measured with respect
to the [O~\textsc{iii}] line peak at $z = 7.2120$. Contours are drawn
at ($\pm1$, $\pm2$, $\pm3$, ...)$\times \sigma$ for the high-velocity
components and ($\pm2$, $\pm3$)$\times \sigma$ for the narrow
component for clarity. Negative contours are shown by the dotted
lines. The background image is the Subaru narrowband Ly$\alpha$ image
\cite{shibuya12}. The ellipse at the bottom-left corner represents
the naturally-weighted beam size of ALMA.
\end{minipage}
\label{fig-017_kin}
\end{center}
\end{figure}
A possible explanation of these marginal high-velocity component is
rotating motion of gravitationally-bounded gas, which is often observed
in high-$z$ massive galaxies \cite{wang13,venemans15}. A dynamical mass
is an order of
$M_{\rm dyn}\sim1\times10^5~v_{\rm circ}^2 D$ M$_{\odot}$, where
$v_{\rm circ} = 0.75~\Delta v~(\sin{i})^{-1}$ is the circular velocity
in units of km~s$^{-1}$ ($\Delta v$ and $i$ are, respectively, the line
FWHM and the inclination angle) and $D$ is the diameter of the galaxy
measured in kpc. If the possible red/blue-shifted components of
the [O~\textsc{iii}] line is produced by a rotating disk with a
$D\simeq2$ kpc and a velocity width of $\Delta v=400$ km~s$^{-1}$
(FWHM), the dynamical mass is estimated to be
$M_{\rm dyn}\sim5\times10^{10}~M_{\odot}$, where we assume that
the galaxy is a circular disk and the intrinsic source size gives the
inclination angle, i.e., $i = \cos^{-1}{(0.''3/0.''4)}$. The dynamical
mass is 2 orders of magnitude larger than the best-fit stellar mass
obtained from the SED fitting (\S3.5). However, it can not be excluded
that a passive stellar population of $<5\times10^{10}$ M$_\odot$
coexists with a young starburst in SXDF-NB1006-2 (Figure~S11). In
a cosmological simulation \cite{shimizu14,shimizu15}, galaxies at
$z=7.2$ with similar UV luminosities to SXDF-NB1006-2 have a stellar mass
of a few $\times10^{10}$ M$_\odot$ (see \S4, Figure~S10) and a dark
halo mass of a few $\times10^{11}$ M$_{\odot}$. Therefore, the
high-velocity component could be explained by rotational motion.
Another interpretation is violent motions such as outflows driven by
stellar winds and supernovae in the star forming region of
SXDF-NB1006-2, which is probably seen as the narrow component
(FWHM of $\approx80$ km s$^{-1}$). Yet another possibility is turbulent
motion driven by, for example, a merger event which might mimic the
spatial offset, although we find a single unresolved component in
the $J$ band image tracing the rest-frame UV continuum (Figure~2A).
However, we can not conclude the origin of the possible
high-velocity components because of the limited SNR for the moment.
\section{Optical-to-near infrared data}
The target galaxy, SXDF-NB1006-2, is in the Subaru/XMM-Newton Deep
Survey Field (SXDF) \cite{furusawa08} where multi-wavelength deep
observations have been carried out. We have gathered archival
optical-to-near infrared (NIR) deep images available in the SXDF:
Subaru/Suprime-Cam broadband $z'$ \cite{furusawa16} and narrowband
$NB1006$ \cite{shibuya12}, UKIRT/WFCAM broadband $J$, $H$, and $K$ taken
in the UKIRT Infrared Deep Sky Survey (UKIDSS) Ultra-Deep Survey (UDS)
\cite{lawrence07}, and {\it Spitzer}/IRAC $3.6~\mu$m and $4.5~\mu$m
taken in the {\it Spitzer} Extended Deep Survey (SEDS)
\cite{ashby13}. We measured the point spread functions (PSFs) using
stellar objects in these images, resulting in FWHMs of $1.''0$ ($z'$),
$0.''4$ ($NB1006$), $0.''8$ ($J$, $H$, and $K$), and $1.''8$ ($3.6\mu$m
and $4.5\mu$m). Except for the $NB1006$ image, the photometry was
performed using $2\times$PSF (FWHM) apertures because the object is
almost unresolved or not detected. For the $NB1006$ image, we performed
Kron photometry \cite{kron80} with the parameter $k=2$ to obtain a total
flux density from the spatially extended Ly$\alpha$ emission. The
photometric measurements are summarized in Table~S3. The magnitudes are
the AB system \cite{oke90}.
\begin{table}[htb]
\begin{center}
Table~S3: \textsf{Photometric data of SXDF-NB1006-2.}
\begin{tabular}{cccc}
\hline \hline
Band & Wavelength ($\mu$m) & PSF FWHM ($''$) & Magnitude (AB) \\
\hline
$z'$ & 0.91 & $1.''0$ & $>27.06$ $^*$ \\
$NB1006$ & 1.00 & $0.''4$ & $24.50\pm0.22$ $^\dag$ \\
$J$ & 1.26 & $0.''8$ & $25.46\pm0.18$ $^\ddag$\\
$H$ & 1.65 & $0.''8$ & $>25.64$ $^*$\\
$K$ & 2.23 & $0.''8$ & $>25.84$ $^*$\\
$IRAC3.6$ & 3.54 & $1.''8$ & $>24.64$ $^*$\\
$IRAC4.5$ & 4.49 & $1.''8$ & $>24.26$ $^*$\\
\hline
\end{tabular}
\\
$^*$ 3$\sigma$ lower limit in $2\times$PSF circular aperture.\\
$^\dag$ Kron magnitude with a $2.''64\times1.''32$ ellipse aperture.\\
$^\ddag$ $2\times$PSF circular aperture.\\
\end{center}
\end{table}
\subsection{Velocity offset between Ly$\alpha$ and [O~{\sc iii}] lines}
We have performed a profile fitting of the Ly$\alpha$ line with an
asymmetric Gaussian function \cite{shibuya14b}:
\begin{equation}
F_\lambda = A\exp\left[
\frac{-(\lambda-\lambda_0)^2}{2\{\sigma+a(\lambda-\lambda_0)\}^2}
\right]\,,
\end{equation}
where $A$ is the peak flux, $\lambda_0$ is the peak wavelength, $\sigma$
is the line width, and $a$ is the asymmetric parameter. If $a>0$, the
blue part of the line profile is weakened as the Ly$\alpha$ line observed
in high-$z$ \cite{kashikawa06}. The usual Gaussian function is recovered
with $a=0$. First, we have made a fitting with a Gaussian function and
obtained the following results: $A=(1.67\pm0.15)\times10^{-18}$ erg
s$^{-1}$ cm$^{-2}$ \AA$^{-1}$, $\lambda_0=9987.51\pm0.822$ \AA, and
$\sigma=4.71\pm0.42$ \AA\ (Figure~S6). The corresponding redshift is
$z_{\rm Ly\alpha}=7.2156\pm0.0007$ (Gaussian fit). Next, we have made
a fitting with an asymmetric Gaussian function with a fixed $\sigma=4.71$
\AA\ from the Gaussian fit, which is also consistent with the observed
FWHM of the line (11.5 \AA\ \cite{shibuya12}). The results are
$A=(1.65\pm0.16)\times10^{-18}$ erg s$^{-1}$ cm$^{-2}$ \AA$^{-1}$,
$\lambda_0=9986.67\pm0.967$ \AA, and $a=0.169\pm0.065$ (Figure~S6). The
corresponding redshift is $z_{\rm Ly\alpha}=7.2150\pm0.0008$
(asymmetric Gaussian fit). Assuming that the [O~{\sc iii}] 88 $\mu$m
line at $z_{\rm [OIII]}=7.2120\pm0.0003$ traces the systemic redshift,
we have obtained the velocity offset of the Ly$\alpha$ line
$\Delta v_{\rm Ly\alpha}=+(1.1\pm0.3)\times10^2$ km s$^{-1}$, where we
have corrected the Ly$\alpha$ redshift for the heliocentric motion of
Earth at the observing date ($+4$ km s$^{-1}$). Note that the ALMA
spectrum of the [O~{\sc iii}] line is already corrected for the Earth's
motion in the data reduction process.
\begin{figure}
\begin{center}
\includegraphics[width=8cm]{FigS6.eps}
\vspace{0.5cm}\\
\begin{minipage}{14cm}
Figure~S6: \textsf{Ly$\alpha$ line profile fitting results.}
The black solid line is the observed spectrum and the red solid
line is the best-fit profile with an asymmetric Gaussian function
expressed in equation (1). The cyan dashed line is the best-fit
result with a simple Gaussian function for a reference. The central
wavelengths with their $\pm1\sigma$ uncertainties are shown by the
upward arrows with error-bars. The wavelength range used in the
fitting is 9975--9998 \AA. The top horizontal axis is the velocity
shift relative to the systemic redshift $z=7.2120$ measured from the
[O~{\sc iii}] 88 $\mu$m line and corrected for the heliocentric
motion of Earth.
\end{minipage}
\end{center}
\end{figure}
\subsection{Empirical SFR estimation}
We now estimate the SFR of SXDF-NB1006-2 with empirical relations.
We assume a Salpeter initial mass function (IMF) \cite{salpeter55} with
the mass range of 0.1--100 M$_\odot$ throughout this paper. There
is a good correlation between the [O~{\sc iii}] 88 $\mu$m line
luminosity and the SFR derived from the sum of the FUV and IR
luminosities based on a large compilation of various kinds of galaxies
including nearby low-metallicity dwarfs, ULIRGs, AGNs, and high-$z$
dusty starbursts \cite{delooze14}. The [O~{\sc iii}]--SFR relations
for specific kinds of galaxies are slightly different from each other.
If we assume the relation derived from the entire sample of
\cite{delooze14}, we obtain a SFR $>100$ M$_\odot$ yr$^{-1}$
for SXDF-NB1006-2 (Figure~S7). On the other hand, the $J$ band
(i.e. rest-frame $\approx1500$ \AA) luminosity of this galaxy
indicates a SFR $\sim10$ M$_\odot$ yr$^{-1}$ with a standard FUV--SFR
conversion \cite{kennicutt98}. This conversion assumes a constant SFR
more than a few 100 Myr, while it actually depends on the duration
of star formation. If the star formation age is $\sim1$ Myr, we indeed
obtain $\sim100$ M$_\odot$ yr$^{-1}$ which is consistent with the
estimation based on the [O~{\sc iii}] line. This suggests that the
target galaxy is in a young violent star formation phase.
\begin{figure}
\begin{center}
\includegraphics[width=8cm]{FigS7.eps}
\vspace{0.5cm}\\
\begin{minipage}{14cm}
Figure~S7: \textsf{Empirical relation between the SFR and
the [O~{\sc iii}] 88 $\mu$m line luminosity \cite{delooze14}.}
The best-fit relation with $\pm1\sigma$ standard deviation for a
large compilation of the data of various kinds of galaxies including
low-metallicity nearby dwarf galaxies (circles) and $z\sim3$--4 dusty
starburst galaxies (diamonds) is shown by the yellow line with the
gray shade. The [O~{\sc iii}] line luminosity with $\pm1\sigma$
uncertainty of SXDF-NB1006-2 at $z=7.2$ is shown by the horizontal
blue line with the orange shade. The crosses show SFRs estimated from
the observed FUV luminosity of the target galaxy, under various
assumptions on the duration of star formation. All the SFRs are
calibrated to ones with the Salpeter IMF \cite{salpeter55} with
0.1--100 M$_\odot$.
\end{minipage}
\end{center}
\end{figure}
\section{Spectral energy distribution modeling}
In order to derive physical properties of the galaxy, SXDF-NB1006-2,
we have performed a spectral energy distribution (SED) fitting
\cite{sawicki98,ono10}.
This is based on a standard $\chi^2$ minimization method:
\begin{equation}
\chi^2 = \sum_{i=1}^N \left(
\frac{F_{i,{\rm model}}-F_{i,{\rm obs}}}{\sigma_{i,{\rm obs}}}
\right)^2 \,,
\end{equation}
where $F_{i,{\rm model}}$, $F_{i,{\rm obs}}$, and $\sigma_{i,{\rm obs}}$
are the model flux density (or flux), the observed flux density (or
flux) and the observed uncertainty of $i$th data point, respectively. We
have used not only the broadband photometric data ($J$, $H$, $K$, $IRAC3.6$
and $IRAC4.5$) but also the narrowband $NB1006$ photometry, the [O~{\sc
iii}] 88 $\mu$m line flux and the total IR flux upper limit as
constraints. For non-detection bands and the IR flux, we simply set
$F_{i,{\rm obs}}=0$ and take their $3\sigma$ limits as
$\sigma_{i,{\rm obs}}$. This treatment makes the fitting favor
$F_{i,{\rm model}}$ below the $3\sigma$ limits for the non-detection
data. There are other choices to manage the non-detection data, for
example, taking $F_{i,{\rm obs}}=\sigma_{i,{\rm obs}}=1.5\sigma$
limit (option 2 of \cite{bolzonella00}) or a modification
of equation~(2) to treat the upper limits \cite{sawicki12}. We have
tried these two methods and found that the best-fit parameters do not
change but their $1\sigma$ ranges tend to be smaller. This is because
the latter two methods put a larger weight on the non-detection data.
Thus, our approach above is more conservative than the latter two
methods. The $z'$ band which is severely affected by the intergalactic
attenuation is omitted because we have fixed the redshift to that of the
[O~{\sc iii}] line ($z=7.212$) and the non-detection in the $z'$ band
does not have much information. Therefore, the number of the constraints
is $N=8$.
\subsection{Stellar continuum}
We have adopted theoretical spectra generated with a public stellar
population synthesis code {\sc PEGASE ver.~2} \cite{pegase}.
We assume metallicities of $Z=0.0004$, 0.001, 0.002, 0.004, 0.008, 0.02,
and 0.05 with a classical solar metallicity of $Z=0.02$
\cite{anders89}. The stellar IMF is assumed to be a standard Salpeter
one \cite{salpeter55} with the range of 0.1--100 M$_\odot$. A constant
star formation history is also assumed for simplicity. In this
case, the obtained age and stellar mass are regarded as those of the
most recent star formation episode. If the galaxy has previous star
formation episodes, the true age and stellar mass are larger than those
obtained here. On the other hand, for instantaneous quantities such
as the SFR and dust attenuation, the assumption of a constant SFR is
valid in the sense of an average during the star formation episode.
We have set a lower limit of 1 Myr in the age.
Metallicity evolution, gas infall, outflow, nebular emission,
and dust extinction have not been considered at this stage.
\subsection{Nebular emission}
The spectra of young star-forming galaxies are significantly affected by
emission from ionized gas, so-called nebular emission \cite{schaerer09}.
We have added the nebular continuum (two-photon, bound-free, and
free-free continua) and 119 UV-to-optical ($\lambda<1$ $\mu$m in
the source rest-frame) emission lines to the model spectra following
the prescription of \cite{inoue10,inoue11}. This emission line
model is based on a large set of calculations of H~{\sc ii} regions
using a public photoionization code, {\sc cloudy} \cite{ferland13} and
reproduces the observed strengths of several prominent emission lines
such as [O~{\sc iii}] $\lambda$5007 relative to the hydrogen H$\beta$
line very well. For the [O~{\sc iii}] 88 $\mu$m line, the model
presented in \cite{inoue14} is adopted. This [O~{\sc iii}] line
model is also made with {\sc cloudy} and agrees with the available
observations of the [O~{\sc iii}] 88 $\mu$m line very well (Figure~S8).
\begin{figure}
\begin{center}
\includegraphics[width=8cm]{FigS8.eps}
\vspace{0.5cm}\\
\begin{minipage}{14cm}
Figure~S8: \textsf{Emissivity of the [O~{\sc iii}] 88 $\mu$m line per
a unit SFR as a function of the oxygen abundance.} The horizontal
axis is the oxygen abundance relative to that of the Sun
($[{\rm O/H}]=\log_{10}(n_{\rm O}/n_{\rm H})-\log_{10}(n_{\rm
O}/n_{\rm H})_\odot$ with $12+\log_{10}(n_{\rm O}/n_{\rm
H})_\odot=8.69$) \cite{asplund09}. The data of nearby dwarf galaxies
come from observations with the {\it Herschel} satellite
\cite{madden13,delooze14,cormier15}. The data of LMC and nearby spiral
galaxies as well as model predictions are taken from \cite{inoue14}.
We show the models with the ionization parameter $\log_{10}U$ and the
hydrogen atom density $\log_{10}n_{\rm H}$ noted in the panel. The
SFR is calibrated to ones with the Salpeter IMF with
0.1--100 M$_\odot$.
\end{minipage}
\end{center}
\end{figure}
We allow the escape of hydrogen ionizing photons (wavelength
$\lambda<912$ \AA\ in the source rest-frame; Lyman continuum) from
H~{\sc ii} regions and the surrounding ISM in the galaxy to the IGM. Not
only the stellar ionizing photons but also nebular bound-free ionizing
photons can escape to the IGM \cite{inoue10}. We assume that both escapes
happen with the same escape fraction, $f_{\rm esc}$, defined as the
number fraction of the escaped photons among the produced photons.
\subsection{Dust attenuation and IR luminosity}
We assume the Calzetti attenuation law \cite{calzetti94,calzetti01}
for the attenuation by dust particles in the ISM as a standard manner
found in literature, while recent studies may suggest deviations
from the Calzetti law in high-$z$ galaxies \cite{capak15,reddy15}.
Fortunately, the obtained attenuation amount for the target galaxy is
small and the shape of the attenuation law would not affect the
conclusions of this paper.
The Calzetti law predicts about a factor of 2 higher attenuation for the
nebular emission than for the stellar continuum. This is because
H~{\sc ii} regions producing the nebular emission are more deeply
embeded in gas and dust clouds than stars observed in the UV
wavelength. On the other hand, such a difference between nebular and
stellar emissions may not be supported by observations of young
star-forming galaxies \cite{kashino13,hashimoto15}. We then introduce a
parameter, $R_{\rm gs}\equiv (E_{B-V})_{\rm gas}/(E_{B-V})_{\rm star}$,
to describe this effect and assume $R_{\rm gs}=1$ or 2. The original
Calzetti law predicts $R_{\rm gs}=2.3$.
We assume that the radiation energy attenuated by dust is finally
absorbed by dust in the ISM and thermally re-emitted in
the IR (i.e., we assume the energy scattered out to the IGM to be
negligible). This energy is compared to the total IR luminosity
estimated with the dust temperature of 40 K (Table~1), assuming the
total dust emission comes only from the star-forming regions of
interest.
\subsection{Ly$\alpha$ emission line}
The narrowband $NB1006$ photometry is mainly determined by the
Ly$\alpha$ emission line although it also contains information of the UV
continuum. Since Ly$\alpha$ photons suffer from resonant scattering
by neutral hydrogen, the transfer in the ISM is complex. In addition to
the ISM, Ly$\alpha$ photons are also scattered by neutral hydrogen in
the IGM. Therefore, the observed Ly$\alpha$ flux becomes $F_{\rm
Ly\alpha}^{\rm obs}=F_{\rm Ly\alpha}^{\rm int}T_{\rm Ly\alpha}^{\rm
IGM}e^{-\tau^{\rm ISM}_{\rm Ly\alpha}}$, where
$F_{\rm Ly\alpha}^{\rm int}$ is the intrinsic Ly$\alpha$ flux,
$\tau_{\rm Ly\alpha}^{\rm ISM}$ and $T_{\rm Ly\alpha}^{\rm IGM}$ are,
respectively, the ISM optical depth and the IGM transmission for
Ly$\alpha$ photons. Recent studies of LAEs suggest that a simple recipe
like the Calzetti law with $R_{\rm gs}\simeq1$
(i.e. $(E_{B-V})_{\rm gas}=(E_{B-V})_{\rm stars}$) reasonably
explains the Ly$\alpha$ optical depth in the ISM inferred from the
Ly$\alpha$ line profile \cite{hashimoto15}. The IGM effect is more
severe at higher redshift due to a higher neutral fraction in the
IGM. At redshift $z\simeq7.2$, the neutral fraction, $x_{\rm HI}$, in
the IGM is estimated to be $\sim0.5$ \cite{robertson15} although it
is still uncertain. According to a cosmological radiative transfer
simulation, the Ly$\alpha$ transmission through the IGM with an average
$x_{\rm HI}=0.5$ is $T_{\rm Ly\alpha}^{\rm IGM}=0.35_{-0.15}^{+0.10}$
\cite{jensen13}. Since this $T_{\rm Ly\alpha}^{\rm IGM}$ range
also covers a wide range of $x_{\rm HI}=0.2$--0.8 \cite{jensen13},
we assume that the $T_{\rm Ly\alpha}^{\rm IGM}$ range encloses the
uncertainty of $x_{\rm HI}$.
\subsection{Results}
We use $N=8$ observational constraints: 6 photometric data ($NB1006$,
$J$, $H$, $K$, $IRAC3.6$ and $IRAC4.5$) and the [O~{\sc iii}] 88 $\mu$m
line flux and the total IR flux. On the other hand, there are 7 model
parameters: the metallicity $Z$, the IGM Ly$\alpha$ transmission
$T_{\rm Ly\alpha}^{\rm IGM}$, the dust attenuation ratio of nebular to
stellar emissions $R_{\rm gs}$, the SFR, the age, the stellar dust
attenuation $(E_{B-V})_{\rm star}$, and the escape fraction of ionizing
photons $f_{\rm esc}$. For the first 3 parameters, we fixed the values:
$Z=0.0004$, 0.001, 0.002, 0.004, 0.008, 0.02($=Z_\odot)$, or 0.05,
$T_{\rm Ly\alpha}^{\rm IGM}=0.20$, 0.30, 0.35, 0.40, or 0.45,
and $R_{\rm gs}=1$ or 2. We then searched for the best set of the rest 4
parameters by a standard $\chi^2$ method. The resultant best-fit values
and their 68.4\% ranges (i.e. $\Delta \chi^2<1$) are summarized in
Table~S4, where we only show the cases with $R_{\rm gs}=1$ but the
$R_{\rm gs}=2$ cases are not very different because of a very small dust
attenuation.
We find that the minimum $\chi^2$ value is obtained with the metallicity
$Z=0.002(=0.1Z_\odot)$ but the $0.001\leq Z\leq0.02$ cases give equally
good fit results. On the other hand, the $Z=0.0004$ and 0.05 cases
are rejected at a $>95\%$ confidence level, except for the case of
$Z=0.0004$ and $T_{\rm Ly\alpha}^{\rm IGM}=0.20$ which can be rejected
at a $\sim90\%$. Therefore, the galaxy, SXDF-NB1006-2, is likely to
have a metallicity of $0.05\leq Z/Z_\odot\leq1$.
The best-fit model is $Z=0.002$, $T_{\rm Ly\alpha}^{\rm IGM}=0.40$,
$\log_{10}(SFR/{\rm M}_\odot~{\rm yr}^{-1})=2.54$, the age of 1 Myr,
no dust attenuation, and $f_{\rm esc}=0.54$, which is shown in
Figures~2B--2D. The obtained 1 Myr age is in fact the lower limit of the
population synthesis model. This shortest age is favored by the very
blue UV color of the galaxy ($J-H<-0.18$ corresponding to the UV slope
$\beta<-2.6$ [$3\sigma$]). This blue UV color also favors small dust
attenuation but the upper limit of the dust IR luminosity gives a
stronger constraint on the dust attenuation (see Figure~S9 and a
discussion below). If we fix $Z$ and
$T_{\rm Ly\alpha}^{\rm IGM}\geq0.30$, a non-zero $f_{\rm esc}$ tends to
be favored. However, there are many sets of $Z$ and
$T_{\rm Ly\alpha}^{\rm IGM}$ giving $\chi^2$ as good as the best-fit
case statistically. We derive the 68.4\% ranges of each parameters
(i.e. $\Delta \chi^2<1$) among all cases examined with $R_{\rm gs}=1$.
The results are as follows:
$1.83\leq \log_{10}(SFR~[{\rm M_\odot~yr^{-1}}])\leq2.71$,
$6.00\leq \log_{10}(t~[{\rm yr}]) \leq7.00$,
$0.00\leq (E_{B-V})_{\rm star}<0.04$, and
$0.00\leq f_{\rm esc} \leq 0.71$. For the stellar mass, we find
$\log_{10}(M_{\rm star}/{\rm M}_\odot)=8.53$ as the best-fit and the
68.4\% range of
$8.32\leq \log_{10}(M_{\rm star}/{\rm M}_\odot)<9.33$
as a joint constraint of $\log_{10}(SFR~[{\rm M_\odot~yr^{-1}}])$
and $\log_{10}(t~[{\rm yr}])$ (i.e. $\Delta \chi^2<2.3$).
\begin{table}
\begin{center}
Table~S4: \textsf{A summary of SED fitting results.}
\small
\begin{tabular}{ccccccc}
\hline \hline
$Z$ & $T_{\rm Ly\alpha}^{\rm IGM}$ &
log$_{10}(SFR~{\rm [M_\odot~yr^{-1}]})$ & log$_{10}(t~{\rm [yr]})$
& $(E_{B-V})_{\rm star}$ & $f_{\rm esc}$ & $\chi^2_{\rm min}$ \\
\hline
0.0004 & 0.45 & $2.83_{-0.12}^{+0.10}$ & $6.00_{-0.00}^{+0.00}$
& $0.08_{-0.03}^{+0.02}$ & $0.44_{-0.16}^{+0.14}$ & 9.376 \\
0.0004 & 0.40 & $2.83_{-0.14}^{+0.09}$ & $6.00_{-0.00}^{+0.00}$
& $0.08_{-0.03}^{+0.01}$ & $0.40_{-0.18}^{+0.14}$ & 8.417 \\
0.0004 & 0.35 & $2.79_{-0.12}^{+0.11}$ & $6.00_{-0.00}^{+0.00}$
& $0.07_{-0.02}^{+0.02}$ & $0.34_{-0.17}^{+0.15}$ & 7.331 \\
0.0004 & 0.30 & $2.75_{-0.12}^{+0.13}$ & $6.00_{-0.00}^{+0.00}$
& $0.06_{-0.02}^{+0.02}$ & $0.28_{-0.18}^{+0.16}$ & 6.258 \\
0.0004 & 0.20 & $2.65_{-0.11}^{+0.15}$ & $6.00_{-0.00}^{+0.00}$
& $0.04_{-0.02}^{+0.02}$ & $0.09_{-0.09}^{+0.20}$ & 4.035 \\
\hline
0.0010 & 0.45 & $2.74_{-0.15}^{+0.10}$ & $6.00_{-0.00}^{+0.00}$
& $0.05_{-0.03}^{+0.01}$ & $0.52_{-0.14}^{+0.10}$ & 3.304 \\
0.0010 & 0.40 & $2.70_{-0.14}^{+0.11}$ & $6.00_{-0.00}^{+0.00}$
& $0.04_{-0.02}^{+0.02}$ & $0.49_{-0.15}^{+0.10}$ & 2.855 \\
0.0010 & 0.35 & $2.66_{-0.14}^{+0.13}$ & $6.00_{-0.00}^{+0.00}$
& $0.03_{-0.02}^{+0.02}$ & $0.44_{-0.15}^{+0.13}$ & 2.481 \\
0.0010 & 0.30 & $2.61_{-0.14}^{+0.15}$ & $6.00_{-0.00}^{+0.00}$
& $0.02_{-0.02}^{+0.02}$ & $0.38_{-0.15}^{+0.15}$ & 2.166 \\
0.0010 & 0.20 & $2.53_{-0.25}^{+0.14}$ & $6.00_{-0.00}^{+0.30}$
& $0.00_{-0.00}^{+0.03}$ & $0.26_{-0.26}^{+0.16}$ & 1.871 \\
\hline
0.0020 & 0.45 & $2.58_{-0.18}^{+0.14}$ & $6.00_{-0.00}^{+0.30}$
& $0.01_{-0.01}^{+0.03}$ & $0.57_{-0.14}^{+0.11}$ & 1.695 \\
0.0020 & 0.40 & $2.54_{-0.20}^{+0.14}$ & $6.00_{-0.00}^{+0.30}$
& $0.00_{-0.00}^{+0.04}$ & $0.54_{-0.18}^{+0.12}$ & {\bf 1.629} \\
0.0020 & 0.35 & $2.54_{-0.28}^{+0.13}$ & $6.00_{-0.00}^{+0.30}$
& $0.00_{-0.00}^{+0.03}$ & $0.52_{-0.25}^{+0.11}$ & 1.694 \\
0.0020 & 0.30 & $2.34_{-0.25}^{+0.30}$ & $6.30_{-0.30}^{+0.30}$
& $0.01_{-0.01}^{+0.03}$ & $0.34_{-0.29}^{+0.27}$ & 1.890 \\
0.0020 & 0.20 & $2.09_{-0.11}^{+0.31}$ & $6.48_{-0.18}^{+0.22}$
& $0.00_{-0.00}^{+0.02}$ & $0.02_{-0.02}^{+0.38}$ & 2.075 \\
\hline
0.0040 & 0.45 & $2.30_{-0.30}^{+0.29}$ & $6.30_{-0.30}^{+0.40}$
& $0.00_{-0.00}^{+0.03}$ & $0.50_{-0.30}^{+0.21}$ & 1.701 \\
0.0040 & 0.40 & $2.31_{-0.46}^{+0.12}$ & $6.30_{-0.00}^{+0.70}$
& $0.00_{-0.00}^{+0.03}$ & $0.50_{-0.50}^{+0.11}$ & 1.710 \\
0.0040 & 0.35 & $2.11_{-0.30}^{+0.29}$ & $6.48_{-0.18}^{+0.60}$
& $0.00_{-0.00}^{+0.03}$ & $0.31_{-0.31}^{+0.28}$ & 1.875 \\
0.0040 & 0.30 & $2.11_{-0.30}^{+0.25}$ & $6.48_{-0.18}^{+0.52}$
& $0.00_{-0.00}^{+0.02}$ & $0.28_{-0.28}^{+0.27}$ & 1.955 \\
0.0040 & 0.20 & $1.94_{-0.10}^{+0.24}$ & $6.70_{-0.22}^{+0.26}$
& $0.00_{-0.00}^{+0.01}$ & $0.03_{-0.03}^{+0.34}$ & 2.981 \\
\hline
0.0080 & 0.45 & $2.31_{-0.30}^{+0.30}$ & $6.30_{-0.30}^{+0.40}$
& $0.01_{-0.01}^{+0.03}$ & $0.43_{-0.35}^{+0.25}$ & 1.847 \\
0.0080 & 0.40 & $2.27_{-0.34}^{+0.31}$ & $6.30_{-0.30}^{+0.48}$
& $0.00_{-0.00}^{+0.03}$ & $0.39_{-0.39}^{+0.26}$ & 1.779 \\
0.0080 & 0.35 & $2.27_{-0.37}^{+0.27}$ & $6.30_{-0.30}^{+0.54}$
& $0.00_{-0.00}^{+0.03}$ & $0.37_{-0.37}^{+0.23}$ & 1.790 \\
0.0080 & 0.30 & $2.07_{-0.17}^{+0.29}$ & $6.48_{-0.18}^{+0.30}$
& $0.00_{-0.00}^{+0.02}$ & $0.13_{-0.13}^{+0.34}$ & 1.954 \\
0.0080 & 0.20 & $1.99_{-0.07}^{+0.32}$ & $6.60_{-0.30}^{+0.10}$
& $0.00_{-0.00}^{+0.01}$ & $0.00_{-0.00}^{+0.40}$ & 2.974 \\
\hline
0.0200 & 0.45 & $2.44_{-0.38}^{+0.12}$ & $6.00_{-0.00}^{+0.48}$
& $0.00_{-0.00}^{+0.03}$ & $0.52_{-0.43}^{+0.11}$ & 1.667 \\
0.0200 & 0.40 & $2.45_{-0.50}^{+0.09}$ & $6.00_{-0.00}^{+0.70}$
& $0.00_{-0.00}^{+0.03}$ & $0.51_{-0.51}^{+0.10}$ & 1.845 \\
0.0200 & 0.35 & $2.20_{-0.27}^{+0.30}$ & $6.30_{-0.30}^{+0.40}$
& $0.00_{-0.00}^{+0.03}$ & $0.26_{-0.26}^{+0.32}$ & 1.771 \\
0.0200 & 0.30 & $2.02_{-0.08}^{+0.45}$ & $6.48_{-0.48}^{+0.22}$
& $0.00_{-0.00}^{+0.02}$ & $0.01_{-0.01}^{+0.52}$ & 1.937 \\
0.0200 & 0.20 & $2.04_{-0.07}^{+0.22}$ & $6.48_{-0.18}^{+0.12}$
& $0.00_{-0.00}^{+0.01}$ & $0.00_{-0.00}^{+0.32}$ & 3.001 \\
\hline
0.0500 & 0.45 & $2.60_{-0.11}^{+0.10}$ & $6.00_{-0.00}^{+0.00}$
& $0.08_{-0.03}^{+0.02}$ & $0.79_{-0.06}^{+0.05}$ & 13.145 \\
0.0500 & 0.40 & $2.60_{-0.12}^{+0.10}$ & $6.00_{-0.00}^{+0.00}$
& $0.08_{-0.03}^{+0.02}$ & $0.77_{-0.08}^{+0.06}$ & 12.067 \\
0.0500 & 0.35 & $2.60_{-0.12}^{+0.09}$ & $6.00_{-0.00}^{+0.00}$
& $0.08_{-0.03}^{+0.02}$ & $0.74_{-0.08}^{+0.07}$ & 10.855 \\
0.0500 & 0.30 & $2.57_{-0.10}^{+0.11}$ & $6.00_{-0.00}^{+0.00}$
& $0.07_{-0.02}^{+0.02}$ & $0.71_{-0.09}^{+0.07}$ & 9.498 \\
0.0500 & 0.20 & $2.54_{-0.12}^{+0.09}$ & $6.00_{-0.00}^{+0.00}$
& $0.06_{-0.02}^{+0.02}$ & $0.62_{-0.10}^{+0.08}$ & 6.311 \\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{figure}
\begin{center}
\includegraphics[width=7cm]{FigS9.eps}
\vspace{0.5cm}\\
\begin{minipage}{14cm}
Figure~S9: \textsf{Visualizations of SED fitting results.}
(A) The SFR ranges within $\Delta \chi^2<2.30$ (a $\pm1\sigma$ range
for two parameters) from the best-fit models whose locations are
shown by the crosses. The black, red, green and blue colors represent
the cases using only broadband (BB) photometries ($J$, $H$, $K$,
$IRAC3.6$, and $IRAC4.5$), BB + the infrared (IR) luminosity limit,
BB + IR + the narrowband (NB) photometry ($NB1006$), and BB + IR + NB
+ the [O~{\sc iii}] line, respectively. The most metal-poor and
metal-rich models were rejected in the last case. (B) The same as A
but for the star formation age. (C) The same as A but for the dust
attenuation for stars. (D) The same as A but for the escape
fraction of ionizing photons.
\end{minipage}
\end{center}
\end{figure}
The [O~{\sc iii}] 88 $\mu$m line flux is the most powerful constraint in
the SED modeling of SXDF-NB1006-2 (Figure~S9). Without the [O~{\sc
iii}] line, we cannot obtain any meaningful constraint on the
metallicity, whereas using the [O~{\sc iii}] line, we can reject the
most metal-poor and metal-rich cases examined (i.e. $Z=0.0004$
and 0.05). The [O~{\sc iii}] line also improves the constraints on
other parameters dramatically, although the best-fit values of the
SFR, age, and dust attenuation are not very different regardless of the
sets of the data used in the fitting. Generally, less data give a weaker
convergence around the best-fit values as expected. When we use only the
broadband data as constraints, only a few models are rejected. The
IR luminosity limit greatly improves the constraint on the dust
attenuation (Figure~S9C). Using the narrowband $NB1006$ which
includes the Ly$\alpha$ line slightly improves overall constraints.
\section{Comparison with a cosmological simulation}
We now compare SXDF-NB1006-2 with galaxies in a large
cosmological hydrodynamic simulation of galaxy formation and evolution
\cite{shimizu14,shimizu15} which reproduces the observed UV
luminosity functions and UV colors of Lyman break galaxies at
$z\sim7$--10 very well. Comparing SXDF-NB1006-2 with the galaxies taken
from the simulation output at z=7.22, we discuss implications on the
physical and chemical properties of the target galaxy. A brief
explanation of the simulation is as follows, while further details of the
simulation are found in \cite{shimizu14,shimizu15}. The simulation code
we used is {\sc gadget-3}: an updated version of {\sc gadget-2}
\cite{springel05}. The physical recipes describing the star formation,
chemical evolution, supernovae and radiation feedback
\cite{okamoto08,okamoto10,okamoto14} are implemented. We employ
$2\times640^3$ particles for dark matter and gas in a comoving volume of
$100h^{-1}$ Mpc cube. The mass of a dark matter particle is
$2.84\times10^8h^{-1}$ M$_\odot$ and the mass of a gas particle is
initially $5.17\times10^7h^{-1}$ M$_\odot$. The softening length for the
gravitational force is set to be $6.0 h^{-1}$ comoving kpc. The gas
particles may form star particles if the star formation criteria are
satisfied. We also implement the emission line model
\cite{inoue11,inoue14} into the simulation, assuming the zero escape of
ionizing photons.
\begin{figure}
\begin{center}
\includegraphics[width=12cm]{FigS10.eps}
\vspace{0.5cm}\\
\begin{minipage}{14cm}
Figure~S10: \textsf{Comparisons of SXDF-NB1006-2 with galaxies in a
cosmological hydrodynamic simulation.} Panels A--C show the [O~{\sc
iii}] 88 $\mu$m line luminosity, the dust IR luminosity, and the
oxygen abundance (or metallicity), respectively, as a function of the
absolute UV magnitude uncorrected for the dust attenuation. Panels
D--F are the same as A--C but as a function of the stellar mass. The
circles with error-bars are the data of SXDF-NB1006-2; we take a
$3\sigma$ upper limit for the dust IR luminosity with an dust
temperature of 40 K and an emissivity index of 1.5. The stellar
mass obtained from the SED fitting should be regarded as a lower
limit because of our simple constant star formation history. The plus
marks are galaxies at $z=7.22$ taken from the simulation
\cite{shimizu15}.
\end{minipage}
\end{center}
\end{figure}
Comparisons of SXDF-NB1006-2 with the $z=7.22$ galaxies taken from
the simulation (Figure~S10) show that there are five galaxies with
similar UV luminosities to that of SXDF-NB1006-2 (panels A--C). The
[O~{\sc iii}] line luminosity of SXDF-NB1006-2 is very close to the two
highest ones among the five. The SFRs of the two simulated galaxies are
51 and 92 M$_\odot$ yr$^{-1}$, whereas the SFR of SXDF-NB1006-2 is
estimated at $\sim300$ M$_\odot$ yr$^{-1}$ from the SED modeling. The
best-fit SED model suggests a $\sim50\%$ escape of ionizing photons,
indicating a factor of $\sim2$ reduction of the [O~{\sc iii}] line
luminosity per SFR in SXDF-NB1006-2. This partly accounts for the SFR
difference in spite of similar [O~{\sc iii}] line luminosities. In any
case, SXDF-NB1006-2 seems to be in an intense starburst phase which
enhances the [O~{\sc iii}] line luminosity.
We also find that the average of the IR luminosities of the five
simulated galaxies are 0.5 dex higher than the $3\sigma$ upper limit of
SXDF-NB1006-2 (panel B), suggesting that the target galaxy has much less
dust than the simulated galaxies. In fact, these simulated galaxies have
$(E_{B-V})_{\rm star}=0.15$, whereas that of SXDF-NB1006-2 is less than
0.04 mag from the SED fitting. On the other hand, the metallicity of
SXDF-NB1006-2 is similar to or higher than those of the five (panel C).
These indicates that SXDF-NB1006-2 has a much less dust-to-metal mass
ratio than the simulated galaxies where we have assumed the ratio to be
0.5 as in the ISM of the Milky Way \cite{draine11} and of the Solar
neighborhood \cite{kimura03}.
\begin{figure}
\begin{center}
\includegraphics[width=8cm]{FigS11.eps}
\vspace{0.5cm}\\
\begin{minipage}{12cm}
Figure~S11: \textsf{Comparison of spectral energy distributions.}
The circles with the error-bars are the data of SXDF-NB1006-2. The
green solid line is the best-fit model and the asterisks are those
convolved with the filter response curves. The magenta solid line
shows a maximally possible passive stellar population
($M_*=5\times10^{10}$ M$_\odot$ and age of 700 Myr). The squares
connected with lines are the five simulated galaxies having a
similar UV luminosity to SXDF-NB1006-2.
\end{minipage}
\end{center}
\end{figure}
The stellar mass of SXDF-NB1006-2 obtained from the SED fitting is an
order of magnitude smaller than those expected in the simulation (panels
D--F). This small stellar mass is mainly due to the very short age
($\sim1$ Myr) which is constrained from the blue UV color of the galaxy.
Indeed, the observed UV color of SXDF-NB1006-2 is much bluer than those
of the five simulated galaxies (Figure~S11). However, it is
still possible that in addition to the $\sim1$ Myr starburst,
SXDF-NB1006-2 has an underlying passive stellar population with a
stellar mass of $<5\times10^{10}$ M$_\odot$ and an age of 700 Myr
($\approx$ the age of the Universe at $z=7.2$) without violating
the Spitzer data at 3.6 $\mu$m and 4.5 $\mu$m (Figure~S11). Even in
this case, the blue UV color of SXDF-NB1006-2 is not affected by this
passive stellar population and the conclusion that SXDF-NB1006-2 is a
very young starburst and has little dust content is preserved.
\section{Ionizing photon emission efficiency}
There is an important quantity in the context of cosmic reionization:
the ionizing photon injection rate into the IGM per unit UV luminosity
density of galaxies \cite{bouwens15b}. This can be expressed as
\begin{equation}
f_{\rm esc}\xi_{\rm ion}=f_{\rm esc}
\left(\frac{Q_{\rm H}}{SFR}\right)
\frac{SFR}{L_{\nu_{1500}}^{\rm obs}}\,,
\end{equation}
where $(Q_{\rm H}/SFR)$ is the intrinsic production rate of hydrogen
ionizing photons ($\lambda<912$ \AA) per a unit SFR and
$L_{\nu_{1500}}^{\rm obs}$ is the luminosity density at UV
($\lambda\sim1500$ \AA\ in the source rest-frame). We estimate this rate
for SXDF-NB1006-2. For the stellar population corresponding to
the best-fit model with $Z=0.002$ and the 1 Myr age,
$(Q_{\rm H}/SFR)=2.58\times10^{52}$ s$^{-1}$.
Different metallicities cause only 0.03 dex variation.
However, the star formation age affects the production rate; a constant
SFR of 10 Myr ($+1\sigma$ age) gives a 0.40 dex larger production rate.
If we adopt the best-fit stellar population and uncertainties of the SFR
and $f_{\rm esc}$ in the case of $T_{\rm Ly\alpha}^{\rm IGM}=0.40$
(Table~S4), we obtain
$\log_{10}(f_{\rm esc}\xi_{\rm ion}/{\rm Hz~erg^{-1}})=25.44^{+0.18}_{-0.26}$
by using an error propagation formula for the logarithm. If we adopt the
uncertainties of the final estimates of the SFR and $f_{\rm esc}$
(Table~1) and take into account the 0.40 dex upward uncertainty caused
by the star formation age, we obtain
$\log_{10}(f_{\rm esc}\xi_{\rm ion}/{\rm Hz~erg^{-1}})=25.44^{+0.46}_{-0.84}$.
We compare the obtained ionizing photon injection rate per UV luminosity
density for SXDF-NB1006-2 with that required to reproduce the comoving
volume emissivity of ionizing photons of
$\log_{10}(\dot N_{\rm ion} {\rm [s^{-1}~Mpc^{-3}]})=50.79\pm0.06$ at
$z\sim7$ which is estimated from various observational constraints on
cosmic reionization with a parametric expression of the
$\dot N_{\rm ion}$ evolution \cite{bouwens15b}. Since the photon
injection rate is given by
$\log_{10}(f_{\rm esc}\xi_{\rm ion})=
\log_{10}(\dot N_{\rm ion})-\log_{10}(\rho_{\rm UV})$, where
$\rho_{\rm UV}$ is the comoving UV luminosity density, we need to
integrate a UV luminosity function. Here we consider two UV luminosity
functions at $z\sim7$ reported by \cite{bouwens15} and
\cite{finkelstein15}. We have performed a set of Monte Carlo
realizations of Schechter function fits to the luminosity functions
fluctuated based on the quoted uncertainties and obtained a set of
$\rho_{\rm UV}$ as a function of a faint-limit of $M_{\rm UV}$ by
integrating the best-fit Schechter function in each realization. We have
also taken into account the uncertainty of the ionizing photon
emissivity, $\log_{10}(\dot N_{\rm ion})$, in the procedure to obtain
the emission efficiency, $\log_{10}(f_{\rm esc}\xi_{\rm ion})$. Finally,
we have calculated the mean and standard deviation of
$\log_{10}(f_{\rm esc}\xi_{\rm ion})$ among the Monte Carlo
realizations as a function of the faint $M_{\rm UV}$ limit.
From this comparison (Figure~S12), we find that it is difficult to
reproduce the ionizing photon emissivity at $z\sim7$ only by galaxies
brighter than SXDF-NB1006-2 ($M_{\rm UV}=-21.53$), even if these
galaxies emit ionizing photons as strong as that galaxy, because of the
small number density of such bright galaxies. On the other hand, if
galaxies with $M_{\rm UV}<-17$, which are already detected in deep HST
surveys, have an ionizing photon emission efficiency similar to
SXDF-NB1006-2, the ionizing photon emissivity is likely to be achieved
or even exceeded by 0.4--0.6-dex. However, if objects emitting ionizing
photons as strong as SXDF-NB1006-2 are rare among galaxies with
$M_{\rm UV}<-17$, fainter, currently undetected galaxies should
contribute to the cosmic ionizing photon emissivity.
\begin{figure}
\begin{center}
\includegraphics[width=8cm]{FigS12.eps}
\vspace{0.5cm}\\
\begin{minipage}{12cm}
Figure~S12: \textsf{Comparison of ionizing photon injection rates
into the IGM per UV luminosity density.} The red and green shades
show the rates required to match the cosmic ionizing photon
emissivity estimated at $z\sim7$ \cite{bouwens15b}, when we integrate
the UV luminosity function of \cite{bouwens15} (denoted as B15LF) or
\cite{finkelstein15} (denoted as F15LF), respectively, down to the
faint UV magnitude limit indicated on the horizontal axis.
The rate obtained from SXDF-NB1006-2 is shown by a five-pointed-star
with error-bars. The smaller error-bars show the case with the
best-fit stellar population and $T_{\rm Ly\alpha}^{\rm IGM}=0.40$,
but the larger error-bars show the case considering all uncertainties
in our estimates. The vertical dotted line at $M_{\rm UV}=-17$
indicates a detection limit with HST/WFC3 \cite{bouwens15}. The open
star with error-bars shows the data of SXDF-NB1006-2 at the HST
detection limit for a reference.
\end{minipage}
\end{center}
\end{figure}
\end{document}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,640 |
Download Prime Suspect 2 series
John Strickland
Crime / Drama / Mystery / Romance / Thriller
5 wins&5 nominations
Detective Chief Inspector Jane Tennison now deals with a racially charged murder. The long-dead body of a young black woman is discovered in a district recently convulsed by police brutality and which now is in the midst of a highly-charged political campaign. Her investigation is hampered by the hostility of the local populace, and the clumsy methods of some of her subordinates and irresponsible journalists make things worse. Can she solve the case before a race riot breaks out? Her job is complicated when a former lover is assigned to her... more
Detective Chief Inspector Jane Tennison now deals with a racially charged murder. The long-dead body of a young black woman is discovered in a district recently convulsed by police brutality and which now is in the midst of a highly-charged political campaign. Her investigation is hampered by the hostility of the local populace, and the clumsy methods of some of her subordinates and irresponsible journalists make things worse. Can she solve the case before a race riot breaks out? Her job is complicated when a former lover is assigned to her command as a subordinate.
Episode #1.2
In Series 2, Tennison has proven...
In Series 2, Tennison has proven herself and won her colleagues' grudging respect. When she is assigned to head a murder inquiry in the African Caribbean community, it seems further evidence that her star is on the rise. Suddenly she's in the thick of a racial controversy, working with a black detective who is also her former lover (guest star Colin Salmon, Keen Eddie), and dealing with prejudice in the field and on her team. Tennison begins to wonder if she's been handed a plum case or hung out to dry.
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Prime Suspect 229 November 2011
This is possibly even better than the Prime Suspect 1. DCI Tennison has to find a murderer when the body of an unknown teenage girl is dug up from a suburban garden. The theme of tensions between the Afro-Caribbean community and the Metropolitan Police is added to the mix of Tennisons personal... more
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Q: How to close a window group programmatically in SwiftUI macOS app? Here's the example:
@main
struct ExampleApp: App {
var body: some Scene {
MainScene()
SecondScene()
}
}
struct MainScene: Scene {
var body: some Scene {
WindowGroup(id: "mainscene") {
Text("Main")
Button("close") {
// how to close current scene
}
}
}
}
I don't know how to get current WindowGroup instance, so don't have any idea how to close current window programmatically.
A: Access the current window by using: NSApp.windows.first? or NSApplication.shared.keyWindow?, but NSApplication.shared.keyWindow? will return nil when the window is not active.
struct MainScene: Scene {
var body: some Scene {
WindowGroup(id: "mainscene") {
Button("close") {
NSApplication.shared.keyWindow?.close()
}
.task {
NSApp.windows.first?.close()
}
}
}
}
| {
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Q: Dependency error when running a Spark application that instantiates a NiFi receiver I've been trying to resolve this issue for a while, but I can't seem to find an answer. I am writing a simple Spark application in Scala which instantiates a NiFi receiver, and although it builds successfully with SBT, I receive the following error when I try to run the application using spark-submit:
Exception in thread "main" java.lang.NoClassDefFoundError: org/apache/nifi/spark/NiFiReceiver
at <app name>.main(main.scala)
at sun.reflect.NativeMethodAccessorImpl.invoke0(Native Method)
at sun.reflect.NativeMethodAccessorImpl.invoke(Unknown Source)
at sun.reflect.DelegatingMethodAccessorImpl.invoke(Unknown Source)
at java.lang.reflect.Method.invoke(Unknown Source)
at org.apache.spark.deploy.SparkSubmit$.org$apache$spark$deploy$SparkSubmit$$runMain(SparkSubmit.scala:731)
at org.apache.spark.deploy.SparkSubmit$.doRunMain$1(SparkSubmit.scala:181)
at org.apache.spark.deploy.SparkSubmit$.submit(SparkSubmit.scala:206)
at org.apache.spark.deploy.SparkSubmit$.main(SparkSubmit.scala:121)
at org.apache.spark.deploy.SparkSubmit.main(SparkSubmit.scala)
Caused by: java.lang.ClassNotFoundException: org.apache.nifi.spark.NiFiReceiver
at java.net.URLClassLoader.findClass(Unknown Source)
at java.lang.ClassLoader.loadClass(Unknown Source)
at java.lang.ClassLoader.loadClass(Unknown Source)
... 10 more
I have tried a few variations, but this is my build.sbt file:
name := "<application name here>"
version := "1.0"
scalaVersion := "2.10.5"
libraryDependencies += "org.apache.spark" %% "spark-core" % "1.6.2" % "provided"
libraryDependencies += "org.apache.spark" %% "spark-streaming" % "1.6.2" % "provided"
libraryDependencies += "org.apache.nifi" % "nifi-spark-receiver" % "0.7.0"
libraryDependencies += "org.apache.nifi" % "nifi-site-to-site-client" % "0.7.0"
It should be noted that if I change the two nifi lines to use the Scala equivalents (i.e. the first percent sign in each line is replaced with two percent signs), I actually receive the following error when I run "sbt package":
[error] (*:update) sbt.ResolveException: unresolved dependency: org.apache.nifi#nifi-spark-receiver_2.10;0.7.0: not found
[error] unresolved dependency: org.apache.nifi#nifi-site-to-site-client_2.10;0.7.0: not found
As I mentioned before, with the single percentage signs (and therefore using the Java dependencies) I get no error on build, but I do at runtime.
The relevant part of my application (with certain names removed) is as follows:
import org.apache.spark.SparkContext
import org.apache.spark.SparkContext._
import org.apache.spark.SparkConf
import java.time
import java.time._
import org.apache.nifi._
import java.nio.charset._
import org.apache.nifi.spark._
import org.apache.nifi.remote.client._
import org.apache.spark._
import org.apache.nifi.events._
import org.apache.spark.streaming._
import org.apache.spark.streaming.StreamingContext._
import org.apache.nifi.remote._
import org.apache.nifi.remote.protocol._
import org.apache.spark.streaming.receiver._
import org.apache.spark.storage._
import java.io._
import org.apache.spark.serializer._
object <app name> {
def main(args: Array[String]) {
val nifiUrl = "<nifi url>"
val nifiReceiverConfig = new SiteToSiteClient.Builder()
.url(nifiUrl)
.portName("Data for Spark")
.buildConfig()
val conf = new SparkConf().setAppName("<app name>")
val ssc = new StreamingContext(conf, Seconds(10))
val packetStream = ssc.receiverStream(new NiFiReceiver(nifiReceiverConfig, StorageLevel.MEMORY_ONLY))
The error is referring to the last line here, where the NifiReceiver is instantiated - it can't seem to find that class name anywhere.
I have do far tried a number of approaches including the following (separately):
1) Finding the jar files for nifi-spark-receiver and nifi-site-to-site-client and adding them into a lib directory in my project
2) Following this post https://community.hortonworks.com/articles/12708/nifi-feeding-data-to-spark-streaming.html. I ended up making a copy of spark-default.conf.template in my Spark conf directory, renaming it to spark-defaults.conf and adding the two lines in step 1 at that link into the file (substituting for the actual names and locations of the files in question). I then ensured that I had all the necessary import statements that were used in the two code examples on that page
3) Created a project directory at the root of my application directory, and then created a file called assembly.sbt inside it. I then added the following line inside (as referenced here: https://github.com/sbt/sbt-assembly):
addSbtPlugin("com.eed3si9n" % "sbt-assembly" % "0.14.3")
After that I then ran "sbt assembly" instead of "sbt package" to have the application create an uber jar, but this then failed as well with the same error as when running "sbt package" with the Scala dependencies in the build.sbt file:
[error] (*:update) sbt.ResolveException: unresolved dependency: org.apache.nifi#nifi-spark-receiver_2.10;0.7.0: not found
[error] unresolved dependency: org.apache.nifi#nifi-site-to-site-client_2.10;0.7.0: not found
Please let me know if any further information is required. Thanks in advance for any help.
A: Okay, so I've managed to resolve this, and here is the answer for anyone who may be interested:
The answer was to go back down the uber-jar route and use "sbt assembly" instead of "sbt package" to include the necessary dependency jars in my uber-jar.
1) Create a directory called "project" under the root and place a file called assembly.sbt in there containing the following (the addition here from my original attempt is the resolvers line):
resolvers += Resolver.url("sbt-plugin-releases-scalasbt", url("http://repo.scala-sbt.org/scalasbt/sbt-plugin-releases/"))
addSbtPlugin("com.eed3si9n" % "sbt-assembly" % "0.14.3")
2) In the build.sbt file at the root of the project, use following dependency references (i.e. not spark version specific):
libraryDependencies += "org.apache.nifi" % "nifi-spark-receiver" % "0.7.0"
libraryDependencies += "org.apache.nifi" % "nifi-site-to-site-client" % "0.7.0"
I also marked spark-core and spark-streaming as "provided", i.e.
libraryDependencies += "org.apache.spark" %% "spark-core" % "1.6.2" % "provided"
libraryDependencies += "org.apache.spark" %% "spark-streaming" % "1.6.2" % "provided"
This means you'll need to provide spark separately, but it stops making the uber-jar even larger.
3) In the same file, add the following code to deal with merges when pulling the dependencies in (this is important):
mergeStrategy in assembly <<= (mergeStrategy in assembly) { (old) =>
{
case PathList("META-INF", xs @ _*) => MergeStrategy.discard
case x => MergeStrategy.first
}
}
4) Ensure that the relevant import statements are present in your scala file, e.g.
import org.apache.nifi._
import org.apache.nifi.spark._
etc.
Then when you run "sbt assembly" it should build successfully - just reference this jar when calling "spark-submit", i.e.
spark-submit --class "<application class name>" --master "<local or url>" "<path to uber-jar from project root directory>"
It should be noted that the following post was a massive help in finding this solution: java.lang.NoClassDefFoundError: org/apache/spark/streaming/twitter/TwitterUtils$ while running TwitterPopularTags
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,914 |
\section{Introduction}
Distributed information processing and distributed representations were proposed in the 1980s for solving optimization and factorization problems. For example, associative networks~\cite{hopfield1982neural} can solve optimization problems like the traveling salesman problem~\cite{hopfield1985neural}.
This research direction continues to draw interest for many reasons.
First, it can provide insights into how biological neural circuits solve optimization problems occurring in cognitive tasks.
Second, distributed representations are well-suited for implementation in unconventional parallel hardware~\cite{JaegerComputing2020, MoughanHDParallel2021}, such as neuromorphic hardware~\cite{TrueNorth14, Loihi18, NNkNN20}, potentially providing scalable and low-energy solutions to challenging computing problems~\cite{DaviesAdvancingLoihi2021}.
In this paper, we use a framework for forming structure-sensitive distributed representations that can flexibly encode compositional data structures~\cite{RachkovskijStructures2001} known as Vector Symbolic Architectures (VSA) a.k.a. Hyperdimensional Computing~\cite{KanervaHyperdimensional2009,GaylerJackendoff2003, FradySDR2020}.
In VSAs, the essential operation for forming distributed representations of compositional data structures is the binding operation~\cite{KleykoComputingParadigm2021}.
However, many VSA data structures require parsing, which amounts to a challenging combinatorial factorization problem that must be solved.
Recently, resonator networks~\cite{FradyResonator2020} were proposed that can efficiently factor compositional representations into their constituents \cite{KentResonatorNetworks2020}.
Here, we demonstrate how the VSA technique \emph{fractional power encoding} (FPE) \cite{PlateNested1994, FradyFunctions2021, FradyFunctionsNICE2022} can be used to represent integers as vectors and we show how the problem of factorizing integers can be expressed as the problem of factorizing vectors. We then explain how resonator networks can be extended to solve prime factorization of integers, and we measure the performance and scaling of our method.
\section{Methods}
\label{sect:concepts}
\subsection{Vector Symbolic Architectures and Fourier Holographic Reduced Representations}
\label{sect:supp:vsa}
First in this section, we provide a brief overview of the required components from VSAs~\cite{KleykoComputingParadigm2021}. Please consult~\cite{KleykoSurveyVSA2021Part1,KleykoSurveyVSA2021Part2} for a comprehensive survey.
The key components of any VSA model are: a high-dimensional vector space where random vectors are pseudo-orthogonal ($n$ denotes the dimensionality); symbol representations with randomized atomic vectors (a.k.a. hypervectors; bold lowercase letters, e.g., $\mathbf{a}$); item memory for storing atomic hypervectors and performing auto-associative search (matrices indicated by bold uppercase letters, e.g., $\mathbf{A}$).
Here, we are utilizing a version of VSA known as Fourier Holographic Reduced Representations (FHRR)~\cite{PlateNested1994}.
In FHRR~\cite{PlateNested1994, PlateHolographic1995},
the atomic hypervectors are complex-valued random vectors, where each vector component can be considered as an angle (phasor) randomly and independently selected from the uniform distribution over $(0, 2\pi]$ and with magnitude of one.
The similarity measure is
expressed by the normalized inner product between two phasor hypervectors ($\mathbf{a}$ and $\mathbf{b}$) as $
\frac{1}{n} \Re (\mathbf{a}^\dagger \mathbf{b})$,
where $\mathbf{a}^{\dagger}$ is the complex conjugate transpose, and $\Re$ denotes the real part of the inner product.
Each VSA model defines key operations used to manipulate atomic hypervectors. In FHRR, these are:
\emph{binding} (denoted as $\odot$), which is implemented as component-wise multiplication (Hadamard product); \emph{inverse and unbinding}, which in FHRR corresponds to taking the complex conjugate of the vector to unbind (inverse $\overline{\mathbf{a}}$/$\mathbf{a}^{-1}$) and applying the Hadamard product ($\mathbf{b} \odot \overline{\mathbf{a}}$); \emph{superposition}, (a.k.a. bundling, denoted as $+$) which is implemented as component-wise complex addition, possibly followed by some normalization function; and \emph{permutation}, which can be implemented through a convolution operation or permutation (denoted as $\rho$ but we do not use it here).
\subsection{Fractional Power Encoding}
\label{sec:fpe}
In standard VSA methods, randomized atomic vectors act as symbols and can be manipulated like traditional symbolic representations.
However, when sub-symbolic data or continuous values need to be represented (e.g., to solve machine learning problems~\cite{RahimiBiosignal2019, GeClassificationReview2020,KleykoDensityEncoding2020}), it is important to form a similarity-preserving encoding.
Fractional power encoding (FPE) is a method for such a similarity-preserving encoding, originally proposed in~\cite{PlateNested1994} (see Section 5.6) as a generalization of the fractional power vector~\cite{PlateRecurrent1992}.
This approach has recently received renewed interest for representing continuous manifolds, such as location in an environment \cite{KomerNavigation2020}, and has been connected to kernel methods for describing continuous functions \cite{FradyFunctions2021, FradyFunctionsNICE2022}.
The idea behind the fractional power encoding starts with a single atomic hypervector $\mathbf{z}$.
This vector can then be used to represent different integers through self-binding, where each self-binding step creates a new
hypervector that is dissimilar to all the others. For instance, the value of 2 is represented by $\mathbf{z} \odot \mathbf{z}$; 3 is represented by $\mathbf{z} \odot \mathbf{z} \odot \mathbf{z}$, and so on.
This can be expressed as exponentiating the vector (component-wise) with the integer value, i.e.
the hypervector representation of integer $i$ is formed as: $\mathbf{z}(i) = \mathbf{z}^i$.
It was recognized \cite{PlateNested1994} that this exponentiation process can be defined continuously when using complex-valued FHRR vectors. Thus, the same scheme can be easily generalized to encoding any scalar $x$ as: $\mathbf{z}(x) = \mathbf{z}^{\beta x}$,
where we introduced a parameter $\beta$ that controls the similarity-preserving properties of the resulting encoding.
Hypervectors obtained with FPE preserve similarity between nearby values of $x$, while values further away have reduced similarity. The exact shape of this similarity metric defines the similarity kernel, and $\beta$ regulates the width of this similarity kernel.
\begin{comment}
\begin{figure}[tb
\centering
\includegraphics[width=1.0\columnwidth]{img/Example_bandwidth}
\caption{
Similarity kernels for FPE with base vectors sampled from uniform phase distributions.
Panels illustrate the effect of the bandwidth parameter $\beta$ on the shape of the kernel.
Gray and blue curves show the similarity kernels at two different locations $x=6.0$ and $x=10.0$, respectively to demonstrate the fact that the resulting similarity kernels are translation-invariant.
The resulting similarity kernels approximate the sinc kernel.
The reported values are averages computed from $30$ simulation runs; $n=512$.
The bars denote the standard deviations.
}
\label{fig:bandwidth}
\end{figure}
\end{comment}
Another important property of FPE that we leverage is that it defines a systematic relationship between the binding operation and FPEs of scalars.
In particular, binding the hypervectors representing two scalars $x$ and $y$ results in a hypervector representing $x+y$:
\noindent
\begin{equation}
\mathbf{z}(x) \odot \mathbf{z}(y) = \mathbf{z}^{\beta x} \odot \mathbf{z}^{\beta y} = \mathbf{z}^{\beta (x+y)} = \mathbf{z}(x+y).
\label{eq:fpe:sum}
\end{equation}
There is much more to be said about uses of FPE, including the representation of functions, the shape of the similarity kernel, and representations of multi-dimensional numerical data. We kindly refer interested readers to a recent thorough treatment of FPE in~\cite{FradyFunctions2021, FradyFunctionsNICE2022}.
\subsection{Resonator Networks}
\label{sect:fac:rn}
In VSA, the representation of a conjunction of two or more hypervectors (e.g., $\mathbf{a}$, and $\mathbf{b}$) is achieved by binding: $\mathbf{s} = \mathbf{a} \odot \mathbf{b}$.
The resulting hypervector $\mathbf{s}$ is pseudo-orthogonal to the argument hypervectors (factors), and every combination of arguments results in a unique $\mathbf{s}$.
The binding operation is invertible; when
given all but one factor ($\mathbf{b}$), one can simply compute the unknown factor ($\mathbf{a}$) from the bound representation by unbinding:
$\mathbf{s} \odot \overline{\mathbf{b}} = \mathbf{a} \odot \cancel{\mathbf{b} \odot \overline{\mathbf{b}}} = \mathbf{a}$.
However, if none of the factors are given, then while decoding the vector is still feasible, it becomes a combinatorial search problem whose complexity grows exponentially with the number of factors.
For instance, if there are 100 possibilities for factor $\mathbf{a}$ and 100 for $\mathbf{b}$, then the challenge is to search over all 10,000 combinations of two factors.
Recent work~\cite{FradyResonator2020,KentResonatorNetworks2020} proposed an elegant mechanism, called the resonator network, to address the challenge of factorizing $\mathbf{s}$ into its arguments.
To factor the representation from the input hypervector $\mathbf{s}$, the resonator network uses multiple populations, $\hat{\mathbf{a}}(t)$, and $\hat{\mathbf{b}}(t)$, each of which tries to infer a particular factor from the input hypervector.
Each factor that is a possibility is stored in a separate factor item memory ($\mathbf{A}$, $\mathbf{B}$).
Each population, called a resonator, communicates with the input hypervector and the other populations using the following dynamics:
\noindent
\begin{equation}
\begin{split}
&\mathbf{\hat{a}}(t+1)= f_n \Big( \mathbf{A} \Re \Big( \mathbf{A}^{\dagger} (\mathbf{s} \odot \overline{\mathbf{\hat{b}}}(t)) \Big) \Big); \\
&\mathbf{\hat{b}}(t+1)= f_n \Big( \mathbf{B} \Re \Big( {\mathbf{B}}^{\dagger} (\mathbf{s} \odot \overline{\mathbf{\hat{a}}}(t)) \Big) \Big).
\end{split}
\label{eq:resnet:text}
\end{equation}
\noindent
\begin{figure*}[t
\centering
\includegraphics[width=2.0\columnwidth]{img/Example_superposition_upd}
\caption{
An example of varying behavior of the superposition of FPEs corresponding to a set of scalars: $\{\log(2), \log(3), \log(5), \log(11)\}$; $y$-axis depicts the average cosine similarity (thick lines) between the superposition hypervector and FPEs of scalars in $x$-axis.
Thin vertical lines correspond to the locations of the elements of the considered set.
The corresponding panels show cosine similarities for FPEs formed with different values of $\beta$.
The reported values are averages computed from $30$ simulation runs; $n=512$.
}
\label{fig:superposition}
\end{figure*}
The process is iterative and progresses in discrete time steps, $t$.
In essence at time $t$, each population can hold multiple weighted estimates for one of the factors through the VSA principle of superposition~\cite{FradyCapacity2018, KleykoPerceptron2020}.
This allows a population to test multiple guesses for factor identity simultaneously.
Each resonator uses the current estimates from the other populations to invert the input hypervector and infer the factor of interest.\footnote{
Note that in~(\ref{eq:resnet:text}) the update is synchronous, i.e., estimates at $t+1$ are based on the estimates from the previous $t$th iteration.
It is possible to update estimates asynchronously.
We use this asynchronous mode to perform the evaluation (see Section~\ref{fact:setup}).
}
The cost of superposition is crosstalk noise, making the inference step noisy when many estimates are tested at once.
Therefore, the next step uses the factor item memory to remove the extraneous estimates.
The estimate
for each factor is \emph{cleaned up} by constraining the resonator activity only to the allowed atomic hypervectors stored in the corresponding factor item memory.
Finally, a regularization step (denoted as $f_n(*)$) limiting the values of components of new estimates is needed.
Successive iterations of this inference and clean up procedure (\ref{eq:resnet:text}) eliminate the noise as the factors become identified and find their place in the input vector.
When the factors are fully identified, the resonator network reaches a stable equilibrium, and the factors can be deduced from the stable activity pattern.
\section{Results}
\subsection{Logarithmic FPE for encoding integers}
\label{fact:setup}
We will use the problem of factorizing semiprimes (composite integers with exactly 2 prime factors) to demonstrate how FPE and resonator networks can interact together to solve this problem.
A semiprime $s$ is obtained simply by multiplying two primes $x$ and $y$:
\noindent
\begin{equation}
\label{eq:semiprime}
s=xy.
\end{equation}
\noindent
We use $\mathcal{P}(s)$ to denote the set of all primes that are potential factors of $s$ (since the minimum prime factor is two, all primes less than or equal to $s/2$ are possible factors). The $i$th element of the set is denoted as $\mathcal{P}(s)_i$.
When setting up the factorization search, the set $\mathcal{P}(s)$ should be known.
The set is used to form the item memory $\mathbf{\Phi}$ containing hypervectors for all primes in $\mathcal{P}(s)$.
The hypervector of $\mathcal{P}(s)_i$ (denoted as $\mathbf{\Phi}_i$) is formed with FPE (see Section~\ref{sec:fpe}) as:
\noindent
\begin{equation}
\mathbf{\Phi}_i = \mathbf{z}(\log(\mathcal{P}(s)_i)) = \mathbf{z}^{\beta \log(\mathcal{P}(s)_i)}.
\label{eq:fpe:codebook}
\end{equation}
\noindent
Critically, the combination of the properties of the log transformation and FPE (see (\ref{eq:fpe:sum})) results in the following behavior of hypervectors: \noindent
\begin{equation}
\begin{split}
& \mathbf{z}(\log(s)) = \mathbf{z}^{\beta \log(s)} = \mathbf{z}^{\beta \log(xy)}
= \mathbf{z}^{\beta (\log(x) + \log(y) )}\\
& = \mathbf{z}^{\beta \log(x)} \odot \mathbf{z}^{\beta \log(y)}
= \mathbf{z}(\log(x)) \odot \mathbf{z}(\log(y)).
\end{split}
\end{equation}
\noindent
In other words, the log transformation combined with FPE results in the binding between vectors being equivalent to the FPE vector of the product.
This allows one to express the problem of semiprime factorization in terms of vector factorization, which can be solved through the resonator network formulation as in Section~\ref{sect:fac:rn}, where each resonator uses the codebook $\mathbf{\Phi}$.
\subsection{FPEs in superposition}
\label{sec:fpe:pecul}
Traditionally in VSA, the most straightforward use of the superposition operation is to represent a set of elements~\cite{KanervaHyperdimensional2009,KleykoABF2020}.
As with any other VSA representation, FPEs can also be used with the superposition operation to, e.g., represent a set of scalar values.
However, due to the similarity-preserving properties of FPEs, the hypervector resulting from the superposition of several FPEs might exhibit counter-intuitive ``hybrid'' behavior, which is neither fully symbolic nor fully subsymbolic.
When utilizing the resonator network to solve the factorization problem, we need to account for the behavior of FPE superpositions.
Consider an example of the following set: \\ $\{\log(2)$, $\log(3)$, $\log(5)$, $\log(11)\}$, where the hypervector $\mathbf{s}$ representing the set is formed using the FPEs corresponding to the elements of the set as:
\noindent
\begin{equation}
\label{eq:set}
\mathbf{s} = \mathbf{z}^{\beta \log(2)} + \mathbf{z}^{\beta \log(3)} +\mathbf{z}^{\beta \log(5)} +\mathbf{z}^{\beta \log(11)}.
\end{equation}
Fig.~\ref{fig:superposition} presents the average cosine similarity between $\mathbf{s}$ and the corresponding FPEs of scalars along the $x$-axis for different values of $\beta$.
The first thing to notice in Fig.~\ref{fig:superposition} is that the choice of $\beta$ profoundly affects the obtained similarity distributions.
When $\beta=2.1$, FPEs of $\{\log(2), \log(3), \log(5)\}$ are similar to each other, and so when superimposed together, they interact in such a way that there is one large peak of similarity near their mean.
Increasing $\beta$ to $3.1$ splits the large peak into two: one for $\log(5)$ and one in between $\log(2)$ and $\log(3)$.
This is intuitive since $\log(2)$ and $\log(3)$ are closer to each other than $\log(3)$ and $\log(5)$.
Finally, once $\beta$ is large enough (e.g., $\beta=5.0$) all four peaks become clearly distinct and they correspond to the values of the elements in the set.
\begin{figure*}[t
\begin{minipage}[h]{0.35\linewidth}
\center{\includegraphics[width=1.0\linewidth]{img/resonator_network} \\ A}
\end{minipage}
\hfill
\begin{minipage}[h]{0.64\linewidth}
\center{\includegraphics[width=1.0\linewidth]{img/Waterfall} \\ B}
\end{minipage}
\caption{
Left panel (A): an example of a resonator network with two factors for integer factorization according to~(\ref{eq:resnet:semiprimes}).
Each resonator uses the estimate from the other resonator to infer one of the factors (e.g. $\mathbf{z}(\log(s)) \odot \overline{\hat{\mathbf{y}}}$), these estimates are then cleaned-up by limiting them to the span of the codebook ($\mathbf{\Phi} \Re \mathbf{\Phi}^\dagger$), and finally the vector elements are restored to unit magnitude phasors ($f_n(x_i) = x_i/|x_i|$).
Right panel (B): an example of convergence of a resonator network. Color values correspond to the normalized inner product between $\mathbf{\hat{x}}$ (Factor \# 1) and $\mathbf{\hat{y}}$ (Factor \# 2) and the entries of the codebook $\mathbf{\Phi}$ corresponding to the primes depicted on $x$-axis. A small range of primes are shown for visualization. The dynamics are initially very chaotic until around iteration 30 where the network identifies the solution and quickly reaches a stable equilibrium.
}
\label{fig:resonator}
\end{figure*}
Thus, setting the value of $\beta$ allows traversing between two extremes: when $\beta$ is very small, FPEs of scalars that are far away from each other are still very similar (subsymbolic behavior) while when $\beta$ is very large, FPEs of scalars that are near each other are dissimilar (symbolic behavior).
In other contexts, we expect that different applications might favor different modes.
For example, when working with clustering problems~\cite{BandaragodaTrajectoryTraffic2019, ImaniHDCluster2019,KleykoBoostingSOM2019, HernandezClustering2021, OsipovHyperSeed2021}, there is potential that subsymbolic merging would be useful for generating centroids and computing means.
On the other hand, for integer factorization, it is important to choose $\beta$ such that we only operate in the symbolic mode, as we desire that each different value is treated as a distinct alternative within the solution space of the factorization problem.
Practically, this means we have chosen $\beta$ to be large so that the inner product between the hypervectors of any two adjacent primes would be close to zero.
This was achieved by setting $\beta$ as :
\begin{equation}
\beta=\frac{10^4}{\min_i(\log(\mathcal{P}(s)_{i+1}) - \log(\mathcal{P}(s)_{i})) }
\label{eqn:beta}
\end{equation}
The extra factor of $10^{4}$ was added to ensure that $\beta$ was always sufficiently large.
\subsection{Factorization of semiprimes with the resonator network}
In this case, the dynamics of the resonator network factorizing $s$ into $x$ and $y$ is described as follows:
\noindent
\begin{equation}
\begin{split}
&\mathbf{\hat{x}}(t+1)= f_n \Big( \mathbf{\Phi} \Re \Big( {\mathbf{\Phi}}^{\dagger} (\mathbf{z}(\log(s)) \odot \overline{\mathbf{\hat{y}}}(t)) \Big) \Big); \\
&\mathbf{\hat{y}}(t+1)= f_n \Big( \mathbf{\Phi} \Re \Big( {\mathbf{\Phi}}^{\dagger} (\mathbf{z}(\log(s)) \odot \overline{\mathbf{\hat{x}}}(t+1) ) \Big) \Big),
\end{split}
\label{eq:resnet:semiprimes}
\end{equation}
\noindent
where $\hat{\mathbf{x}}(t)$ and $\hat{\mathbf{y}}(t)$ denote the hypervectors corresponding to the current estimates of the resonator network for $\mathbf{z}(\log(x))$ and $\mathbf{z}(\log(y))$;
$f_n(x_i) = x_i/|x_i|$ normalizes each component to unit magnitude.
Note that in (\ref{eq:resnet:semiprimes}) both factors use the same item memory $\mathbf{\Phi}$ since both $x$ and $y$ are present in $\mathcal{P}(s)$.
Once the resonator network converges or reaches the maximum number of iterations, the most recent estimates of the resonator network are used to obtain the predictions $\hat{x}$ and $\hat{y}$ corresponding to the primes in $\mathcal{P}(s)$ whose hypervectors in $\mathbf{\Phi}$ are the most similar to $\hat{\mathbf{x}}(t)$ and $\hat{\mathbf{y}}(t)$.
In the experiments, the maximum number of iterations was $100$.
A schematic overview of the resonator is shown in Fig.~\ref{fig:resonator}.
Let us walk-through an example of the factorization process described above.
Assume that we would like to factorize semiprime $s=603,329$ into its factors ($x=757$ and $y=797$).
First, we need to define all the primes in $\mathcal{P}(s)$ that will be used to form the item memory $\mathbf{\Phi}$.
In this case, the cardinality of $\mathcal{P}(s)$ will be $|\mathcal{P}(s)|=26,135$ with the smallest prime being $2$ and the largest one being $301,657$.
Once $\mathcal{P}(s)$ is fixed, we can use our equation for $\beta$ (\ref{eqn:beta}) to calculate $\beta\approx\num{1.5e9}$.
Next, we need to choose a suitable dimensionality of hypervectors $n$ (see the next section for performance evalutation of different values of $n$).
Then, a random $n$-dimensional base vector $\mathbf{z}$ is generated.
The base vector $\mathbf{z}$ is used to populate the item memory $\mathbf{\Phi}$ with hypervectors corresponding to FPEs of logarithms of elements of $\mathcal{P}(s)$ according to (\ref{eq:fpe:codebook}).
We also form the FPE of the given semiprime $s$ as $\mathbf{z}(\log(s)) = \mathbf{z}^{\beta \log(s)}$.
The final step is to setup and run the resonator network according to~(\ref{eq:resnet:semiprimes}) using the obtained $\mathbf{z}(\log(s))$ and $\mathbf{\Phi}$.
The initial estimates for $\hat{\mathbf{x}}(0)$ and $\hat{\mathbf{y}}(0)$ are set to the normalized superposition of all hypervectors in $\mathbf{\Phi}$.
If $n$ is large enough, then after several iterations with high probability the resonator network will converge. The final state of $\hat{\mathbf{x}}(t)$ and $\hat{\mathbf{y}}(t)$ can be matched to the closest hypervectors in $\mathbf{\Phi}$, which will correspond to primes $757$ and $797$ (Fig. \ref{fig:resonator}). Note that in this configuration, $\mathbf{\hat{x}}$ and $\mathbf{\hat{y}}$ can converge to either one of the primes.
\begin{figure*}[tb
\centering
\includegraphics[width=1.99\columnwidth]{img/Factorization_fixed_dim_scaling}
\caption{
Left panel (A): the accuracy of semiprimes factorization against the cardinality of $\mathcal{P}(s)$.
The reported values are averages computed from $4,000$ random semiprimes.
Central panel (B):
average minimal dimensionality of hypervectors ($y$-axis) required to achieve at least 95\% successful factorization for the given cardinality of $\mathcal{P}(s)$ ($x$-axis).
Thin black dashed line depicts the linear relation between $n$ and $|\mathcal{P}(s)|$.
The colored lines correspond to different starting points used to form $\mathcal{P}(s)$.
The reported values are averages computed from $10$ simulation runs.
During each simulation run, $1,000$ randomly chosen semiprimes were used to assess the factorization accuracy for every considered value of $n$.
Right panel (C): number of iterations used by the resonator network to either converge to a solution or to reach the maximum number of iterations (set to $100$).
}
\label{fig:fact:fixed:dim}
\end{figure*}
\subsection{Empirical evaluation of performance and scaling }
\label{sect:empirical}
In this section, we report the results of the empirical evaluation of the proposed approach.
In the experiments below, we need to measure the success of the factorization by the average accuracy of correctly factorizing many semiprimes $s=xy$ (where $x$ and $y$ are chosen randomly from $\mathcal{P}(s)$).
First, we examine the factorization accuracy against the number of elements in $\mathcal{P}(s)$ and as a function of dimensionality, reported in left panel in Fig.~\ref{fig:fact:fixed:dim}.
For each value of $n$, we can identify three regimes with respect to the accuracy:
high-fidelity, where the accuracy is nearly perfect, low-fidelity, where the accuracy is not perfect but above chance, and random guessing, where the accuracy is effectively chance.
It is evident that with increased $n$, the maximum size of $\mathcal{P}(s)$ within the high-fidelity regime also increased.
It is also important to estimate how the complexity of the approach scales. To do so, we consider the dimensionality of hypervectors required to perform the factorization successfully for the given cardinality of $\mathcal{P}(s)$.
We have defined the successful factorization as the accuracy that is greater than or equal to $0.95$.
The scaling of the required dimensionality of hypervectors converges to a line with slope of approximately 1 with respect to cardinality of $\mathcal{P}(s)$ (central panel in Fig. \ref{fig:fact:fixed:dim}).
This observation is in line with the experiments reported in~\cite{KentResonatorNetworks2020}, where random hypervectors were used to form an abstract factorization problem.
Practically, this also means that for large $|\mathcal{P}(s)|$, reasonable values of $n$ would be sufficient to perform the factorization.
Recall that in Section~\ref{fact:setup}, we discussed the choice of suitable $\beta$ for a given $\mathcal{P}(s)$ to continue operating in the symbolic mode.
Intuitively, the potential issue with scaling $\beta$ is that when $\mathcal{P}(s)$ contains large primes, $\beta$ will also be large due to the use of the log transformation. Large values of $\beta$ could cause numerical issues when performing the FPE. In order to demonstrate the potential role of $\beta$ on the factorization performance, Fig.~\ref{fig:fact:fixed:dim}B also depicts the required dimensionalities of hypervectors for primes in $\mathcal{P}(s)$ that were picked using different starting points, which following (\ref{eqn:beta}) leads to different values for $\beta$.
First, the results suggest that the ranges of primes requiring the use of larger values of $\beta$ did not incur a drastic increase in dimensionality of hypervectors, so factorization performance is mainly limited by the capacity of the resonator.
Second, since larger values of $\beta$ are not an issue, the proposed approach can handle varying ranges of primes.
\begin{figure*}[tb
\centering
\includegraphics[width=1.99\columnwidth]{img/Factorization_fixed_dim_3Factors}
\caption{
Left panel: the accuracy of $3$-almost primes factorization against the cardinality of $\mathcal{P}(s)$.
Right panel: number of iterations used by the resonator network to either converge to a solution or to reach the maximum number of iterations (set to $100$).
The values are averages from $1,000$ randomly chosen $3$-almost primes.
}
\label{fig:fact:fixed:dim:3Factors}
\end{figure*}
In addition to the factorization accuracy, it is also worth looking at the average number of iterations used by the resonator network to converge (right panel in Fig.~\ref{fig:fact:fixed:dim}).
There is a clear correspondence between the accuracy and the number of iterations of the resonator network.
When the resonator network was in the high-fidelity regime, only a few iterations were required to find a solution, and $|\mathcal{P}(s)|$ increased the number of iterations also increased.
Finally, once $|\mathcal{P}(s)|$ was too large for a chosen $n$, the resonator converged to the wrong answer or reached the iteration limit.
\begin{comment}
\begin{figure}[tb
\centering
\includegraphics[width=1.0\columnwidth]{img/Required_Dim_2_factors}
\caption{
Average minimal dimensionality of hypervectors ($y$-axis) required to achieve at least 95\% successful factorization for the given cardinality of $\mathcal{P}(s)$ ($x$-axis); $|\mathcal{P}(s)| \in~2^{[3:0.5:14.5]}$.
Thin black dashed line depicts the linear relation between $n$ and $|\mathcal{P}(s)|$.
The colored lines correspond to different starting points used to form $\mathcal{P}(s)$.
The reported values are averages computed from $10$ simulation runs.
During each simulation run, $1,000$ randomly chosen semiprimes were used to assess the factorization accuracy for every considered value of $n$.
}
\label{fig:fact:scaling}
\end{figure}
\end{comment}
\subsection{Factoring composite numbers beyond semiprimes}
The proposed approach is not limited to semiprimes.
In principle, it can be applied on any $k$-almost prime.
In Fig.~\ref{fig:fact:fixed:dim:3Factors}, we report the case of factorization of integers with three factors ($s=abc$) using a similar setup as above. Now there are three resonators, and each resonator is designed for three factors. They each have a similar update dynamics, for instance for the first factor:
\begin{equation}
\mathbf{\hat{a}}(t+1)= f_n \Big( \mathbf{\Phi} \Re \Big( {\mathbf{\Phi}}^{\dagger} (\mathbf{z}(\log(s)) \odot \overline{\mathbf{\hat{b}}}(t)) \odot \overline{\mathbf{\hat{c}}}(t) \Big) \Big).
\end{equation}
The observed results are consistent with that obtained for the case of semiprimes in Fig.~\ref{fig:fact:fixed:dim}.
Note that compared to the case of semiprimes, for $3$-almost primes larger values of $n$ were required to get to the high-fidelity regime. This is expected since for the semiprimes the search space grows as $|\mathcal{P}(s)|^2$ while for $3$-almost primes it grows much faster as $|\mathcal{P}(s)|^3$.
The approach can be extended to other composite numbers by including more resonators in the network. Again, the capacity and likelihood of solving the factorization problem depend on the combinatorics of the factors, and this grows exponentially with number of factors. Further, the identity vector (vector of all $1$s, a.k.a. $\mathbf{z}(\log(1)) = \mathbf{z}^{0}$) can be added to $\mathbf{\Phi}$ to enable solving problems with unknown number of factors.
\section{Discussion}
\label{sect:discussion}
\subsection{Summary of the study}
Our goal was to demonstrate that while Vector Symbolic Architectures (VSA)~\cite{KanervaHyperdimensional2009, KleykoComputingParadigm2021} were originally proposed to solve problems in cognition, VSAs are a highly flexible and powerful framework for expressing challenging computational problems in high-dimensional vector spaces.
We used integer factorization to showcase both the expressiveness of VSAs and novel techniques of representing numbers in high-dimensional vectors.
VSAs are now well-known as frameworks for many novel computational devices that are designed for highly efficient and parallel computations \cite{rahimi2007random,RahimiNanoscalable2017,DaviesAdvancingLoihi2021}.
Using VSAs to express challenging computational problems that can be solved by neural network architectures, like the resonator network, brings out the potential of utilizing neuromorphic hardware. It is relatively straight-forward to scale VSA algorithms like the resonator network -- this simply means expanding the dimensionality of the vector representations. This ease of scalability is compatible with large scale meshes of neuromorphic chips \cite{NNkNN20}. While we did not execute these experiments on neuromorphic hardware, there are several previous models \cite{EliasmithSPAUN2012} and novel proposals \cite{FradyTPAM2019, FradySDR2020} that neuromorphically perform VSA computations.
One of our main contributions was expanding our understanding of how number systems can be expressed in vector spaces.
Recently the technique of fractional power encoding (FPE) has been gaining new attention as a way to represent geometrical spaces, maps, manifolds and functions~\cite{PlateNested1994,KomerContinuous2019, FradyFunctions2021, FradyFunctionsNICE2022}.
Integers are the ordinal data type, and, therefore, their distributed representation should preserve the data topology, which provides a proper setup for the use of fractional power encoding.
Previously integers were easily represented by the integer powers of an FPE, but in this formulation the binding operation results in the FPE of the integer sum.
By expressing the FPE with the logarithms of integers, we enabled a representation of integer values where the binding operation now leads to the FPE of the integer product.
With this formulation for representing integers, we could then express the integer factorization problem as the problem of vector factorization, which can be solved using resonator networks \cite{FradyResonator2020}.
This FPE representation of logarithmic integers meant we needed to examine the consequences of superpositions of FPE vectors. The resonator network uses the principle of \emph{search in superposition}, and for it to successfully solve the factorization problem, the individual factors need to be uniquely identifiable by their FPE vectors.
We used the parameter $\beta$ to rescale the FPE vectors so that the logarithmic FPE representations were sufficiently spaced such that their similarity kernels were not overlapping.
There is a simple strategy for scaling $\beta$ based on the minimum log distance between neighboring primes, but we also showed that there are few side-effects for making $\beta$ much larger, and generally factorization performance is not too dependent on $\beta$ as long as it is sufficiently large.
The way the factorization problem is solved by the resonator network is best described via the concept of \emph{search in superposition}. During this process, many number combinations may be considered simultaneously, something that is not possible with conventional digital number representations.
We believe that the extended idea of \emph{computing in superposition} ~\cite{KleykoComputingParadigm2021} is a particularly important aspect of VSAs that should be investigated further.
It is also worth emphasizing, that the resonator network can be used beyond semiprimes (i.e., with more than two factors; see Section~\ref{sect:empirical}).
\subsection{Related work}
\label{sect:disc:realted}
Our main contribution is a formalism for using VSAs to solve integer factorization, providing a way to solve classical factorization problems with distributed representations.
Though our formalism benefits from the properties of VSAs (such as being distributed, robust, and computing in superposition), it was not our goal to demonstrate the superiority to other algorithms.
The promise of our approach is really the potential of scalable and efficient execution of such an algorithm on neuromorphic hardware.
It is important to keep in mind that the proposed approach should not be considered as a panacea in terms of providing a straight-forward polynomial solution to the semiprimes factorization problem.
The number of primes grows exponentially w.r.t. the number of bits used to represent a number, which also implies the exponential growth of the resonator.
This work expands on previous efforts to solve optimization and factorization problems with neural attractor networks and physics-based architectures.
Early work proposed to design neural attractor networks based on matrix-type auto-associative memories~\cite{palm1980associative, hopfield1982neural, FrolovWillshaw2002, FrolovTime2006, knoblauch2010memory} so that their dynamics is governed by a Lyapunov function that represents the objective of a particular optimization problem, for example, the path length in a traveling-salesman problem~\cite{hopfield1985neural}. The fixed point attractor dynamics of such networks searches the solution space and settles at an approximate solution.
The resonator network similarly settles at a solution if there is one. However, its dynamics is not governed by an energy landscape/Hamiltonian, if there is no solution, it can converge to limit cycles or chaotic orbits. It has been shown empirically that this richer dynamic repertoire accelerates the search and, as a result, outperforms gradient-based optimization significantly \cite{KentResonatorNetworks2020}.
Quantum computing has been proposed as a physics-based method for solving combinatorial optimization problems~\cite{apolloni1989quantum, shor1999polynomial}.
There are some similarities between quantum algorithms and the working principles of the resonator network used in this paper.
The resonator network operates on sums of complex numbers to solve factorization problems, which can be regarded a classical analog of the superposition principle used in quantum computing.
Adiabatic quantum annealing~\cite{apolloni1989quantum} methods solve an optimization problem by using the quantum tunneling effect. The optimization problem can be mapped to the Hamiltonian of a quantum system. The optimal solution or global minimum of the problem objective is found by slowly evolving the potential from an initial easy Hamiltonian to a more complicated Hamiltonian.
Due to the challenge of building reliable and large quantum computers, there is a renewed interest in classical physics-based solvers, such as networks of coupled oscillators~\cite{wang2019oim, ahmed2021probabilistic}.
The variables in the described resonator networks are complex-valued phasors, which can be represented by oscillators or by spiking neural networks~\cite{FradyTPAM2019}.
\subsection{Future work}
\label{sect:disc:future}
There are several extensions of our approach, including subsequent analyses, that seem especially promising:
While we presented the algorithmic approach and its realization on the conventional parallel hardware (GPU), the real promise is in the implementing VSAs in neuromorphic hardware.
To this date, it is still an open question how to implement a VSA system in such hardware in full.
Probably the closest mapping to spiking hardware is provided via the Neural Engineering Framework~\cite{EliasmithNeural2003, BekolayNengo2014}, although the potential challenge of this approach is spike efficiency.
An alternative mapping proposal that is spike efficient is via representing FHRR phasors with spike times~\cite{FradyTPAM2019}, but this approach is not yet extended to account for all VSA operations.
Another promising hardware direction is in-memory computing~\cite{KarunaratneInMemory2020, KarunaratneHDAugmented2021}.
Since the above hardware is inherently noisy, it would provide a natural setup for demonstrating the robustness of our proposed factorization approach to noise (as expected from simulations performed in \cite{KentResonatorNetworks2020}).
Another important direction is to design mapping of other difficult problems, such as the subset-sum problem and other combinatorial optimization problems, to resonator networks with FHRR.
A particular challenge for designing mappings for other problems is the absence of strict guidelines directing mapping development, so for each problem the mapping has to be done ad hoc.
Another limitation is that although resonator networks are well-suited for finding exact solutions (i.e., solving equality problems), it is less obvious how to formulate the problem of finding a maximum or minimum.
\begin{acks}
FTS, BAO, CB, and DK were supported by Intel's THWAI.
BAO and DK were supported by AFOSR FA9550-19-1-0241.
DK was supported by the MSCA Fellowship (grant 839179).
CJK was supported by the DoD through the NDSEG Fellowship.
FTS was supported by Intel and NIH R01-EB026955.
\end{acks}
\bibliographystyle{ACM-Reference-Format}
\section{Introduction}
ACM's consolidated article template, introduced in 2017, provides a
consistent \LaTeX\ style for use across ACM publications, and
incorporates accessibility and metadata-extraction functionality
necessary for future Digital Library endeavors. Numerous ACM and
SIG-specific \LaTeX\ templates have been examined, and their unique
features incorporated into this single new template.
If you are new to publishing with ACM, this document is a valuable
guide to the process of preparing your work for publication. If you
have published with ACM before, this document provides insight and
instruction into more recent changes to the article template.
The ``\verb|acmart|'' document class can be used to prepare articles
for any ACM publication --- conference or journal, and for any stage
of publication, from review to final ``camera-ready'' copy, to the
author's own version, with {\itshape very} few changes to the source.
\section{Template Overview}
As noted in the introduction, the ``\verb|acmart|'' document class can
be used to prepare many different kinds of documentation --- a
double-blind initial submission of a full-length technical paper, a
two-page SIGGRAPH Emerging Technologies abstract, a ``camera-ready''
journal article, a SIGCHI Extended Abstract, and more --- all by
selecting the appropriate {\itshape template style} and {\itshape
template parameters}.
This document will explain the major features of the document
class. For further information, the {\itshape \LaTeX\ User's Guide} is
available from
\url{https://www.acm.org/publications/proceedings-template}.
\subsection{Template Styles}
The primary parameter given to the ``\verb|acmart|'' document class is
the {\itshape template style} which corresponds to the kind of publication
or SIG publishing the work. This parameter is enclosed in square
brackets and is a part of the {\verb|documentclass|} command:
\begin{verbatim}
\documentclass[STYLE]{acmart}
\end{verbatim}
Journals use one of three template styles. All but three ACM journals
use the {\verb|acmsmall|} template style:
\begin{itemize}
\item {\verb|acmsmall|}: The default journal template style.
\item {\verb|acmlarge|}: Used by JOCCH and TAP.
\item {\verb|acmtog|}: Used by TOG.
\end{itemize}
The majority of conference proceedings documentation will use the {\verb|acmconf|} template style.
\begin{itemize}
\item {\verb|acmconf|}: The default proceedings template style.
\item{\verb|sigchi|}: Used for SIGCHI conference articles.
\item{\verb|sigchi-a|}: Used for SIGCHI ``Extended Abstract'' articles.
\item{\verb|sigplan|}: Used for SIGPLAN conference articles.
\end{itemize}
\subsection{Template Parameters}
In addition to specifying the {\itshape template style} to be used in
formatting your work, there are a number of {\itshape template parameters}
which modify some part of the applied template style. A complete list
of these parameters can be found in the {\itshape \LaTeX\ User's Guide.}
Frequently-used parameters, or combinations of parameters, include:
\begin{itemize}
\item {\verb|anonymous,review|}: Suitable for a ``double-blind''
conference submission. Anonymizes the work and includes line
numbers. Use with the \verb|\acmSubmissionID| command to print the
submission's unique ID on each page of the work.
\item{\verb|authorversion|}: Produces a version of the work suitable
for posting by the author.
\item{\verb|screen|}: Produces colored hyperlinks.
\end{itemize}
This document uses the following string as the first command in the
source file:
\begin{verbatim}
\documentclass[sigconf]{acmart}
\end{verbatim}
\section{Modifications}
Modifying the template --- including but not limited to: adjusting
margins, typeface sizes, line spacing, paragraph and list definitions,
and the use of the \verb|\vspace| command to manually adjust the
vertical spacing between elements of your work --- is not allowed.
{\bfseries Your document will be returned to you for revision if
modifications are discovered.}
\section{Typefaces}
The ``\verb|acmart|'' document class requires the use of the
``Libertine'' typeface family. Your \TeX\ installation should include
this set of packages. Please do not substitute other typefaces. The
``\verb|lmodern|'' and ``\verb|ltimes|'' packages should not be used,
as they will override the built-in typeface families.
\section{Title Information}
The title of your work should use capital letters appropriately -
\url{https://capitalizemytitle.com/} has useful rules for
capitalization. Use the {\verb|title|} command to define the title of
your work. If your work has a subtitle, define it with the
{\verb|subtitle|} command. Do not insert line breaks in your title.
If your title is lengthy, you must define a short version to be used
in the page headers, to prevent overlapping text. The \verb|title|
command has a ``short title'' parameter:
\begin{verbatim}
\title[short title]{full title}
\end{verbatim}
\section{Authors and Affiliations}
Each author must be defined separately for accurate metadata
identification. Multiple authors may share one affiliation. Authors'
names should not be abbreviated; use full first names wherever
possible. Include authors' e-mail addresses whenever possible.
Grouping authors' names or e-mail addresses, or providing an ``e-mail
alias,'' as shown below, is not acceptable:
\begin{verbatim}
\author{Brooke Aster, David Mehldau}
\email{dave,judy,steve@university.edu}
\email{firstname.lastname@phillips.org}
\end{verbatim}
The \verb|authornote| and \verb|authornotemark| commands allow a note
to apply to multiple authors --- for example, if the first two authors
of an article contributed equally to the work.
If your author list is lengthy, you must define a shortened version of
the list of authors to be used in the page headers, to prevent
overlapping text. The following command should be placed just after
the last \verb|\author{}| definition:
\begin{verbatim}
\renewcommand{\shortauthors}{McCartney, et al.}
\end{verbatim}
Omitting this command will force the use of a concatenated list of all
of the authors' names, which may result in overlapping text in the
page headers.
The article template's documentation, available at
\url{https://www.acm.org/publications/proceedings-template}, has a
complete explanation of these commands and tips for their effective
use.
Note that authors' addresses are mandatory for journal articles.
\section{Rights Information}
Authors of any work published by ACM will need to complete a rights
form. Depending on the kind of work, and the rights management choice
made by the author, this may be copyright transfer, permission,
license, or an OA (open access) agreement.
Regardless of the rights management choice, the author will receive a
copy of the completed rights form once it has been submitted. This
form contains \LaTeX\ commands that must be copied into the source
document. When the document source is compiled, these commands and
their parameters add formatted text to several areas of the final
document:
\begin{itemize}
\item the ``ACM Reference Format'' text on the first page.
\item the ``rights management'' text on the first page.
\item the conference information in the page header(s).
\end{itemize}
Rights information is unique to the work; if you are preparing several
works for an event, make sure to use the correct set of commands with
each of the works.
The ACM Reference Format text is required for all articles over one
page in length, and is optional for one-page articles (abstracts).
\section{CCS Concepts and User-Defined Keywords}
Two elements of the ``acmart'' document class provide powerful
taxonomic tools for you to help readers find your work in an online
search.
The ACM Computing Classification System ---
\url{https://www.acm.org/publications/class-2012} --- is a set of
classifiers and concepts that describe the computing
discipline. Authors can select entries from this classification
system, via \url{https://dl.acm.org/ccs/ccs.cfm}, and generate the
commands to be included in the \LaTeX\ source.
User-defined keywords are a comma-separated list of words and phrases
of the authors' choosing, providing a more flexible way of describing
the research being presented.
CCS concepts and user-defined keywords are required for for all
articles over two pages in length, and are optional for one- and
two-page articles (or abstracts).
\section{Sectioning Commands}
Your work should use standard \LaTeX\ sectioning commands:
\verb|section|, \verb|subsection|, \verb|subsubsection|, and
\verb|paragraph|. They should be numbered; do not remove the numbering
from the commands.
Simulating a sectioning command by setting the first word or words of
a paragraph in boldface or italicized text is {\bfseries not allowed.}
\section{Tables}
The ``\verb|acmart|'' document class includes the ``\verb|booktabs|''
package --- \url{https://ctan.org/pkg/booktabs} --- for preparing
high-quality tables.
Table captions are placed {\itshape above} the table.
Because tables cannot be split across pages, the best placement for
them is typically the top of the page nearest their initial cite. To
ensure this proper ``floating'' placement of tables, use the
environment \textbf{table} to enclose the table's contents and the
table caption. The contents of the table itself must go in the
\textbf{tabular} environment, to be aligned properly in rows and
columns, with the desired horizontal and vertical rules. Again,
detailed instructions on \textbf{tabular} material are found in the
\textit{\LaTeX\ User's Guide}.
Immediately following this sentence is the point at which
Table~\ref{tab:freq} is included in the input file; compare the
placement of the table here with the table in the printed output of
this document.
\begin{table}
\caption{Frequency of Special Characters}
\label{tab:freq}
\begin{tabular}{ccl}
\toprule
Non-English or Math&Frequency&Comments\\
\midrule
\O & 1 in 1,000& For Swedish names\\
$\pi$ & 1 in 5& Common in math\\
\$ & 4 in 5 & Used in business\\
$\Psi^2_1$ & 1 in 40,000& Unexplained usage\\
\bottomrule
\end{tabular}
\end{table}
To set a wider table, which takes up the whole width of the page's
live area, use the environment \textbf{table*} to enclose the table's
contents and the table caption. As with a single-column table, this
wide table will ``float'' to a location deemed more
desirable. Immediately following this sentence is the point at which
Table~\ref{tab:commands} is included in the input file; again, it is
instructive to compare the placement of the table here with the table
in the printed output of this document.
\begin{table*}
\caption{Some Typical Commands}
\label{tab:commands}
\begin{tabular}{ccl}
\toprule
Command &A Number & Comments\\
\midrule
\texttt{{\char'134}author} & 100& Author \\
\texttt{{\char'134}table}& 300 & For tables\\
\texttt{{\char'134}table*}& 400& For wider tables\\
\bottomrule
\end{tabular}
\end{table*}
Always use midrule to separate table header rows from data rows, and
use it only for this purpose. This enables assistive technologies to
recognise table headers and support their users in navigating tables
more easily.
\section{Math Equations}
You may want to display math equations in three distinct styles:
inline, numbered or non-numbered display. Each of the three are
discussed in the next sections.
\subsection{Inline (In-text) Equations}
A formula that appears in the running text is called an inline or
in-text formula. It is produced by the \textbf{math} environment,
which can be invoked with the usual
\texttt{{\char'134}begin\,\ldots{\char'134}end} construction or with
the short form \texttt{\$\,\ldots\$}. You can use any of the symbols
and structures, from $\alpha$ to $\omega$, available in
\LaTeX~\cite{Lamport:LaTeX}; this section will simply show a few
examples of in-text equations in context. Notice how this equation:
\begin{math}
\lim_{n\rightarrow \infty}x=0
\end{math},
set here in in-line math style, looks slightly different when
set in display style. (See next section).
\subsection{Display Equations}
A numbered display equation---one set off by vertical space from the
text and centered horizontally---is produced by the \textbf{equation}
environment. An unnumbered display equation is produced by the
\textbf{displaymath} environment.
Again, in either environment, you can use any of the symbols and
structures available in \LaTeX\@; this section will just give a couple
of examples of display equations in context. First, consider the
equation, shown as an inline equation above:
\begin{equation}
\lim_{n\rightarrow \infty}x=0
\end{equation}
Notice how it is formatted somewhat differently in
the \textbf{displaymath}
environment. Now, we'll enter an unnumbered equation:
\begin{displaymath}
\sum_{i=0}^{\infty} x + 1
\end{displaymath}
and follow it with another numbered equation:
\begin{equation}
\sum_{i=0}^{\infty}x_i=\int_{0}^{\pi+2} f
\end{equation}
just to demonstrate \LaTeX's able handling of numbering.
\section{Figures}
The ``\verb|figure|'' environment should be used for figures. One or
more images can be placed within a figure. If your figure contains
third-party material, you must clearly identify it as such, as shown
in the example below.
\begin{figure}[h]
\centering
\includegraphics[width=\linewidth]{sample-franklin}
\caption{1907 Franklin Model D roadster. Photograph by Harris \&
Ewing, Inc. [Public domain], via Wikimedia
Commons. (\url{https://goo.gl/VLCRBB}).}
\Description{A woman and a girl in white dresses sit in an open car.}
\end{figure}
Your figures should contain a caption which describes the figure to
the reader.
Figure captions are placed {\itshape below} the figure.
Every figure should also have a figure description unless it is purely
decorative. These descriptions convey what's in the image to someone
who cannot see it. They are also used by search engine crawlers for
indexing images, and when images cannot be loaded.
A figure description must be unformatted plain text less than 2000
characters long (including spaces). {\bfseries Figure descriptions
should not repeat the figure caption – their purpose is to capture
important information that is not already provided in the caption or
the main text of the paper.} For figures that convey important and
complex new information, a short text description may not be
adequate. More complex alternative descriptions can be placed in an
appendix and referenced in a short figure description. For example,
provide a data table capturing the information in a bar chart, or a
structured list representing a graph. For additional information
regarding how best to write figure descriptions and why doing this is
so important, please see
\url{https://www.acm.org/publications/taps/describing-figures/}.
\subsection{The ``Teaser Figure''}
A ``teaser figure'' is an image, or set of images in one figure, that
are placed after all author and affiliation information, and before
the body of the article, spanning the page. If you wish to have such a
figure in your article, place the command immediately before the
\verb|\maketitle| command:
\begin{verbatim}
\begin{teaserfigure}
\includegraphics[width=\textwidth]{sampleteaser}
\caption{figure caption}
\Description{figure description}
\end{teaserfigure}
\end{verbatim}
\section{Citations and Bibliographies}
The use of \BibTeX\ for the preparation and formatting of one's
references is strongly recommended. Authors' names should be complete
--- use full first names (``Donald E. Knuth'') not initials
(``D. E. Knuth'') --- and the salient identifying features of a
reference should be included: title, year, volume, number, pages,
article DOI, etc.
The bibliography is included in your source document with these two
commands, placed just before the \verb|\end{document}| command:
\begin{verbatim}
\bibliographystyle{ACM-Reference-Format}
\section{Introduction}
ACM's consolidated article template, introduced in 2017, provides a
consistent \LaTeX\ style for use across ACM publications, and
incorporates accessibility and metadata-extraction functionality
necessary for future Digital Library endeavors. Numerous ACM and
SIG-specific \LaTeX\ templates have been examined, and their unique
features incorporated into this single new template.
If you are new to publishing with ACM, this document is a valuable
guide to the process of preparing your work for publication. If you
have published with ACM before, this document provides insight and
instruction into more recent changes to the article template.
The ``\verb|acmart|'' document class can be used to prepare articles
for any ACM publication --- conference or journal, and for any stage
of publication, from review to final ``camera-ready'' copy, to the
author's own version, with {\itshape very} few changes to the source.
\section{Template Overview}
As noted in the introduction, the ``\verb|acmart|'' document class can
be used to prepare many different kinds of documentation --- a
double-blind initial submission of a full-length technical paper, a
two-page SIGGRAPH Emerging Technologies abstract, a ``camera-ready''
journal article, a SIGCHI Extended Abstract, and more --- all by
selecting the appropriate {\itshape template style} and {\itshape
template parameters}.
This document will explain the major features of the document
class. For further information, the {\itshape \LaTeX\ User's Guide} is
available from
\url{https://www.acm.org/publications/proceedings-template}.
\subsection{Template Styles}
The primary parameter given to the ``\verb|acmart|'' document class is
the {\itshape template style} which corresponds to the kind of publication
or SIG publishing the work. This parameter is enclosed in square
brackets and is a part of the {\verb|documentclass|} command:
\begin{verbatim}
\documentclass[STYLE]{acmart}
\end{verbatim}
Journals use one of three template styles. All but three ACM journals
use the {\verb|acmsmall|} template style:
\begin{itemize}
\item {\verb|acmsmall|}: The default journal template style.
\item {\verb|acmlarge|}: Used by JOCCH and TAP.
\item {\verb|acmtog|}: Used by TOG.
\end{itemize}
The majority of conference proceedings documentation will use the {\verb|acmconf|} template style.
\begin{itemize}
\item {\verb|acmconf|}: The default proceedings template style.
\item{\verb|sigchi|}: Used for SIGCHI conference articles.
\item{\verb|sigchi-a|}: Used for SIGCHI ``Extended Abstract'' articles.
\item{\verb|sigplan|}: Used for SIGPLAN conference articles.
\end{itemize}
\subsection{Template Parameters}
In addition to specifying the {\itshape template style} to be used in
formatting your work, there are a number of {\itshape template parameters}
which modify some part of the applied template style. A complete list
of these parameters can be found in the {\itshape \LaTeX\ User's Guide.}
Frequently-used parameters, or combinations of parameters, include:
\begin{itemize}
\item {\verb|anonymous,review|}: Suitable for a ``double-blind''
conference submission. Anonymizes the work and includes line
numbers. Use with the \verb|\acmSubmissionID| command to print the
submission's unique ID on each page of the work.
\item{\verb|authorversion|}: Produces a version of the work suitable
for posting by the author.
\item{\verb|screen|}: Produces colored hyperlinks.
\end{itemize}
This document uses the following string as the first command in the
source file:
\begin{verbatim}
\documentclass[sigconf]{acmart}
\end{verbatim}
\section{Modifications}
Modifying the template --- including but not limited to: adjusting
margins, typeface sizes, line spacing, paragraph and list definitions,
and the use of the \verb|\vspace| command to manually adjust the
vertical spacing between elements of your work --- is not allowed.
{\bfseries Your document will be returned to you for revision if
modifications are discovered.}
\section{Typefaces}
The ``\verb|acmart|'' document class requires the use of the
``Libertine'' typeface family. Your \TeX\ installation should include
this set of packages. Please do not substitute other typefaces. The
``\verb|lmodern|'' and ``\verb|ltimes|'' packages should not be used,
as they will override the built-in typeface families.
\section{Title Information}
The title of your work should use capital letters appropriately -
\url{https://capitalizemytitle.com/} has useful rules for
capitalization. Use the {\verb|title|} command to define the title of
your work. If your work has a subtitle, define it with the
{\verb|subtitle|} command. Do not insert line breaks in your title.
If your title is lengthy, you must define a short version to be used
in the page headers, to prevent overlapping text. The \verb|title|
command has a ``short title'' parameter:
\begin{verbatim}
\title[short title]{full title}
\end{verbatim}
\section{Authors and Affiliations}
Each author must be defined separately for accurate metadata
identification. Multiple authors may share one affiliation. Authors'
names should not be abbreviated; use full first names wherever
possible. Include authors' e-mail addresses whenever possible.
Grouping authors' names or e-mail addresses, or providing an ``e-mail
alias,'' as shown below, is not acceptable:
\begin{verbatim}
\author{Brooke Aster, David Mehldau}
\email{dave,judy,steve@university.edu}
\email{firstname.lastname@phillips.org}
\end{verbatim}
The \verb|authornote| and \verb|authornotemark| commands allow a note
to apply to multiple authors --- for example, if the first two authors
of an article contributed equally to the work.
If your author list is lengthy, you must define a shortened version of
the list of authors to be used in the page headers, to prevent
overlapping text. The following command should be placed just after
the last \verb|\author{}| definition:
\begin{verbatim}
\renewcommand{\shortauthors}{McCartney, et al.}
\end{verbatim}
Omitting this command will force the use of a concatenated list of all
of the authors' names, which may result in overlapping text in the
page headers.
The article template's documentation, available at
\url{https://www.acm.org/publications/proceedings-template}, has a
complete explanation of these commands and tips for their effective
use.
Note that authors' addresses are mandatory for journal articles.
\section{Rights Information}
Authors of any work published by ACM will need to complete a rights
form. Depending on the kind of work, and the rights management choice
made by the author, this may be copyright transfer, permission,
license, or an OA (open access) agreement.
Regardless of the rights management choice, the author will receive a
copy of the completed rights form once it has been submitted. This
form contains \LaTeX\ commands that must be copied into the source
document. When the document source is compiled, these commands and
their parameters add formatted text to several areas of the final
document:
\begin{itemize}
\item the ``ACM Reference Format'' text on the first page.
\item the ``rights management'' text on the first page.
\item the conference information in the page header(s).
\end{itemize}
Rights information is unique to the work; if you are preparing several
works for an event, make sure to use the correct set of commands with
each of the works.
The ACM Reference Format text is required for all articles over one
page in length, and is optional for one-page articles (abstracts).
\section{CCS Concepts and User-Defined Keywords}
Two elements of the ``acmart'' document class provide powerful
taxonomic tools for you to help readers find your work in an online
search.
The ACM Computing Classification System ---
\url{https://www.acm.org/publications/class-2012} --- is a set of
classifiers and concepts that describe the computing
discipline. Authors can select entries from this classification
system, via \url{https://dl.acm.org/ccs/ccs.cfm}, and generate the
commands to be included in the \LaTeX\ source.
User-defined keywords are a comma-separated list of words and phrases
of the authors' choosing, providing a more flexible way of describing
the research being presented.
CCS concepts and user-defined keywords are required for for all
articles over two pages in length, and are optional for one- and
two-page articles (or abstracts).
\section{Sectioning Commands}
Your work should use standard \LaTeX\ sectioning commands:
\verb|section|, \verb|subsection|, \verb|subsubsection|, and
\verb|paragraph|. They should be numbered; do not remove the numbering
from the commands.
Simulating a sectioning command by setting the first word or words of
a paragraph in boldface or italicized text is {\bfseries not allowed.}
\section{Tables}
The ``\verb|acmart|'' document class includes the ``\verb|booktabs|''
package --- \url{https://ctan.org/pkg/booktabs} --- for preparing
high-quality tables.
Table captions are placed {\itshape above} the table.
Because tables cannot be split across pages, the best placement for
them is typically the top of the page nearest their initial cite. To
ensure this proper ``floating'' placement of tables, use the
environment \textbf{table} to enclose the table's contents and the
table caption. The contents of the table itself must go in the
\textbf{tabular} environment, to be aligned properly in rows and
columns, with the desired horizontal and vertical rules. Again,
detailed instructions on \textbf{tabular} material are found in the
\textit{\LaTeX\ User's Guide}.
Immediately following this sentence is the point at which
Table~\ref{tab:freq} is included in the input file; compare the
placement of the table here with the table in the printed output of
this document.
\begin{table}
\caption{Frequency of Special Characters}
\label{tab:freq}
\begin{tabular}{ccl}
\toprule
Non-English or Math&Frequency&Comments\\
\midrule
\O & 1 in 1,000& For Swedish names\\
$\pi$ & 1 in 5& Common in math\\
\$ & 4 in 5 & Used in business\\
$\Psi^2_1$ & 1 in 40,000& Unexplained usage\\
\bottomrule
\end{tabular}
\end{table}
To set a wider table, which takes up the whole width of the page's
live area, use the environment \textbf{table*} to enclose the table's
contents and the table caption. As with a single-column table, this
wide table will ``float'' to a location deemed more
desirable. Immediately following this sentence is the point at which
Table~\ref{tab:commands} is included in the input file; again, it is
instructive to compare the placement of the table here with the table
in the printed output of this document.
\begin{table*}
\caption{Some Typical Commands}
\label{tab:commands}
\begin{tabular}{ccl}
\toprule
Command &A Number & Comments\\
\midrule
\texttt{{\char'134}author} & 100& Author \\
\texttt{{\char'134}table}& 300 & For tables\\
\texttt{{\char'134}table*}& 400& For wider tables\\
\bottomrule
\end{tabular}
\end{table*}
Always use midrule to separate table header rows from data rows, and
use it only for this purpose. This enables assistive technologies to
recognise table headers and support their users in navigating tables
more easily.
\section{Math Equations}
You may want to display math equations in three distinct styles:
inline, numbered or non-numbered display. Each of the three are
discussed in the next sections.
\subsection{Inline (In-text) Equations}
A formula that appears in the running text is called an inline or
in-text formula. It is produced by the \textbf{math} environment,
which can be invoked with the usual
\texttt{{\char'134}begin\,\ldots{\char'134}end} construction or with
the short form \texttt{\$\,\ldots\$}. You can use any of the symbols
and structures, from $\alpha$ to $\omega$, available in
\LaTeX~\cite{Lamport:LaTeX}; this section will simply show a few
examples of in-text equations in context. Notice how this equation:
\begin{math}
\lim_{n\rightarrow \infty}x=0
\end{math},
set here in in-line math style, looks slightly different when
set in display style. (See next section).
\subsection{Display Equations}
A numbered display equation---one set off by vertical space from the
text and centered horizontally---is produced by the \textbf{equation}
environment. An unnumbered display equation is produced by the
\textbf{displaymath} environment.
Again, in either environment, you can use any of the symbols and
structures available in \LaTeX\@; this section will just give a couple
of examples of display equations in context. First, consider the
equation, shown as an inline equation above:
\begin{equation}
\lim_{n\rightarrow \infty}x=0
\end{equation}
Notice how it is formatted somewhat differently in
the \textbf{displaymath}
environment. Now, we'll enter an unnumbered equation:
\begin{displaymath}
\sum_{i=0}^{\infty} x + 1
\end{displaymath}
and follow it with another numbered equation:
\begin{equation}
\sum_{i=0}^{\infty}x_i=\int_{0}^{\pi+2} f
\end{equation}
just to demonstrate \LaTeX's able handling of numbering.
\section{Figures}
The ``\verb|figure|'' environment should be used for figures. One or
more images can be placed within a figure. If your figure contains
third-party material, you must clearly identify it as such, as shown
in the example below.
\begin{figure}[h]
\centering
\includegraphics[width=\linewidth]{sample-franklin}
\caption{1907 Franklin Model D roadster. Photograph by Harris \&
Ewing, Inc. [Public domain], via Wikimedia
Commons. (\url{https://goo.gl/VLCRBB}).}
\Description{A woman and a girl in white dresses sit in an open car.}
\end{figure}
Your figures should contain a caption which describes the figure to
the reader.
Figure captions are placed {\itshape below} the figure.
Every figure should also have a figure description unless it is purely
decorative. These descriptions convey what's in the image to someone
who cannot see it. They are also used by search engine crawlers for
indexing images, and when images cannot be loaded.
A figure description must be unformatted plain text less than 2000
characters long (including spaces). {\bfseries Figure descriptions
should not repeat the figure caption – their purpose is to capture
important information that is not already provided in the caption or
the main text of the paper.} For figures that convey important and
complex new information, a short text description may not be
adequate. More complex alternative descriptions can be placed in an
appendix and referenced in a short figure description. For example,
provide a data table capturing the information in a bar chart, or a
structured list representing a graph. For additional information
regarding how best to write figure descriptions and why doing this is
so important, please see
\url{https://www.acm.org/publications/taps/describing-figures/}.
\subsection{The ``Teaser Figure''}
A ``teaser figure'' is an image, or set of images in one figure, that
are placed after all author and affiliation information, and before
the body of the article, spanning the page. If you wish to have such a
figure in your article, place the command immediately before the
\verb|\maketitle| command:
\begin{verbatim}
\begin{teaserfigure}
\includegraphics[width=\textwidth]{sampleteaser}
\caption{figure caption}
\Description{figure description}
\end{teaserfigure}
\end{verbatim}
\section{Citations and Bibliographies}
The use of \BibTeX\ for the preparation and formatting of one's
references is strongly recommended. Authors' names should be complete
--- use full first names (``Donald E. Knuth'') not initials
(``D. E. Knuth'') --- and the salient identifying features of a
reference should be included: title, year, volume, number, pages,
article DOI, etc.
The bibliography is included in your source document with these two
commands, placed just before the \verb|\end{document}| command:
\begin{verbatim}
\bibliographystyle{ACM-Reference-Format}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,094 |
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